Common Statistical Methods for Clinical Research with SAS Examples, Second Edition

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Common Statistical Methods for Clinical Research: with SAS® Examples

Glenn A. Walker

SAS Publishing

More praise for this new edition “This book is an extension and improvement to the excellent first edition, which appeared in 1996. Each chapter has an introduction covering a specific statistical method and gives examples of where the method is used. The introduction is followed by a synopsis presenting the necessary and significant information for the reader to get a good understanding of the statistical method. Excellent examples are given, followed by the SAS code for the example. Detailed annotations make the SAS output easy to understand. The book also gives numerous extensions for the methods. Dr. Walker has certainly used his many years of consulting experience, his teaching experience, and his understanding of SAS to produce an even better book that is equally understandable and helpful to the novice as well as the experienced statistician. Each will benefit from this insightful journey into statistics and the use of SAS, whether as a teaching tool or a refresher. I can recommend this book wholeheartedly.” Stephan Ogenstad, Ph.D. Senior Director, Biometrics Vertex Pharmaceuticals Incorporated

SECOND EDITION

CommonStatistical Methods for Clinical Research with SAS Examples ®

Glenn A. Walker

58086

The correct bibliographic citation for this manual is as follows: Walker, Glenn A. 2002. Common Statistical Methods ® for Clinical Research with SAS Examples, Second Edition. Cary, NC: SAS Institute Inc. ®

Common Statistical Methods for Clinical Research with SAS Examples, Second Edition Copyright © 2002 by SAS Institute Inc., Cary, NC, USA ISBN 1-59047-040-0 All rights reserved. Printed in the United States of America. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, or otherwise, without the prior written permission of the publisher, SAS Institute Inc. U.S. Government Restricted Rights Notice: Use, duplication, or disclosure of this software and related documentation by the U.S. government is subject to the Agreement with SAS Institute and the restrictions set forth in FAR 52.227-19, Commercial Computer Software-Restricted Rights (June 1987). SAS Institute Inc., SAS Campus Drive, Cary, North Carolina 27513. 1st printing, July 2002 SAS Publishing provides a complete selection of books and electronic products to help customers use SAS software to its fullest potential. For more information about our e-books, e-learning products, CDs, and hardcopy books, visit the SAS Publishing Web site at www.sas.com/pubs or call 1-800-727-3228. ®

SAS and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc. in the USA and other countries. ® indicates USA registration. Other brand and product names are trademarks of their respective companies.

Table of Contents

Preface to the Second Edition .........................................................................................................xi Preface to the First Edition ............................................................................................................xiii Chapter 1 - Introduction & Basics 1.1 1.2 1.3 1.4 1.5 1.6

Statistics—the Field .................................................................................................................... 1 Probability Distributions ........................................................................................................... 4 Study Design Features ............................................................................................................... 9 Descriptive Statistics ................................................................................................................ 13 Inferential Statistics ................................................................................................................. 16 Summary ................................................................................................................................... 21

Chapter 2 – Topics in Hypothesis Testing 2.1 2.2 2.3 2.4 2.5 2.6 2.7

Significance Levels .................................................................................................................... 23 Power ......................................................................................................................................... 25 One-Tailed and Two-Tailed Tests ............................................................................................ 26 p-Values ..................................................................................................................................... 27 Sample Size Determination ..................................................................................................... 27 Multiple Testing ........................................................................................................................ 30 Summary ................................................................................................................................... 38

Chapter 3 – The Data Set TRIAL 3.1 3.2 3.3 3.4 3.5

Introduction............................................................................................................................... 39 Data Collection ......................................................................................................................... 39 Creating the Data Set TRIAL.................................................................................................. 42 Statistical Summarization ....................................................................................................... 45 Summary.................................................................................................................................... 54

Chapter 4 – The One-Sample t-Test 4.1 4.2 4.3 4.4

Introduction............................................................................................................................... 55 Synopsis ..................................................................................................................................... 55 Examples ................................................................................................................................... 56 Details & Notes ......................................................................................................................... 64

Chapter 5 – The Two-Sample t-Test 5.1 5.2 5.3 5.4

Introduction............................................................................................................................... 67 Synopsis ..................................................................................................................................... 67 Examples ................................................................................................................................... 68 Details & Notes ......................................................................................................................... 73

iv Table of Contents

Chapter 6 – One-Way ANOVA 6.1 6.2 6.3 6.4

Introduction............................................................................................................................... 77 Synopsis ..................................................................................................................................... 77 Examples ................................................................................................................................... 80 Details & Notes ......................................................................................................................... 86

Chapter 7 – Two-Way ANOVA 7.1 7.2 7.3 7.4

Introduction............................................................................................................................... 91 Synopsis ..................................................................................................................................... 91 Examples ................................................................................................................................... 94 Details & Notes ....................................................................................................................... 107

Chapter 8 – Repeated Measures Analysis 8.1 8.2 8.3 8.4

Introduction............................................................................................................................. 111 Synopsis ................................................................................................................................... 111 Examples ................................................................................................................................. 115 Details & Notes ....................................................................................................................... 148

Chapter 9 – The Crossover Design 9.1 9.2 9.3 9.4

Introduction............................................................................................................................. 157 Synopsis ................................................................................................................................... 158 Examples ................................................................................................................................. 160 Details & Notes ....................................................................................................................... 171

Chapter 10 – Linear Regression 10.1 10.2 10.3 10.4

Introduction............................................................................................................................. 175 Synopsis ................................................................................................................................... 175 Examples ................................................................................................................................. 178 Details & Notes ....................................................................................................................... 193

Chapter 11 – Analysis of Covariance 11.1 11.2 11.3 11.4

Introduction............................................................................................................................. 199 Synopsis ................................................................................................................................... 200 Examples ................................................................................................................................. 203 Details & Notes ....................................................................................................................... 221

Chapter 12 – The Wilcoxon Signed-Rank Test 12.1 12.2 12.3 12.4

Introduction............................................................................................................................. 227 Synopsis ................................................................................................................................... 227 Examples ................................................................................................................................. 228 Details & Notes ....................................................................................................................... 234

Chapter 13 – The Wilcoxon Rank-Sum Test 13.1 13.2 13.3 13.4

Introduction............................................................................................................................. 237 Synopsis ................................................................................................................................... 237 Examples ................................................................................................................................. 239 Details & Notes ....................................................................................................................... 244

Table of Contents

v

Chapter 14 – The Kruskal-Wallis Test 14.1 14.2 14.3 14.4

Introduction............................................................................................................................. 247 Synopsis ................................................................................................................................... 247 Examples ................................................................................................................................. 249 Details & Notes ....................................................................................................................... 253

Chapter 15 – The Binomial Test 15.1 15.2 15.3 14.4

Introduction............................................................................................................................. 257 Synopsis ................................................................................................................................... 257 Examples ................................................................................................................................. 259 Details & Notes ....................................................................................................................... 261

Chapter 16 – The Chi-Square Test 16.1 16.2 16.3 16.4

Introduction............................................................................................................................. 265 Synopsis ................................................................................................................................... 265 Examples ................................................................................................................................. 267 Details & Notes........................................................................................................................ 279

Chapter 17 – Fisher’s Exact Test 17.1 17.2 17.3 17.4

Introduction............................................................................................................................. 285 Synopsis ................................................................................................................................... 285 Examples ................................................................................................................................. 286 Details & Notes........................................................................................................................ 289

Chapter 18 – McNemar’s Test 18.1 18.2 18.3 18.4

Introduction............................................................................................................................. 293 Synopsis ................................................................................................................................... 293 Examples ................................................................................................................................. 294 Details & Notes........................................................................................................................ 301

Chapter 19 – The Cochran-Mantel-Haenszel Test 19.1 19.2 19.3 19.4

Introduction............................................................................................................................. 305 Synopsis ................................................................................................................................... 305 Examples ................................................................................................................................. 307 Details & Notes........................................................................................................................ 313

Chapter 20 – Logistic Regression 20.1 20.2 20.3 20.4

Introduction............................................................................................................................. 317 Synopsis ................................................................................................................................... 318 Examples ................................................................................................................................. 322 Details & Notes........................................................................................................................ 340

Chapter 21 – The Log-Rank Test 21.1 21.2 21.3 21.4

Introduction............................................................................................................................. 349 Synopsis ................................................................................................................................... 350 Examples.................................................................................................................................. 351 Details & Notes........................................................................................................................ 359

vi Table of Contents

Chapter 22 – The Cox Proportional Hazards Model 22.1 22.2 22.3 22.4

Introduction............................................................................................................................. 365 Synopsis ................................................................................................................................... 366 Examples.................................................................................................................................. 367 Details & Notes........................................................................................................................ 376

Chapter 23 – Exercises 23.1 23.2 23.3 23.4

Introduction............................................................................................................................. 381 Exercises .................................................................................................................................. 381 Appropriateness of the Methods ........................................................................................... 383 Summary.................................................................................................................................. 387

Appendix A – Probability Tables A.1 A.2 A.3

Probabilities of the Standard Normal Distribution............................................................. 390 Critical Values of the Student t-Distribution....................................................................... 391 Critical Values of the Chi-Square Distribution ................................................................... 392

Appendix B – Common Distributions Used in Statistical Inference B.1 B.2 B.3 B.4

Notation.................................................................................................................................... 393 Properties................................................................................................................................. 394 Results ...................................................................................................................................... 395 Distributional Shapes.............................................................................................................. 397

Appendix C – Basic ANOVA Concepts C.1 C.2 C.3

Within- vs. Between-Group Variation.................................................................................. 399 Noise Reduction by Blocking ................................................................................................. 401 Least Squares Mean (LS-mean) ............................................................................................ 405

Appendix D – SS Types I, II, III, and IV Methods for an Unbalanced Two-Way Layout D.1 D.2 D.3 D.4 D.5

SS Types Computed by SAS .................................................................................................. 409 How to Determine the Hypotheses Tested ........................................................................... 411 Empty Cells.............................................................................................................................. 416 More Than Two Treatment Groups..................................................................................... 417 Summary.................................................................................................................................. 419

Appendix E – Multiple Comparison Methods E.1 E.2 E.3

Multiple Comparisons of Means ........................................................................................... 421 Multiple Comparisons of Binomial Proportions.................................................................. 432 Summary.................................................................................................................................. 436

Appendix F – Data Transformations F.1 F.2

Introduction............................................................................................................................. 437 The Log Transformation........................................................................................................ 438

Appendix G – SAS Code for Exercises in Chapter 23.................................................... 441 References .............................................................................................................................................. 447 Index ................................................................................................................................................... 453

Examples

4.1 4.2 5.1 6.1 7.1 7.2 8.1 8.2 8.3 9.1 9.2 10.1 10.2 11.1 11.2 12.1 13.1 14.1 15.1 16.1 16.2 16.3 17.1 18.1 18.2 19.1 20.1 20.2 20.3 21.1 22.1 22.2

Body-Mass Index ......................................................................................................................56 Paired-Difference in Weight Loss ..........................................................................................60 FEV1 Changes ..........................................................................................................................68 HAM-A Scores in GAD ...........................................................................................................80 Hemoglobin Changes in Anemia ............................................................................................94 Memory Function ...................................................................................................................103 Arthritic Discomfort Following Vaccine ..............................................................................115 Treadmill Walking Distance in Intermittent Claudication................................................127 Disease Progression in Alzheimer’s Trial.............................................................................137 Diaphoresis Following Cardiac Medication ........................................................................160 Antibiotic Blood Levels Following Aerosol Inhalation.......................................................165 Anti-Anginal Response vs. Disease History .........................................................................178 Symptomatic Recovery in Pediatric Dehydration...............................................................185 Triglyceride Changes Adjusted for Glycemic Control.......................................................203 ANCOVA in Multi-Center Hypertension Study.................................................................214 Rose-Bengal Staining in KCS................................................................................................228 Global Evaluations of Seroxatene in Back Pain..................................................................239 Psoriasis Evaluation in Three Groups..................................................................................249 Genital Wart Cure Rate.........................................................................................................259 ADR Frequency with Antibiotic Treatment ........................................................................267 Comparison of Dropout Rates for 4 Dose Groups..............................................................274 Active vs. Placebo Comparisons of Degree of Response ....................................................277 CHF Incidence in CABG after ARA ....................................................................................286 Bilirubin Abnormalities Following Drug Treatment..........................................................294 Symptom Frequency Before and After Treatment .............................................................300 Dermotel Response in Diabetic Ulcers..................................................................................307 Relapse Rate Adjusted for Remission Time in AML..........................................................322 Symptom Relief in Gastroparesis..........................................................................................331 Intercourse Success Rate in Erectile Dysfunction...............................................................336 HSV-2 Episodes Following gD2 Vaccine..............................................................................351 HSV-2 Episodes with gD2 Vaccine (continued) ..................................................................367 Hyalurise in Vitreous Hemorrhage.......................................................................................370

viii Common Statistical Methods for Clinical Research with SAS Examples

Figures

1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 2.1. 3.1. 3.2. 3.3. 7.1. 7.2. 8.1. 8.2. 8.3. 8.4. 10.1. 10.2. 11.1. 20.1. 20.2. 20.3. 20.4. B.1. C.1.

Probability Distribution for Z = X1 .................................................................................... 7 Probability Distribution for Z = X1+X2 .............................................................................. 7 Probability Distribution for Z = X1+X2+X3 ...................................................................... 8 Probability Distribution for Z = X1+X2+X3+X4+X5+X6+X7+X8 ...................................... 9 Histogram of Height Measurements (n=25) .................................................................... 13 Histogram of Height Measurements (n=300) ................................................................. 14 Power Curve for the Z-Test .............................................................................................. 26 Sample Data Collection Form 1 (Dichotomous Response) ............................................. 40 Sample Data Collection Form 2 (Categorical Response)................................................ 41 Sample Data Collection Form 3 (Continuous Numeric Response) ................................ 41 No Interaction (a,b,c) and Interaction (d,e,f) Effects.................................................... 102 Interaction in Example 7.2 .............................................................................................. 106 Sample Profiles of Drug Response.................................................................................. 114 Response Profiles (Example 8.1) ..................................................................................... 123 Mean Walking Distances for Example 8.2 ..................................................................... 136 ADAS-cog Response Profiles (Example 8.3) .................................................................. 139 Simple Linear Regression of y on x ................................................................................ 176 95% Confidence Bands for Example 10.1...................................................................... 196 ANCOVA-Adjusted Means for Example 11.1 ............................................................... 208 Logistic Probability Function ......................................................................................... 319 Probability of Relapse (Px) vs. Remission Time (X) for Example 20.1........................ 325 Plot of Transformation from P to the Logit(P).............................................................. 343 Plot of Estimated Relapse Probabilities vs. Time since Prior Remission (X).............. 346 Distributional Shapes ...................................................................................................... 398 Plot of Mean Weights Shows No Treatment-by-Gender Interaction .......................... 405

x Common Statistical Methods for Clinical Research with SAS Examples

Prreeffaaccee ttoo tthhee SSeeccoonndd EEddiittiioonn This second edition expands the first edition with the inclusion of new sections, examples, and extensions of statistical methods described in the first edition only in their most elementary forms. New methods sections include analysis of crossover designs (Chapter 9) and multiple comparison methods (Appendix D). Chapters that present repeated measures analysis (Chapter 8), linear regression (Chapter 10), analysis of covariance (Chapter 11), the chi-square test (Chapter 16), and logistic regression (Chapter 20) have been notably expanded, and 50% more examples have been added throughout the book. A new chapter of exercises has also been added to give the reader practice in applying the various methods presented. Also new in this edition is an introduction to α-adjustments for interim analyses (Chapter 2). Although many of the new features will have wide appeal, some are targeted to the more experienced data analyst. These include discussion of the proportional odds model, the clustered binomial problem, collinearity in multiple regression, the use of time-dependent covariates with Cox regression, and the use of generalized estimating equations in repeated measures analysis. These methods, which are based on more advanced concepts than those found in most of the book, are routinely encountered in data analysis applications of clinical investigations and, as such, fit the description of ‘common statistical methods for clinical research’. However, so as not to overwhelm the less experienced reader, these concepts are presented only briefly, usually by example, along with references for further reading. First and foremost, this is a statistical methods book. It is designed to have particular appeal to those involved in clinical research, biometrics, epidemiology, and other health or medical related research applications. Unlike other books in the SAS Books by Users (BBU) library, SAS is not the primary focus of this book. Rather, SAS is presented as an indispensable tool that greatly simplifies the analyst’s task. While consulting for dozens of companies over 25 years of statistical application to clinical investigation, I have never seen a successful clinical program that did not use SAS. Because of its widespread use within the pharmaceutical industry, I include SAS here as the ‘tool’ of choice to illustrate the statistical methods. The examples have been updated to Version 8 of SAS, however, the programming statements used have been kept ‘portable’, meaning that most can be used in earlier versions of SAS as they appear in the examples, unless otherwise noted. This includes the use of portable variable and data set names, despite accommodation for use of long names beginning with Version 8. Because SAS is

xii Common Statistical Methods for Clinical Research with SAS Examples

not the main focus of this book but is key to efficient data analysis, programming details are not included here, but they can be found in numerous references cited throughout the book. Many of these references are other books in the Books by Users program at SAS, which provide the details, including procedure options, use of ODS, and the naming standards that are new in Version 8. For statistical programming, my favorites include Categorical Data Analysis Using the SAS System, Second Edition, by Stokes, Davis, and Koch (2000) and Survival Analysis Using the SAS System, A Practical Guide, by Paul Allison (1995). I welcome and appreciate reader comments and feedback through the SAS Publications Web site.

Glenn A. Walker July 2002

Preface

xiii

Prreeffaaccee ttoo tthhee FFiirrsstt EEddiittiioonn This book was written for those involved in clinical research and who may, from time to time, need a guide to help demystify some of the most commonly used statistical methods encountered in our profession. All too often, I have heard medical directors of clinical research departments express frustration at seemingly cryptic statistical methods sections of protocols which they are responsible for approving. Other nonstatisticians, including medical monitors, investigators, clinical project managers, medical writers and regulatory personnel, often voice similar sentiment when it comes to statistics, despite the profound reliance upon statistical methods in the success of the clinical program. For these people, I offer this book (sans technical details) as a reference guide to better understand statistical methods as applied to clinical investigation and the conditions and assumptions under which they are applied. For the clinical data analyst and statistician new to clinical applications, the examples from a clinical trials setting may help in making the transition from other statistical fields to that of clinical trials. The discussions of 'Least-Squares' means, distinguishing features of the various SAS® types of sums-of-squares, and relationships among various tests (such as the Chi-Square Test, the CochranMantel-Haenszel Test and the Log-Rank Test) may help crystalize the analyst's understanding of these methods. Analysts with no prior SAS experience should benefit by the simplifed SAS programming statements provided with each example as an introduction to SAS analyses. This book may also aid the SAS programmer with limited statistical knowledge in better grasping an overall picture of the clinical trials process. Many times knowledge of the hypotheses being tested and appropriate interpretation of the SAS output relative to those hypotheses will help the programmer become more efficient in responding to the requests of other clinical project team members. Finally, the medical student will find the focused presentation on the specific methods presented to be of value while proceeding through a first course in biostatistics. For all readers, my goal was to provide a unique approach to the description of commonly used statistical methods by integrating both manual and computerized solutions to a wide variety of examples taken from clinical research. Those who learn best by example should find this approach rewarding. I have found no other book which demonstrates that the SAS output actually does have the same results as the manual solution of a problem using the calculating formulas. So ever reassuring this is for the student of clinical data analysis!

xiv Common Statistical Methods for Clinical Research with SAS Examples

Each statistical test is presented in a separate chapter, and includes a brief, nontechnical introduction, a synopsis of the test, one or two examples worked manually followed by an appropriate solution using the SAS statistical package, and finally, a discussion with details and relevant notes. Chapters 1 and 2 are introductory in nature, and should be carefully read by all with no prior formal exposure to statistics. Chapter 1 provides an introduction to statistics and some of the basic concepts involved in inference-making. Chapter 2 goes into more detail with regard to the main aspects of hypothesis testing, including significance levels, power and sample size determination. For those who use analysis-of-variance, Appendix C provides a non-technical introduction to ANOVA methods. The remainder of the book may be used as a text or reference. As a reference, the reader should keep in mind that many of the tests discussed in later chapters rely on concepts presented earlier in the book, strongly suggesting prerequisite review. This book focuses on statistical hypothesis testing as opposed to other inferential techniques. For each statistical method, the test summary is clearly provided, including the null hypothesis tested, the test statistic and the decision rule. Each statistical test is presented in one of its most elementary forms to provide the reader with a basic framework. Many of the tests discussed have extensions or variations which can be used with more complex data sets. The 18 statistical methods presented here (Chapters 3-20) represent a composite of those which, in my experience, are most commonly used in the analysis of clinical research data. I can't think of a single study I've analyzed in nearly 20 years which did not use at least one of these tests. Furthermore, many of the studies I've encountered have used exclusively the methods presented here, or variations or extensions thereof. Thus, the word 'common' in the title. Understanding of many parts of this book requires some degree of statistical knowledge. The clinician without such a background may skip over many of the technical details and still come away with an overview of the test's applications, assumptions and limitations. Basic algebra is the only prerequisite, as derivations of test procedures are omitted, and matrix algebra is mentioned only in an appendix. My hope is that the statistical and SAS analysis aspects of the examples would provide a springboard for the motivated reader, both to go back to more elementary texts for additional background and to go forward to more advanced texts for further reading. Many of the examples are based on actual clinical trials which I have analyzed. In all cases, the data are contrived, and in many cases fictitious names are used for different treatments or research facilities. Any resemblence of the data or the tests' results to actual cases is purely coincidental.

Glenn A. Walker May 1996

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42

Common Statistical Methods for Clinical Research with SAS Examples

&UHDWLQJWKH'DWD6HW75,$/ Create and store the data set TRIAL in the directory bookfiles\examples\sas on Drive C. The SAS code for creating this data set is shown below. All three response types are input as follows: • • •

The dichotomous response variable is named RESP. It has coded values of 0 (if symptoms are absent) and 1 (if symptoms are present); The categorical response variable is named SEV. It is coded as 0, 1, 2, or 3 for ‘none’, ‘mild’, ‘moderate’, and ‘severe’; The continuous numeric response variable is named SCORE. Its values can range between 0 and 100. These values represent the percentage of the greatest possible severity perceived by the patient.

Because these data are used for illustrative purposes only, the values of the SCORE variable å are entered and, for ease in constructing the data set, the other response variables (RESP and SEV) are computed rather than entered manually. RESP is set to 0 or 1, based on whether SCORE is 0 or >0, respectively . The severity category (SEV) field is created as follows: SCORE values of 1 to 30 are defined as ‘Mild’; 31 to 69, ‘Moderate’; and 70 to 100, ‘Severe’ ê.

6$6&RGHIRU&UHDWLQJWKH'DWD6HW75,$/ LIBNAME EXAMP ’c:\bookfiles\examples\sas’; DATA EXAMP.TRIAL; INPUT TRT $ CENTER PAT SEX $ AGE SCORE @@; å RESP = (SCORE GT 0); /* RESP=0 (symptoms are absent), =1 (symptoms are present) IF (SCORE = 0) THEN SEV = 0; /* “No Symptoms” IF ( 1 LE SCORE LE 30) THEN SEV = 1; /* “Mild Symptoms” IF (31 LE SCORE LE 69) THEN SEV = 2; /* “Moderate Symptoms” IF (SCORE GE 70) THEN SEV = 3; /* “Severe Symptoms” DATALINES; A 1 101 M 55 5 A 1 104 F 27 0 A 1 106 M 31 35 A 1 107 F 44 21 A 1 109 M 47 15 A 1 111 F 69 70 A 1 112 F 31 10 A 1 114 F 50 0 A 1 116 M 32 20 A 1 118 F 39 25 A 1 119 F 54 0 A 1 121 M 70 38 A 1 123 F 57 55 A 1 124 M 37 18 A 1 126 F 41 0 A 1 128 F 48 8 A 1 131 F 35 0 A 1 134 F 28 0 A 1 135 M 27 40 A 1 138 F 42 12 A 2 202 M 58 68 A 2 203 M 42 22 A 2 206 M 26 30 A 2 207 F 36 0 A 2 210 F 35 25 A 2 211 M 51 0 A 2 214 M 51 60 A 2 216 F 42 15 A 2 217 F 50 50 A 2 219 F 41 35 A 2 222 F 59 0 A 2 223 F 38 10 A 2 225 F 32 0 A 2 226 F 28 16 A 2 229 M 42 48 A 2 231 F 51 45 A 2 234 F 26 90 A 2 235 M 42 0 A 3 301 M 38 28 A 3 302 M 41 20 A 3 304 M 65 75 A 3 306 F 64 0 A 3 307 F 30 30 A 3 309 F 64 5 A 3 311 M 39 80

*/ */ */ */ */

ê

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A A A B B B B B B B B B B B B B B B B ;

3 3 3 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3

314 319 325 105 113 120 127 132 137 201 208 213 220 227 232 305 312 317 323

F F F M M F F M F F F M F F M M M M F

57 34 56 45 61 19 53 49 57 63 48 24 45 41 37 46 52 52 51

85 26 35 20 75 55 40 20 95 10 12 88 72 0 32 15 40 60 35

A A B B B B B B B B B B B B B B B B

3 3 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3

315 321 102 108 115 122 129 133 139 204 209 215 221 228 233 308 313 320

M M M F M F M F F M F M F F F M F F

61 39 19 44 45 38 48 28 47 36 42 40 27 24 33 59 33 32

12 10 68 65 83 0 0 0 40 49 40 59 55 65 0 82 40 2

A A B B B B B B B B B B B B B B B B

3 3 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3

318 324 103 110 117 125 130 136 140 205 212 218 224 230 303 310 316 322

F M F M F M F F M M F M M M M F M F

45 27 51 32 21 37 36 34 29 36 32 31 56 44 40 62 62 43

95 0 10 25 0 72 80 45 0 16 0 24 70 30 26 38 87 0

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PROC SORT DATA = EXAMP.TRIAL; BY PAT; PROC PRINT DATA = EXAMP.TRIAL; VAR PAT TRT CENTER SEX AGE RESP SEV SCORE; TITLE ‘Printout of Data Set TRIAL, Sorted by PAT’; RUN;

2873875HVXOWVIURP352&35,177KH'DWD6HW75,$/ Printout of Data Set TRIAL, Sorted by PAT

Obs

PAT

TRT

CENTER

SEX

AGE

RESP

SEV

SCORE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122

A B B A B A A B A B A A B A B A B A A B A B

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

M M F F M M F F M M F F M F M M F F F F M F

55 19 51 27 45 31 44 44 47 32 69 31 61 50 45 32 21 39 54 19 70 38

1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 1 1 0

1 2 1 0 1 2 1 2 1 1 3 1 3 0 3 1 0 1 0 2 2 0

5 68 10 0 20 35 21 65 15 25 70 10 75 0 83 20 0 25 0 55 38 0

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23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 301 302 303 304 305

A A B A B A B B A B B A A B B A B B B A A B B A A B B A A B B A B A A B A B B A A B A A B B A B A B B A A A A B A B

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3

F M M F F F M F F M F F M F F F F M F M M M M M F F F F M F M M M F F M F F F F F M F F F F M M F M F F M M M M M M

57 37 37 41 53 48 48 36 35 49 28 28 27 34 57 42 47 29 63 58 42 36 36 26 36 48 42 35 51 32 24 51 40 42 50 31 41 45 27 59 38 56 32 28 41 24 42 44 51 37 33 26 42 38 41 40 65 46

1 1 1 0 1 1 0 1 0 1 0 0 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1

2 1 3 0 2 1 0 3 0 1 0 0 2 2 3 1 2 0 1 2 1 2 1 1 0 1 2 1 0 0 3 2 2 1 2 1 2 3 2 0 1 3 0 1 0 2 2 1 2 2 0 3 0 1 1 1 3 1

55 18 72 0 40 8 0 80 0 20 0 0 40 45 95 12 40 0 10 68 22 49 16 30 0 12 40 25 0 0 88 60 59 15 50 24 35 72 55 0 10 70 0 16 0 65 48 30 45 32 0 90 0 28 20 26 75 15

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306 307 308 309 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325

A A B A A B B A A B B A A B A B B A A

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

F F M F M M F F M M M F F F M F F M F

64 30 59 64 39 52 33 57 61 62 52 45 34 32 39 43 51 27 56

0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1

0 1 3 1 3 2 2 3 1 3 2 3 1 1 1 0 2 0 2

0 30 82 5 80 40 40 85 12 87 60 95 26 2 10 0 35 0 35

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)LUVWXVH352&0($16VHH287387 PROC SORT DATA = EXAMP.TRIAL; BY TRT; PROC MEANS MEAN STD N MIN MAX DATA = EXAMP.TRIAL; BY TRT; VAR SCORE AGE; TITLE “Summary Statistics for ‘SCORE’ and ‘AGE’ Variables”; RUN;

1RZXVH352&81,9$5,$7(VHH287387 PROC UNIVARIATE DATA = EXAMP.TRIAL; BY TRT; VAR SCORE; TITLE “Expanded Summary Statistics for ‘SCORE’” ; RUN;

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1H[W ZULWH WKH PHDQ DQG PHGLDQ YDOXHV IRU 6&25( IRU HDFK WUHDWPHQW JURXS DQG VWXG\ FHQWHU FRPELQDWLRQ WR DQ RXWSXW GDWD VHW WKHQ SULQW WKH UHVXOWV VHH 287387 PROC SORT DATA = EXAMP.TRIAL; BY TRT CENTER; PROC UNIVARIATE NOPRINT DATA = EXAMP.TRIAL; BY TRT CENTER; VAR SCORE; OUTPUT OUT = SUMMRY N = NUM MEAN = AVESCORE MEDIAN = MEDSCORE; RUN;

PROC PRINT DATA = SUMMRY; TITLE “Summary Statistics for ‘SCORE’ by Treatment Group & Study Center”; RUN;

9LVXDOL]HWKHGLVWULEXWLRQRIVFRUHVXVLQJDURXJKKLVWRJUDPVHH287387 PROC CHART DATA = EXAMP.TRIAL; VBAR SCORE / MIDPOINTS = 10 30 50 70 90 GROUP = TRT; TITLE “Distribution of ‘SCORE’ by Treatment Group”; RUN;

2EWDLQUHVSRQVHUDWHVE\WUHDWPHQWJURXSXVLQJ352&)5(4VHH287387 PROC FORMAT; VALUE RSPFMT RUN;

0 = ‘0=Abs.’

1 = ‘1=Pres’;

PROC FREQ DATA = EXAMP.TRIAL; TABLES TRT*RESP / NOCOL NOPCT; FORMAT RESP RSPFMT.; TITLE ‘Summary of Response Rates by Treatment Group’; RUN;

2EWDLQ WKH IUHTXHQF\ WDEOHV IRU WKH VHYHULW\ FDWHJRULHV XVLQJ 352& )5(4 VHH287387 PROC FORMAT; VALUE SEVFMT

0 1 2 3

= = = =

‘0=None’ ‘1=Mild’ ‘2=Mod.’ ‘3=Sev.’

;

RUN; PROC FREQ DATA = EXAMP.TRIAL; TABLES TRT*SEV / NOCOL NOPCT; FORMAT SEV SEVFMT.; TITLE ‘Severity Distribution by Treatment Group’; RUN;

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PROC FREQ DATA = EXAMP.TRIAL; TABLES SEX*TRT*SEV / NOCOL NOPCT; FORMAT SEV SEVFMT.; TITLE ‘Severity Distribution by Treatment Group and Sex’; RUN;

2EWDLQDKLVWRJUDPRIVHYHULW\IRUHDFKWUHDWPHQWJURXSVHH287387 PROC CHART DATA = EXAMP.TRIAL; VBAR SEV / MIDPOINTS = 0 1 2 3 GROUP = TRT; TITLE “Distribution of ‘SEV’ by Treatment Group”; RUN;

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2873875HVXOWVIURP352&0($16 Summary Statistics for ‘SCORE’ and ‘AGE’ Variables ------------------------------ TRT=A --------------------------------The MEANS Procedure Variable Mean Std Dev N Minimum Maximum ---------------------------------------------------------------------SCORE 26.6730769 26.9507329 52 0 95.0000000 AGE 43.7307692 12.2187824 52 26.0000000 70.0000000 --------------------------------------------------------------------------------------------------- TRT=B ---------------------------------

Variable Mean Std Dev N Minimum Maximum ---------------------------------------------------------------------SCORE 38.333333 29.6937083 48 0 95.0000000 AGE 41.3333333 11.7949382 48 19.0000000 63.0000000 ----------------------------------------------------------------------

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2873875HVXOWVIURP352&81,9$5,$7( Expanded Summary Statistics for ‘SCORE’ ---------------------------- TRT=A -----------------------------------The UNIVARIATE Procedure Variable: SCORE Moments N Mean Std Deviation Skewness Uncorrected SS Coeff Variation

Location Mean Median Mode

Test

52 26.6730769 26.9507329 1.02297616 74039 101.04096

Sum Weights Sum Observations Variance Kurtosis Corrected SS Std Error Mean

52 1387 726.342006 0.13964333 37043.4423 3.73739421

Basic Statistical Measures Variability

26.67308 20.00000 0.00000

Std Deviation Variance Range Interquartile Range

26.95073 726.34201 95.00000 36.50000

Tests for Location: Mu0=0 -Statistic-----p Value------

Student's t Sign Signed Rank

t M S

7.136811 19.5 390

Pr > |t| Pr >= |M| Pr >= |S|

<.0001 <.0001 <.0001

Quantiles (Definition 5) Quantile Estimate 100% Max 99% 95% 90% 75% Q3 50% Median 25% Q1 10% 5% 1% 0% Min

95.0 95.0 85.0 70.0 39.0 20.0 2.5 0.0 0.0 0.0 0.0

Extreme Observations ----Lowest-------Highest---

Value

Obs

Value

Obs

0 0 0 0 0

51 42 38 33 31

75 80 85 90 95

41 45 46 37 48

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2873875HVXOWVIURP352&81,9$5,$7(FRQWLQXHG Expanded Summary Statistics for ‘SCORE’ ----------------------------- TRT=B -------------------------------Moments N Mean Std Deviation Skewness Uncorrected SS Coeff Variation

48 38.3333333 29.6937083 0.21433323 111974 77.4618477

Location Mean Median Mode

Sum Weights Sum Observations Variance Kurtosis Corrected SS Std Error Mean

48 1840 881.716312 -1.2051435 41440.6667 4.28591762

Basic Statistical Measures Variability

38.33333 39.00000 0.00000

Std Deviation Variance Range Interquartile Range

29.69371 881.71631 95.00000 54.00000

Tests for Location: Mu0=0 Test

-Statistic-

-----p Value------

Student's t Sign Signed Rank

t M S

Pr > |t| Pr >= |M| Pr >= |S|

8.94402 19.5 390

Quantiles (Definition 5) Quantile Estimate 100% Max 99% 95% 90% 75% Q3 50% Median 25% Q1 10% 5% 1% 0% Min

95 95 87 82 65 39 11 0 0 0 0

Extreme Observations ----Lowest-------Highest---

Value

Obs

Value

Obs

0 0 0 0 0

47 37 33 26 20

82 83 87 88 95

40 7 44 27 18

<.0001 <.0001 <.0001

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2873876XPPDU\6WDWLVWLFVLQDQ2XWSXW'DWD6HW Summary Statistics for ‘SCORE’ by Treatment Group & Study Center

Obs

TRT

CENTER

NUM

AVESCORE

MEDSCORE

1 2 3 4 5 6

A A A B B B

1 2 3 1 2 3

20 18 14 20 17 11

18.6000 28.5556 35.7857 39.6500 36.5882 38.6364

13.5 23.5 27.0 40.0 32.0 38.0

2873875HVXOWVIURP352&&+$57 Distribution of ‘SCORE’ by Treatment Group Frequency 25 + *** | *** 24 + *** | *** 23 + *** | *** 22 + *** | *** 21 + *** | *** 20 + *** | *** 19 + *** | *** 18 + *** | *** 17 + *** | *** 16 + *** | *** 15 + *** *** | *** *** 14 + *** *** *** | *** *** *** 13 + *** *** *** | *** *** *** 12 + *** *** *** | *** *** *** 11 + *** *** *** | *** *** *** 10 + *** *** *** *** | *** *** *** *** 9 + *** *** *** *** *** | *** *** *** *** *** 8 + *** *** *** *** *** *** | *** *** *** *** *** *** 7 + *** *** *** *** *** *** | *** *** *** *** *** *** 6 + *** *** *** *** *** *** *** | *** *** *** *** *** *** *** 5 + *** *** *** *** *** *** *** *** | *** *** *** *** *** *** *** *** 4 + *** *** *** *** *** *** *** *** *** *** | *** *** *** *** *** *** *** *** *** *** 3 + *** *** *** *** *** *** *** *** *** *** | *** *** *** *** *** *** *** *** *** *** 2 + *** *** *** *** *** *** *** *** *** *** | *** *** *** *** *** *** *** *** *** *** 1 + *** *** *** *** *** *** *** *** *** *** | *** *** *** *** *** *** *** *** *** *** •------------------------------------------------------------10 30 50 70 90 10 30 50 70 90 SCORE Midpoint |--------- A ---------|

|--------- B ---------|

TRT

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2873875HVXOWVIURP352&)5(4'LFKRWRPRXV5HVSRQVH )UHTXHQFLHV Summary of Response Rates by Treatment Group The FREQ Procedure Table of TRT by RESP TRT

RESP

Frequency| Row Pct |0=Abs. |1=Pres | ---------+--------+--------+ A | 13 | 39 | | 25.00 | 75.00 | ---------+--------+--------+ B | 9 | 39 | | 18.75 | 81.25 | ---------+--------+--------+ Total 22 78

Total 52

48

100

2873875HVXOWVIURP352&)5(46HYHULW\)UHTXHQFLHVE\ 7UHDWPHQW*URXS Severity Distribution by Treatment Group The FREQ Procedure Table of TRT by SEV TRT

SEV

Frequency| Row Pct |0=None |1=Mild |2=Mod. |3=Sev. | ---------+--------+--------+--------+--------+ A | 13 | 22 | 11 | 6 | | 25.00 | 42.31 | 21.15 | 11.54 | ---------+--------+--------+--------+--------+ B | 9 | 12 | 17 | 10 | | 18.75 | 25.00 | 35.42 | 20.83 | ---------+--------+--------+--------+--------+ Total 22 34 28 16

Total 52

48

100

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SEV

Frequency| Row Pct |0=None |1=Mild |2=Mod. |3=Sev. | ---------+--------+--------+--------+--------+ A | 10 | 12 | 5 | 4 | | 32.26 | 38.71 | 16.13 | 12.90 | ---------+--------+--------+--------+--------+ B | 7 | 4 | 11 | 3 | | 28.00 | 16.00 | 44.00 | 12.00 | ---------+--------+--------+--------+--------+ Total 17 16 16 7

Total 31

25

56

Table 2 of TRT by SEV Controlling for SEX=M TRT

SEV

Frequency| Row Pct |0=None |1=Mild |2=Mod. |3=Sev. | ---------+--------+--------+--------+--------+ A | 3 | 10 | 6 | 2 | | 14.29 | 47.62 | 28.57 | 9.52 | ---------+--------+--------+--------+--------+ B | 2 | 8 | 6 | 7 | | 8.70 | 34.78 | 26.09 | 30.43 | ---------+--------+--------+--------+--------+ Total 5 18 12 9

Total 21

23

44

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TRT

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CHHAAPPTTEERR 4 The One-Sample t-Test 4.1 4.2 4.3 4.4

Introduction........................................................................................................ 55 Synopsis .............................................................................................................. 55 Examples............................................................................................................. 56 Details & Notes................................................................................................... 64

4.1

Introduction The one-sample t-test is used to infer whether an unknown population mean differs from a hypothesized value. The test is based on a single sample of ‘n’ measurements from the population. A special case of the one-sample t-test is often used to determine if a mean response changes under different experimental conditions by using paired observations. Paired measurements most often arise from repeated observations for the same patient, either at two different time points (e.g., pre- and post-study) or from two different treatments given to the same patient. Because the analyst uses the changes (differences) in the paired measurements, this test is often referred to as the ‘paired-difference’ t-test. A hypothesized mean difference of 0 might be interpreted as ‘no clinical response’. The one-sample t-test, along with its two-sample counterpart presented in the next chapter, must surely be the most frequently cited inferential tests used in statistics.

4.2

Synopsis A sample of n data points, y1, y2, ..., yn, is randomly selected from a normally distributed population with unknown mean, m. This mean is estimated by the sample mean, y. You hypothesize the mean to be some value, m0. The greater the deviation between y and m0, the greater the evidence that the hypothesis is untrue. The test statistic t is a function of this deviation, standardized by the standard error of y, namely, s/ n . Large values of t will lead to rejection of the null hypothesis. When H0 is true, t has the Student-t probability distribution with n–1 degrees of freedom.

56

Common Statistical Methods for Clinical Research with SAS Examples

The one-sample t-test is summarized as follows:

null hypothesis: alt. hypothesis:

H0: m = m0 HA: m =/ m0 t=

test statistic: decision rule:

y −µ 0 s/ n

reject H0 if | t | > ta/2,n–1

The value ta/2,n–1 represents the ‘critical t-value’ of the Student t-distribution at a two-tailed significance level, a, and n–1 degrees of freedom. This value can be obtained from Appendix A.2 or from SAS, as described later (see 4.4.3). The one-sample t-test is often used in matched-pairs situations, such as testing for pre- to post-study changes. The goal is to determine whether a difference exists between two time points or between two treatments based on data values from the same patients for both measures. The ‘paired-difference’ t-test is conducted using the procedures outlined for the one-sample t-test with m=the mean difference (sometimes denoted md), m0= 0 , and the yi’s = the paired-differences. Examples 4.1 and 4.2 demonstrate application of the one-sample t-test in non-matched and matched situations, respectively.

4.3

Examples

p Example 4.1 –Body-Mass Index

In a number of previous Phase I and II studies of male, non-insulin-dependent diabetic (NIDDM) patients conducted by Mylitech Biosystems, Inc., the mean body mass index (BMI) was found to be 28.4. An investigator has 17 male NIDDM patients enrolled in a new study and wants to know if the BMI from this sample is consistent with previous findings. BMI is computed as the ratio of weight in kilograms to the square of the height in meters. What conclusion can the investigator make based on the following height and weight data from his 17 patients?

Chapter 4: The One-Sample t-Test 57

TABLE 4.1. Raw Data for Example 4.1 Patient Number

Height (cm)

1

178

2

Weight (kg)

Patient Number

Height (cm)

Weight (Kg)

101.7

10

183

97.8

170

97.1

11

n/a

78.7

3

191

114.2

12

172

77.5

4

179

101.9

13

183

102.8

5

182

93.1

14

169

81.1

6

177

108.1

15

177

102.1

7

184

85.0

16

180

112.1

8

182

89.1

17

184

89.7

9

179

95.8

Solution First, calculate BMI as follows: convert height in centimeters to meters by dividing by 100, square this quantity, then divide it into the weight. BMI data (yi) are shown below, along with the squares and summations. Because no height measurement is available for Patient No. 11, BMI cannot be computed and is considered a missing value. TABLE 4.2. Computation of Sum of Squares / Example 4.1 Patient Number

BMI (y)

1

32.1

2

2

2

Patient Number

BMI (y)

BMI 2 (y )

1030.30

10

29.2

852.85

33.6

1128.87

11

.

.

3

31.3

979.94

12

26.2

686.26

4

31.8

1011.43

13

30.7

942.28

5

28.1

789.98

14

28.4

806.30

6

34.5

1190.58

15

32.6

1062.08

7

25.1

630.33

16

34.6

1197.07

8

26.9

723.55

17

26.5

701.96

9

29.9

893.96

BMI 2 (y )

TOTALS

Σyi

= 481.5 2

Σ(yi) = 14627.74

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Chapter 4: The One-Sample t-Test 59

The output automatically shows the BMI summary statistics, namely, the mean and the standard deviation . The standard error and the upper and lower confidence limits associated with a 95% confidence interval are also shown for both the mean and standard deviation. In SAS Releases 6.12 and earlier, PROC TTEST cannot be used to conduct the one-sample t-test. Instead, you first transform the response variable in the DATA step by subtracting the null value (e.g., BMI0 = BMI-28.4), then use PROC MEANS with the T and PRT options on this new variable. The SAS statements for this example would be: PROC MEANS T PRT DATA = DIAB; VAR BMI0; RUN;

SAS Code for Example 4.1 DATA DIAB; INPUT PATNO WT_KG HT_CM @@; BMI = WT_KG / ((HT_CM/100)**2); DATALINES; 1 101.7 178 2 97.1 170 3 114.2 191 4 101.9 179 5 93.1 182 6 108.1 177 7 85.0 184 8 89.1 182 9 95.8 179 10 97.8 183 11 78.7 . 12 77.5 172 13 102.8 183 14 81.1 169 15 102.1 177 16 112.1 180 17 89.7 184 ;

PROC PRINT DATA = DIAB; VAR PATNO HT_CM WT_KG BMI; FORMAT BMI 5.1; TITLE1 'One-Sample t-Test'; TITLE2 'EXAMPLE 4.1: Body-Mass Index Data'; RUN; PROC TTEST H0=28.4 DATA=DIAB; VAR BMI; RUN;

60

Common Statistical Methods for Clinical Research with SAS Examples

Output 4.1. SAS Output for Example 4.1 One-Sample t-Test EXAMPLE 4.1: Body-Mass Index Data Obs

PATNO

HT_CM

WT_KG

BMI

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

178 170 191 179 182 177 184 182 179 183 . 172 183 169 177 180 184

101.7 97.1 114.2 101.9 93.1 108.1 85.0 89.1 95.8 97.8 78.7 77.5 102.8 81.1 102.1 112.1 89.7

32.1 33.6 31.3 31.8 28.1 34.5 25.1 26.9 29.9 29.2 . 26.2 30.7 28.4 32.6 34.6 26.5

The TTEST Procedure

Variable BMI

Lower CL N Mean 16

28.478

Variable BMI

Statistics Upper CL Lower CL Mean Mean Std Dev

30.093

31.709

2.24

Statistics Std Err Minimum 0.7581

25.106

Variable

T-Tests DF t Value

BMI

15

2.23

Std Dev

Upper CL Std Dev

3.0323

4.693

Maximum 34.599

Pr > |t| 0.0411

p Example 4.2 -- Paired-Difference in Weight Loss

Mylitech is developing a new appetite suppressing compound for use in weight reduction. A preliminary study of 35 obese patients provided the following data on patients' body weights (in pounds) before and after 10 weeks of treatment with the new compound. Does the new treatment look at all promising?

Chapter 4: The One-Sample t-Test 61

TABLE 4.3. Raw Data for Example 4.2 Weight In Pounds Subject Number

Pre-

1

165

2

Weight In Pounds

Post-

Subject Number

Pre-

Post-

160

19

177

171

202

200

20

181

170

3

256

259

21

148

154

4

155

156

22

167

170

5

135

134

23

190

180

6

175

162

24

165

154

7

180

187

25

155

150

8

174

172

26

153

145

9

136

138

27

205

206

10

168

162

28

186

184

11

207

197

29

178

166

12

155

155

30

129

132

13

220

205

31

125

127

14

163

153

32

165

169

15

159

150

33

156

158

16

253

255

34

170

161

17

138

128

35

145

152

18

287

280

Solution First, calculate the weight loss for each patient (yi) by subtracting the ‘post'- weight from the ‘pre-’ weight (see Output 4.2). The mean and standard deviation of these differences are yd = 3.457 and s = 6.340 based on n = 35 patients. A test for significant mean weight loss is equivalent to testing whether the mean loss, md, is greater than 0. You use a one-tailed test because the interest is in weight loss (not weight ‘change’). The hypothesis test is summarized as follows: null hypothesis:

H0: md = 0

alt. hypothesis:

HA: md > 0

test statistic:

t=

decision rule:

reject H0 if t > t0.05,34 = 1.691

conclusion:

Because 3.226 > 1.691, you reject H0 at a significance level of 0.05 and conclude that a significant mean weight loss occurs following treatment.

yd sd / n

=

3.457 = 3.226 6.340/ 35

62

Common Statistical Methods for Clinical Research with SAS Examples

SAS Analysis of Example 4.2 The SAS code and output for analyzing this data set are shown below. The program computes the paired differences (variable=WTLOSS) as the data are read into the data set OBESE. The program prints a listing of the data by using PROC PRINT. A part of the listing is shown in Output 4.2 11 . As in Example 4.1, PROC TTEST without a CLASS statement can be used to conduct the paired t-test. In this case, you tell SAS to compute the differences, WTLOSS, so you can simply include the VAR WTLOSS statement with PROC TTEST. Alternatively, you may omit the VAR statement and include the PAIRED statement followed by the pair of variables being compared, separated by an asterisk (*). This eliminates the need to compute the differences in a separate DATA step. This example illustrates the PAIRED statement by including the two variables, WTPRE and WTPST 12 . SAS will automatically compute the paired differences (WTPRE–WTPST), then conduct the t-test. The output shows a t-statistic of 3.23 13 , which corroborates the manual computations. Because SAS prints the twotailed p-value 14 , this value must be halved to obtain the one-tailed result, p= 0.0028/2 =0.0014. You reject the null hypothesis and conclude that a significant weight loss occurs because this p-value is less than the nominal significance level of a = 0.05. In SAS Releases 6.12 and earlier, PROC TTEST cannot be used to conduct paired t-tests. Instead, use PROC MEANS with the T and PRT options on the variable WTLOSS. The SAS statements for this example would be: PROC MEANS MEAN STD N T PRT DATA=OBESE; VAR WTLOSS; RUN;

SAS Code for Example 4.2 DATA OBESE; INPUT SUBJ WTPRE WTPST @@; WTLOSS = WTPRE – WTPST; DATALINES; 1 165 160 2 202 200 3 256 5 135 134 6 175 162 7 180 9 136 138 10 168 162 11 207 13 220 205 14 163 153 15 159 17 138 128 18 287 280 19 177 21 148 154 22 167 170 23 190 25 155 150 26 153 145 27 205 29 178 166 30 129 132 31 125 33 156 158 34 170 161 35 145 ;

259 187 197 150 171 180 206 127 152

4 8 12 16 20 24 28 32

155 174 155 253 181 165 186 165

156 172 155 255 170 154 184 169

Chapter 4: The One-Sample t-Test 63

11 PROC PRINT DATA = OBESE; VAR SUBJ WTPRE WTPST WTLOSS; TITLE1 'One-Sample t-Test'; TITLE2 'EXAMPLE 4.2: Paired-Difference in Weight Loss'; RUN; PROC TTEST DATA = OBESE; 12 PAIRED WTPRE*WTPST; RUN;

Output 4.2. SAS Output for Example 4.2 One-Sample t-Test EXAMPLE 4.2: Paired-Difference in Weight Loss OBS

SUBJ

WTPRE

WTPST

1 2 3 4 5 6 7 8 . . . 33 34 35

1 2 3 4 5 6 7 8 . . . 33 34 35

165 202 256 155 135 175 180 174 . . . 156 170 145

160 200 259 156 134 162 187 172 . . . 158 161 152

WTLOSS 5 2 -3 -1 1 13 -7 2 . . . -2 9 -7

11

The TTEST Procedure Statistics

Variable WTPRE – WTPST

N 35

Lower CL Mean 1.2792

Upper CL Mean Mean 3.4571 5.635

Lower CL Std Dev 5.1283

Std Dev 6.3401

Statistics Variable Std Err WTPRE - WTPST 1.0717

Minimum -7

Maximum 15

T-Tests Variable WTPRE - WTPST

DF 34

t Value 3.23 13

Pr > |t| 0.0028 14

Upper CL Std Dev 8.3068

64

Common Statistical Methods for Clinical Research with SAS Examples

4.4

Details & Notes The main difference between the Z-test (Chapter 2) and the t-test 4.4.1. is that the Z-statistic is based on a known standard deviation, s, while the tstatistic uses the sample standard deviation, s, as an estimate of s. With the assumption of normally distributed data, the variance s2 is more closely estimated by the sample variance s2 as n gets large. It can be shown that the t-test is equivalent to the Z-test for infinite degrees of freedom. In practice, a ‘large’ sample is usually considered n ³ 30. The distributional relationship between the Z- and t-statistics is shown in Appendix B. If the assumption of normally distributed data cannot be made, the mean might not represent the best measure of central tendency. In such cases, a nonparametric rank test, such as the Wilcoxon signed-rank test (Chapter 12), might be more appropriate. The UNIVARIATE procedure in SAS can be used to test for normality. Because Example 4.1 tests for any difference from a hypothesized 4.4.2. value, a two-tailed test is used. A one-tailed test would be used when you want to test whether the population mean is strictly greater than or strictly less than the hypothesized threshold level (m0), such as in Example 4.2. We use the rejection region according to the alternative hypothesis as follows:

Type of Test

Alternative Hypothesis

Corresponding Rejection Region

two-tailed

HA: m =/ m0

reject H0 if t > ta/2,n–1 or t < –ta/2,n–1

one-tailed (right)

HA: m > m0

reject H0 if t > ta,n–1

one-tailed (left)

HA: m < m0

reject H0 if t < -ta,n–1

Most statistics text books have tables of the t-distribution in an 4.4.3. Appendix. These tables often provide the critical t-values for levels of a = 0.10, 0.05, 0.025, 0.01, and 0.005, as in Appendix A.2. Critical t-values can also be found from many statistical programs. In SAS, you use the function TINV(1–a,n–1), where a = a for a one-tailed test and a = a/2 for a two-tailed test. In Example 4.1, the critical t-value of 2.131 with n=16 (15 degrees of freedom) can be found with the SAS function TINV(0.975,15).

Chapter 4: The One-Sample t-Test 65

Alternatively, the p-value associated with the test statistic, t, based on υ degrees of freedom, can be found by using the SAS function PROBT(t,υ), which gives the cumulative probability of the Student-t distribution. Because you want the tail probabilities, the associated two-tailed p-value is 2H(1–PROBT(t, υ)). In Example 4.1, the p-value 0.041 can be confirmed by using the SAS expression 2H(1–PROBT(2.23,15)). In Example 4.2, the onetailed p-value 0.0014 can be confirmed by using the SAS expression 1–PROBT(3.226,34). A non-significant result does not necessarily imply that the null 4.4.4. hypothesis is true, only that insufficient evidence exists to contradict it. Larger sample sizes are often needed and an investigation of the power curve (see Chapter 2) is necessary for ‘equivalency studies’. Significance does not imply causality. In Example 4.2, concluding 4.4.5. that the treatment caused the significant weight loss would be presumptuous. Causality can better be investigated using a concurrent untreated or control group in the study, and strictly controlling other experimental conditions. In controlled studies, comparison of responses among groups is carried out with tests such as the two-sample t-test (Chapter 5) or analysis of variance methods (Chapters 6-8). Known values of n measurements uniquely determine the sample 4.4.6. mean y . But given y, the n measurements cannot uniquely be determined. In fact, n–1 of the measurements can be freely selected, with the nth determined by y. Thus the term n–1 degrees of freedom. Note: The number of degrees of freedom, often denoted by the Greek letter υ (nu), is a parameter of the Student-t probability distribution.

66

Common Statistical Methods for Clinical Research with SAS Examples

CHHAAPPTTEERR 5 The Two-Sample t-Test 5.1 5.2 5.3 5.4

Introduction........................................................................................................ 67 Synopsis .............................................................................................................. 67 Examples............................................................................................................. 68 Details & Notes................................................................................................... 73

5.1

Introduction The two-sample t-test is used to compare the means of two independent populations, denoted m1 and m2. This test has ubiquitous application in the analysis of controlled clinical trials. Examples might include the comparison of mean decreases in diastolic blood pressure between two groups of patients receiving different antihypertensive agents, or estimating pain relief from a new treatment relative to that of a placebo based on subjective assessment of percentimprovement in two parallel groups. Assume that the two populations are normally distributed and have the same variance (s2).

5.2

Synopsis Samples of n1 and n2 observations are randomly selected from the two populations, with the measurements from Population i denoted by yi1, yi2, ..., yini (for i = 1, 2). The unknown means, m1 and m2, are estimated by the sample means, y1 and y2 , respectively. The greater the difference between the sample means, the greater the evidence that the hypothesis of equality of population means (H0) is untrue. The test statistic, t, is a function of this difference standardized by its standard error, namely s(1/n1 + 1/n2)1/2. Large values of t will lead to rejection of the null hypothesis. When H0 is true, t has the Student-t distribution with N–2 degrees of freedom, where N = n1+n2.

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Chapter 5: The Two-Sample t-Test 69

TABLE 5.1. Raw Data for Example 5.1 ABC-123 Group

Placebo Group

Patient Number

Baseline

Week 6

Patient Number

Baseline

Week 6

101

1.35

n/a

102

3.01

3.90

103

3.22

3.55

104

2.24

3.01

106

2.78

3.15

105

2.25

2.47

108

2.45

2.30

107

1.65

1.99

109

1.84

2.37

111

1.95

n/a

110

2.81

3.20

112

3.05

3.26

113

1.90

2.65

114

2.75

2.55

116

3.00

3.96

115

1.60

2.20

118

2.25

2.97

117

2.77

2.56

120

2.86

2.28

119

2.06

2.90

121

1.56

2.67

122

1.71

n/a

124

2.66

3.76

123

3.54

2.92

Solution Let m1 and m2 represent the mean increases in FEV1 for the ABC-123 and the placebo groups, respectively. The first step is to calculate each patient's increase in FEV1 from baseline to Week 6 (shown in Output 5.1). Patients 101, 111, and 122 are excluded from this analysis because no Week-6 measurements are available. The FEV1 increases (in liters) are summarized by treatment group as follows: TABLE 5.2. Treatment Group Summary Statistics for Example 5.1

ABC-123

Placebo

Mean

( yi )

0.503

0.284

SD

(si)

0.520

0.508

Sample Size

(ni)

11

10

The pooled variance is

s2p =

(11 − 1) ⋅ 0.5202 + (10 − 1) ⋅ 0.5082 = 0.265 21 − 2

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Chapter 5: The Two-Sample t-Test 71

SAS Code for Example 5.1 DATA FEV; INPUT PATNO TRTGRP $ FEV0 FEV6 @@; CHG = FEV6 - FEV0; IF CHG = . THEN DELETE; DATALINES; 101 A 1.35 . 103 A 3.22 3.55 106 108 A 2.45 2.30 109 A 1.84 2.37 110 113 A 1.90 2.65 116 A 3.00 3.96 118 120 A 2.86 2.28 121 A 1.56 2.67 124 102 P 3.01 3.90 104 P 2.24 3.01 105 107 P 1.65 1.99 111 P 1.95 . 112 114 P 2.75 2.55 115 P 1.60 2.20 117 119 P 2.06 2.90 122 P 1.71 . 123 ;

A A A A P P P P

2.78 2.81 2.25 2.66 2.25 3.05 2.77 3.54

3.15 3.20 2.97 3.76 2.47 3.26 2.56 2.92

PROC FORMAT; VALUE $TRT 'A' = 'ABC-123' 'P' = 'PLACEBO'; RUN;

PROC PRINT DATA = FEV; VAR PATNO TRTGRP FEV0 FEV6 CHG; FORMAT TRTGRP $TRT. FEV0 FEV6 CHG 5.2; TITLE1 'Two-Sample t-Test'; TITLE2 'EXAMPLE 5.1: FEV1 Changes'; RUN; PROC MEANS MEAN STD N T PRT DATA = FEV; BY TRTGRP; VAR FEV0 FEV6 CHG; FORMAT TRTGRP $TRT.; RUN; PROC TTEST DATA = FEV; CLASS TRTGRP; VAR CHG; FORMAT TRTGRP $TRT.; RUN;

72

Common Statistical Methods for Clinical Research with SAS Examples

OUTPUT 5.1. SAS Output for Example 5.1 Two-Sample t-Test EXAMPLE 5.1: FEV1 Changes Obs

PATNO

TRTGRP

FEV0

FEV6

CHG

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

103 106 108 109 110 113 116 118 120 121 124 102 104 105 107 112 114 115 117 119 123

ABC-123 ABC-123 ABC-123 ABC-123 ABC-123 ABC-123 ABC-123 ABC-123 ABC-123 ABC-123 ABC-123 PLACEBO PLACEBO PLACEBO PLACEBO PLACEBO PLACEBO PLACEBO PLACEBO PLACEBO PLACEBO

3.22 2.78 2.45 1.84 2.81 1.90 3.00 2.25 2.86 1.56 2.66 3.01 2.24 2.25 1.65 3.05 2.75 1.60 2.77 2.06 3.54

3.55 3.15 2.30 2.37 3.20 2.65 3.96 2.97 2.28 2.67 3.76 3.90 3.01 2.47 1.99 3.26 2.55 2.20 2.56 2.90 2.92

0.33 0.37 -0.15 0.53 0.39 0.75 0.96 0.72 -0.58 1.11 1.10 0.89 0.77 0.22 0.34 0.21 -0.20 0.60 -0.21 0.84 -0.62

Two-Sample t-Test EXAMPLE 5.1: FEV1 Changes ------------------- TRTGRP=ABC-123 ------------------------------The MEANS Procedure

Variable Mean Std Dev N t Value Pr > |t| ----------------------------------------------------------------FEV0 2.4845455 0.5328858 11 15.46 <.0001 FEV6 2.9872727 0.5916095 11 16.75 <.0001 CHG 0.5027273 0.5198286 11 3.21 0.0094 --------------------------------------------- ---------------------------------------- TRTGRP=PLACEBO ------------------------------Variable Mean Std Dev N t Value Pr > |t| -----------------------------------------------------------------FEV0 2.4920000 0.6355365 10 12.40 <.0001 FEV6 2.7760000 0.5507006 10 15.94 <.0001 CHG 0.2840000 0.5077882 10 1.77 0.1107 ------------------------------------------------------------------

Chapter 5: The Two-Sample t-Test 73

OUTPUT 5.1. SAS Output for Example 5.1 (continued)

Two-Sample t-Test EXAMPLE 5.1: FEV1 Changes The TTEST Procedure Statistics

N

Lower Mean

ABC-123 11 PLACEBO 10 Diff (1-2)

0.1535 -0.079 -0.251

Variable CHG CHG CHG

Class

CL Upper CL Mean Mean 0.5027 0.284 0.2187

0.852 0.6472 0.6889

Lower CL Upper CL Std Dev Std Dev Std Dev 0.3632 0.3493 0.391

0.5198 0.5078 0.5142

0.9123 0.927 0.751

Statistics Variable CHG CHG CHG

Class

ABC-123 PLACEBO Diff (1-2)

Std Err

Minimum

Maximum

0.1567 0.1606 0.2247

-0.58 -0.62

1.11 0.89

T-Tests Variable CHG CHG

Method

Variances

DF

t Value

Pooled Satterthwaite

Equal Unequal

19 18.9

0.97 0.97

Variable CHG

5.4

Method Folded F

Equality of Variances Num DF Den DF 10

9

Pr > |t|

0.3425 0.3420

F Value

Pr > F

1.05

0.9532

Details & Notes The assumption of equal variances can be tested using the F-test. 5.4.1. An F-test generally arises as a ratio of variances. When the hypothesis of equal variances is true, the ratio of sample variances should be about 1. The probability distribution of this ratio is known as the F-distribution (which is widely used in the analysis of variance).

74

Common Statistical Methods for Clinical Research with SAS Examples

The test for comparing two variances can be made as follows: 2

2

null hypothesis: alt. hypothesis:

H0: s1 = s2 2 2 HA: s1 =/ s2

test statistic:

2 2 F = sU / sL

decision rule:

reject H0 if F > Fc(α/2)

The subscripts U and L denote ‘upper’ and ‘lower’. The sample with the larger 2 sample variance (sU ) is considered the ‘upper’ sample, and the sample with the 2 smaller sample variance (sL ) is the ‘lower’. Large values of F indicate a large disparity in sample variances and would lead to rejection of H0. The critical F value, Fc, can be obtained from the F-tables in most elementary statistics books or many computer packages, based on nU-1 upper and nL-1 lower degrees of freedom. In SAS, the critical F value is found by using the FINV function:

Fc(α/2) = FINV(1– α/2, nU–1, nL–1) 2

2

In Example 5.1, compute F = 0.520 / 0.508 = 1.048, which leads to non-rejection of the null hypothesis of equal variances at α=0.05 based on Fc(0.025) = 3.96. As noted in Output 5.1, the p-value associated with this preliminary test is 0.9532 , which can also be obtained using the PROBF function in SAS: 2*(1–PROBF(1.048,10,9)). You can proceed with the t-test assuming equal variances because the evidence fails to contradict that assumption. If the hypothesis of equal variances is rejected, the t-test might give erroneous results. In such cases, a modified version of the t-test proposed by Satterthwaite is often used. The ‘Satterthwaite adjustment’ consists of using the statistic similar to the t-test but with an approximate degrees of freedom, carried out as follows:

test statistic:

decision rule:

t′ =

y1 – y 2 2

2

s1 + s 2 n1 n 2

reject H 0 if

t ¢ > t α/2, q

Chapter 5: The Two-Sample t-Test 75

where q represents the approximate degrees of freedom computed as follows 2 (with wi = si / ni ):

q=

(w 1 + w 2 )2 w12 w22 + (n 1 – 1) (n 2 – 1)

Although Satterthwaite’s adjustment is not needed for Example 5.1, it can be used to illustrate the computational methods as follows. Compute 2

w1 = 0.520 / 11 = 0.0246 and

2

w2 = 0.508 / 10 = 0.0258 so that q=

(0.0246 + 0.0258) 2 = 18.9 0.0246 2 0.0258 2 + 10 9

and

t′ =

0.503 – 0.284 0.520 2 0.508 2 + 11 10

= 0.975

SAS computes these quantities automatically in the output of PROC TTEST (See Output 5.1). Satterthwaite’s t-value and approximate degrees of freedom are given for Unequal under Variances in the SAS output. (See in Output 5.1.) These results should be used when the F-test for equal variances is significant, or when it is known that the population variances differ. The significance level of a statistical test will be altered if it is 5.4.2. conditioned on the results of a preliminary test. Therefore, if a t-test depends on the results of a preliminary F-test for variance homogeneity, the actual significance level might be slightly different than what is reported. This difference gets smaller as the significance level of the preliminary test increases. With this in mind, you might want to conduct the preliminary F-test for variance homogeneity at a significance level greater than 0.05, usually 0.10, 0.15, or even 0.20. For example, if 0.15 were used, Satterthwaites adjustment would be used if the F-test were significant at p < 0.15.

76

Common Statistical Methods for Clinical Research with SAS Examples

The assumption of normality can be tested by using the Shapiro-Wilk 5.4.3. test or the Kolmogorov-Smirnov test executed with the NORMAL option in PROC UNIVARIATE in SAS. Rejection of the assumption of normality in the small sample case precludes the use of the t-test. As n1 and n2 become large (generally³30), you don’t need to rely so heavily on the assumption that the data have an underlying normal distribution in order to apply the two-sample t-test. However, with non-normal data, the mean might not represent the most appropriate measure of central tendency. With a skewed distribution, for example, the median might be more representative of the distributional center than the mean. In such cases, a rank test, such as the Wilcoxon rank-sum test (Chapter 13) or the log-rank test (Chapter 21), should be considered. Note that, if within-group tests are used for the analysis instead of 5.4.4. the two-sample t-test, the researcher might reach a different, erroneous conclusion. In Example 5.1, you might hastily conclude that a significant treatment-related increase in mean FEV1 exists just by looking at the withingroup results. The two-sample t-test, however, is the more appropriate test for the comparison of between-group changes because the control group response must be factored out of the response from the active group. In interpreting the changes, you might argue that the mean change in FEV1 for the active group is comprised of two additive effects, namely the placebo effect plus a therapeutic benefit. If it can be established that randomization to the treatment groups provides effectively homogeneous groups, you might conclude that the FEV1 mean change for the active group in Example 5.1 can be broken down as follows: 0.503 = 0.284 + 0.219 (Total Effect) (Placebo Effect) (Therapeutic Effect) To validate such interpretations of the data, you would first establish baseline comparability of the two groups. The two-sample t-test can also be applied for this purpose by analyzing the baseline values in the same way the changes were analyzed in Example 5.1. While larger p-values (> 0.10) give greater credence to the null 5.4.5. hypothesis of equality, such a conclusion should not be made without considering the test’s power function, especially with small sample sizes. Statistical power is the probability of rejecting the null hypothesis when it is not true. This probability is based on the value of the parameter assumed as the alternative and, with many alternatives, a power function can be constructed. This concept is discussed briefly in Chapter 2. In checking for differences between means in either direction, 5.4.6. Example 5.1 uses a two-tailed test. If interest is restricted to differences in only one direction, a one-tailed test can be applied in the same manner as described for the one-sample t-test (Chapter 4). The probability given in the SAS output, Pr>|t|, can be halved to obtain the corresponding one-tailed p-value.

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6$6&RGHIRU([DPSOH DATA GAD; INPUT PATNO DOSEGRP $ HAMA @@; DATALINES; 101 LO 21 104 LO 18 106 LO 19 110 LO . 112 LO 28 116 LO 22 120 LO 30 121 LO 27 124 LO 28 125 LO 19 130 LO 23 136 LO 22 137 LO 20 141 LO 19 143 LO 26 148 LO 35 152 LO . 103 HI 16 105 HI 21 109 HI 31 111 HI 25 113 HI 23 119 HI 25 123 HI 18 127 HI 20 128 HI 18 131 HI 16 135 HI 24 138 HI 22 140 HI 21 142 HI 16 146 HI 33 150 HI 21 151 HI 17 102 PB 22 107 PB 26 108 PB 29 114 PB 19 115 PB . 117 PB 33 118 PB 37 122 PB 25 126 PB 28 129 PB 26 132 PB . 133 PB 31 134 PB 27 139 PB 30 144 PB 25 145 PB 22 147 PB 36 149 PB 32 ; PROC SORT DATA = GAD; BY DOSEGRP; PROC MEANS MEAN STD N DATA = GAD; BY DOSEGRP; VAR HAMA; TITLE1 ’One-Way ANOVA’; TITLE2 ’EXAMPLE 6.1: HAM-A Scores in GAD’; RUN;

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PROC GLM DATA = GAD; CLASS DOSEGRP; MODEL HAMA = DOSEGRP; MEANS DOSEGRP/T DUNNETT(‘PB’); ñ CONTRAST ‘ACTIVE vs. PLACEBO’ DOSEGRP 0.5 0.5 –1; RUN;

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Mean Std Dev N ---------------------------------21.5882353 4.9630991 17 ------------------------------------------------------------ DOSEGRP=LO -----------------------------Analysis Variable : HAMA Mean Std Dev N ---------------------------------23.8000000 4.9742192 15 ------------------------------------------------------------ DOSEGRP=PB -----------------------------Analysis Variable : HAMA Mean Std Dev N ---------------------------------28.0000000 5.0332230 16 ----------------------------------

The GLM Procedure Class Level Information Class DOSEGRP

Levels 3

Values HI LO PB

Number of observations

52

NOTE: Due to missing values, only 48 observations can be used in this analysis.

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Source

Sum of Squares

DF

Model 2 Error 45 Corrected Total 47

Mean Square

347.149020 1120.517647 1467.666667

R-Square 0.236531

173.574510 24.900392

Coeff Var 20.43698

F Value

Pr > F

6.97

0.0023

Root MSE 4.990029

HAMA Mean 24.41667

Source DOSEGRP

DF 2

Type I SS 347.1490196

Mean Square 173.5745098

F Value 6.97

Pr > F 0.0023

Source DOSEGRP

DF 2

Type III SS 347.1490196

Mean Square 173.5745098

F Value 6.97

Pr > F 0.0023

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t Tests (LSD) for HAMA

NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.

Alpha 0.05 Error Degrees of Freedom 45 Error Mean Square 24.90039 Critical Value of t 2.01410

Comparisons significant at the 0.05 level are indicated by ***.

DOSEGRP Comparison PB PB LO LO HI HI

-

LO HI PB HI PB LO

Difference Between Means 4.200 6.412 -4.200 2.212 -6.412 -2.212

95% Confidence Limits 0.588 2.911 -7.812 -1.349 -9.912 -5.772

7.812 9.912 -0.588 5.772 -2.911 1.349

*** *** *** ***

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Dunnett’s t Tests for HAMA NOTE: This test controls the Type I experimentwise error for comparisons of all treatments against a control. Alpha 0.05 Error Degrees of Freedom 45 Error Mean Square 24.90039 Critical Value of Dunnett’s t 2.28361 Comparisons significant at the 0.05 level are indicated by ***.

DOSEGRP Comparison LO HI

- PB - PB

Difference Between Means -4.200 -6.412

Simultaneous 95% Confidence Limits -8.295 -10.381

-0.105 -2.443

*** ***

Dependent Variable: HAMA Contrast ACTIVE vs. PLACEBO

Mean Square

F Value

Pr > F

1 299.9001075

299.9001075

12.04

0.0012

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------------------------ TRT=ACT TYPE=CERVICAL ---------------------Mean Std Dev N ---------------------------------1.3125000 0.9876921 8 --------------------------------------------------------- TRT=ACT TYPE=PROSTATE ---------------------Mean Std Dev N ---------------------------------2.2000000 1.0042766 8 ------------------------------------------

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DF

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Root MSE

HGBCH Mean

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F Value

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DF 2 1 2

Source DF DOSE 2 CENTER 1 CENTER*DOSE 2

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Coeff Var 26.17421

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Y Mean 8.033898

Type I SS 256.6302710 3.1777983 39.7683260 Type III SS 251.4197307 2.2272995 39.7683260

Mean Square 128.3151355 3.1777983 19.8841630 Mean Square 125.7098653 2.2272995 19.8841630

F Value 29.02 0.72 4.50

F Value 28.43 0.50 4.50

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Y LSMEAN

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LSMEAN Number

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<.0001 <.0001 <.0001

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Example 8.1:

Repeated-Measures ANOVA Arthritic Discomfort Following Vaccine

Obs

VACGRP

PAT

VISIT

SCORE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

ACT ACT ACT ACT ACT ACT ACT ACT ACT ACT ACT ACT PBO PBO PBO PBO PBO PBO PBO PBO PBO PBO PBO PBO

101 101 101 103 103 103 104 104 104 107 107 107 102 102 102 105 105 105 106 106 106 108 108 108

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

6 3 0 7 3 1 4 1 2 8 4 3 6 5 5 9 4 6 5 3 4 6 2 3

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Example 8.1:

Repeated-Measures ANOVA Arthritic Discomfort Following Vaccine The GLM Procedure

Class Level Information Class

Levels

VACGRP PAT VISIT

Values

2 8 3

ACT PBO 101 102 103 104 105 106 107 108 1 2 3

Number of observations

24

Dependent Variable: SCORE DF

Sum of Squares

Model 11 Error 12 Corrected Total 23

103.1666667 12.1666667 115.3333333

Source

R-Square 0.894509

Source VACGRP PAT(VACGRP) VISIT VACGRP*VISIT

Source VACGRP

PAT(VACGRP) VISIT VACGRP*VISIT

Coeff Var 24.16609

Mean Square 9.3787879 1.0138889

Root MSE 1.006920

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Pr > F

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SCORE Mean 4.166667

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F Value

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0.0070

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Tests of Hypotheses Using the Type III MS for PAT(VACGRP) as an Error Term

Source

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Type III SS

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F Value

VACGRP

1

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2.53

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Repeated-Measures ANOVA Arthritic Discomfort Following Vaccine [Multivariate Approach]

Obs

VACGRP

PAT

MO1

MO2

MO3

1 2 3 4 5 6 7 8

ACT ACT ACT ACT PBO PBO PBO PBO

101 103 104 107 102 105 106 108

6 7 4 8 6 9 5 6

3 3 1 4 5 4 3 2

0 1 2 3 5 6 4 3

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Levels 2

Values ACT PBO

Number of observations 8 Dependent Variable: MO1 Sum of Source DF Squares Mean Square Model Error Corrected Total

1 6 7

R-Square 0.006993

0.12500000 17.75000000 17.87500000 Coeff Var 26.98009

0.12500000 2.95833333

Root MSE 1.719981

F Value

Pr > F

0.04

0.8439

MO1 Mean 6.375000

Source

DF

Type III SS

Mean Square

F Value

Pr > F

VACGRP

1

0.12500000

0.12500000

0.04

0.8439

Dependent Variable: MO2 Source Model Error Corrected Total

DF

Sum of Squares

1 6 7

1.12500000 9.75000000 10.87500000

R-Square 0.103448

Source VACGRP

Coeff Var 40.79216

DF 1

Type III SS 1.12500000

Mean Square

F Value

Pr > F

1.12500000 1.62500000

0.69

0.4372

Root MSE 1.274755

MO2 Mean 3.125000

Mean Square 1.12500000

F Value 0.69

Pr > F 0.4372

Dependent Variable: MO3

Source Model Error Corrected Total

DF

Sum of Squares

1 6 7

18.00000000 10.00000000 28.00000000

R-Square 0.642857

Coeff Var 43.03315

Mean Square

F Value

Pr > F

18.00000000 1.66666667

10.80

0.0167

Root MSE 1.290994

MO3 Mean 3.000000

Source

DF

Type III SS

Mean Square

F Value

Pr > F

VACGRP

1

18.00000000

18.00000000

10.80

0.0167

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Repeated-Measures ANOVA Arthritic Discomfort Following Vaccine [Multivariate Approach] The GLM Procedure

Repeated Measures Analysis of Variance Repeated Measures Level Information Dependent Variable Level of VISIT

MO1 1

MO2 2

MO3 3

Sphericity Tests

Variables Transformed Variates Orthogonal Components

DF

Mauchly’s Criterion

2 2

0.8962875 0.9547758

Chi-Square

Pr > ChiSq

0.5474703 0.2313939

0.7605 0.8907

ù

Manova Test Criteria and Exact F Statistics for the Hypothesis of no VISIT Effect H = Type III SSCP Matrix for VISIT E = Error SSCP Matrix S=1

M=0

Statistic

Value

Wilks’ Lambda Pillai’s Trace Hotelling-Lawley Trace Roy’s Greatest Root

0.10207029 0.89792971 8.79716981 8.79716981

11

N=1.5

F Value

Num DF

21.99 21.99 21.99 21.99

Den DF

2 2 2 2

5 5 5 5

Manova Test Criteria and Exact F Statistics for the Hypothesis of no VISIT*VACGRP Effect H = Type III SSCP Matrix for VISIT*VACGRP

Pr > F 0.0033 0.0033 0.0033 0.0033

12

E = Error SSCP Matrix S=1 Statistic Wilks’ Lambda Pillai’s Trace Hotelling-Lawley Trace Roy’s Greatest Root

M=0

N=1.5

Value

F Value

0.44444444 0.55555556 1.25000000 1.25000000

3.13 3.13 3.13 3.13

Num DF 2 2 2 2

Den DF 5 5 5 5

Pr > F 0.1317 0.1317 0.1317 0.1317

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2873876$62XWSXWIRU([DPSOH±µ0XOWLYDULDWH¶$SSURDFK FRQWLQXHG Example 8.1:

Repeated-Measures ANOVA Arthritic Discomfort Following Vaccine [Multivariate Approach] The GLM Procedure

Repeated Measures Analysis of Variance Tests of Hypotheses for Between Subjects Effects Source

DF

Type III SS

Mean Square

VACGRP Error

1 6

10.66666667 25.33333333

10.66666667 4.22222222

F Value

Pr > F

2.53 13

0.1631

Univariate Tests of Hypotheses for Within Subject Effects Source

DF

Type III SS

Mean Square

F Value

Pr > F

VISIT VISIT*VACGRP Error(VISIT)

2 2 12

58.58333333 8.58333333 12.16666667

29.29166667 4.29166667 1.01388889

28.89 4.23

<.0001 0.0406

Source VISIT VISIT*VACGRP Error(VISIT)

Adj Pr > F G - G H - F <.0001 0.0433

Greenhouse-Geisser Epsilon Huynh-Feldt Epsilon

<.0001 0.0406

0.9567 14 1.6282

Analysis of Variance of Contrast Variables VISIT_N represents the nth successive difference in VISIT Contrast Variable: VISIT_1 Source

DF

Type III SS

Mean Square

F Value

Mean VACGRP Error

1 1 6

84.50000000 0.50000000 11.00000000

84.50000000 0.50000000 1.83333333

46.09 0.27

Pr > F 0.0005 0.6202 15

Contrast Variable: VISIT_2

Source

DF

Type III SS

Mean Square

F Value

Mean VACGRP Error

1 1 6

0.12500000 10.12500000 10.75000000

0.12500000 10.12500000 1.79166667

0.07 5.65

Pr > F 0.8005 0.0550 15

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Example 8.2:

Repeated-Measures ANOVA Treadmill Walking Distance in Intermittent Claudication The GLM Procedure Class Level Information Class Levels Values TREATMNT CENTER

2 2

ACT PBO 1 2

Number of observations

38

Dependent Variable: WD0 Source

DF

Model Error Corrected Total

3 34 37

R-Square 0.034023 Source TREATMNT CENTER TREATMNT*CENTER

Sum of Squares Mean Square 2190.13085 62181.76389 64371.89474

730.04362 1828.87541

F Value

Pr > F

0.40

0.7545

Coeff Var 25.01668

Root MSE 42.76535

WD0 Mean 170.9474

DF

Type III SS

Mean Square

F Value

1 1 1

0.280018 45.591846 2088.387545

0.280018 45.591846 2088.387545

0.00 0.02 1.14

Pr > F 0.9902 0.8755 18 0.2928

Dependent Variable: WD1 Source

DF

Model Error Corrected Total

3 34 37

R-Square 0.071590 Source TREATMNT CENTER TREATMNT*CENTER

Sum of Squares Mean Square 3825.59284 49611.98611 53437.57895

1275.19761 1459.17606

F Value

Pr > F

0.87

0.4642

Coeff Var 21.59429

Root MSE 38.19916

WD1 Mean 176.8947

DF

Type III SS

Mean Square

F Value

1 1 1

3215.520161 788.946685 201.720878

3215.520161 788.946685 201.720878

2.20 0.54 0.14

Pr > F 0.1469 0.4672 18 0.7123

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2873876$62XWSXWIRU([DPSOHFRQWLQXHG Example 8.2:

Repeated-Measures ANOVA Treadmill Walking Distance in Intermittent Claudication The GLM Procedure

Dependent Variable: WD2 Source

DF

Sum of Squares

Model Error Corrected Total

3 34 37

7128.96857 74577.34722 81706.31579

R-Square 0.087251 Source TREATMNT CENTER TREATMNT*CENTER

Mean Square

F Value

Pr > F

2376.32286 2193.45139

1.08

0.3693

Coeff Var 24.88399

Root MSE 46.83430

WD2 Mean 188.2105

DF

Type III SS

Mean Square

F Value

1 1 1

6860.021953 504.355287 225.215502

6860.021953 504.355287 225.215502

3.13 0.23 0.10

Pr > F 0.0860 0.6346 18 0.7506

Dependent Variable: WD3 Source

DF

Model Error Corrected Total

3 34 37

R-Square 0.115061 Source TREATMNT CENTER TREATMNT*CENTER

Sum of Squares 7086.08918 54499.30556 61585.39474

Mean Square 2362.02973 1602.92075

F Value

Pr > F

1.47

0.2391

Coeff Var 20.90104

Root MSE 40.03649

WD3 Mean 191.5526

DF

Type III SS

Mean Square

F Value

1 1 1

3542.716846 2512.286738 771.211470

3542.716846 2512.286738 771.211470

2.21 1.57 0.48

Pr > F 0.1463 0.2191 18 0.4926

Dependent Variable: WD4 Source

DF

Model Error Corrected Total

3 34 37

R-Square 0.113554 Source TREATMNT CENTER TREATMNT*CENTER

Sum of Squares 8437.19591 65863.77778 74300.97368

Mean Square 2812.39864 1937.16993

F Value

Pr > F

1.45

0.2450

Coeff Var 22.45275

Root MSE 44.01329

WD4 Mean 196.0263

DF

Type III SS

Mean Square

F Value

1 1 1

5975.405018 1141.225806 1319.290323

5975.405018 1141.225806 1319.290323

3.08 0.59 0.68

Pr > F 0.0880 0.4481 18 0.4150

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2873876$62XWSXWIRU([DPSOHFRQWLQXHG Example 8.2:

Repeated-Measures ANOVA Treadmill Walking Distance in Intermittent Claudication The GLM Procedure

Repeated Measures Level Information Dependent Variable Level of MONTH

WD0 1

WD1 2

WD2 3

WD3 4

WD4 5

Partial Correlation Coefficients from the Error SSCP Matrix / Prob > |r| DF = 34

WD0

WD1

WD2

WD3

WD4

WD0

1.000000

0.881446 <.0001

0.778029 <.0001

0.781755 <.0001

0.742002 <.0001

WD1

0.881446 <.0001

1.000000

0.703789 <.0001

0.663442 <.0001

0.754124 <.0001

WD2

0.778029 <.0001

0.703789 <.0001

1.000000

0.695239 <.0001

0.724997 <.0001

WD3

0.781755 <.0001

0.663442 <.0001

0.695239 <.0001

1.000000

0.826616 <.0001

WD4

0.742002 <.0001

0.754124 <.0001

0.724997 <.0001

0.826616 <.0001

1.000000

Sphericity Tests

Variables Transformed Variates Orthogonal Components

DF

Mauchly’s Criterion

Chi-Square

9 9

0.3190188 0.5139265

37.036211 21.578963

Pr > ChiSq <.0001 0.0103 19

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2873876$62XWSXWIRU([DPSOHFRQWLQXHG Example 8.2:

Repeated-Measures ANOVA Treadmill Walking Distance in Intermittent Claudication Manova Test Criteria and Exact F Statistics for the Hypothesis of no MONTH Effect H = Type III SSCP Matrix for MONTH E = Error SSCP Matrix S=1

Statistic Wilks’ Lambda Pillai’s Trace Hotelling-Lawley Trace Roy’s Greatest Root

M=1

Value 0.56579676 0.43420324 0.76741911 0.76741911

N=14.5

F Value 5.95 5.95 5.95 5.95

Num DF 4 4 4 4

Den DF 31 31 31 31

Pr > F 0.0011 0.0011 23 0.0011 0.0011

Manova Test Criteria and Exact F Statistics for the Hypothesis of no MONTH*TREATMNT Effect H = Type III SSCP Matrix for MONTH*TREATMNT E = Error SSCP Matrix S=1 Statistic Wilks’ Lambda Pillai’s Trace Hotelling-Lawley Trace Roy’s Greatest Root

M=1

Value 0.68930532 0.31069468 0.45073593 0.45073593

N=14.5

F Value 3.49 3.49 3.49 3.49

Num DF 4 4 4 4

Den DF 31 31 31 31

Pr > F 0.0182 0.0182 20 0.0182 0.0182

Manova Test Criteria and Exact F Statistics for the Hypothesis of no MONTH*CENTER Effect H = Type III SSCP Matrix for MONTH*CENTER E = Error SSCP Matrix S=1 Statistic Wilks’ Lambda Pillai’s Trace Hotelling-Lawley Trace Roy’s Greatest Root

M=1

Value 0.81294393 0.18705607 0.23009714 0.23009714

N=14.5

F Value 1.78 1.78 1.78 1.78

Num DF 4 4 4 4

Den DF 31 31 31 31

Pr > F 0.1574 0.1574 0.1574 0.1574

Manova Test Criteria and Exact F Statistics for the Hypothesis of no MONTH*TREATMNT*CENTER Effect H = Type III SSCP Matrix for MONTH*TREATMNT*CENTER E = Error SSCP Matrix S=1 Statistic Wilks’ Lambda Pillai’s Trace Hotelling-Lawley Trace Roy’s Greatest Root

M=1

Value 0.88498645 0.11501355 0.12996081 0.12996081

N=14.5

F Value 1.01 1.01 1.01 1.01

Num DF 4 4 4 4

Den DF 31 31 31 31

Pr > F 0.4188 0.4188 0.4188 0.4188

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2873876$62XWSXWIRU([DPSOHFRQWLQXHG Example 8.2:

Repeated-Measures ANOVA Treadmill Walking Distance in Intermittent Claudication

The GLM Procedure Repeated Measures Analysis of Variance Tests of Hypotheses for Between Subjects Effects Source TREATMNT CENTER TREATMNT*CENTER Error

DF 1 1 1 34

Type III SS 15215.6775 141.5815 3864.2911 245350.6639

Mean Square 15215.6775 141.5815 3864.2911 7216.1960

F Value 2.11 0.02 0.54

Pr > F 0.1556 21 0.8894 0.4693 22

Analysis of Variance of Contrast Variables MONTH_N represents the contrast between the nth level of MONTH and the 1st Contrast Variable: MONTH_2 Source DF Type III SS

Mean Square

F Value

1035.12545 3275.81362 455.22581 992.00000 13878.44444

1035.12545 3275.81362 455.22581 992.00000 408.18954

2.54 8.03 1.12 2.43

Contrast Variable: MONTH_3 Source DF Type III SS

Mean Square

F Value

9854.88351 6947.95878 246.66846 941.98029 30794.47222

9854.88351 6947.95878 246.66846 941.98029 905.71977

10.88 7.67 0.27 1.04

Contrast Variable: MONTH_4 Source DF Type III SS

Mean Square

F Value

Mean TREATMNT CENTER TREATMNT*CENTER Error

13127.49507 3605.98970 3234.75314 321.41980 25663.01389

13127.49507 3605.98970 3234.75314 321.41980 754.79453

17.39 4.78 4.29 0.43

Contrast Variable: MONTH_5 Source DF Type III SS

Mean Square

F Value

Mean TREATMNT CENTER TREATMNT*CENTER Error

20601.23701 6057.49507 1643.02195 87.92518 972.78717

Mean TREATMNT CENTER TREATMNT*CENTER Error

Mean TREATMNT CENTER TREATMNT*CENTER Error

1 1 1 1 34

1 1 1 1 34

1 1 1 1 34

1 1 1 1 34

20601.23701 6057.49507 1643.02195 87.92518 33074.76389

21.18 6.23 1.69 0.09

Pr > F 0.1205 0.0077 24 0.2984 0.1283

Pr > F 0.0023 0.0090 24 0.6051 0.3150

Pr > F 0.0002 0.0358 24 0.0461 0.5184

Pr > F <.0001 0.0176 24 0.2025 0.7655

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PAT ADAS02 ADAS04 ADAS06 ADAS08 ADAS10 ADAS12; 33 37 46

28 31 52

30 35 48

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ODS SELECT GLM.Repeated.PartialCorr GLM.Repeated.MANOVA.Model.Error.Sphericity GLM.Repeated.MANOVA.Model.MONTH.MultStat GLM.Repeated.MANOVA.Model.MONTH_TREAT.MultStat GLM.Repeated.WithinSubject.ModelANOVA; PROC GLM DATA = ALZHMRS; CLASS TREAT; MODEL ADAS02 ADAS04 ADAS06 ADAS08 ADAS10 ADAS12 = TREAT / NOUNI; 26 REPEATED MONTH / PRINTE SUMMARY; TITLE1 ’Repeated-Measures ANOVA’; TITLE2 ‘Example 8.3: Disease Progression in Alzheimer’s; TITLE3 ' '; TITLE4 'Multivariate Approach Using GLM'; QUIT; RUN;

25

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µXQLYDULDWH¶ WHVW IRU WKH 0RQWKE\7UHDWPHQW LQWHUDFWLRQ 29 SURYLGHV WKH UHVXOW S \RXZRXOGJHWE\XVLQJWKHµXQLYDULDWH¶DSSURDFKWRUHSHDWHGPHDVXUHV DIWHU H[FOXGLQJ DOO SDWLHQWV ZKR KDYH RQH RU PRUH PLVVLQJ YDOXHV DQG DVVXPLQJ FRPSRXQGV\PPHWU\%HFDXVHLWZDVDOUHDG\FRQFOXGHGWKDWWKLVZRXOGQRWEHDQ DSSURSULDWHPHWKRGRIDQDO\VLV\RXPLJKWORRNDWWKH*UHHQKRXVH*HLVHUDGMXVWHG YDOXHV VHH 6HFWLRQ 7KH SYDOXH IRU WKH LQWHUDFWLRQ HIIHFW XVLQJ WKH *UHHQKRXVH*HLVHUDGMXVWPHQWLV 30 2873876$62XWSXWIRU([DPSOH±352&*/0 Repeated-Measures ANOVA Example 8.3: Disease Progression in Alzheimer’s Multivariate Approach Using GLM The GLM Procedure Repeated Measures Analysis of Variance

Partial Correlation Coefficients from the Error SSCP Matrix / Prob > |r| DF = 53

ADAS02

ADAS04

ADAS06

ADAS08

ADAS10

ADAS12

ADAS02

1.000000

0.892847 <.0001

0.715396 <.0001

0.743266 <.0001

0.734904 <.0001

0.703999 <.0001

ADAS04

0.892847 <.0001

1.000000

0.750344 <.0001

0.745911 <.0001

0.731188 <.0001

0.653269 <.0001

ADAS06

0.715396 <.0001

0.750344 <.0001

1.000000

0.852443 <.0001

0.690418 <.0001

0.690489 <.0001

ADAS08

0.743266 <.0001

0.745911 <.0001

0.852443 <.0001

1.000000

0.847485 <.0001

0.784974 <.0001

ADAS10

0.734904 <.0001

0.731188 <.0001

0.690418 <.0001

0.847485 <.0001

1.000000

0.900155 <.0001

ADAS12

0.703999 <.0001

0.653269 <.0001

0.690489 <.0001

0.784974 <.0001

0.900155 <.0001

1.000000

Sphericity Tests 27

Variables

DF

Mauchly’s Criterion

Transformed Variates Orthogonal Components

14 14

0.022696 0.2211441

Chi-Square

Pr > ChiSq

193.4424 77.106867

<.0001 <.0001

42

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Manova Test Criteria and Exact F Statistics for the Hypothesis of no MONTH Effect H = Type III SSCP Matrix for MONTH E = Error SSCP Matrix S=1 Statistic Wilks’ Lambda Pillai’s Trace Hotelling-Lawley Trace Roy’s Greatest Root

M=1.5

N=23.5

Value

F Value

Num DF

Den DF

Pr > F

0.38811297 0.61188703 1.57656938 1.57656938

15.45 15.45 15.45 15.45

5 5 5 5

49 49 49 49

<.0001 <.0001 <.0001 <.0001

Manova Test Criteria and F Approximations for the Hypothesis of no MONTH*TREAT Effect H = Type III SSCP Matrix for MONTH*TREAT E = Error SSCP Matrix S=2 Statistic Wilks’ Lambda Pillai’s Trace Hotelling-Lawley Trace Roy’s Greatest Root

M=1

N=23.5

Value

F Value

Num DF

1.18 1.18 1.18 1.96

10 10 10 5

0.79719596 0.21044586 0.24481085 0.19587117

Den DF

98 100 70.804 50

Pr > F 0.3163 0.3160 0.3162 0.1012

NOTE: F Statistic for Roy’s Greatest Root is an upper bound. NOTE: F Statistic for Wilks’ Lambda is exact.

Univariate Tests of Hypotheses for Within Subject Effects Source MONTH MONTH*TREAT Error(MONTH)

DF

Type III SS

Mean Square

F Value

5 10 265

1529.760451 222.315729 4366.776533

305.952090 22.231573 16.478402

18.57 1.35

Source MONTH MONTH*TREAT Error(MONTH)

Pr > F <.0001 0.2044 29

Adj Pr > F G - G H - F <.0001 0.2337

<.0001 0.2271

30

28

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OUTPUT; OUTPUT; OUTPUT; OUTPUT; OUTPUT; OUTPUT;

ODS SELECT Mixed.RCorr Mixed.Dimensions Mixed.FitStatistics Mixed.Tests3;

PROC MIXED DATA = UNIALZ; CLASS TREAT MONTH PAT; MODEL ADASCOG = TREAT MONTH TREAT*MONTH; 31 REPEATED / TYPE=UN SUBJECT=PAT(TREAT) RCORR; 32 TITLE4 ’PROC MIXED Using Unstructured Covariance (UN)’; RUN;

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21 28 0 80 6 454 26 480

Estimated R Correlation Matrix for PAT(TREAT) 2 H 41

Row 1 2 3 4 5 6

Col1

Col2

Col3

Col4

Col5

Col6

1.0000 0.9005 0.7703 0.8002 0.7763 0.7773

0.9005 1.0000 0.8179 0.7995 0.7521 0.7261

0.7703 0.8179 1.0000 0.8738 0.7223 0.7418

0.8002 0.7995 0.8738 1.0000 0.8628 0.8273

0.7763 0.7521 0.7223 0.8628 1.0000 0.9206

0.7773 0.7261 0.7418 0.8273 0.9206 1.0000

34

Fit Statistics -2 Res Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better)

2649.9 2691.9 2694.1 2741.9

Type 3 Tests of Fixed Effects

Effect TREAT MONTH TREAT*MONTH

Num DF

Den DF

F Value

Pr > F

2 5 10

77 77 77

0.94 21.55 2.08

0.3967 <.0001 0.0357

33

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… 37 REPEATED / TYPE=ARMA(1,1) SUBJECT=PAT(TREAT); TITLE4 'PROC MIXED Using 1st-Order Auto-Regressive Moving Average Covariance (ARMA(1,1))';

… 38 REPEATED / TYPE=CS SUBJECT=PAT(TREAT); TITLE4 'PROC MIXED Using Compound Symmetric Covariance (CS)'; RUN;

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Repeated-Measures ANOVA Example 8.3: Disease Progression in Alzheimer’s PROC MIXED Using First-Order Auto-Regressive Covariance (AR(1)) Fit Statistics -2 Res Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better)

2698.2 2702.2 2702.2 2706.9

Type 3 Tests of Fixed Effects 35

Effect TREAT MONTH TREAT*MONTH

Num DF

Den DF

F Value

Pr > F

2 5 10

77 359 359

0.94 11.21 1.43

0.3952 <.0001 0.1671

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2873876$62XWSXWIRU([DPSOH±352&0,;('FRQWLQXHG Repeated-Measures ANOVA Example 8.3: Disease Progression in Alzheimer’s PROC MIXED Using Toeplitz Covariance (TOEP) Fit Statistics -2 Res Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better)

2671.1 2683.1 2683.3 2697.4

Type 3 Tests of Fixed Effects

36

Effect TREAT MONTH TREAT*MONTH

Num DF

Den DF

F Value

Pr > F

2 5 10

77 359 359

0.91 19.65 2.06

0.4077 <.0001 0.0266

PROC MIXED Using 1st-Order Auto-Regressive-Moving Average Covariance (ARMA(1,1)) Fit Statistics -2 Res Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better)

2691.2 2697.2 2697.2 2704.3

Type 3 Tests of Fixed Effects 37

Num Effect

Den DF

TREAT MONTH TREAT*MONTH

2 5 10

DF

F Value

77 359 359

0.90 14.02 1.47

Pr > F 0.4116 <.0001 0.1478

PROC MIXED Using Compound Symmetric Covariance (CS) Fit Statistics -2 Res Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better)

2743.5 2747.5 2747.5 2752.2

Type 3 Tests of Fixed Effects 38

Effect

DF

Num DF

TREAT MONTH TREAT*MONTH

2 5 10

77 359 359

Den F Value

Pr > F

0.88 26.69 2.30

0.4172 <.0001 0.0125

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2 5 10

2.02 46.14 21.76

0.3639 <.0001 0.0164

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CHHAAPPTTEERR 9 The Crossover Design 9.1 9.2 9.3 9.4

Introduction...................................................................................................... 157 Synopsis ............................................................................................................ 158 Examples........................................................................................................... 160 Details & Notes................................................................................................. 171

9.1

Introduction The crossover design is used to compare the mean responses of two or more treatments when each patient receives each treatment over successive time periods. Typically, patients are randomized to treatment sequence groups which determine the order of treatment administration. While responses among treatments in a parallel study are considered independent, responses among treatments in a crossover study are correlated because they are measured on the same patient, much like the repeated measures setup discussed in Chapter 8. In fact, the crossover is a special case of a repeated measures design. Procedures for treatment comparisons must consider this correlation when analyzing data from a crossover design. In addition to treatment comparisons, the crossover analysis can also investigate period effects, sequence effects, and carryover effects. Because the crossover study features within-patient control among treatment groups, fewer patients are generally required than with a parallel study. This advantage is often offset, however, by an increase in the length of the study. For example, when there are two treatments, the study will be at least twice as long as its parallel-design counterpart because treatments are studied in each of two successive periods. The addition of a wash-out period between treatments, which is usually required in a crossover study, tends to further lengthen the trial duration. The crossover design is normally used in studies with short treatment periods, most frequently in pre-clinical and early-phase clinical studies, such as bioavailability, dose-ranging, bio-equivalence, and pharmacokinetic trials. If it’s thought that the patient’s condition will not be the same at the beginning of each treatment period, the crossover design should be avoided. Lengthier treatment periods provide

158

Common Statistical Methods for Clinical Research with SAS Examples

increased opportunities for changing clinical conditions and premature termination. For the same reasons, crossover designs should limit the number of study periods as much as possible. The focus in this chapter is on the 2-period crossover design, although an example that uses 4 periods is also presented. Responses are assumed to be normally distributed.

9.2

Synopsis For the two-way crossover design, each patient receives two treatments, A and B, in one of two sequences, that is, A followed by B (A-B) or B followed by A (B-A). Samples of n1 and n2 patients are randomly assigned to each of the two sequence groups, A-B and B-A, respectively (usually, n1 = n2). Response measurements (yijk) are taken for each patient following each treatment for a total th of N = 2n1 + 2n2 measurements, where yijk is the measurement for the k patient in the sequence group for which Treatment i (i = 1 for Treatment A, i = 2 for Treatment B) is given in Period j (j = 1, 2). A sample layout is shown in Table 9.1. TABLE 9.1. Crossover Layout Sequence

Patient Number

Period 1

Period 2

(Treatment A)

(Treatment B)

A-B

1 3 6 7 . . .

y111 y112 y113 y114 . . .

y221 y222 y223 y224 . . .

B-A

2 4 5 8 . . .

(Treatment B) y211 y212 y213 y214 . . .

(Treatment A) y121 y122 y123 y124 . . .

The data are analyzed using analysis of variance methods with the following factors: Treatment Group, Period, Sequence, and Patient-within-Sequence. Computations proceed as shown in Chapters 6, 7, and 8 to obtain the ANOVA summary shown in Table 9.2.

Chapter 9: The Crossover Design 159

TABLE 9.2. ANOVA Summary Table for a 2-Period Crossover Source

df

SS

MS

F

Treatment (T)

1

SST

MST

FT = MST / MSE

Period (P)

1

SSP

MSP

FP = MSP / MSE

Sequence (S)

1

SSS

MSS

FS = MSS / MSP(S)

Patient-withinSeqence (P(S))

n1+n2–2

SSP(S)

MSP(S)

Error

n1+n2–2

SSE

MSE

Total

N–1

TOT(SS)

The Treatment sum of squares, SST, is proportional to the sum of squared deviations of each Treatment mean from the overall mean. Likewise, SSP is computed from the sum of squared deviations of each Period mean from the overall mean, etc. The mean squares (MS) are found by dividing the sums of squares (SS) by their respective degrees of freedom (df). Computations are shown in Example 9.1. The hypothesis of equal treatment means is tested at a two-tailed significance level, α, by comparing the F-test statistic, FT, with the critical F-value based on 1 upper 1 and n1+n2–2 lower degrees of freedom, denoted by Fn1 + n 2 − 2 (α). The test summary for no treatment effect is

null hypothesis: alt. hypothesis:

H0: µA = µB HA: mA =/ mB

test statistic:

MST FT = ____ MSE

decision rule:

1 reject H0 if FT > Fn1 + n 2 − 2 (α)

The Period and Sequence effects can also be tested in a similar manner. As in the repeated measures ANOVA (Chapter 8), the test for Sequence effect uses the mean square for Patient-within-Sequence in the denominator because patients are nested within sequence groups. When the hypothesis of no Sequence Group effect is true, variation between sequence groups simply reflects patient-to-patient variation.

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Common Statistical Methods for Clinical Research with SAS Examples

9.3

Examples

p Example 9.1 -- Diaphoresis Following Cardiac Medication

A preliminary study was conducted on 16 normal subjects to examine the time to perspiration following administration of a single dose of a new cardiac medication with known diaphoretic effects. Subjects were asked to walk on a treadmill at the clinic following dosing with the new medication (A), and again on a separate clinic visit following placebo (B). A two-period crossover design was used with 8 subjects randomized to each of the two sequence groups. Time in minutes until the appearance of perspiration beads on the forehead was recorded as shown in Table 9.3. Is there any difference in perspiration times? TABLE 9.3. Raw Data for Example 9.1 Sequence A-B

B-A

Subject Number 1 3 5 6 9 10 13 15

2 4 7 8 11 12 14 16

Period 1

Period 2

(A)

(B)

6 8 12 7 9 6 11 8

4 7 6 8 10 4 6 8

(B)

(A)

7 6 11 7 8 4 9 13

5 9 7 4 9 5 8 9

Solution Using the two-way crossover ANOVA, you want to compare mean perspiration times between treatment groups, controlling for treatment period and sequence (order) of administration. For a balanced design, the sum of squares for each of these sources can be computed in the same manner as demonstrated previously (Chapters 6, 7, 8). First, obtain the marginal means, as shown in Table 9.4.

Chapter 9: The Crossover Design 161

TABLE 9.4. Summary of Marginal Means for Example 9.1 Sequence

Subject Number

Subject Means

Period 1

Period 2

(A)

(B)

6 8 12 7 9 6 11 8

4 7 6 8 10 4 6 8

5.0 7.5 9.0 7.5 9.5 5.0 8.5 8.0

8.375

6.625

7.5000

(B) 5 9 7 4 9 5 8 9

(A) 7 6 11 7 8 4 9 13

6.0 7.5 9.0 5.5 8.5 4.5 8.5 11.0

Sequence Means

7.000

8.125

7.5625

Overall Means

7.6875

7.3750

7.53125

A-B

1 3 5 6 9 10 13 15

Sequence Means B-A

2 4 7 8 11 12 14 16

Treatment A Mean: 8.2500

Treatment B Mean: 6.8125

Now, you can compute the sum of squares as follows: 2

SS(Treatment)

= (16 · (8.2500 – 7.53125) ) + 2 (16 · (6.8125 – 7.53125) ) = 16.53125

SS(Period)

= (16 · (7.6875 – 7.53125) ) + 2 (16 · (7.3750 – 7.53125) ) = 0.78125

SS(Sequence)

= (16 · (7.5000 – 7.53125) ) + 2 (16 · (7.5625 – 7.53125) ) = 0.03125

TOTAL(SS)

= ((6 – 7.53125) + 2 (8 – 7.53125) + … + 2 (13 – 7.53125) ) = 167.96875

2

2

2

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Common Statistical Methods for Clinical Research with SAS Examples

SS(Patients-within-Sequence) 2

2

2

= ((5.0 – 7.5000) + (7.5 – 7.5000) + … + ( 8.0 – 7.5000) + 2 2 2 (6.0 – 7.5625) + (7.5 – 7.5625) + … + (11.0 – 7.5625) ) = 103.4375 Because this is a balanced design, the error sum of squares can be found by subtraction, SS(Error)

= TOTAL(SS) – (SS(TRT) + SS(PD) + SS(SEQ) + SS(PAT(SEQ))) = 167.96875 – (16.53125 + 0.78125 + 0.03125 + 103.4375) = 47.1875

Completing the ANOVA table, you obtain Table 9.5. ANOVA Summary for Example 9.1 Source

df

SS

MS

F

Treatment (T)

1

16.53125

16.53125

4.90

Period (P)

1

0.78125

0.78125

0.23

Sequence (S)

1

0.03125

0.03125

0.004

Patient-within Sequence (P(S))

14

103.4375

7.38839

Error

14

47.1875

3.37054

Total

31

167.96875

The test summary for Treatment effect becomes null hypothesis:

H0: mA = mB

alt. hypothesis:

HA: mA =/ mB

test statistic:

FT = 4.90

decision rule:

reject H0 if FT > F14 (0.05) = 4.60

conclusion:

Because 4.90 > 4.60, you reject H0 and conclude that the new medication and placebo have different diaphoretic profiles, based on a significance level of a = 0.05.

1

Chapter 9: The Crossover Design 163

SAS Analysis of Example 9.1 The SAS code for analyzing this data set and resulting output are shown below. All four sources of variation (Sequence Group, Treatment, Period, and Patient) must appear in the CLASS statement in PROC GLM. Note that the MODEL statement in PROC GLM is very similar to that of the repeated measures ANOVA (Chapter 8) . The Sequence Group (SEQ) represents the two randomized groups, and patients are nested within Sequence Group. Therefore, a TEST statement is needed to conduct the appropriate F-test for the Sequence Group effect using the PAT(SEQ) as the error term . The SS3 option in the MODEL statement requests printing the SAS Type III results only. Because this data set is balanced, the Type I and III results would be identical. The ANOVA results shown in Output 9.1 agree with the manual calculations. The Treatment effect has a significant p-value of 0.0439 , which

indicates a departure from the null hypothesis of equal treatment means. Neither the Sequence effect (which tests for differences in the order of administration) nor the Period effect (which tests for differences between periods) are significant (p=0.9491 and 0.6376, respectively).

SAS Code for Example 9.1 DATA XOVER; INPUT PAT SEQ DATALINES; 1 AB A 1 6 3 9 AB A 1 9 10 1 AB B 2 4 3 9 AB B 2 10 10 2 BA A 2 7 4 11 BA A 2 8 12 2 BA B 1 5 4 11 BA B 1 9 12 ;

$ TRT $ PD Y @@; AB AB AB AB BA BA BA BA

A A B B A A B B

1 1 2 2 2 2 1 1

8 6 7 4 6 4 9 5

5 13 5 13 7 14 7 14

AB AB AB AB BA BA BA BA

A A B B A A B B

1 12 1 11 2 6 2 6 2 11 2 9 1 7 1 8

6 15 6 15 8 16 8 16

AB AB AB AB BA BA BA BA

A A B B A A B B

1 7 1 8 2 8 2 8 2 7 2 13 1 4 1 9

PROC GLM DATA = XOVER; CLASS SEQ TRT PD PAT; MODEL Y = SEQ PAT(SEQ) TRT PD / SS3; TEST H=SEQ E=PAT(SEQ); TITLE1 'Crossover Design'; TITLE2 ‘Example 9.1: Diaphoresis Following Cardiac Medication’; RUN;

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Common Statistical Methods for Clinical Research with SAS Examples

OUTPUT 9.1. SAS Output for Example 9.1 Crossover Design Example 9.1: Diaphoresis Following Cardiac Medication The GLM Procedure Class Level Information Class

Levels

SEQ TRT PD PAT

Values

2 2 2 16

AB BA A B 1 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Number of observations

32

Dependent Variable: Y Source

DF

Sum of Squares

Mean Square

F Value

Pr > F

Model

17

120.7812500

7.1047794

2.11

0.0824

Error

14

47.1875000

3.3705357

Corrected Total

31

167.9687500

Source SEQ PAT(SEQ) TRT PD

R-Square

Coeff Var

Root MSE

Y Mean

0.719070

24.37712

1.835902

7.531250

DF 1 14 1 1

Type III SS 0.0312500 103.4375000 16.5312500 0.7812500

Mean Square

F Value

Pr > F

0.0312500 7.3883929 16.5312500 0.7812500

0.01 2.19 4.90 0.23

0.9247 0.0771 0.0439 0.6376

Tests of Hypotheses Using the Type III MS for PAT(SEQ) as an Error Term Source SEQ

DF

Type III SS

Mean Square

F Value

Pr > F

1

0.03125000

0.03125000

0.00

0.9491

The next example illustrates the analysis of a 4-period crossover design, including a test for carryover effect using SAS.

Chapter 9: The Crossover Design 165

p Example 9.2 -- Antibiotic Blood Levels Following Aerosol Inhalation

Normal subjects were enrolled in a pilot study to compare blood levels among 4 doses (A, B, C, and D) of an antibiotic using a new aerosol delivery formulation. The response measure is area-under-the-curve (AUC) up to 6 hours after dosing. A 4-period crossover design was used with a 3-day washout period between doses. Three subjects were randomized to each of four sequence groups, as shown in Table 9.6 along with the data (given as logAUC’s). Are there any differences in blood levels among the 4 doses? TABLE 9.6. Raw Data for Example 9.2 Dosing Sequence

Subject Number

Period 1

Period 2

Period 3

Period 4

A-B-D-C

102

2.31

3.99

11.75

4.78

106

3.95

2.07

7.00

4.20

109

4.40

6.40

9.76

6.12

104

6.81

8.38

1.26

10.56

105

9.05

6.85

4.79

4.86

111

7.02

5.70

3.14

7.65

101

6.00

4.79

2.35

3.81

108

5.25

10.42

5.68

4.48

112

2.60

6.97

3.60

7.54

103

8.15

3.58

8.79

4.94

107

12.73

5.31

4.67

5.84

110

6.46

2.42

4.58

1.37

B-C-A-D

C-D-B-A

D-A-C-B

Solution Treatment, Period, and Sequence effects are computed in the same way as presented for Example 9.1. Going directly to the SAS analysis of these data, a test for any treatment carryover effects is included. The carryover effect is somewhat entangled with the treatment effect. To separate them, you need to compute both the adjusted and unadjusted treatment effects and include a carry-over effect in the ANOVA model. In SAS, this can be done by obtaining both the Type I and Type III results from PROC GLM, as shown in the SAS analysis that follows.

166

Common Statistical Methods for Clinical Research with SAS Examples

SAS Analysis of Example 9.2 The data are input in data lines according to Sequence Group (SEQGRP) and Patient (PAT), as shown in the INHAL1 data set in the SAS code for Example 9.2. The Dose Group (DOSE) and Period (PER) variables are added in the data set INHAL2 using the OUTPUT statement in the DATA step. The final data set for analysis, INHAL, is created to include a character-valued variable for the carryover effect, CO , whose values represent the dose group from the previous period. A value of 0 is assigned to CO for the first period. Three additional carryover effects are included, one for each pairwise difference. Select Dose D as the ‘reference’ dose and define COA, COB, and COC as the difference in carryover effect between Doses A and D, B and D, and C and D, respectively. Thus, COA will take the value of +1 if Dose A was given in the previous period, –1 if Dose D was given in the previous period, and 0 otherwise. Similarly for COB and COC. The printout of the data set INHAL shows the values of each of these carryover effects . PROC GLM is used to analyze the data. Initially, the overall carryover effect, CO, is included in the crossover analysis (this is referred to as “Model 1”). The MODEL statement includes effects for Sequence Group (SEQGRP), Dose Group (DOSE), Period (PER), Patient-within-Sequence Group (PAT(SEQGRP)), and the overall carryover (CO). The carryover effect has a sum of squares of 30.936 and a p-value of 0.0687 , which deserves further analysis because of its marginal significance. You can run a second GLM model that includes the individual dose carryover effects (relative to Dose D) by including COA, COB, and COC in the MODEL statement (this is referred to as “Model 2”) 11 . This second GLM model separates the overall carryover effect sum of squares into the components due to each dose comparison. Note that the Type I sums of squares for COA, COB, and COC in Model 2 12 add up to the sum of squares for CO in Model 1. The largest component of carryover is clearly due to Dose B, which has a significant p-value of 0.0267 13 . The Dose Group effect adjusted for carryover is highly significant in both Models 1 14 and 2 15 (p=0.0001). No Sequence Group (p=0.6526) 16 or Period effects (p=0.7947) 17 are evident. SAS Code for Example 9.2 DATA INHAL1; INPUT SEQGRP $ PAT AUC1 AUC2 AUC3 AUC4 DATALINES; ABDC 102 2.31 3.99 11.75 4.78 ABDC 106 3.95 2.07 7.00 4.20 ABDC 109 4.40 6.40 9.76 6.12 BCAD 104 6.81 8.38 1.26 10.56 BCAD 105 9.05 6.85 4.79 4.86 BCAD 111 7.02 5.70 3.14 7.65 CDBA 101 6.00 4.79 2.35 3.81

@@;

Chapter 9: The Crossover Design 167

CDBA CDBA DACB DACB DACB ;

108 5.25 10.42 112 2.60 6.97 103 8.15 3.58 107 12.73 5.31 110 6.46 2.42

5.68 3.60 8.79 4.67 4.58

4.48 7.54 4.94 5.84 1.37

DATA INHAL2; SET INHAL1; DOSE = SUBSTR(SEQGRP,1,1); DOSE = SUBSTR(SEQGRP,2,1); DOSE = SUBSTR(SEQGRP,3,1); DOSE = SUBSTR(SEQGRP,4,1); RUN;

PER PER PER PER

= = = =

1; 2; 3; 4;

AUC AUC AUC AUC

= = = =

AUC1; AUC2; AUC3; AUC4;

PROC SORT DATA = INHAL2; BY PAT PER; DATA INHAL; SET INHAL2; KEEP PAT SEQGRP DOSE PER AUC CO COA COB COC; CO = LAG(DOSE); IF PER = 1 THEN CO = '0'; COA = 0; COB = 0; COC = 0; IF CO = 'A' THEN COA = 1; IF CO = 'B' THEN COB = 1; IF CO = 'C' THEN COC = 1; IF CO = 'D' THEN DO; COA = -1; COB = -1; COC = -1; END; RUN;

OUTPUT; OUTPUT; OUTPUT; OUTPUT;

PROC SORT DATA = INHAL; BY SEQGRP PER PAT; PROC PRINT DATA = INHAL; TITLE1 'CrossOver Design'; TITLE2 'Example 9.2: Antibiotic Blood Levels Following Aerosol Inhalation' ; RUN; /* Model 1 */ PROC GLM DATA = INHAL; CLASS SEQGRP DOSE PER PAT CO; MODEL AUC = SEQGRP PAT(SEQGRP) DOSE PER CO; TEST H=SEQGRP E=PAT(SEQGRP); TITLE3 ‘Model 1’; RUN;

/* Model 2 */ PROC GLM DATA = INHAL; CLASS SEQGRP DOSE PER PAT; MODEL AUC = SEQGRP PAT(SEQGRP) DOSE PER COA COB COC; TEST H=SEQGRP E=PAT(SEQGRP); TITLE3 ‘Model 2’; RUN;

11

168

Common Statistical Methods for Clinical Research with SAS Examples

OUTPUT 9.2. SAS Output for Example 9.2 CrossOver Design Example 9.2: Antibiotic Blood Levels Following Aerosol Inhalation Obs

PAT

SEQGRP

DOSE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

102 106 109 102 106 109 102 106 109 102 106 109 104 105 111 104 105 111 104 105 111 104 105 111 101 108 112 101 108 112 101 108 112 101 108 112 103 107 110 103 107 110 103 107 110 103 107 110

ABDC ABDC ABDC ABDC ABDC ABDC ABDC ABDC ABDC ABDC ABDC ABDC BCAD BCAD BCAD BCAD BCAD BCAD BCAD BCAD BCAD BCAD BCAD BCAD CDBA CDBA CDBA CDBA CDBA CDBA CDBA CDBA CDBA CDBA CDBA CDBA DACB DACB DACB DACB DACB DACB DACB DACB DACB DACB DACB DACB

A A A B B B D D D C C C B B B C C C A A A D D D C C C D D D B B B A A A D D D A A A C C C B B B

PER 1 1 1 2 2 2 3 3 3 4 4 4 1 1 1 2 2 2 3 3 3 4 4 4 1 1 1 2 2 2 3 3 3 4 4 4 1 1 1 2 2 2 3 3 3 4 4 4

AUC 2.31 3.95 4.40 3.99 2.07 6.40 11.75 7.00 9.76 4.78 4.20 6.12 6.81 9.05 7.02 8.38 6.85 5.70 1.26 4.79 3.14 10.56 4.86 7.65 6.00 5.25 2.60 4.79 10.42 6.97 2.35 5.68 3.60 3.81 4.48 7.54 8.15 12.73 6.46 3.58 5.31 2.42 8.79 4.67 4.58 4.94 5.84 1.37

CO

COA

COB

0 0 0 A A A B B B D D D 0 0 0 B B B C C C A A A 0 0 0 C C C D D D B B B 0 0 0 D D D A A A C C C

0 0 0 1 1 1 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 -1 -1 1 1 1 0 0 0

0 0 0 0 0 0 1 1 1 -1 -1 -1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 1 0 0 0 -1 -1 -1 0 0 0 0 0 0

COC 0 0 0 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 -1 -1 -1 0 0 0 0 0 0 -1 -1 -1 0 0 0 1 1 1

Chapter 9: The Crossover Design 169

OUTPUT 9.2. SAS Output for Example 9.2 (continued) CrossOver Design Example 9.2: Antibiotic Blood Levels Following Aerosol Inhalation Model 1 The GLM Procedure

Class

Class Level Information Values

Levels

SEQGRP

4

ABDC BCAD CDBA DACB

DOSE

4

A B C D

PER

4

1 2 3 4

PAT

12

CO

5

101 102 103 104 105 106 107 108 109 110 111 112 0 A B C D Number of observations

48

Dependent Variable: AUC Source

DF

Sum of Squares

Model Error Corrected Total

20 27 47

225.1866862 104.8902450 330.0769312

Mean Square

F Value

11.2593343 3.8848239

2.90

Pr > F 0.0053

R-Square

Coeff Var

Root MSE

AUC Mean

0.682225

34.38658

1.970996

5.731875

DF

Type I SS

Mean Square

F Value

Pr > F

3 8 3 3 3

7.1111896 48.6932667 134.4534729 3.9931229 30.9356342

2.3703965 6.0866583 44.8178243 1.3310410 10.3118781

0.61 1.57 11.54 0.34 2.65

0.6142 0.1815 <.0001 0.7947 0.0687

Pr > F

Source SEQGRP PAT(SEQGRP) DOSE PER CO

Source SEQGRP PAT(SEQGRP) DOSE PER CO

DF

Type III SS

Mean Square

F Value

3 8 3 2 3

10.3361626 48.6932667 118.0792959 0.0628167 30.9356342

3.4453875 6.0866583 39.3597653 0.0314083 10.3118781

0.89 1.57 10.13 0.01 2.65

0.4604 0.1815 0.0001 14 0.9920 0.0687

Tests of Hypotheses Using the Type III MS for PAT(SEQGRP) as an Error Term Source

DF

Type III SS

Mean Square

F Value

Pr > F

SEQGRP

3

10.33616258

3.44538753

0.57

0.6526

170

Common Statistical Methods for Clinical Research with SAS Examples

OUTPUT 9.2. SAS Output for Example 9.2 (continued)

CrossOver Design Example 9.2: Antibiotic Blood Levels Following Aerosol Inhalation Model 2 The GLM Procedure

Class

Class Level Information Values

Levels

SEQGRP DOSE PER PAT

4 4 4 12

ABDC BCAD CDBA DACB A B C D 1 2 3 4 101 102 103 104 105 106 107 108 109 110 111 112 Number of observations

48

Dependent Variable: AUC Source

DF

Sum of Squares

Model Error Corrected Total

20 27 47

225.1866862 104.8902450 330.0769312

Mean Square

F Value

Pr > F

11.2593343 3.8848239

2.90

0.0053

R-Square

Coeff Var

Root MSE

AUC Mean

0.682225

34.38658

1.970996

5.731875

Source

DF

Type I SS

3 8 3 3 1 1 1

7.1111896 48.6932667 134.4534729 3.9931229 0.0870204 21.3521113 12 9.4965025

SEQGRP PAT(SEQGRP) DOSE PER COA COB COC Source

Mean Square

F Value

2.3703965 6.0866583 44.8178243 1.3310410 0.0870204 21.3521113 9.4965025

0.61 1.57 11.54 0.34 0.02 5.50 2.44

DF

Type III SS

Mean Square

F Value

3 8 3 3 1 1 1

10.3361626 48.6932667 118.0792959 3.9931229 1.9374669 28.9850625 9.4965025

3.4453875 6.0866583 39.3597653 1.3310410 1.9374669 28.9850625 9.4965025

0.89 1.57 10.13 0.34 0.50 7.46 2.44

SEQGRP PAT(SEQGRP) DOSE PER COA COB COC

Pr > F 0.6142 0.1815 <.0001 0.7947 0.8821 0.0267 13 0.1296 Pr > F 0.4604 0.1815 0.0001 15 0.7947 17 0.4861 0.0110 0.1296

Tests of Hypotheses Using the Type III MS for PAT(SEQGRP) as an Error Term Source

DF

Type III SS

Mean Square

F Value

SEQGRP

3

10.33616258

3.44538753

0.57

Pr > F 0.6526 16

Chapter 9: The Crossover Design 171

9.4

Details & Notes Study resources might be wasted by conducting crossover studies 9.4.1 that are likely to yield inconsistent treatment differences among periods. This scenario could arise from a carryover effect of certain treatments, improper wash-out periods between treatments, or the inability of patients to be restored to their original baseline conditions at the start of each period. Designs balanced for residual effects should be used whenever possible with crossover studies. Such schemes have the property that each treatment is preceded by every other treatment the same number of times. The presence of treatment carryover effects can easily be investigated with these types of designs. The layout used in Example 9.2 is that of a 4-treatment design balanced for residuals. In this design, four sequence groups are used, and each treatment is preceded by each of the other treatments exactly once. An example of a 3-treatment (A, B, and C), 3-period crossover design balanced for residuals is shown in Table 9.5. Six sequence groups are used, and each treatment is preceded by every other treatment exactly twice. Note that such balance would not be possible with only three sequence groups. In general, if k represents the number of treatments, the minimum number of sequence groups needed for complete residual balance is k, if k is even, and 2k, if k is odd. TABLE 9.5. A 3-Period Crossover Design Balanced for Residuals Sequence

Period 1

Period 2

Period 3

1 2 3 4 5 6

A B C A B C

B C A C A B

C A B B C A

Study design features should also include an adequate wash-out duration between treatment periods. As previously mentioned, the length of the trial and the number of periods should also be limited to minimize changing clinical conditions of the patients during the study. In a two-period crossover design under the assumption of no 9.4.2 carryover effect (such as in Example 9.1), the Period effect is completely confounded with the Treatment-by-Sequence Group interaction. In other words, any difference between Sequence Groups in the Treatment comparisons are indistinguishable from Period differences. The results of Example 9.1 would be identical if the PD effect is replaced with the TRT*SEQ interaction in the MODEL statement in PROC GLM.

172

Common Statistical Methods for Clinical Research with SAS Examples

Similarly, a Treatment effect implies that response differences between Period 1 and Period 2 are in opposite directions for the two Sequence Groups, i.e., a Period-by-Sequence Group interaction. The Treatment (TRT) effect in the MODEL statement in PROC GLM used in Example 9.1 could be replaced with this interaction (PD*SEQ) with the same results. When the MODEL statement is written in this way, it is identical to the form discussed for the general repeated measures ANOVA of Chapter 8. The preceding interactions cannot be used interchangeably with main effects in crossovers that involve more than two periods or those in which a carryover effect might exist. In the crossover analysis, the presence of a Treatment-by-Period 9.4.3 interaction generally requires separate analyses within each period using, for example, the two-sample t-test (Chapter 5). In such a situation, many analysts accept only the results of the first period. Dropouts following the first period will create bias in the analysis 9.4.4 of the crossover study. For example, in the two-way crossover, the ‘withinpatient control’ advantage of the crossover is lost when a patient has data from the first period, but not from the second. Whether that patient is dropped from the analysis or the analysis is adjusted for the missing value (e.g., use of imputation), the results will likely have some unknown bias. If there are many dropouts, the primary analysis might use just the data from the first period. Because dropout rates usually increase with lengthier studies, this problem can be controlled by using a small number of short treatment periods whenever a crossover design is contemplated. In Example 9.2, the SAS Type I sums of squares for Dose Group 9.4.5 (DOSE), Period (PER), and Sequence Group (SEQGRP) are unadjusted for carryover effects. These are the same SS’s that would be obtained if the carryover terms (CO or COA, COB, COC) were omitted from the model (Model 2). Tests for a significant Dose Group effect would typically use the Type III sum of squares, which is adjusted for residuals. However, when a significant carryover effect is found, caution must be used in the interpretation of the Treatment effects (see Section 9.4.7). The term ‘balanced residuals’ implies that, if the carryover effect 9.4.6 is the same for each treatment, these effects will be cancelled when looking at differences in treatment means, and thus, the true treatment difference is free of any carryover effects. Biased estimates of treatment differences will result if the residuals are not balanced across treatment groups. To check for different degrees of carryover among treatments, a 9.4.7 preliminary test for carryover effects can be made in the manner demonstrated in Example 9.2. Such preliminary tests for model assumptions are often conducted at a significance level greater than 0.05, perhaps in the 0.10 to 0.20 range. If non-significant, the carryover effect can be dropped from the model.

Chapter 9: The Crossover Design 173

The p-value for the overall carryover effect of 0.0687 in Example 9.2 is small enough to raise concern regarding carryover. Significant carryover effects might provide useful information for designing future studies (e.g., using a longer observation period and/or a longer wash-out period). Although this chapter introduces the crossover design for normal 9.4.8 response data, crossover designs can also be performed with non-normal responses, which include categorical and rank data. Using binary responses and non-parametric approaches to the analysis of non-normal responses in crossovers are discussed in Jones and Kenward (1989), and examples using SAS are included in Stokes, Davis, and Koch (2000).

174

Common Statistical Methods for Clinical Research with SAS Examples

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Obs

PAT

X_DUR

Y_IMPR

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

7 1 19 11 5 2 4 9 13 14 15 18 3 16 8 12 17 6 20 10

1 1 1 1 1 1 2 2 2 2 3 3 3 3 4 4 5 5 6 6

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Variable Mean Std Dev N ---------------------------------------------X_DUR 2.8000000 1.7044833 20 Y_IMPR 38.0000000 35.0338182 20 ----------------------------------------------

The GLM Procedure Number of observations

20

Dependent Variable: Y_IMPR

Source

DF

Model Error Corrected Total

1 18 19

R-Square 0.241880 Source X_DUR

ô

Sum of Squares 5640.65217 17679.34783 23320.00000

Coeff Var 82.47328

Mean Square

F Value

Pr > F

5640.65217 982.18599

5.74

0.0276

Root MSE 31.33985

Y_IMPR Mean 38.00000

DF

Type I SS

Mean Square

F Value

1

5640.652174

5640.652174

5.74

Parameter

Estimate

Intercept X_DUR

66.30434783 -10.10869565

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ê

13.73346931 4.21820157

t Value 4.83 -2.40

Pr > F

ò

0.0276

Pr > |t| 0.0001 0.0276

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Observation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Observed

95% Confidence Limits for Residual Mean Predicted Value

Predicted

10.0000000 40.0000000 40.0000000 60.0000000 80.0000000 90.0000000 30.0000000 50.0000000 50.0000000 110.0000000 20.0000000 20.0000000 30.0000000 70.0000000 -10.0000000 0.0000000 -30.0000000 60.0000000 0.0000000 40.0000000

56.1956522 -46.1956522 34.4879982 56.1956522 - 16.1956522 34.4879982 56.1956522 -16.1956522 34.4879982 56.1956522 3.8043478 34.4879982 56.1956522 23.8043478 34.4879982 56.1956522 33.8043478 34.4879982 46.0869565 -16.0869565 29.7460282 46.0869565 3.9130435 29.7460282 46.0869565 3.9130435 29.7460282 46.0869565 63.9130435 29.7460282 35.9782609 -15.9782609 21.1491101 35.9782609 -15.9782609 21.1491101 35.9782609 -5.9782609 21.1491101 35.9782609 34.0217391 21.1491101 25.8695652 -35.8695652 7.7076388 25.8695652 -25.8695652 7.7076388 15.7608696 -45.7608696 -8.6702890 15.7608696 44.2391304 -8.6702890 5.6521739 -5.6521739 -26.3006276 5.6521739 34.3478261 -26.3006276

Sum of Residuals Sum of Squared Residuals Sum of Squared Residuals - Error SS PRESS Statistic First Order Autocorrelation Durbin-Watson D

-0.00000 17679.34783 0.00000 22187.33121 -0.05364 1.91983

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6$6&RGHIRU([DPSOH DATA DEHYD; INPUT PAT Y DOSE AGE WT @@; DATALINES; 1 77 0.0 4 28 2 65 1.5 5 35 4 63 1.0 9 76 5 75 0.5 5 31 7 70 1.0 6 35 8 90 2.5 6 47 10 72 3.0 8 50 11 67 2.0 7 50 13 75 1.5 4 33 14 58 3.0 8 59 16 55 0.0 8 58 17 80 1.0 7 55 19 44 0.5 9 66 20 62 1.0 6 43 22 75 2.5 7 50 23 77 1.5 5 29 25 68 3.0 9 61 26 71 2.5 10 71 28 80 2.0 3 27 29 70 0.0 9 56 31 88 1.0 3 22 32 68 0.5 5 37 34 90 3.0 5 45 35 79 1.5 8 53 ;

3 75 2.5 8 55 6 82 2.0 5 27 9 49 0.0 9 59 12 100 2.5 7 46 15 58 1.5 6 40 18 55 2.0 10 76 21 60 1.0 6 48 24 80 2.5 11 64 27 90 1.5 4 26 30 58 2.5 8 57 33 60 0.5 6 44 36 90 2.0 4 29

PROC PRINT DATA = DEHYD; ù VAR PAT Y DOSE AGE WT; TITLE1 ’Multiple Linear Regression’; TITLE2 ’Example 10.2: Recovery in Pediatric Dehydration’; RUN; PROC REG CORR MODEL Y = MODEL Y = MODEL Y =

11 DATA = DEHYD; DOSE AGE WT / SS1 SS2 VIF COLLINOINT; DOSE ; 17 DOSE AGE ;

MODEL Y = DOSE WT RUN;

;

18

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Obs

PAT

Y

DOSE

AGE

WT

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

77 65 75 63 75 82 70 90 49 72 67 100 75 58 58 55 80 55 44 62 60 75 77 80 68 71 90 80 70 58 88 68 60 90 79 90

0.0 1.5 2.5 1.0 0.5 2.0 1.0 2.5 0.0 3.0 2.0 2.5 1.5 3.0 1.5 0.0 1.0 2.0 0.5 1.0 1.0 2.5 1.5 2.5 3.0 2.5 1.5 2.0 0.0 2.5 1.0 0.5 0.5 3.0 1.5 2.0

4 5 8 9 5 5 6 6 9 8 7 7 4 8 6 8 7 10 9 6 6 7 5 11 9 10 4 3 9 8 3 5 6 5 8 4

28 35 55 76 31 27 35 47 59 50 50 46 33 59 40 58 55 76 66 43 48 50 29 64 61 71 26 27 56 57 22 37 44 45 53 29

ù

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2873876$62XWSXWIRU([DPSOHFRQWLQXHG Multiple Linear Regression Example 10.2: Recovery in Pediatric Dehydration The REG Procedure 11

Correlation Variable DOSE AGE WT Y

DOSE

AGE

WT

Y

1.0000 0.1625 0.1640 0.3595

0.1625 1.0000 0.9367 -0.4652

0.1640 0.9367 1.0000 -0.5068

0.3595 -0.4652 -0.5068 1.0000

The REG Procedure Model: MODEL1 Dependent Variable: Y Analysis of Variance

Source

DF

Sum of Squares

Mean Square

Model Error Corrected Total

3 32 35

2667.66870 3151.22019 5818.88889

889.22290 98.47563

Root MSE Dependent Mean Coeff Var

9.92349 71.55556 13.86823

R-Square Adj R-Sq

F Value

Pr > F

9.03 12

0.4584 0.4077

0.0002

15

Parameter Estimates

Variable

DF

Parameter Estimate 13

Standard Error

Intercept DOSE AGE WT

1 1 1 1

85.47636 6.16969 0.27695 -0.54278

5.96528 1.79081 2.28474 0.32362

t Value 14 14.33 3.45 0.12 -1.68

Pr>|t| Type I SS <.0001 0.0016 0.9043 0.1032

Parameter Estimates

Variable

DF

Type II SS

Variance Inflation

Intercept DOSE AGE WT

1 1 1 1

20219 1168.84352 1.44701 277.00687

0 1.02833 8.16330 8.16745

184327 752.15170 638.51013 277.00687

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2873876$62XWSXWIRU([DPSOHFRQWLQXHG Multiple Linear Regression Example 10.2: Recovery in Pediatric Dehydration The REG Procedure Model: MODEL1 21

Collinearity Diagnostics(intercept adjusted)

Number

Eigenvalue

1 2 3

1.99054 0.94617 0.06329

Condition Index 1.00000 1.45044 5.60804

-----Proportion of Variation--DOSE AGE WT 0.02518 0.97480 0.00002134

0.02918 0.00337 0.96745

0.02918 0.00330 0.96752

Model: MODEL2 Dependent Variable: Y Analysis of Variance

Source

DF

Sum of Squares

Mean Square

Model Error Corrected Total

1 34 35

752.15170 5066.73719 5818.88889

752.15170 149.02168

Root MSE Dependent Mean Coeff Var

12.20744 71.55556 17.06009

R-Square Adj R-Sq

F Value

Pr > F

5.05

0.0313

0.1293 0.1037

16

Parameter Estimates

Variable

DF

Parameter Estimate

Standard Error

t Value

Pr > |t|

Intercept DOSE

1 1

63.89576 4.88058

3.97040 2.17242

16.09 2.25

<.0001 0.0313

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2873876$62XWSXWIRU([DPSOHFRQWLQXHG Multiple Linear Regression Example 10.2: Recovery in Pediatric Dehydration The REG Procedure Model: MODEL3 Dependent Variable: Y 17

Analysis of Variance

Source

DF

Sum of Squares

Model Error Corrected Total

2 33 35

2390.66183 3428.22706 5818.88889

1195.33092 103.88567

10.19243 71.55556 14.24408

R-Square Adj R-Sq

Root MSE Dependent Mean Coeff Var

Mean Square

F Value

Pr > F

11.51

0.0002

19

0.4108 0.3751

Parameter Estimates

Variable

DF

Parameter Estimate

Standard Error

t Value

Pr > |t|

Intercept DOSE AGE

1 1 1

84.07228 6.06709 -3.30580

6.06631 1.83827 0.83240

13.86 3.30 -3.97

<.0001 0.0023 0.0004

Model: MODEL4 Dependent Variable: Y 18

Analysis of Variance

Source

DF

Sum of Squares

Mean Square

Model Error Corrected Total

2 33 35

2666.22169 3152.66720 5818.88889

1333.11084 95.53537

Root MSE Dependent Mean Coeff Var

9.77422 71.55556 13.65962

R-Square Adj R-Sq

F Value

Pr > F

13.95

<.0001

0.4582 0.4254

20

Parameter Estimates

Variable

DF

Parameter Estimate

Standard Error

t Value

Pr > |t|

Intercept DOSE WT

1 1 1

85.59416 6.17526 -0.50610

5.79705 1.76329 0.11307

14.77 3.50 -4.48

<.0001 0.0013 <.0001

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6$6&RGHIRU([DPSOH DATA TRI; INPUT TRT $ PAT HGBA1C TRICHG DATALINES; FIB 2 7.0 5 FIB 4 6.0 10 FIB 8 8.6 –20 FIB 11 6.3 0 FIB 16 6.6 10 FIB 17 7.4 –10 FIB 21 6.5 –15 FIB 23 6.2 5 FIB 27 8.5 –40 FIB 28 9.2 –25 FIB 33 7.0 –10 GEM 1 5.1 10 GEM 5 7.2 –15 GEM 6 6.4 5 GEM 10 6.0 –15 GEM 12 5.6 -5 GEM 15 6.7 –20 GEM 18 8.6 –40 GEM 22 6.0 –10 GEM 25 9.3 –40 GEM 29 7.9 -35 GEM 31 7.4 0 GEM 34 6.5 –10 ; PROC SORT DATA = TRI; BY TRT HGBA1C TRICHG;

@@; FIB FIB FIB FIB FIB GEM GEM GEM GEM GEM GEM

7 13 19 24 30 3 9 14 20 26 32

7.1 -5 7.5 –15 5.3 20 7.8 0 5.0 25 6.0 15 5.5 10 5.5 –10 6.4 -5 8.5 –20 5.0 0

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/* Print data set */ PROC PRINT DATA = TRI; å VAR TRT PAT HGBA1C TRICHG; TITLE1 ’Analysis of Covariance’; TITLE2 ’Example 11.1: Triglyceride Changes Adjusted for Glycemic Control’; RUN; /* Plot data, by group */ PROC PLOT VPERCENT=45 DATA = TRI; PLOT TRICHG*HGBA1C = TRT; RUN;

/* Obtain summary statistics for each group */ PROC MEANS MEAN STD N DATA = TRI; BY TRT; VAR HGBA1C TRICHG; RUN; /* Use glycemic control as covariate */ PROC GLM DATA = TRI; CLASS TRT; MODEL TRICHG = TRT HGBA1C / SOLUTION; LSMEANS TRT/PDIFF STDERR; RUN;

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/* Compare groups with ANOVA, ignoring the covariate */ PROC GLM DATA = TRI; õ CLASS TRT; MODEL TRICHG = TRT / SS3; RUN;

2873876$62XWSXWIRU([DPSOH Analysis of Covariance Example 11.1: Triglyceride Changes Adjusted for Glycemic Control

Obs

TRT

PAT

HGBA1C

TRICHG

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

FIB FIB FIB FIB FIB FIB FIB FIB FIB FIB FIB FIB FIB FIB FIB

30 19 4 23 11 21 16 33 2 7 17 13 24 27 8

5.0 5.3 6.0 6.2 6.3 6.5 6.6 7.0 7.0 7.1 7.4 7.5 7.8 8.5 8.6

25 20 10 5 0 -15 10 -10 5 -5 -10 -15 0 -40 -20

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FIB GEM GEM GEM GEM GEM GEM GEM GEM GEM GEM GEM GEM GEM GEM GEM GEM GEM GEM

28 32 1 14 9 12 10 22 3 20 6 34 15 5 31 29 26 18 25

9.2 5.0 5.1 5.5 5.5 5.6 6.0 6.0 6.0 6.4 6.4 6.5 6.7 7.2 7.4 7.9 8.5 8.6 9.3

-25 0 10 -10 10 -5 -15 -10 15 -5 5 -10 -20 -15 0 -35 -20 -40 -40

Analysis of Covariance Example 11.1: Triglyceride Changes Adjusted for Glycemic Control Plot of TRICHG*HGBA1C.

Symbol is value of TRT.

TRICHG | | 25 +F 20 + F 15 + G 10 + G G F F 5 + F G F 0 +G F G F -5 + G G F -10 + G G G F F -15 + G F G F -20 + G GF -25 + F -30 + -35 + G -40 + FG G | •+------------+-------------+-------------+-------------+-------+ 5 6 7 8 9 10 HGBA1C

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2873876$62XWSXWIRU([DPSOHFRQWLQXHG The MEANS Procedure ------------------------------- TRT=FIB ----------------------------Variable Mean Std Dev N ---------------------------------------------HGBA1C 7.0000000 1.1587349 16 TRICHG -4.0625000 16.9527530 16 ----------------------------------------------

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----------------------- ----- TRT=GEM ----------------------------Variable Mean Std Dev N ---------------------------------------------HGBA1C 6.6444444 1.2594220 18 TRICHG -10.2777778 16.4023153 18 ----------------------------------------------

Analysis of Covariance Example 11.1: Triglyceride Changes Adjusted for Glycemic Control The GLM Procedure Class Level Information Class TRT

Levels 2

Values FIB GEM

Number of observations

34

Dependent Variable: TRICHG Source Model Error Corrected Tot

Sum of Squares

DF 2 31 33

8.075283 2903.689422 9211.764706

Mean Square

F Value

Pr > F

3154.037642 93.667401

33.67

<.0001

R-Square

Coeff Var

Root MSE

TRICHG Mean

0.684785

-131.6234

9.678192

-7.352941

Source

DF

Type I SS

Mean Square

F Value

Pr > F

TRT HGBA1C

1 1

327.216095 5980.859189

327.216095 5980.859189

3.49 63.85

0.0711 <.0001

Source

DF

Type III SS

Mean Square

F Value

Pr > F

TRT HGBA1C

1 1

865.363546 5980.859189

865.363546 5980.859189

9.24 63.85

0.0048 <.0001

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2873876$62XWSXWIRU([DPSOHFRQWLQXHG Parameter Intercept TRT FIB TRT GEM HGBA1C

Standard Error

Estimate

ò

t Value

Pr > |t|

6.70 3.04 . -7.99

<.0001 0.0048 . <.0001

64.59251309 B 9.64331471 10.22171475 B 3.36293670 0.00000000 B . -11.26810398 ñ 1.41014344

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NOTE: The X’X matrix has been found to be singular, and a generalized inverse was used to solve the normal equations. Terms whose estimates are followed by the letter ’B’ are not uniquely estimable. Least Squares Means

TRT FIB GEM

TRICHG LSMEAN -1.9414451 -12.1631599

Standard Error 2.4340646 11 2.2933414

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H0:LSMean1= LSMean2 Pr > |t|

H0:LSMEAN=0 Pr > |t| 0.4312 <.0001

0.0048

Analysis of Covariance Example 11.1: Triglyceride Changes Adjusted for Glycemic Control The GLM Procedure Class Level Information Class TRT

Levels 2

Values FIB GEM

Number of observations

Dependent Variable: TRICHG Sum of Source DF Squares Model Error Corrected Tot

Source TRT

1 32 33

327.216095 8884.548611 9211.764706

34

Mean Square

F Value

Pr > F

327.216095 277.642144

1.18

0.2858

õ

R-Square

Coeff Var

Root MSE

TRICHG Mean

0.035522

-226.6113

16.66260

-7.352941

DF

Type III SS

Mean Square

F Value

Pr > F

1

327.2160948

327.2160948

1.18

0.2858

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BPDIACH @@;

68 115 -10 69 107 -5 58 99 -4

...

COMB 423 66 115 -14

12 PROC PRINT DATA = BP; TITLE1 ’Analysis of Covariance’; TITLE2 ’Example 11.2: ANCOVA in Multi-Center Hypertension Study’; RUN;

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PROC SORT DATA = BP; BY TRT CENTER; 13

PROC MEANS MEAN STD N MIN MAX DATA = BP; BY TRT; VAR AGE BPDIA0 BPDIACH; RUN;

PROC GLM DATA = BP; 14 CLASS TRT CENTER; MODEL BPDIACH = TRT CENTER AGE BPDIA0 / SOLUTION; LSMEANS TRT / STDERR PDIFF; TITLE3 ’GLM Model Including Effects: TRT, CENTER, AGE, BPDIA0’; RUN;

2873876$62XWSXWIRU([DPSOH Analysis of Covariance Example 11.2: ANCOVA in Multi-Center Hypertension Study

Obs

TRT

PAT

AGE

BPDIA0

BPDIACH

CENTER

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 . . . 70 71 72 73 74 75

SOLE SOLE SOLE SOLE SOLE SOLE SOLE SOLE SOLE SOLE SOLE SOLE SOLE SOLE SOLE

101 103 104 106 109 202 203 204 206 208 211 212 214 216 217

55 68 45 69 54 58 62 51 47 61 40 36 58 64 55

102 115 97 107 115 99 119 107 96 98 110 103 109 119 104

-6 -10 -2 -5 -7 -4 -11 -9 0 -6 5 -3 10 -12 4

1 1 1 1 1 2 2 2 2 2 2 2 2 2 2

COMB COMB COMB COMB COMB COMB

417 419 420 422 423 426

53 43 77 55 66 38

109 96 100 103 115 105

3 -14 -5 -8 -14 -3

4 4 4 4 4 4

12

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2873876$62XWSXWIRU([DPSOHFRQWLQXHG The MEANS Procedure --------------------------- TRT=COMB -----------------------------Variable Mean Std Dev N Minimum Maximum ------------------------------------------------------------------AGE 55.3846154 11.0706235 39 34.0000000 79.0000000 BPDIA0 106.3333333 7.0012530 39 93.0000000 119.0000000 BPDIACH -6.8205128 6.2318021 39 -20.0000000 7.0000000 ------------------------------------------------------------------13

---------------------------- TRT=SOLE ----------------------------Variable Mean Std Dev N Minimum Maximum ------------------------------------------------------------------AGE 54.1944444 9.4923110 36 36.0000000 73.0000000 BPDIA0 106.6666667 7.9677923 36 95.0000000 119.0000000 BPDIACH -4.0555556 5.9997354 36 -15.0000000 10.0000000 -------------------------------------------------------------------

Analysis of Covariance Example 11.2: ANCOVA in Multi-Center Hypertension Study GLM Model Including Effects: TRT, CENTER, AGE, BPDIA0 The GLM Procedure Class Level Information Class Levels Values TRT 2 COMB SOLE CENTER 4 1 2 3 4 Number of observations

75

Dependent Variable: BPDIACH

Source

DF

Sum of Squares

Model Error Corrected Total

6 68 74

641.585250 2237.161416 2878.746667

R-Square 0.222870

Coeff Var -104.4139

Mean Square

F Value

Pr > F

106.930875 32.899433

3.25

0.0072

Root MSE 5.735803

BPDIACH Mean -5.493333

15

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DF

Type I SS

Mean Square

F Value

Pr > F

TRT CENTER AGE BPDIA0

1 3 1 1

143.1141880 49.0195734 195.3204109 254.1310779

143.1141880 16.3398578 195.3204109 254.1310779

4.35 0.50 5.94 7.72

0.0408 0.6858 0.0175 0.0070

Source

DF

Type III SS

Mean Square

F Value

Pr > F

TRT CENTER AGE BPDIA0

1 3 1 1

139.2345407 13.1006818 92.6596679 254.1310779

139.2345407 4.3668939 92.6596679 254.1310779

4.23 0.13 2.82 7.72

0.0435 0.9403 0.0979 0.0070

Parameter Intercept TRT COMB TRT SOLE CENTER 1 CENTER 2 CENTER CENTER AGE BPDIA0

Estimate 30.88660621 -2.74007804 0.00000000 -0.96419474 -0.65014780 -1.03110593 0.00000000 -0.11500993 -0.26398254

B B B B B B B

Standard Error

t Value

Pr > |t|

10.04435860 1.33193688 . 2.23622128 1.63467283 1.86433408 . 0.06853055 0.09498183

3.08 -2.06 . -0.43 -0.40 -0.55 . -1.68 -2.78

0.0030 0.0435 . 0.6677 0.6921 0.5820 . 0.0979 0.0070

NOTE: The X’X matrix has been found to be singular, and a generalized inverse was used to solve the normal equations. Terms whose estimates are followed by the letter ’B’ are not uniquely estimable.

Analysis of Covariance Example 11.2: ANCOVA in Multi-Center Hypertension Study GLM Model Including Effects: TRT, CENTER, AGE, BPDIA0

Least Squares Means

TRT

BPDIACH LSMEAN

Standard Error

H0:LSMEAN=0 Pr > |t|

COMB SOLE

-6.93129242 -4.19121438

0.98013179 0.99307483

<.0001 <.0001

H0:LSMean1= LSMean2 Pr > |t| 0.0435

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TRT TRT TRT TRT TRT TRT TRT

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DF

Type III SS

Mean Square

F Value

Pr > F

TRT 1 143.1141880 143.1141880 3.82 0.0545 ------------------------------------------------------------------(2) Source DF Type III SS Mean Square F Value Pr > F TRT 1 146.5980349 146.5980349 3.82 0.0547 CENTER 3 49.0195734 16.3398578 0.43 0.7352 ------------------------------------------------------------------(3) Source DF Type III SS Mean Square F Value Pr > F TRT 1 153.8692890 153.8692890 4.72 0.0331 BPDIA0 1 388.0563898 388.0563898 11.90 0.0009 ------------------------------------------------------------------(4) Source DF Type III SS Mean Square F Value Pr > F TRT 1 122.7314895 122.7314895 3.51 0.0649 AGE 1 220.8161121 220.8161121 6.32 0.0142 ------------------------------------------------------------------(5) Source DF Type III SS Mean Square F Value Pr > F TRT 1 137.3094860 137.3094860 4.33 0.0410 AGE 1 97.3139906 97.3139906 3.07 0.0840 BPDIA0 1 264.5542683 264.5542683 8.35 0.0051 ------------------------------------------------------------------(6) Source DF Type III SS Mean Square F Value Pr > F TRT 1 157.4152735 157.4152735 4.66 0.0343 CENTER 3 17.7550045 5.9183348 0.18 0.9128 BPDIA0 1 356.7918209 356.7918209 10.57 0.0018 ------------------------------------------------------------------(7) Source DF Type III SS Mean Square F Value Pr > F TRT CENTER AGE

1 3 1

124.5076559 23.5238722 195.3204109

124.5076559 7.8412907 195.3204109

3.45 0.22 5.41

0.0676 0.8842 0.0230

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CHHAAPPTTEERR 12 The Wilcoxon Signed-Rank Test 12.1 12.2 12.3 12.4

Introduction...................................................................................................... 227 Synopsis ............................................................................................................ 227 Examples........................................................................................................... 228 Details & Notes................................................................................................. 234

12.1 Introduction The Wilcoxon signed-rank test is a non-parametric analog of the one-sample t-test (Chapter 4). The signed-rank test can be used to make inferences about a population mean or median without requiring the assumption of normally distributed data. As its name implies, the Wilcoxon signed-rank test is based on the ranks of the data. One of the most common applications of the signed-rank test in clinical trials is to compare responses between correlated or paired data, such as testing for significant pre- to post-study changes of a non-normally distributed evaluation measurement. The layout is the same as that for the paired-difference t-test (Chapter 4).

12.2 Synopsis Given a sample, y1, y2, ..., yn, of n non-zero differences or changes (zeroes are ignored), let Ri represent the rank of | yi | (when ranked lowest to highest). Let R(+) represent the sum of the ranks associated with the positive values of the yi’s, and let R(–) represent the sum of the ranks associated with the negative values of the yi’s. The test statistic is based on the smaller of R(+) and R(–). The hypothesis of ‘no mean difference’ is rejected if this statistic is larger than the critical value found in a special set of tables. To avoid the requirement of special table values, you can use an approximation to this test based on the t-distribution, as shown next.

228

Common Statistical Methods for Clinical Research with SAS Examples

Let (+)

S = (R

(–)

– R )/2

and V = (n(n + 1)(2n + 1)) / 24 The approximate test can be summarized as

null hypothesis: alt. hypothesis: test statistic:

decision rule:

H0: θ = 0 HA: θ =/ 0 T=

S ⋅ n–1 n ⋅ V – S2

reject H0 if |T| > tα/2,n–1

θ represents the unknown population mean, median, or other location parameter. When H0 is true, T has an approximate Student-t distribution with n–1 degrees of freedom. Widely available t-tables can be used to determine the rejection region. When there are tied data values, the averaged rank is assigned to the corresponding Ri values. A small adjustment to V for tied data values is appropriate, as follows: suppose there are g groups of tied data values. For the jth group, compute cj = m(m-1)(m+1), where m is the number of tied data values for that group. The correction factor of C = c1 + c2 + ... + cg is used to adjust V, as follows:

V = (n (n + 1) (2n + 1) – C / 2) / 24 12.3 Examples p Example 12.1 – Rose-Bengal Staining in KCS

A study enrolling 24 patients diagnosed with keratitis sicca (KCS) was conducted to determine the effect of a new ocular wetting agent, Oker-Rinse, for improving ocular dryness based on Rose-Bengal staining scores. Patients were instructed to use Oker-Rinse in one eye and Hypotears®, as a control, in the other eye 4 times a day for 3 weeks. Rose-Bengal staining scores were measured as 0, 1, 2, 3, or 4 in each of four areas of the eye: cornea, limbus, lateral conjunctiva, and medial conjunctiva. Higher scores represent a greater number of devitalized cells, a condition associated with KCS. The overall scores for each eye (see Table 12.1) were obtained by adding the scores over the four areas. Is there evidence of a difference between the two test preparations in the overall Rose-Bengal staining scores?

Chapter12: The Wilcoxon Signed-Rank Test 229

TABLE 12.1. Raw Data for Example 12.1 Patient Number

Hypotears

Oker-Rinse

Patient Number

Hypotears

Oker-Rinse

1

15

8

13

8

10

2

10

3

14

10

2

3

6

7

15

11

4

4

5

13

16

13

7

5

10

2

17

6

1

6

15

12

18

6

11

7

7

14

19

9

3

8

5

8

20

5

5

9

8

13

21

10

2

10

12

3

22

9

8

11

4

9

23

11

5

12

13

3

24

8

8

Solution Because each patient receives both treatments, you have a matched-pairs setup. The analysis is performed on the differences in Rose-Bengal scores between the treatment groups. The data can be plotted to reveal that the differences have a bimodal distribution (two peaks), which is not consistent with an assumption of normality. Indeed, a formal test for normality (see SAS results later) indicates a significant departure from this assumption. Therefore, use of the Wilcoxon signedrank test is preferable to using the paired-difference t-test. You want to test the hypothesis that θ, the ‘average’ difference in Rose-Bengal scores between the treatment groups, is 0. The differences in scores and ranks of their absolute values are shown in Table 12.2. The average rankings for tied differences are used. For example, a difference of ±1 occurs twice (Patients 3 and 22) so the average of the ranks 1 and 2 (= 1.5) is used. Notice that the 0 differences are omitted from the rankings, leaving n = 22, not 24.

230

Common Statistical Methods for Clinical Research with SAS Examples

TABLE 12.2. Ranks of the Data for Example 12.1

Patient Number

Difference

Rank

Patient Number

Difference

Rank

1 2 3 4 5 6 7 8 9 10 11 12

7 7 -1 -8 8 3 -7 -3 -5 9 -5 10

14.5 14.5 1.5 18.5 18.5 4.5 14.5 4.5 7.5 21 7.5 22

13 14 15 16 17 18 19 20 21 22 23 24

-2 8 7 6 5 -5 6 0 8 1 6 0

3 18.5 14.5 11 7.5 7.5 11 -18.5 1.5 11 --

Compute R(+) = (14.5 + 14.5 + 18.5 + 4.5 + 21 + 22 + 18.5 + 14.5 + 11 + 7.5 + 11 + 18.5 + 1.5 + 11) = 188.5 and R(–) = (1.5 + 18.5 + 14.5 + 4.5 + 7.5 + 7.5 + 3 + 7.5) = 64.5 so that S = (188.5 – 64.5) / 2 = 62.0 To apply the correction factor for ties, you record 6 groups of ties with the frequencies (m's): 2, 2, 4, 3, 4, and 4, (see Table 12.3). Thus, c1 = 2 (1) (3) = 6, c2 = 6, c3 = 4 (3) (5) = 60, c4 = 3 (2) (4) = 24, c5 = 60, and c6 = 60, yielding C = 216. TABLE 12.3. Calculation of Correction for Ties for Example 12.1

Tie Group (i)

|Difference|

Number of Ties (m)

ci =m(m-1)(m+1)

1 2 3 4 5 6

1 3 5 6 7 8

2 2 4 3 4 4

6 6 60 24 60 60 C=216

Chapter12: The Wilcoxon Signed-Rank Test 231

Finally, V

= (n (n + 1) (2n + 1) – C / 2) / 24

= (22 (23) (45) – 108) / 24 = 944.25 The test is summarized as follows: null hypothesis: alt. hypothesis:

H0: θ = 0 HA: θ =/ 0 T=

test statistic: =

S ⋅ n –1 n ⋅ V – S2 62 ⋅ 21 (22 ⋅ 944.25) – 622

= 2.184

decision rule:

reject H0 if |T| > t0.025,21 = 2.080

conclusion:

Because 2.184 > 2.080, reject H0, and conclude that there is a significant difference in RoseBengal scores between treatments.

SAS Analysis of Example 12.1 In the test summary just presented, the SAS function PROBT can be used to obtain the p-value associated with a t-statistic of 2.184 based on 21 degrees of freedom, namely, p = 0.0405 (found by using the SAS expression 2*(1–PROBT(2.184, 21))). The SAS code and output for executing this analysis are shown on the next pages. The Wilcoxon signed-rank test can be performed in SAS using PROC UNIVARIATE. The NORMAL option is used to provide a test for normality based on the Shapiro-Wilk test. This test is rejected with a p-value of 0.0322 , which indicates that the data cannot be assumed to have come from a normal distribution. This result makes the paired-difference one-sample t-test (Chapter 4) inappropriate for this analysis. In Output 12.1, the signed-rank statistic is confirmed to be 62 with a p-value of 0.0405 . SAS uses the t-approximation to the Wilcoxon signed-rank test if n is greater than 20. For n £ 20, SAS computes the exact probability.

232

Common Statistical Methods for Clinical Research with SAS Examples

SAS Code for Example 12.1 DATA KCS; INPUT PAT HYPOTEAR OKERINSE @@; DIFF = HYPOTEAR - OKERINSE; DATALINES; 1 15 8 2 10 3 3 6 7 5 10 2 6 15 12 7 7 14 9 8 13 10 12 3 11 4 9 13 8 10 14 10 2 15 11 4 17 6 1 18 6 11 19 9 3 21 10 2 22 9 8 23 11 5 ;

4 5 13 8 5 8 12 13 3 16 13 7 20 5 5 24 8 8

PROC UNIVARIATE NORMAL DATA = KCS; VAR DIFF; TITLE1 'The Wilcoxon-Signed-Rank Test'; TITLE2 'Example 12.1: Rose Bengal Staining in KCS'; RUN;

OUTPUT 12.1. SAS Output for Example 12.1 The Wilcoxon-Signed-Rank Test Example 12.1: Rose Bengal Staining in KCS The UNIVARIATE Procedure Variable: DIFF Moments N Mean Std Deviation Skewness Uncorrected SS Coeff Variation

24 2.29166667 5.66821164 -0.4011393 865 247.340144

Sum Weights Sum Observations Variance Kurtosis Corrected SS Std Error Mean

24 55 32.1286232 -1.2859334 738.958333 1.15701886

Basic Statistical Measures Location Mean Median Mode

2.29167 4.00000 -5.00000

Variability Std Deviation Variance Range Interquartile Range

5.66821 32.12862 18.00000 9.50000

NOTE: The mode displayed is the smallest of 4 modes with a count of 3.

Chapter12: The Wilcoxon Signed-Rank Test 233

OUTPUT 12.1 SAS Output for Example 12.1 (continued) The Wilcoxon-Signed-Rank Test Example 12.1: Rose Bengal Staining in KCS The UNIVARIATE Procedure Variable: DIFF

Tests for Location: Mu0=0 Test

-Statistic-

-----p Value------

Student's t Sign Signed Rank

t M S

Pr > |t| Pr >= |M| Pr >= |S|

1.980665 3 62

0.0597 0.2863 0.0405

Tests for Normality Test

--Statistic---

-----p Value------

Shapiro-Wilk Kolmogorov-Smirnov Cramer-von Mises Anderson-Darling

W D W-Sq A-Sq

Pr Pr Pr Pr

0.908189 0.201853 0.145333 0.859468

< > > >

Quantiles (Definition 5) Quantile 100% Max 99% 95% 90% 75% Q3 50% Median 25% Q1 10% 5% 1% 0% Min

Estimate 10.0 10.0 9.0 8.0 7.0 4.0 -2.5 -5.0 -7.0 -8.0 -8.0

Extreme Observations ----Lowest----

----Highest---

Value

Obs

Value

Obs

-8 -7 -5 -5 -5

4 7 18 11 9

8 8 8 9 10

5 14 21 10 12

W D W-Sq A-Sq

0.0322 0.0125 0.0248 0.0235

234

Common Statistical Methods for Clinical Research with SAS Examples

12.4 Details & Notes 12.4.1 . The Wilcoxon signed-rank test is considered a non-parametric test because it makes no assumptions regarding the distribution of the population from which the data are collected. This appears to be a tremendous advantage over the t-test, so why not use the signed-rank test in all situations? First, the t-test has been shown to be a more powerful test in detecting true differences when the data are normally distributed. Because the normal distribution occurs quite often in nature, the t-test is the method of choice for a wide range of applications. Secondly, analysts often feel more comfortable reporting a t-test whenever it is appropriate, especially to a non-statistician, because many clinical research professionals have become familiar with the terminology. One reason the t-test has enjoyed popular usage is its robustness under deviations to the underlying assumptions. Finally, the Wilcoxon signedrank test does require the assumption of a symmetrical underlying distribution. When the data are highly skewed or otherwise non-symmetrical, an alternative to the signed-rank test, such as the sign test (discussed in Chapter 15), can be used. 12.4.2. Let Wi = Ri if yi > 0 and Wi = –Ri if yi < 0. It can be shown that the T statistic discussed is equivalent to performing a one-sample t-test (Chapter 4) on the Wi’s, that is, the ranked data adjusted by the appropriate sign. This is sometimes referred to as performing a t-test on the ‘rank-transformed’ data. 12.4.3. An alternative approximation to the signed-rank test for large samples is the Z-test. The test statistic is taken to be S′ = smaller of (R(+), R(–)). S′ has mean and variance:

µ S′ = and σ

2

S′

=

n ⋅ (n+1) 4 n ⋅ (n+1) ⋅ (2n+1) 24

Under H0,

Z=

S′ – µS′ σS′

has an approximate normal distribution with mean 0 and variance 1. The Z–test statistic is compared with the tabled value of Zα/2 for a two-tailed test (= 1.96 for α = 0.05).

Chapter12: The Wilcoxon Signed-Rank Test 235

To illustrate this approximation for Example 12.1, compute S′ = 64.5 µ S′ = and σ S′ =

22 ⋅ 23 = 126.5 4 22 ⋅ 23 ⋅ 45 = 30.8 24

so that Z = (64.5 – 126.5) / 30.8 = –2.013. At p = 0.05, you reject H0 because 2.013 > 1.96. The SAS function PROBNORM can be used to obtain the actual p-value of 0.044. 12.4.4. The one-sample t-test is often used with larger samples, regardless of the distribution of the data, because the Central Limit Theorem states that the sample mean y is normally distributed for large n, no matter what distribution the underlying data have (see Chapter 1). Usually, for n > 30, we can safely use the one-sample t-test (Chapter 4), and ignore distributional assumptions, providing the distribution is symmetric. The Wilcoxon signedrank test and the t-test become closer as n gets larger. For non-symmetrical distributions, the mean, median, and other measures of central tendency might have widely disparate values. The t-test should be used only if the mean is the appropriate measure of central tendency for the population being studied. 12.4.5. The method shown here is for a two-tailed test. For a one-tailed test, the same methods that are shown in Chapter 4 for the one-sample t-test can be used. If SAS is used, the p-value associated with the signed-rank test can be halved for an approximate one-tailed test.

236

Common Statistical Methods for Clinical Research with SAS Examples

CHHAAPPTTEERR 1133 The Wilcoxon Rank-Sum Test 13.1 13.2 13.3 13.4

Introduction...................................................................................................... 237 Synopsis ............................................................................................................ 237 Examples........................................................................................................... 239 Details & Notes................................................................................................. 244

13.1 Introduction The Wilcoxon rank-sum test is a non-parametric analog of the two-sample t-test (Chapter 5). It is based on ranks of the data, and is used to compare location parameters, such as the mean or median, between two independent populations without the assumption of normally distributed data. Although the Wilcoxon rank-sum test was developed for use with continuous numeric data, the test is also applied to the analysis of ordered categorical data. In clinical data analysis, this test is often useful, for example, for comparing patient global assessments or severity ratings between two treatment groups.

13.2 Synopsis The data are collected as two independent samples of size n1 and n2, denoted by y11, y12, ..., y1n1 and y21, y22, ..., y2n2. The data are ranked, from lowest to highest, over the combined samples, and the test statistic is based on the sum of ranks for the first group. Let r1j = rank of y1j (j = 1, 2, ..., n1) and r2j = rank of y2j (j = 1, 2, ..., n2), and compute n1

R1 = å r1j j=1

and n2

R 2 = å r2j j=1

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Chapter 13: The Wilcoxon Rank-Sum Test 239

The test statistic, using a 0.5 continuity correction, is based on an approximate normal distribution, summarized as follows:

null hypothesis: alt. hypothesis:

H0: θ1 = θ2 HA: θ1 =/ θ2 Z=

test statistic: decision rule:

| R 1 – µ R 1 | – 0.5 σ R1

reject H0 if |Z| > Zα/2

where θ1 and θ2 represent the mean, median, or other location parameters for the two populations.

13.3 Examples p Example 13.1 -- Global Evaluations of Seroxatene in Back Pain

In previous studies of the new anti-depressant, Seroxatene, researchers noticed that patients with low back pain experienced a decrease in radicular pain after 6 to 8 weeks of daily treatment. A new study was conducted in 28 patients to determine whether this phenomenon is a drug-related response or coincidental. Patients with MRI-confirmed disk herniation and symptomatic leg pain were enrolled and randomly assigned to receive Seroxatene or a placebo for 8 weeks. At the end of the study, patients were asked to provide a global rating of their pain, relative to baseline, on a coded improvement scale as follows: -------- Deterioration -------

--------- Improvement -------

Marked

Moderate

Slight

No Change

Slight

Moderate

Marked

-3

-2

-1

0

+1

+2

+3

The data are shown in Table 13.1. Is there evidence that Seroxatene has any effect on radicular back pain?

240

Common Statistical Methods for Clinical Research with SAS Examples

TABLE 13.1. Raw Data for Example 13.1 ------ Seroxatene Group -------

-------- Placebo Group ---------

Patient Number

Score

Patient Number

Score

Patient Number

Score

Patient Number

Score

2

0

16

-1

1

3

15

0

3

2

17

2

4

-1

18

-1

5

3

20

-3

7

2

19

-3

6

3

21

3

9

3

23

-2

8

-2

22

3

11

-2

25

1

10

1

24

0

13

1

28

0

12

3

26

2

14

3

27

-1

Solution With a situation involving a two-sample comparison of means, you might first consider the two-sample t-test or the Wilcoxon rank-sum test. Because the data consist of ordered categorical responses, the assumption of normality is dubious, so you can apply the rank-sum test. Tied data values are identified as shown in Table 13.2. TABLE 13.2. Calculation of Correction for Ties in Example 13.1 Response (y)

Number of Ties (m)

Ranks

Average Rank 1.5

6

4

24

2

ck = m(m -1)

-3

2

1, 2

-2

3

3, 4, 5

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4

6, 7, 8, 9

7.5

60

0

4

10, 11, 12, 13

11.5

60

+1

3

14, 15, 16

15

24

+2

4

17, 18, 19, 20

18.5

60

+3

8

21, 22, 23, 24, 25, 26, 27, 28

24.5

504 C = 738

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Common Statistical Methods for Clinical Research with SAS Examples

The test summary, based on a normal approximation at a significance level of α = 0.05, becomes null hypothesis: alt. hypothesis:

H0: θ1 = θ2 HA: θ1 =/ θ2

test statistic:

Z= =

R1 − µ R1 − 0.5 σ R1 (261 − 232) − 0.5 = 1.346 448.38

decision rule:

reject H0 if |Z| > 1.96

conclusion:

Because 1.346 is not > 1.96, you do not reject H0, and conclude that there is insufficient evidence of a difference between Seroxatene and placebo in global back pain evaluations.

Interpolation of normal probabilities tabulated in Appendix A.1 results in a twotailed p-value of 0.178 for the Z-statistic of 1.346. The p-value can also be found by using the SAS function PROBNORM. In SAS, the two-tailed p-value for a Z-value of 1.346 is (1–PROBNORM(1.346)) = 0.1783.

SAS Analysis of Example 13.1 The Wilcoxon rank-sum test is performed in SAS by using PROC NPAR1WAY with the WILCOXON option, as shown in the SAS code on the next page. The sum of ranks for either group can be used to compute the test statistic. The manual calculations use R1 = 261, while SAS uses R2 =145 . When the smaller value is used, the result is a negative Z-value . In either case, for a two-tailed test, | Z | = 1.346 with a p-value of 0.1783 . The output from PROC NPAR1WAY also gives the results of the analysis using the Kruskal-Wallis chi-square approximation (Chapter 14).

Chapter 13: The Wilcoxon Rank-Sum Test 243

SAS Code for Example 13.1 DATA RNKSM; INPUT TRT $ DATALINES; SER 2 0 SER SER 8 -2 SER SER 16 –1 SER SER 22 3 SER PBO 1 3 PBO PBO 11 –2 PBO PBO 19 -3 PBO ;

PAT SCORE @@; 3 2 10 1 17 2 24 0 4 –1 13 1 23 -2

SER SER SER SER PBO PBO PBO

5 3 12 3 20 -3 26 2 7 2 15 0 25 1

SER SER SER SER PBO PBO PBO

6 3 14 3 21 3 27 –1 9 3 18 -1 28 0

PROC NPAR1WAY WILCOXON DATA = RNKSM; CLASS TRT; VAR SCORE; TITLE1 'The Wilcoxon Rank-Sum Test'; TITLE2 'Example 13.1: Seroxatene in Back Pain'; RUN;

OUTPUT 13.1. SAS Output for Example 13.1 The Wilcoxon Rank-Sum Test Example 13.1: Seroxatene in Back Pain The NPAR1WAY Procedure Wilcoxon Scores (Rank Sums) for Variable SCORE Classified by Variable TRT Sum of Expected Std Dev Mean TRT N Scores Under H0 Under H0 Score ------------------------------------------------------------------SER 16 261.0 232.0 21.175008 16.312500 PBO 12 145.0 174.0 21.175008 12.083333 Average scores were used for ties. Wilcoxon Two-Sample Test Statistic Normal Approximation Z One-Sided Pr < Z Two-Sided Pr > |Z| t Approximation One-Sided Pr < Z Two-Sided Pr > |Z|

145.0000

-1.3459 0.0892 0.1783

0.0948 0.1895

Z includes a continuity correction of 0.5. Kruskal-Wallis Test Chi-Square DF Pr > Chi-Square

1.8756 1 0.1708

244

Common Statistical Methods for Clinical Research with SAS Examples

13.4 Details & Notes 13.4.1. The term ‘mean’ is used loosely in this chapter to refer to the appropriate location parameter. Because the mean is not always the best measure of a distribution’s center, the location parameter can refer to some other measure, such as one of the measures of central tendency described in Chapter 1 (Table 1.3). 13.4.2. For symmetric distributions, the population mean and median are the same. For skewed distributions with long tails to the right, the median is usually smaller than the mean and considered a better measure of the distributional ‘center’ or location. The geometric mean is also smaller than the arithmetic mean and is often used as a location parameter for exponentially distributed data. The Wilcoxon rank-sum test tests for a location or positional shift in distributions without the need to identify the best measure of central tendency. Thus, the parameter θ is simply a symbol generically denoting a distribution’s location. 13.4.3. Although you need not make assumptions regarding the actual distributions of the data, the Wilcoxon rank-sum test does assume that the two population distributions have the same shape and differ only by a possible shift in location. Thus, you assume the same dispersion, which is analogous to the assumption of variance homogeneity required of the two-sample t-test. Unlike the Wilcoxon signed-rank test, you need not assume the data come from a symmetric distribution. 13.4.4. The test statistic, R1, has a symmetric distribution about its mean, n1(N+1) / 2. Because of this symmetry, the one-tailed p-value is easily obtained by halving the two-tailed p-value. 13.4.5. The Mann-Whitney U-test is another non-parametric test for comparing location parameters based on two independent samples. Mathematically, it can be shown that the Mann-Whitney test is equivalent to the Wilcoxon rank-sum test. 13.4.6. Notice that the approximate test statistic, Z, can be computed from R2 instead of R1. When using R2, the mean and standard deviation (µR, σR) are computed by reversing n1 and n2 in the formulas given. 13.4.7. Another approximation to the Wilcoxon rank-sum test is the use of the two-sample t-test on the ranked data. In SAS, use PROC RANK to rank the pooled data from the two samples, then use PROC TTEST. A comparison between using the SAS procedures with the standard Wilcoxon rank-sum test is discussed by Conover and Iman (1981).

Chapter 13: The Wilcoxon Rank-Sum Test 245

13.4.8. The assumption of normality can be tested by using PROC UNIVARIATE in SAS. If this is rejected, the Wilcoxon rank-sum test is preferable to the two-sample t-test. Tests such as the Kolmogorov-Smirnov test, available by specifying the EDF option in PROC NPAR1WAY in SAS, can be used to compare the equality of two distributions. Significance indicates a difference in location, variability, or distributional ‘shapes’ between the two populations.

246

Common Statistical Methods for Clinical Research with SAS Examples

CHHAAPPTTEERR 1144 The Kruskal-Wallis Test 14.1 14.2 14.3 14.4

Introduction...................................................................................................... 247 Synopsis ............................................................................................................ 247 Examples........................................................................................................... 249 Details & Notes................................................................................................. 253

14.1 Introduction The Kruskal-Wallis test is a non-parametric analogue of the one-way ANOVA (Chapter 6). It is used to compare population location parameters (mean, median, etc.) among two or more groups based on independent samples. Unlike an ANOVA, the assumption of normally distributed responses is not necessary. The KruskalWallis test is an extension of the Wilcoxon rank-sum test (Chapter 13) for more than two groups, just as a one-way ANOVA is an extension of the two-sample t-test. The Kruskal-Wallis test is based on the ranks of the data. This test is often useful in clinical trials for comparing responses among three or more dose groups or treatment groups using samples of non-normally distributed response data.

14.2 Synopsis The data are collected as k (³2) independent samples of size n1, n2, ... nk, as shown in Table 14.1. TABLE 14.1. Layout for the Kruskal-Wallis Test

Group 1

Group 2

y11 y12 ... y1n1

y21 y22 ... y2n2

...

Group k

yk1 yk2 ... yknk

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Common Statistical Methods for Clinical Research with SAS Examples

TABLE 14.2. Raw Data for Example 14.1 0.1% Solution

0.2% Solution

Placebo

Patient Number

Category Code

Patient Number

Category Code

Patient Number

Category Code

1

5

3

5

2

5

6

4

5

8

4

3

9

1

7

2

8

7

12

7

10

8

11

1

15

4

14

7

13

2

19

3

18

4

16

4

20

6

22

5

17

2

23

7

26

4

21

1

27

8

28

6

24

4

32

7

31

4

25

5

29

4

30

5

Solution This data set has an ordinal scale response with unequally spaced intervals. If you use a one-way ANOVA, the results might depend on the coding scheme used. Because the codes of 1 to 8 are arbitrary, the Kruskal-Wallis test is appropriate; the results do not depend on the magnitude of the coded values, only their ranks. The data can be summarized in a frequency table format that includes the ranks, as shown in Table 14.3. TABLE 14.3. Calculation of Ranks and Correction for Ties in Example 14.1

Ranks

Average Rank

Total Frequency (m)

c= 2 m(m -1)

2

1-3

2

3

24

1

2

4-6

5

3

24

1

0

1

7-8

7.5

2

6

4

2

3

3

9-16

12.5

8

504

26-50%

5

1

2

3

17-22

19.5

6

210

51-75%

6

1

1

0

23-24

23.5

2

6

76-99%

7

3

1

1

25-29

27

5

120

100%

8

1

2

0

30-32

31

3

24

10

10

12

32

C = 918

-- Frequencies -0.1% 0.2% Placebo

Response Category

Code

<0%

1

1

0

0%

2

0

1-10%

3

11-25%

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Common Statistical Methods for Clinical Research with SAS Examples

SAS Analysis of Example 14.1 In the preceding test summary, the PROBCHI function in SAS can be used to obtain the p-value associated with the test statistic 4.473 based on 2 degrees of freedom namely, p = 0.1068 (=1 – PROBCHI(4.473,2)). The Kruskal-Wallis test can be performed in SAS by using PROC NPAR1WAY with the WILCOXON option, as shown in the SAS code for Example 14.1. SAS will automatically perform the Wilcoxon rank-sum test (Chapter 13) if there are only two groups and the Kruskal-Wallis test if there are more than two groups, as determined by the number of levels in the class variable. In this example, there are three treatment levels in the DOSE variable, which is used in the CLASS statement . The output shows the rank sums for each group and the chi-square statistic , which corroborate the manual calculations demonstrated. The p-value is also printed out by SAS. SAS Code for Example 14.1 DATA PSOR; INPUT DOSE $ PAT SCORE @@; DATALINES; 0.1 1 5 0.1 6 4 0.1 9 0.1 15 4 0.1 19 3 0.1 20 0.1 27 8 0.1 32 7 0.2 3 0.2 7 2 0.2 10 8 0.2 14 0.2 22 5 0.2 26 4 0.2 28 PBO 2 5 PBO 4 3 PBO 8 PBO 13 2 PBO 16 4 PBO 17 PBO 24 4 PBO 25 5 PBO 29 ;

1 6 5 7 6 7 2 4

0.1 0.1 0.2 0.2 0.2 PBO PBO PBO

12 23 5 18 31 11 21 30

7 7 8 4 4 1 1 5

PROC NPAR1WAY WILCOXON DATA = PSOR; CLASS DOSE; VAR SCORE; TITLE1 'The Kruskal-Wallis Test'; TITLE2 'Example 14.1: Psoriasis Evaluation in Three Groups’; RUN;

Chapter 14: The Kruskal-Wallis Test 253

OUTPUT 14.1. SAS Output for Example 14.1 Example 14.1:

The Kruskal-Wallis Test Psoriasis Evaluation in Three Groups The NPAR1WAY Procedure

Wilcoxon Scores (Rank Sums) for Variable SCORE Classified by Variable DOSE Sum of Expected Std Dev Mean DOSE N Scores Under H0 Under H0 Score ------------------------------------------------------------------0.1 10 189.50 165.0 24.249418 18.950000 0.2 10 194.00 165.0 24.249418 19.400000 PBO 12 144.50 198.0 25.327691 12.041667 Average scores were used for ties. Kruskal-Wallis Test Chi-Square DF Pr > Chi-Square

4.4737 2 0.1068

14.4 Details & Notes 14.4.1. When k=2, the Kruskal-Wallis chi-square value has 1 degree of freedom This test is identical to the normal approximation used for the Wilcoxon rank-sum test (Chapter 13). As noted in previous sections, a chisquare with 1 degree of freedom can be represented by the square of a standardized normal random variable (see Appendix B). For k=2, the h-statistic is the square of the Wilcoxon rank-sum Z-test (without the continuity correction). 14.4.2. The effect of adjusting for tied ranks is to slightly increase the value of the test statistic, h. Therefore, omission of this adjustment results in a more conservative test. 14.4.3. For small ni’s (< 5), the chi-square approximation might not be appropriate. Special tables for the exact distribution of h are available in many non-parametric statistics books. These tables can be used to obtain the appropriate rejection region for small samples. (For example, Nonparametric Statistical Methods by Hollander and Wolfe (1973) provides tables of the critical values for the Kruskal-Wallis test, the Wilcoxon rank-sum test, and the Wilcoxon signed-rank test.) Exact tests are also available in SAS by using the EXACT statement in PROC NPAR1WAY (see SAS documentation). Exact tests should be used in special cases when the distribution of the test statistic (h) is not well approximated with the chi-square, such as tests involving very small sample sizes or many ties.

254

Common Statistical Methods for Clinical Research with SAS Examples

14.4.4. When the distribution of the response data is unknown and cannot be assumed to be normal, the median (rather than the mean) is frequently used as a measure of location or central tendency. The median is computed as the middle value (for an odd number of observations) or the average of the two middle values (for an even number of observations). Approximate 95% confidence intervals for the median can be constructed using non-parametric methods developed by Hodges and Lehmann (see Hollander and Wolfe, 1973). 14.4.5. Another approximate test based on the ranks uses a one-way ANOVA on the ranked data. This method is often used instead of the KruskalWallis test because similar results can be expected. Use the following SAS statements to rank the data for Example 14.1, then perform the ANOVA. PROC RANK DATA = PSOR OUT = RNK; VAR SCORE; RANKS RNKSCORE; RUN; PROC GLM DATA = RNK; CLASS DOSE; MODEL RNKSCORE = DOSE / SS3; RUN;

The output (Output 14.2) shows the ANOVA results using the ranked data, which gives the p-value 0.1044 (compared with 0.1068 for the chi-square method). 14.4.6. When the Kruskal-Wallis test is significant, pairwise comparisons can be carried out with the Wilcoxon rank-sum test for each pair of groups. However, the multiplicity problem affecting the overall error rate must be considered for larger values of k. For example, the adjusted p-value method, such as a Bonferroni or Holm’s method using PROC MULTTEST in SAS, can be performed as described in Appendix E. Other techniques for handling multiple comparisons when using non-parametric tests are discussed in Hollander and Wolfe (1973).

Chapter 14: The Kruskal-Wallis Test 255

OUTPUT 14.2. Analysis of Example 14.1 Using a One-Way ANOVA on the Ranks

Example 14.1:

The Kruskal-Wallis Test Psoriasis Evaluation in Three Groups The GLM Procedure

Class Level Information Class DOSE

Levels 3

Values 0.1 0.2 PBO

Number of observations Dependent Variable: RNKSCORE

DF

Sum of Squares

Model 2 Error 29 Corrected Tota1 31

382.645833 2268.854167 2651.500000

Source

Source DOSE

32 Rank for Variable SCORE

Mean Square

F Value

191.322917 78.236351

2.45

Pr > F 0.1044

R-Square

Coeff Var

Root MSE

RNKSCORE Mean

0.144313

53.60686

8.845131

16.50000

DF

Type III SS

Mean Square

F Value

2

382.6458333

191.3229167

2.45

Pr > F 0.1044

14.4.7. When the k levels of the Group factor can be ordered, you might want to test for association between response and increasing levels of Group. The dose-response study is an example. If the primary question is whether increasing response is related to larger doses, the Jonckheere-Terpstra test for ordered alternatives can be used. This is a non-parametric test that shares the same null hypothesis with the Kruskal-Wallis test, but the alternative is more specific:

HA: θ1 ≤ θ2 ≤ … ≤ θk with at least one strict inequality (<).

256

Common Statistical Methods for Clinical Research with SAS Examples

The Jonckheere-Terpstra test can be performed by specifying the JT option in the TABLES statement when using PROC FREQ in SAS. The SAS statements for this test using Example 14.1 are shown in the program that follows. (Notice that you first create a new variable, DOS, so that the dose levels are in ascending order.) DATA JTTEST; SET PSOR; IF DOSE = 'PBO' THEN DOS = 0; ELSE DOS = DOSE; RUN; PROC FREQ DATA = JTTEST; TABLES DOS*SCORE / JT; RUN;

This test results in a p-value of 0.056 (compared with 0.107 for the KruskalWallis test), which illustrates the increased power when using this test for the more targeted alternative. The Jonckheere-Terpstra test uses a chi-square approximation for large samples. Exact tests are available (see Hollander and Wolfe, 1973), and these tests can be performed by using the EXACT statement in PROC FREQ in SAS. 14.4.8. As illustrated in 14.4.5, using ANOVA methods based on the ranktransformed data is a common way to analyze data when the parametric assumptions might not be fulfilled. This approach can also be used for a twoway layout by first ranking the observations from lowest to highest across all treatment groups within each block, then using the usual two-way ANOVA methods (Chapter 7) on these ranks to test for significant treatment effects. Such a technique is often substituted for Friedman's test, which is the nonparametric analog of the two-way ANOVA. This approach assumes multiple observations per cell (nij > 1). Other types of layouts can be analyzed using similar ranking methods (see Conover and Iman, 1981).

CHHAAPPTTEERR 1155 The Binomial Test 15.1 15.2 15.3 15.4

Introduction...................................................................................................... 257 Synopsis ............................................................................................................ 257 Examples........................................................................................................... 259 Details & Notes................................................................................................. 261

15.1 Introduction The binomial test is used to make inferences about a proportion or response rate based on a series of independent observations, each resulting in one of two possible mutually exclusive outcomes. The outcomes can be: response to treatment or no response, cure or no cure, survival or death, or in general, event or non-event. These observations are ‘binomial’ outcomes if the chance of observing the event of interest is the same for each observation. In clinical trials, a common use of the binomial test is for estimating a response rate, p, using the number of patients (X) who respond to an investigative treatment out of a total of n studied. Special cases of the binomial test include two commonly used tests in clinical data analysis, McNemar's test (Chapter 18) and the sign test (discussed later in this chapter).

15.2 Synopsis The experiment consists of observing n independent observations, each with one of two possible outcomes: event or non-event. For each observation, the probability of the event is denoted by p (0 < p < 1). The total number of events in n observations, X, follows the binomial probability distribution (Chapter 1). Intuitively, the sample proportion, X/n, would be a good estimate of the unknown population proportion, p. Statistically, it is the best estimate.

258

Common Statistical Methods for Clinical Research with SAS Examples

The general formula for a binomial probability is: Pr(X=x) =

n! ⋅ p x (1 – p) n – x x!(n – x)!

where x can take integer values 0, 1, 2, ..., n. The symbol ‘!’ is read ‘factorial’ and indicates multiplication by successively smaller integer values down to 1, i.e., a! = (a)·(a-1)·(a-2) ... (3)·(2)·(1). (For example, 5! = 5 x 4 x 3 x 2 x 1 = 120). You want to determine whether the population proportion, p, differs from a hypothesized value, p0. If the unknown proportion, p, equals p0, then the estimated proportion, X/n, should be close to p0, i.e., X should be close to n · p0. When p differs from p0, X might be much larger or smaller than n · p 0. Therefore, you reject the null hypothesis if X is ‘too large’ or ‘too small’. The test summary is: null hypothesis: alt. hypothesis:

H0: p = p0 HA: p =/ p0

test statistic:

X = the number of events in n observations

decision rule:

reject H0 if X £ XL or X ³ XU, where XL and XU are chosen to satisfy Pr(XL < X < XU) ³ 1–α

Tables of binomial probabilities found in most introductory statistical texts or the SAS function PROBBNML can be used to determine XL and XU. This method is demonstrated in Example 15.1. Another option, using a normal approximation, is discussed following Example 15.1.

Chapter 15: The Binomial Test 259

15.3 Examples p Example 15.1 -- Genital Wart Cure Rate

A company markets a therapeutic product for genital warts with a known cure rate of 40% in the general population. In a study of 25 patients with genital warts treated with this product, patients were also given high doses of vitamin C. As shown in Table 15.1, 14 patients were cured. Is this consistent with the cure rate in the general population? TABLE 15.1. Raw Data for Example 15.1 Patient Number

Cured ?

Patient Number

Cured ?

Patient Number

Cured ?

YES

8

NO

9

YES

15

NO

16

YES

22

YES

NO

23

NO

3

YES

10

NO

17

NO

24

YES

4

NO

11

YES

18

YES

25

YES

5

YES

12

NO

19

YES

6

YES

13

YES

20

NO

7

NO

14

NO

21

YES

Patient Number

Cured ?

1 2

Solution Let p represent the probability of a cure in any randomly selected patient with genital warts and treated with Vitamin C concomitantly with the company's product. You want to know if this unknown rate differs from the established rate of p0 = 0.4. Thus, null hypothesis: alt. hypothesis:

H0: p = 0.4 HA: p =/ 0.4

test statistic:

X = 14 (the number cured out of the n = 25 patients treated)

decision rule:

reject H0 if X £ 4 or X ³ 15

conclusion:

Do not reject the null hypothesis. Because 14 does not lie in the rejection region, you conclude that there is insufficient evidence from this study to indicate that concomitant Vitamin C treatment has an effect on the product's cure rate.

260

Common Statistical Methods for Clinical Research with SAS Examples

The decision rule is established by finding a range of X values such that the probability of X being outside that range when the null hypothesis is true is approximately equal to the significance level. With a nominal significance level of α = 0.05, the limits of the rejection region, 4 and 15, can be found from a table of binomial probabilities, satisfying Pr(X £ 4) + Pr(X ³ 15) £ 0.05. The PROBBNML function in SAS can also be used (see Section 15.4.4). The actual significance level is 0.044. Normal Approximation For large values of n and non-extreme values of p, a binomial response, X, can be approximated by a normal distribution with mean n×p and variance n×p×(1–p). The approximation improves as n gets larger or as p gets closer to 0.5. Based on this statistical principle, another way to establish the rejection region for α = 0.05 and large n is by computing ½

and

XL = (n × p0) – 1.96(n × p0 × (1 – p0))

½

XU = (n × p0) + 1.96(n × p0 × (1 – p0))

Applying these formulas to Example 15.1, you obtain

½

XL = (25 · 0.4) – 1.96(25(0.4)(0.6)) = 5.2 and ½

XU = (25 · 0.4) + 1.96(25(0.4)(0.6)) = 14.8 Because X can only take integer values, the rejection region becomes X £ 5 and X ³ 15. Because of the small sample size, this approximation produces a slightly larger lower rejection region and yields an actual significance level of 0.064 (based on exact binomial probabilities). A more common form of the normal approximation to the binomial test uses as the test statistic the actual estimate of p, namely pˆ = X/n. Under the assumption of approximate normality, the test summary becomes:

Chapter 15: The Binomial Test 261

null hypothesis: alt. hypothesis:

H0: p = p0 HA: p =/ p0 1 2n p 0 ⋅ (1 – p 0) n

| pˆ – p 0 | –

test statistic:

Z=

decision rule:

reject H0 if | Z | > Zα/2

Because the binomial distribution is a discrete distribution (i.e., takes integer values), while the normal distribution is continuous, the 1/(2n) in the numerator of the test statistic is used as a ‘continuity correction’ to improve the approximation. When H0 is true, Z has an approximate standard normal distribution. For the commonly used value of α = 0.05, Z0.025 = 1.96. In Example 15.1, the test statistic based on the normal approximation is

14 1 − 0.4 − 25 50 0.140 Z= = = 1.429 0.098 0.4 ⋅ 0.6 25

Because 1.429 is less than 1.96, the null hypothesis is not rejected.

15.4 Details & Notes 15.4.1. The normal approximation to the binomial distribution is a result of the Central Limit Theorem (Chapter 1). Let yi represent a binomial response for Patient i (i = 1, 2, …, n) with numeric values of 0 (‘non-event’) and 1 (‘event’). Note that X, the number of ‘events’ in n trials, is simply the sum of the yi’s. The probability of ‘event’, p, is simply the population mean of the distribution from which the yi’s have been selected. That distribution is the binomial distribution with mean, p, and variance, p·(1–p). Therefore, pˆ = X/n = y has an approximate normal distribution with mean p and variance p·(1-p)/n for large n, according to the Central Limit Theorem. The Z-test statistic shown above uses p0, the hypothesized value of the unknown parameter p because the test is conducted assuming H0 is true.

262

Common Statistical Methods for Clinical Research with SAS Examples

15.4.2. The normal approximation to the binomial is generally a good approximation if

and

( n ⋅ p)

– 2 n ⋅ p ⋅ (1 – p) ≥ 0

( n ⋅ p)

+ 2 n ⋅ p ⋅ (1 – p) ≤ n

or equivalently, if n ≥ 4 ´ max((p/(1–p)), ((1–p)/p)) The minimum sample size, n, satisfying this inequality is shown in Table 15.2 for various values of p. Larger n’s are required for response probabilities (p) closer to 0 or 1. TABLE 15.2. Minimum n Required by Normal Approximation as a Function of p

p

n

0.5

4

0.4 or 0.6

6

0.3 or 0.7

10

0.2 or 0.8

16

0.1 or 0.9

36

15.4.3. An approximate 95% confidence interval for estimating the proportion, p, is given by pˆ ± 1.96 ⋅

ˆ pˆ ⋅ (1 – p) n

where pˆ = X/n. With X/n = 14 / 25 = 0.56 in Example 15.1, a 95% confidence interval for p is 0.56 ± 1.96 ((0.56 x 0.44) / 25)½ = 0.56 ± 0.19 = (0.37 - 0.75)

Chapter 15: The Binomial Test 263

15.4.4. The SAS function PROBBNML can be used to obtain binomial probabilities for specified p and n. The actual significance level 0.044 in Example 15.1 is found by using the SAS statement PROBBNML(0.4,25,4) + (1 – PROBBNML(0.4,25,14)).

The PROBNORM function in SAS can be used with the normal approximation to obtain p-values. The approximate p-value for Example 15.1 with a Z-test statistic of 1.429 is found in SAS as p=2*(1 – PROBNORM(1.429) = 0.136. 15.4.5. When p0 = 0.5, the binomial test is sometimes called the sign test. A common application of the sign test is in testing for pre- to post-treatment changes, given information only about whether a measurement increases or decreases following treatment. The number of increases is a binomial random variable with p = 0.5 when the null hypothesis of no pre- to post-treatment changes is true. The UNIVARIATE procedure in SAS can be used to conduct the sign test (Chapter 18). 15.4.6. Because Example 15.1 tests for a difference from the hypothesized value in either direction, a two-tailed test is used. A one-tailed test would be used when you want to test whether the population proportion, p, is strictly greater than or strictly less than the threshold level, p0. Use the rejection region according to the alternative hypothesis as follows: Alternative Hypothesis

Corresponding Rejection Region

two-tailed

HA: p =/ po

reject H0 if Z > Zα/2 or Z < –Zα/2

one-tailed (right)

HA: p > p0

reject H0 if Z > Zα

one-tailed (left)

HA: p < p0

reject H0 if Z < –Zα

Type of Test

264

Common Statistical Methods for Clinical Research with SAS Examples

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Total 66

52

118

Statistics for Table of GRP by RESP Statistic DF Value Prob -----------------------------------------------------Chi-Square 1 5.0119 ê0.0252 Likelihood Ratio Chi-Square 1 5.0270 0.0250 Continuity Adj. Chi-Square 1 4.2070 0.0403 Mantel-Haenszel Chi-Square 1 4.9694 0.0258 Phi Coefficient -0.2061 Contingency Coefficient 0.2018 Cramer's V -0.2061 Fisher's Exact Test ---------------------------------Cell (1,1) Frequency (F) 22 Left-sided Pr <= F 0.0201 Right-sided Pr >= F 0.9925 Table Probability (P) Two-sided Pr <= P Sample Size = 118

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PROC FREQ DATA = DORATE; TABLES DOSE_MG*RESP / CHISQ CMH TREND NOPERCENT NOCOL; WEIGHT CNT; TITLE1 ’The Chi-Square Test’; TITLE2 ’Example 16.2: Comparison of Dropout Rates for 4 Dose Groups’; RUN;

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Cochran-Armitage Trend Test -------------------------Statistic (Z) 2.0254 One-sided Pr > Z 0.0214 Two-sided Pr > |Z| 0.0428

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CHHAAPPTTEERR 1177 Fisher’s Exact Test 17.1 17.2 17.3 17.4

Introduction...................................................................................................... 285 Synopsis ............................................................................................................ 285 Examples........................................................................................................... 286 Details & Notes................................................................................................. 289

17.1 Introduction Fisher's exact test is an alternative to the chi-square test (Chapter 16) for comparing two independent binomial proportions, p1 and p2. This method is based on computing exact probabilities of observing a given result or a more extreme result, when the hypothesis of equal proportions is true. Fisher's exact test is useful when the normal approximation to the binomial might not be applicable, such as in the case of small cell sizes or extreme proportions.

17.2 Synopsis Using the same notation as in Chapter 16, you observe X1 ‘responders’ out of n1 patients studied in one group and X2 ‘responders’ of n2 patients in a second independent group, as shown in Table 17.1. TABLE 17.1. Layout for Fisher’s Exact Test Number of Responders

Number of Non-Responders

Total

Group 1

X1

n1-X1

n1

Group 2

X2

n2-X2

n2

Combined

X1+X2

N-(X1+X2)

N = n1 + n2

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Chapter 17: Fisher’s Exact Test 287

The summary results are shown in Table 17.2. TABLE 17.2. CHF Frequency Summary for Example 17.1 Active Group (i=1)

Placebo Group (i=2)

Combined

Xi

2

5

7

ni

35

20

55

Estimate of p

0.057

0.250

0.127

The conditions for using the chi-square test (Chapter 16) are not met in this case because of the small cell sizes. However, Fisher's exact test can be used as demonstrated next. The observed table and tables with a more extreme result that have the same row and column totals are shown in Table 17.3. TABLE 17.3. Configuration of ‘Extreme’ Tables for Calculation of Fisher’s Exact Probability Table (i)

Table (ii)

Table (iii)

Row Total

X

n–X

X

n–X

X

n–X

n

ACT

2

33

1

34

0

35

35

PBO

5

15

6

14

7

13

20

Column Total

7

48

7

48

7

48

55

Under the null hypothesis, H0, the probability for Table (i) is found by

prob1 =

(7)! ⋅ (48)! ⋅ (35)! ⋅ (20)! = 0.04546 (55)! ⋅ (2)! ⋅ (5)! ⋅ (33)! ⋅ (15)!

Similarly, the probabilities for Tables (ii) and (iii) can be computed as prob2 = 0.00669 and prob3 = 0.00038, respectively. The exact one-tailed p-value is p = 0.04546 + 0.00669 + 0.00038 = 0.05253. Because this value is greater than 0.05, you would not reject the hypothesis of equal proportions at a significance level of 0.05. (This is close to 0.05, however, and it might encourage a researcher to conduct a larger study).

288

Common Statistical Methods for Clinical Research with SAS Examples

SAS Analysis of Example 17.1 Fisher's exact test can be performed by using the FISHER option in the TABLES statement in PROC FREQ, as shown in the SAS code for Example 17.1 . This example uses the WEIGHT statement because the summary results are input. If you use the data set that contains the observations for individual patients, the WEIGHT statement would be omitted. Notice that, with a 2´2 table, either the FISHER or the CHISQ option in the TABLES statement will print both the chisquare test and the Fisher’s exact test results. Fisher's exact probability is printed in Output 17.1, with a one-tailed value of 0.0525 . Notice that SAS prints a warning about using the chi-square test when cell sizes are too small. SAS Code for Example 17.1

DATA CABG; INPUT GRP RESP $ CNT DATALINES; 1 YES 2 1 _NO 33 2 YES 5 2 _NO 15 ;

@@;

/* GRP 1 = Active Group, GRP 2 = Placebo */ PROC FREQ DATA = CABG; TABLES GRP*RESP / FISHER NOPERCENT NOCOL; WEIGHT CNT; TITLE1 “Fisher's Exact Test”; TITLE2 'Example 17.1: CHF Incidence in CABG after ARA'; RUN;

Chapter 17: Fisher’s Exact Test 289

OUTPUT 17.1. SAS Output for Example 17.1

Example 17.1:

Fisher's Exact Test CHF Incidence in CABG after ARA The FREQ Procedure Table of GRP by RESP

GRP

RESP

Frequency‚ Row Pct |YES |_NO | ---------+--------+--------+ 1 | 2 | 33 | | 5.71 | 94.29 | ---------+--------+--------+ 2 | 5 | 15 | | 25.00 | 75.00 | ---------+--------+--------+ Total 7 48

Total 35

20

55

Statistics for Table of GRP by RESP Statistic DF Value Prob -----------------------------------------------------Chi-Square 1 4.2618 0.0390 Likelihood Ratio Chi-Square 1 4.1029 0.0428 Continuity Adj. Chi-Square 1 2.7024 0.1002 Mantel-Haenszel Chi-Square 1 4.1843 0.0408 Phi Coefficient -0.2784 Contingency Coefficient 0.2682 Cramer's V -0.2784 WARNING: 50% of the cells have expected counts less than 5. Chi-Square may not be a valid test.

Fisher's Exact Test ---------------------------------Cell (1,1) Frequency (F) 2 Left-sided Pr <= F 0.0525 Right-sided Pr >= F 0.9929 Table Probability (P) Two-sided Pr <= P

0.0455 0.0857

Sample Size = 55

17.4 Details & Notes 17.4.1. Fisher's exact test is considered a non-parametric test because it does not rely on any distributional assumptions. 17.4.2. Manual computations for Fisher's exact probabilities can be very tedious if you use a calculator, especially, with larger cell sizes. A statistical program, such as SAS, is recommended to facilitate the computations.

290

Common Statistical Methods for Clinical Research with SAS Examples

17.4.3. Fisher's probabilities are not necessarily symmetric. Although some analysts will double the one-tailed p-value to obtain the two-tailed result, this method is usually overly conservative. To obtain the two-tailed p-value, first compute the probabilities associated with all possible tables that have the same row and column totals. The 2-tailed p-value is then found by adding the probability of the observed table with the sum of the probabilities of each table whose probability is less than that of the observed table. To obtain the two-tailed test for Example 17.1 in this way, first compute the probabilities for each table as follows: TABLE 17.4. Individual Table Probabilities for Example 17.1

Table

Group

Responders

NonResponders

1

Active

0

35

Placebo

7

13

Active

1

34

Placebo

6

14

2 3 4 5 6 7 8

Probabilities

0.0004

+

0.0067

+

0.0455

+

Active

2

33

Placebo

5

15

Active

3

32

Placebo

4

16

Active

4

31

Placebo

3

17

Active

5

30

Placebo

2

18

Active

6

29

Placebo

1

19

Active

7

28

Placebo

0

20

0.0331

TOTAL:

1.0000

(Observed)

0.1563 0.2941 0.3040 0.1600 +

+ = Probability included in two-tailed p-value

The observed Table 3 has probability 0.0455. Tables 1, 2, and 8 have probabilities less than 0.0455 and are included in the p-value for the twotailed alternative. Thus, the two-tailed p-value is P = 0.0004 + 0.0067 + 0.0455 + 0.0331 = 0.0857 As noted in Output 17.1, both the one-tailed and two-tailed results for Fisher's exact test are provided.

Chapter 17: Fisher’s Exact Test 291

17.4.4. Fisher's exact test can be extended to situations that involve more than two treatment groups or more than two response levels. This is sometimes referred to as the generalized Fisher’s exact test or the Freeman-Halton test. With g treatment groups, you would establish a g´2 table of responses by extending the 2´2 table. We may also extend this method to multinomial responses, i.e., responses that can result in one of r possible outcomes (r > 2). With two treatment groups, the 2´2 table would be extended to a 2´r table, or more generally, to a g´r table. An example that illustrates the generalized Fisher’s exact test applied to a 3´6 table is shown in the “SAS/STAT User’s Guide”. The alternative hypothesis that is tested by the generalized Fisher’s exact test is the hypothesis that corresponds to ‘general association’, as discussed in Chapter 16. In SAS, the FISHER or EXACT option must be specified in the TABLES statement in PROC FREQ to perform the Fisher’s exact test when the table has more than two rows or two columns.

292

Common Statistical Methods for Clinical Research with SAS Examples

CHHAAPPTTEERR 18 McNemar’s Test 18.1 18.2 18.3 18.4

Introduction...................................................................................................... 293 Synopsis ............................................................................................................ 293 Examples........................................................................................................... 294 Details & Notes................................................................................................. 301

18.1 Introduction McNemar's test is a special case of the binomial test (Chapter 15) for comparing two proportions using paired samples. McNemar's test is often used in a clinical trials application when dichotomous outcomes are recorded twice for each patient under different conditions. The conditions might represent different treatments or different measurement times, for example. The goal is to compare response rates under the two sets of conditions. Because measurements come from the same patients, the assumption of independent groups, which is needed for the chi-square test and Fisher's exact test, is not met. Typical examples where McNemar's test might be applicable include testing for a shift in the proportion of abnormal responses from before treatment to after treatment in the same group of patients, or comparing response rates of two ophthalmic treatments when both are given to each patient, one treatment in each eye.

18.2 Synopsis In general, there are n patients, each observed under two conditions (time points, treatments, etc.) with each condition resulting in a dichotomous outcome. The results can be partitioned into 4 subgroups: the number of patients who respond under both conditions (A), the number of patients who respond under the first but not the second condition (B), the number of patients who respond under the second but not the first condition (C), and the number of patients who fail to respond under either condition (D), as shown in the 2H2 table which follows.

294

Common Statistical Methods for Clinical Research with SAS Examples

TABLE 18.1. Layout for McNemar’s Test

Condition 2

Condition 1

Number of Responders

Number of NonResponders

Total

Number of Responders

A

B

A+B

Number of NonResponders

C

D

C+D

Total

A+C

B+D

n= A+B+C+D

The hypothesis of interest is the equality of the response proportions, p1 and p2, under conditions 1 and 2, respectively. The test statistic is based on the difference in the discordant cell frequencies (B,C) as shown in the test summary below. This statistic has an approximate chi-square distribution when H0 is true.

null hypothesis: alt. hypothesis:

H0: p1 = p2 HA: p1 =/ p2

test statistic:

χ2 =

decision rule:

reject H0 if χ > χ 21 (α)

(B − C) 2 B+C 2

The rejection region is found by obtaining the critical chi-square value based on 1 degree of freedom, χ 21(α), from chi-square tables or by using the SAS function CINV(1–α, df). For α = 0.05, the critical chi-square value is 3.841 (see Table A.3 in Appendix A.3, or the SAS function CINV(0.95,1)).

18.3 Examples p Example 18.1 -- Bilirubin Abnormalities Following Drug Treatment

Eighty-six patients were treated with an experimental drug for 3 months. Preand post-study clinical laboratory results showed abnormally high total bilirubin values (above the upper limit of the normal range) as indicated in Table 18.2. Is there evidence of a change in the pre- to post-treatment rates of abnormalities?

Chapter 18: McNemar’s Test 295

TABLE 18.2. Raw Data for Example 18.1 Patient Number

Pre-

Post-

Patient Number

Pre-

Post-

Patient Number

Pre-

Post-

1

N

N

31

N

N

61

N

N

2

N

N

32

N

N

62

N

N

3

N

N

33

Y

N

63

N

N

4

N

N

34

N

N

64

N

N

5

N

N

35

N

N

65

N

N

6

N

Y

36

N

N

66

N

N

7

Y

Y

37

N

N

67

N

N

8

N

N

38

N

Y

68

N

N

9

N

N

39

N

Y

69

N

Y

10

N

N

40

N

N

70

N

Y

11

N

N

41

N

N

71

Y

Y

12

Y

N

42

N

Y

72

N

N

13

N

N

43

N

N

73

N

N

14

N

Y

44

Y

N

74

N

Y

15

N

N

45

N

N

75

N

N

16

N

N

46

N

N

76

N

Y

17

N

N

47

Y

Y

77

N

N

18

N

N

48

N

N

78

N

N

19

N

N

49

N

N

79

N

N

20

N

Y

50

N

Y

80

N

N

21

N

N

51

Y

N

81

Y

Y

22

Y

N

52

N

N

82

N

N

23

N

N

53

N

Y

83

N

N

24

N

N

54

N

N

84

N

N

25

Y

N

55

Y

Y

85

N

N

26

N

N

56

N

N

86

N

N

27

N

N

57

N

N

28

Y

Y

58

N

N

29

N

Y

59

N

N

30

N

Y

60

N

N

N = normal, Y = abnormally high

296

Common Statistical Methods for Clinical Research with SAS Examples

Solution Let p1 and p2 represent the proportions of patients who have abnormally high bilirubin values (Y) before and after treatment, respectively. The data are summarized in Table 18.3. TABLE 18.3. Abnormality Frequency Summary for Example 18.1 Post-Treatment

Pre-Treatment

N

Y

Total

N

60

14

74

Y

6

6

12

Total

66

20

86

Y = Total bilirubin above upper limit of normal range

The test summary is null hypothesis: alt. hypothesis:

H0: p1 = p2 HA: p1 =/ p2

test statistic:

2 2 c = (14 – 6) ⁄ (14 + 6)

= 64 ⁄ 20 = 3.20 2

decision rule:

reject H0 if c > 3.841

conclusion:

Because 3.20 is not > 3.841, you do not reject H0, concluding that, at a significance level of 0.05, there is insufficient evidence that a shift in abnormality rates occurs with treatment.

The actual p-value of 0.074 for the chi-square value of 3.20 can be found by using the SAS expression 1–PROBCHI(3.20,1).

Chapter 18: McNemar’s Test 297

SAS Analysis of Example 18.1 Use the AGREE option in the TABLES statement in the FREQ procedure to conduct McNemar's test in SAS. The SAS code and output for Example 18.1 follow. The chi-square value of 3.20 with a p-value of 0.074 are shown in the output. SAS Code for Example 18.1 PROC FORMAT; VALUE RSLTFMT 0 = 'N' RUN; DATA BILI; INPUT PAT PRE PST @@; DATALINES; 1 0 0 2 0 0 3 0 0 4 0 7 1 1 8 0 0 9 0 0 10 0 13 0 0 14 0 1 15 0 0 16 0 19 0 0 20 0 1 21 0 0 22 1 25 1 0 26 0 0 27 0 0 28 1 31 0 0 32 0 0 33 1 0 34 0 37 0 0 38 0 1 39 0 1 40 0 43 0 0 44 1 0 45 0 0 46 0 49 0 0 50 0 1 51 1 0 52 0 55 1 1 56 0 0 57 0 0 58 0 61 0 0 62 0 0 63 0 0 64 0 67 0 0 68 0 0 69 0 1 70 0 73 0 0 74 0 1 75 0 0 76 0 79 0 0 80 0 0 81 1 1 82 0 85 0 0 86 0 0 ;

1 = 'Y';

0 0 0 0 1 0 0 0 0 0 0 1 1 0

5 11 17 23 29 35 41 47 53 59 65 71 77 83

0 0 0 0 0 0 0 1 0 0 0 1 0 0

0 0 0 0 1 0 0 1 1 0 0 1 0 0

6 12 18 24 30 36 42 48 54 60 66 72 78 84

0 1 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 1 0 1 0 0 0 0 0 0 0

ODS EXCLUDE SimpleKappa; PROC FREQ DATA = BILI; TABLES PRE*PST / AGREE NOROW NOCOL; FORMAT PRE PST RSLTFMT.; TITLE1 “McNemar's Test”; TITLE2 'Example 18.1: Bilirubin Abnormalities Following Drug Treatment'; RUN;

298

Common Statistical Methods for Clinical Research with SAS Examples

OUTPUT 18.1. SAS Output for Example 18.1 McNemar's Test Example 18.1: Bilirubin Abnormalities Following Drug Treatment The FREQ Procedure Table of PRE by PST PRE

PST

Frequency| Percent | N | Y | Total ---------+--------+--------+ N | 60 | 14 | 74 | 69.77 | 16.28 | 86.05 ---------+--------+--------+ Y | 6 | 6 | 12 | 6.98 | 6.98 | 13.95 ---------+--------+--------+ Total 66 20 86 76.74 23.26 100.00

Statistics for Table of PRE by PST McNemar's Test ----------------------Statistic (S) 3.2000 DF 1 Pr > S 0.0736

Sample Size = 86

Three Response Categories Many situations arise in clinical data analysis in which the data are paired and the response has three categorical levels. The bilirubin response in Example 18.1, for example, can be classified as ‘normal’, ‘above normal’, or ‘below normal’. Clinical laboratory data are often analyzed using pre- to post-study ‘shift’ tables, which show the frequencies of patients shifting among these three categories from baseline to some point in the study following treatment. Such data can be analyzed using the Stuart-Maxwell test, which is an extension of McNemar’s test. The Stuart-Maxwell test is also a chi-square test based on the cell frequencies from a layout similar to that shown in Table 18.4. The null hypothesis is that the distribution of responses among the three categories is the same under both conditions, in this case, pre- and post-study.

Chapter 18: McNemar’s Test 299

TABLE 18.4. Layout for a Paired Experiment with a Trinomial Response

--------------Post-Study ----------Pre-Study

Low

Normal

High

Low

n11

n12

n13

Normal

n21

n22

n23

High

n31

n32

n33

Compute d1 = (n12 + n13) – (n21 + n31) d2 = (n21 + n23) – (n12 + n32) d3 = (n31 + n32) – (n13 + n23) and, for i ¹ j, n¯ij = (nij + nji) / 2 For i = 1, 2, 3, let pi(1) represent the probability of being classified in Category i under Condition 1 and pi(2) represent the probability of being classified in Category i under Condition 2, the test summary is null hypothesis:

H0: pi(1) = pi(2) for i = 1, 2, and 3

alt. hypothesis:

HA: pi(1) =/ pi(2) for i = 1, 2, or 3

test statistic:

S=

decision rule:

reject H0 if S > χ 22 ( α )

(n 23 ⋅ d12 ) + (n13 ⋅ d 22 ) + (n12 ⋅ d 32 ) 2 ⋅ ((n12 ⋅ n 23 ) + (n12 ⋅ n13 ) + (n13 ⋅ n 23 ))

When H0 is true, S has an approximate chi-square distribution with 2 degrees of freedom. For α = 0.05, the critical chi-square value, χ 2(α), is 5.991 from Table A.3 2

(Appendix A) or by using the SAS function =CINV(0.95,2) .

300

Common Statistical Methods for Clinical Research with SAS Examples

p Example 18.2 -- Symptom Frequency Before and After Treatment

Patients characterized their craving of certain high-fat food products before and two weeks after an experimental diet therapy as ‘never’, ‘occasional’ or ‘frequent’, as summarized in Table 18.5. Does the diet appear to have an effect on the frequency of these cravings? TABLE 18.5. Cell Frequencies for Example 18.2 --------------- Two Weeks --------------

Pre-Study

Never

Occasional

Frequent

Never

14

6

4

Occasional

9

17

2

Frequent

6

12

8

Solution Compute d1 = (6 + 4) – (9 + 6) = –5 d2 = (9 + 2) – (6 + 12) = –7 d3 = (6 + 12) – (4 + 2) = 12 and n 12 = (n12 + n21) / 2 = (6 + 9) / 2 = 7.5 n 13 = (n13 + n31) / 2 = (4 + 6) / 2 = 5 n 23 = (n23 + n32) / 2 = (2 + 12) / 2 = 7

The test statistic is S=

(7 ⋅ ( − 5) 2 ) + (5 ⋅ ( − 7) 2 ) + (7.5 ⋅ (12) 2 ) 2 ⋅ ((7.5 ⋅ 7) + (7.5 ⋅ 5) + (5 ⋅ 7))

= 6.00

which is larger than the critical chi-square value of 5.991. Therefore, there has been a significant shift in the distribution of responses at the α = 0.05 level. The marginal response probabilities are estimated as TABLE 18.6. Marginal Response Probabilities for Example 18.2 Pre-Study

2 Weeks

Never

24 / 78 = 30.8%

29 / 78 = 37.2%

Occasional

28 / 78 = 35.9%

35 / 78 = 44.9%

Frequent

26 / 78 = 33.3%

14 / 78 = 17.9%

Chapter 18: McNemar’s Test 301

18.4 Details & Notes 18.4.1. For McNemar’s test, the estimate of p1 is (A+B) / n, and the estimate of p2 is (A+C) / n. The difference in proportions, p1 – p2, is estimated by ((A+B) / n) – ((A+C) / n) = (B–C) / n. An approximate 95% confidence interval for p1–p2 is given by (B – C) 1 ± 1.96 ⋅ ( ) ⋅ n n

B+C –

(B – C) 2 n

For Example 18.1, the approximate 95% confidence interval for p1 – p2 is

(14 – 6) 1 ± 1.96 ⋅ ( ) ⋅ 86 86

14 + 6 –

(14 – 6) 2 86

= 0.093 ± 0.100 or (–0.007 to 0.193) 18.4.2. Note that the chi-square test statistic is based only on the discordant cell sizes (B, C) and ignores the concordant cells (A, D). However, the estimates of p1, p2 and the size of the confidence interval are based on all cells because they are inversely proportional to the total sample size, n. 18.4.3. follows: χ2=

A continuity correction can be used with McNemar's test as ( | B – C| – 1 )2 B+C

This adjustment results in a more conservative test, and the correction factor may be omitted as B+C gets larger. Re-analyzing the data in Example 18.1 2 with the continuity correction, the test statistic becomes χ = 2.45, which is seen to be smaller and less significant (p=0.118) than the uncorrected value. 18.4.4. A well-known relationship in probability theory is that if a random variable, Z, has a standard normal distribution (i.e., mean 0 and variance 1), 2 then Z has a chi-square distribution with 1 degree of freedom. This principle can be used to show the relationship of the McNemar chi-square to the binomial test using the normal approximation as follows.

302

Common Statistical Methods for Clinical Research with SAS Examples

When the null hypothesis is true, the discordant values, B and C, should be about the same, that is B/(B+C) should be about 1/2. McNemar's test is equivalent to using the binomial test (Chapter 15) to test H0: p = 0.5, where p equals the fraction of the events that fall in 1 of the 2 discordant cells. Using the setup of Chapter 15 with n = B + C and X = B, the normal approximation to the binomial test (expressed as a standard normal Z when H0 is true), is 1 ) |B – C|–1 2(B+C) = 0.5 ⋅ 0.5 B+C (B+C)

( | pˆ – 0.5 | – Z =

pˆ =

because

B (B + C)

Therefore, 2

Z =

( | B – C | – 1 )2 = χ2 (B+C)

18.4.5. Because McNemar's test is identical to the binomial test using a normal approximation, it should be used only when the conditions for the normal approximation apply (see Chapter 15). Using the notation here, these 2 2 conditions can be simplified as B ³ C(4–B) and C ³ B(4–C). Notice that these conditions need to be checked only if either B or C is less than 4. If the conditions are not satisfied, the exact binomial probabilities should be used rather than the normal approximation or the chi-square statistic. 18.4.6. The above application of the binomial test is an example of the sign test mentioned in Chapter 15. In applying the sign test, the number of increases (+) are compared with the number of decreases (–), ignoring tied values. In Example 18.1, B represents the number of increases, C represents the number of decreases, and the concordant values (A and D) represent the tied values, which are ignored. 18.4.7. The p-value in the SAS output corresponds to a two-tailed hypothesis test. The one-tailed p-value can be found by halving this value. 18.4.8. McNemar’s test is often used to detect shifts in response rates between pre- and post-treatment measurement times within a single treatment group. A comparison in shifts can be made between two treatment or dose groups (e.g., Group 1 and Group 2) using a normal approximation as follows. As in Table 18.1, let Ai, Bi, Ci, Di, and ni represent the cell and overall frequencies for Group i (i = 1, 2). The proportional shift for Group i is Di =

Ai + Ci Ai + Bi Ci − Bi − = ni ni ni

Chapter 18: McNemar’s Test 303

To test for a difference between groups in these proportional shifts, you can use D1 − D 2

Z=

2

σD

1

+

2

σD

2

as a test statistic that has an approximate standard normal distribution under the hypothesis of no difference in shifts between groups. Above, the variance of Di is computed as 2

σD

i

=

n i (Bi + Ci ) − (Bi − Ci )2 ni3

18.4.9. The Stuart-Maxwell test is sometimes referred to as a test for ‘marginal homogeneity’ because it tests for equality between the two conditions in the marginal response probabilities. Another alternative hypothesis that might be of interest is that of symmetry. Symmetry occurs when the off-diagonal cell probabilities are the same for each 2H2 sub-table. This implies that changes in response levels between the conditions (e.g., preand post-) occur in both directions with the same probability.

With dichotomous outcomes, the conditions of marginal homogeneity and symmetry are equivalent. With more than two response categories, symmetry is a stronger condition and implies marginal homogeneity. The 3H3 tables in Table 18.6 show a pattern of marginal homogeneity with symmetry (Table i) and without symmetry (Table ii). TABLE 18.6. Examples of Symmetry (i) and non-Symmetry (ii) in a 3 x 3 Table

Table i Condition 1

Condition 2 1

2

3

1

30

15

5

50

2

15

5

10

30

3

5

10

5

20

50

30

20

100

Table ii Condition 1

Condition 2 1

2

3

1

30

0

20

50

2

20

10

0

30

3

0

20

0

20

50

30

20

100

304

Common Statistical Methods for Clinical Research with SAS Examples

Bowker’s test can be used to test for symmetry in 3H3 or larger tables. The test statistic is found by adding the McNemar chi-square statistics for each 2H2 sub-table. For a table with k categories, there are k·(k–1) / 2 2H2 sub-tables. For example, the 3H3 table contains three 2H2 sub-tables with categories 1 and 2, 1 and 3, and 2 and 3. Bowker’s test for k categories is a chi-square test with k·(k–1) / 2 degrees of freedom. Bowker’s test for symmetry can be performed in SAS in the same way that McNemar’s test is run (by using the AGREE option in the TABLES statement in PROC FREQ). If SAS detects k > 2, the test for symmetry is automatically performed. Bowker’s test results for Example 18.2 are found by using the SAS program that follows. The output shows a chi-square test statistic of 8.1429 , which indicates a significant departure from the hypothesis of symmetry (p=0.0431). DATA DIET; INPUT PRE WK2 DATALINES; 0 0 14 0 1 6 0 1 0 9 1 1 17 1 2 0 6 2 1 12 2 ;

CNT @@; 2 4 2 2 2 8

ODS SELECT SymmetryTest; PROC FREQ DATA = DIET; TABLES PRE*WK2 / AGREE NOCOL NOROW; WEIGHT CNT; RUN;

OUTPUT 18.2. Output for Symmetry Test in Example 18.2 Statistics for Table of PRE by WK2 Test of Symmetry ----------------------Statistic (S) 8.1429 DF 3 Pr > S 0.0431

Sample Size = 78

18.4.10. Notice that Section 18.4.8 shows one method for comparing the magnitude of shifts in conditions based on correlated responses among independent groups. If there are more than two groups or more than two response levels, or the design involves other stratification factors, a more complex analysis is required. Stokes, Davis, and Koch (2000) illustrate the analysis of examples that involve repeated measures using PROC CATMOD in SAS.

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The Cochran-Mantel-Haenszel Test Example 19.1: Response to Dermotel in Diabetic Ulcers The FREQ Procedure Table 1 of TRT by RESP Controlling for CNTR=1 TRT

Frequency| Row Pct | YES | NO | ---------+--------+--------+ ACT | 26 | 4 | | 86.67 | 13.33 | ---------+--------+--------+ CTL | 18 | 11 | | 62.07 | 37.93 | ---------+--------+--------+ Total 44 15

RESP

Total 30

29

59

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The Cochran-Mantel-Haenszel Test Example 19.1: Response to Dermotel in Diabetic Ulcers

Table 2 of TRT by RESP Controlling for CNTR=2 TRT

RESP

Frequency| Row Pct | YES | NO | ---------+--------+--------+ ACT | 8 | 3 | | 72.73 | 27.27 | ---------+--------+--------+ CTL | 7 | 5 | | 58.33 | 41.67 | ---------+--------+--------+ Total 15 8

Total 11

12

23

Table 3 of TRT by RESP Controlling for CNTR=3 TRT

RESP

Frequency| Row Pct | YES | NO | ---------+--------+--------+ ACT | 7 | 5 | | 58.33 | 41.67 | ---------+--------+--------+ CTL | 4 | 6 | | 40.00 | 60.00 | ---------+--------+--------+ Total 11 11

Total 12

10

22

Table 4 of TRT by RESP Controlling for CNTR=4 TRT

Frequency| Row Pct | YES | NO | ---------+--------+--------+ ACT | 11 | 6 | | 64.71 | 35.29 | ---------+--------+--------+ CTL | 9 | 5 | | 64.29 | 35.71 | ---------+--------+--------+ Total 20 11

RESP

Total 17

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The Cochran-Mantel-Haenszel Test Example 19.1: Response to Dermotel in Diabetic Ulcers Summary Statistics for TRT by RESP Controlling for CNTR Cochran-Mantel-Haenszel Statistics (Based on Table Scores)

ê

Statistic Alternative Hypothesis DF Value Prob --------------------------------------------------------------1 Nonzero Correlation 1 4.0391 0.0445 2 Row Mean Scores Differ 1 4.0391 0.0445 3 General Association 1 4.0391 0.0445

Breslow-Day Test for Homogeneity of the Odds Ratios -----------------------------Chi-Square 1.8947 DF 3 Pr > ChiSq 0.5946

Total Sample Size = 135 Table of TRT by RESP TRT RESP Frequency| Row Pct | YES | NO | ---------+--------+--------+ ACT | 52 | 18 | | 74.29 | 25.71 | ---------+--------+--------+ CTL | 38 | 27 | | 58.46 | 41.54 | ---------+--------+--------+ Total 90 45

Total 70

65

135

Statistics for Table of RESP by TRT Statistic DF Value Prob -----------------------------------------------------Chi-Square 1 3.7978 0.0513 Likelihood Ratio Chi-Square 1 3.8136 0.0508 Continuity Adj. Chi-Square 1 3.1191 0.0774 Mantel-Haenszel Chi-Square 1 3.7697 0.0522 Phi Coefficient 0.1677 Contingency Coefficient 0.1654 Cramer’s V 0.1677 Fisher’s Exact Test ---------------------------------Cell (1,1) Frequency (F) 52 Left-sided Pr <= F 0.9837 Right-sided Pr >= F 0.0385 Table Probability (P) Two-sided Pr <= P Sample Size = 135

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PROC LOGISTIC DATA = AML DESCENDING; ê MODEL Y = TRT X; TITLE1 ’Logistic Regression’; TITLE2 ’Example 20.1: Relapse Rate Adjusted for Remission Time in AML’; RUN; PROC FREQ DATA = AML; TABLES TRT*Y / CHISQ NOCOL NOPERCENT NOCUM; TITLE3 ’Chi-Square Test Ignoring the Covariate’; RUN;

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Example 20.1:

Logistic Regression Relapse Rate Adjusted for Remission Time in AML The LOGISTIC Procedure

Model Information Data Set WORK.AML Response Variable Y Number of Response Levels 2 Number of Observations 102 Link Function Binary Logit Optimization Technique Fisher’s scoring Response Profile Ordered Value

Y

Total Frequency

1 2

1 0

53 49

Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied. Model Fit Statistics

Criterion

Intercept Only

Intercept and Covariates

143.245 145.870 141.245

129.376 137.251 123.376

AIC SC -2 Log L

Testing Global Null Hypothesis: BETA=0 Test

Chi-Square

DF

Pr > ChiSq

17.8687 16.4848 14.0612

2 2 2

0.0001 0.0003 0.0009

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Analysis of Maximum Likelihood Estimates

Parameter

DF

Estimate

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Intercept TRT X

1 1 1

2.6135 -1.1191 -0.1998

0.7149 0.4669 0.0560

Chi-Square

Pr > ChiSq

13.3662 5.7446 12.7187

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Effect TRT X

Point Estimate 0.327 0.819

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Logistic Regression Relapse Rate Adjusted for Remission Time in AML

Association of Predicted Probabilities and Observed Responses Percent Concordant Percent Discordant Percent Tied Pairs

68.5 23.1 8.4 2597

Somers’ D Gamma Tau-a c

0.454 0.496 0.229 0.727

Chi-Square Test Ignoring the Covariate The FREQ Procedure Table of TRT by Y TRT

Y

Frequency| Row Pct | 0| 1| ---------+--------+--------+ 0 | 20 | 30 | | 40.00 | 60.00 | ---------+--------+--------+ 1 | 29 | 23 | | 55.77 | 44.23 | ---------+--------+--------+ Total 49 53

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Statistics for Table of TRT by Y Statistic DF Value Prob -----------------------------------------------------Chi-Square 1 2.5394 0.1110 Likelihood Ratio Chi-Square 1 2.5505 0.1103 Continuity Adj. Chi-Square 1 1.9469 0.1629 Mantel-Haenszel Chi-Square 1 2.5145 0.1128 Phi Coefficient -0.1578 Contingency Coefficient 0.1559 Cramer’s V -0.1578

Fisher’s Exact Test ---------------------------------Cell (1,1) Frequency (F) 20 Left-sided Pr <= F 0.0813 Right-sided Pr >= F 0.9637 Table Probability (P) Two-sided Pr <= P Sample Size = 102

0.0450 0.1189

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Logistic Regression Example 20.2: Symptom Relief in Severe Gastroparesis The LOGISTIC Procedure Model Information Data Set Response Variable Number of Response Levels Number of Observations Link Function Optimization Technique

WORK.GI RESP 4 55 Cumulative Logit Fisher’s scoring

Response Profile Ordered Value

RESP

Total Frequency

1 2 3 4

4 3 2 1

16 17 13 9

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Class

Value

RX

A B

Design Variables 1 12

1 0

Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied. Score Test for the Proportional Odds Assumption Chi-Square

DF

Pr > ChiSq

2.5521

4

0.6353

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Criterion AIC SC -2 Log L

Intercept Only

Intercept and Covariates

155.516 161.538 149.516

149.907 159.944 139.907

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Logistic Regression Example 20.2: Symptom Relief in Severe Gastroparesis The LOGISTIC Procedure Testing Global Null Hypothesis: BETA=0 Test

Chi-Square

DF

Pr > ChiSq

9.6084 9.0147 8.7449

2 2 2

0.0082 0.0110 0.0126

Likelihood Ratio Score Wald

Type III Analysis of Effects

Effect RX HIST

DF

Wald Chi-Square

Pr > ChiSq

1 1

2.9426 6.4304

0.0863 0.0112

Analysis of Maximum Likelihood Estimates

Parameter

DF

Estimate

Standard Error

Chi-Square

0.5831 0.6001 0.6698 0.5061 0.3841

0.4449 3.2123 13.3018 2.9426 6.4304

Pr > ChiSq

13

Intercept4 Intercept3 Intercept2 RX A HIST

1 1 1 1 1

-0.3890 1.0756 2.4430 0.8682 -0.9741

0.5048 0.0731 0.0003 0.0863 15 0.0112 16

Odds Ratio Estimates

Effect RX A vs B HIST

Point Estimate 2.383 14 0.378 17

95% Wald Confidence Limits 0.884 0.178

6.425 0.802

Association of Predicted Probabilities and Observed Responses Percent Concordant Percent Discordant Percent Tied Pairs

57.6 23.7 18.7 1115

Somers’ D Gamma Tau-a c

0.339 0.417 0.255 0.670

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for reference control */ 19

62 0 4 72 1 6 52 6 8 55 2 5 68 0 0 40 14 20 51 5 8 55 9 11 64 5 9 44 3 3 49 5 8 47 4 5

6 15 25 34 41 2 10 17 23 32 38 47

0 0 0 0 0 1 1 1 1 1 1 1

44 1 2 34 5 15 66 1 7 41 7 9 56 12 17 57 3 8 53 8 10 37 6 8 78 1 3 65 7 18 74 10 15 46 6 7

PROC LOGISTIC DATA = DIARY; MODEL SUCC/ATTPT = TRT AGE / SCALE = WILLIAMS; 20 TITLE1 ’Logistic Regression’; TITLE2 ’Example 20.3: Intercourse Success Rate in Erectile Dysfunction’; RUN;

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Logistic Regression Example 20.3: Intercourse Success Rate in Erectile Dysfunction The LOGISTIC Procedure Model Information Data Set Response Variable (Events) Response Variable (Trials) Number of Observations Weight Variable Sum of Weights Link Function Optimization Technique

WORK.DIARY SUCC ATTPT 46 1 / ( 1 + 0.056638 * (ATTPT - 1) ) 26 268.53902437 binary logit Fisher’s scoring

Response Profile Binary Total Outcome Frequency

Ordered Value 1 2

Event Nonevent

Total Weight

234 183

149.48211 119.05692

Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied. Deviance and Pearson Goodness-of-Fit Statistics Criterion

DF

Value

Value/DF

Pr > ChiSq

Deviance Pearson

43 43

48.2014 42.9997

1.1210 1.0000

0.2706 0.4713

Number of events/trials observations: 46

NOTE: Because the Williams method was used to accommodate 25 overdispersion, the Pearson chi-squared statistic and the deviance can no longer be used to assess the goodness of fit of the model. Model Fit Statistics

Criterion AIC SC -2 Log L

Intercept Only

Intercept and Covariates

370.820 374.853 368.820

364.792 376.891 358.792

Testing Global Null Hypothesis: BETA=0 Test Likelihood Ratio Score Wald

Chi-Square

DF

Pr > ChiSq

10.0277 9.9254 9.6295

2 2 2

0.0066 0.0070 0.0081

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Logistic Regression Example 20.3: Intercourse Success Rate in Erectile Dysfunction The LOGISTIC Procedure Analysis of Maximum Likelihood Estimates

Parameter

DF

Estimate

Standard Error

Chi-Square

Intercept TRT AGE

1 1 1

1.3384 0.5529 -0.0271

0.5732 0.2534 0.0108

5.4515 4.7614 6.2497

Pr > ChiSq 0.0196 0.0291 21 0.0124 23

Odds Ratio Estimates

Effect TRT AGE

Point Estimate 1.738 0.973

95% Wald Confidence Limits 22 24

1.058 0.953

2.856 0.994

Association of Predicted Probabilities and Observed Responses Percent Concordant Percent Discordant Percent Tied Pairs

57.9 38.5 3.6 42822

Somers’ D Gamma Tau-a c

0.194 0.201 0.096 0.597

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6 10 15 23 28 34 40 46 4 11 17 22 29 35 41 47

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7 8 11 13 13 16 13 13 10 7 11 6 10 8 14 15

PROC SORT DATA = HSV; BY VAC PAT; PROC PRINT DATA = HSV; VAR VAC PAT WKS CENS X; TITLE1 ’The Log-Rank Test’; TITLE2 ’Example 21.1: HSV-2 Episodes with gD2 Vaccine’; RUN;

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VAC

PAT

WKS

CENS

X

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 . . . 43 44 45 46 47 48

GD2 GD2 GD2 GD2 GD2 GD2 GD2 GD2 GD2 GD2 GD2 GD2 GD2 GD2 GD2 GD2 GD2

1 3 6 7 8 10 12 14 15 18 20 23 24 26 28 31 33

8 12 52 28 44 14 3 52 35 6 12 7 52 52 36 52 9

0 1 1 0 0 0 0 1 0 0 0 1 1 1 0 1 0

12 10 7 10 6 8 8 9 11 13 7 13 9 12 13 8 10

PBO PBO PBO PBO PBO PBO

37 38 41 43 45 47

32 15 5 35 28 6

1 0 0 0 0 0

8 8 14 13 9 15

The LIFETEST Procedure Stratum 1: VAC = GD2 Summary Statistics for Time Variable WKS Quartile Estimates

Percent

Point Estimate

75 50 25

. 35.0000 13.0000

95% Confidence Interval (Lower Upper) 35.0000 15.0000 8.0000

. . 13 28.0000

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2873876$62XWSXWIRU([DPSOHFRQWLQXHG The Log-Rank Test Example 21.1: HSV-2 Episodes with gD2 Vaccine

Mean

Standard Error 3.2823 14

28.7214

NOTE: The mean survival time and its standard error were underestimated because the largest observation was censored and the estimation was restricted to the largest event time. Stratum 2: VAC = PBO Summary Statistics for Time Variable WKS Quartile Estimates

Percent

Point Estimate

95% Confidence Interval (Lower Upper)

75 50 25

28.0000 15.0000 8.0000

19.0000 10.0000 5.0000

Mean

Standard Error

. 27.0000 13 15.0000

2.5387 14

18.2397

NOTE: The mean survival time and its standard error were underestimated because the largest observation was censored and the estimation was restricted to the largest event time.

Summary of the Number of Censored and Uncensored Values

Stratum

VAC

Total

Failed

Censored

Percent Censored

1 GD2 25 14 11 44.00 2 PBO 23 17 6 26.09 --------------------------------------------------------------Total 48 31 17 35.42

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2873876$62XWSXWIRU([DPSOHFRQWLQXHG The Log-Rank Test Example 21.1: HSV-2 Episodes with gD2 Vaccine The LIFETEST Procedure

S u r v i v a l D i s t r i b u t i o n G

SDF | | 1.0 + | | | | | | 0.8 + | | | | | | 0.6 + | | | | | | 0.4 + | |

Survival Function Estimates **--G | G---G P----P | PPG--G | G-G 12 PP | | G---G | GG P--P | P--P G-G | | P--P GG | G-------G | | | G---------G P---P | | | | G---------G P-----P | | GG PP | | G-----G PP | |

| P--------P F | | u | PP n | P---------P c 0.2 + | t | | i | P o | n | | | 0.0 + ----+-----+-----+-----+-----+-----+-----+------+----+-----+-0 5 10 15 20 25 30 35 40 45 50 WKS Censored Observations Strata P | P P P P P P G | G G G G * ----+-----+-----+-----+-----+-----+-----+------+----+-----+-0 5 10 15 20 25 30 35 40 45 50 WKS Legend for Strata Stratum Symbol VAC 1 G GD2 2 P PBO

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2873876$62XWSXWIRU([DPSOHFRQWLQXHG The Log-Rank Test Example 21.1: HSV-2 Episodes with gD2 Vaccine The LIFETEST Procedure

Testing Homogeneity of Survival Curves for WKS over Strata Rank Statistics VAC

Log-Rank

GD2 PBO

-5.1556 5.1556

Wilcoxon

ô

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Covariance Matrix for the Log-Rank Statistics VAC

GD2

GD2 PBO

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PBO

õ

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Test

VAC

GD2

PBO

GD2 PBO

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-7136.46 7136.46

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CHHAAPPTTEERR 22 The Cox Proportional Hazards Model 22.1 22.2 22.3 22.4

Introduction...................................................................................................... 365 Synopsis ............................................................................................................ 366 Examples........................................................................................................... 367 Details & Notes................................................................................................. 376

22.1 Introduction Like the log-rank test discussed in the previous chapter, the Cox proportional hazards model (sometimes referred to as ‘Cox regression’) is used to analyze event or ‘survival’ times. This is a modeling procedure that adjusts for censoring, may include numeric covariates, one or more of which may be time-dependent, and requires no assumption about the distribution of event times. An example of where the Cox proportional hazards model might be useful in clinical trials is a comparison of time to death among treatment groups with adjustments for age, duration of disease, and family disease history. Informally, the inverse of the time to event occurrence is called the ‘hazard’, given a suitable time unit. For example, if an event is expected to occur in 6 months, the hazard is 1/6 (‘events per month’), assuming the unit of time is ‘months’. Of chief concern when using the Cox proportional hazards model is the way in which the hazard changes over time. The hazard function is discussed in more detail later in this chapter (see Section 22.4.2). The hazard of some events might be likely to increase with the passage of time, such as death due to certain types of cancer. Other types of events might become less frequent over longer periods of time. For example, reinfarction following coronary artery bypass can have the greatest risk in the weeks immediately following surgery, then risk might decrease as time passes. Still other types of events might be subject to fluctuating risks, both increasing and decreasing, over time.

366

Common Statistical Methods for Clinical Research with SAS Examples

Theoretically, survival analysis would be pretty easy if you could assume that the hazard remains constant over time. Recognizing the impracticalities of such an assumption, the Cox proportional hazards model adopts the more reasonable assumption that the event hazard rate does change over time, but that the ratio of event hazards between two individuals is constant. This is known as the ‘proportional hazards’ assumption, which requires that the ratio of hazards between any two values of a covariate not vary with time. If Age is a covariate used in the analysis, the proportional hazards assumption would require, for example, that the ratio of the hazard for 60-year-old patients to the hazard for 35-year-old patients be constant over time.

22.2 Synopsis The Cox proportional hazards model is used to test the effect of a specified set of k covariates, X1, X2, ..., Xk on the event times. The Xi’s can be numeric-valued covariates or numerically coded categorical responses that have some natural ordering. The Xi’s can also be dummy variables that represent treatment groups in a comparative trial or levels of a nominal categorical factor as described for logistic regression (see Section 20.4.6). The model for the hazard function of the Cox proportional hazards method has the form

h(t) = λ(t) × e (β1X1 + β 2X 2 + ... + β k X k ) where h(t) represents the hazard function, the Xi’s represent the covariates, the βi’s represent the parameter coefficients of the Xi’s, and λ(t) represents an unspecified initial hazard function. As with regression analysis, the magnitudes of the βi’s reflect the importance of the covariates in the model, and inference about these parameters is the focus here. D. R. Cox, after whom the model is named, developed a method for estimating βi (i=1,2,...,k) by bi based on a ‘maximum partial likelihood’ approach, which is a modification of the maximum likelihood method mentioned in Chapter 20. Manual computations are impractical, and solutions generally require numerical techniques using high-speed computers. When using SAS, the PHREG procedure is recommended for fitting the Cox proportional hazards model.

Chapter 22: The Cox Proportional Hazards Model 367

For large samples, these estimates (bi) have an approximate normal distribution. If sb represents the standard error of the estimate bi, then bi/sb has an approximate standard normal distribution under the null hypothesis that βi = 0, and its square has the chi-square distribution with 1 degree of freedom. The test summary for each model parameter, βi, can be summarized as follows:

null hypothesis: alternative hypothesis:

H0: βi = 0 HA: βi =/ 0

test statistic:

æ bi ö 2 χ w = çç ÷÷ è sb ø

decision rule:

reject H0 if χ

2

2 w

2

> χ (α) 1

The magnitude of the effect of a covariate is often expressed as a hazard ratio, similar to the odds ratio discussed in Chapter 20. Because the hazard at any time, t, for a specified set of covariate values, x1, x2, …, xk, is given by

h(t) = λ(t) × e (β1x 1 + β 2 x 2 + ... + β k x k ) the ratio of hazards for a 1-unit increase in Xi (all other covariates held constant) is β e i. The estimated hazard ratio associated with the covariate Xi is found by simply exponentiating the parameter estimate, bi . Example 22.1, which follows, illustrates the Cox proportional hazards model with a simple example using one dichotomous group variable and one numeric covariate.

22.3 Examples p Example 22.1 -- HSV-2 Episodes with gD2 Vaccine (continued)

(See Example 21.1) The data in Example 21.1 include a history of HSV-2 lesional episodes during the year prior to the study (X). Does the covariate X have any impact on the comparison of HSV-2 recurrence times between treatment groups?

368

Common Statistical Methods for Clinical Research with SAS Examples

Solution Although covariates can be included in the TEST or STRATA statements when using PROC LIFETEST in SAS (Chapter 21), it is often easier to use PROC PHREG to model survival distributions that include one or more covariates. The SAS analysis below shows how to use PROC PHREG to fit the Cox proportional hazards model for this example.

SAS Analysis of Example 22.1 PROC PHREG requires all covariates included in the model to be numeric. The number of episodes in the prior 12 months (X) is a numeric variable. The vaccine group (VAC), however, must be converted. You can do this conversion with the new variable, TRT, which takes the value 1 for the gD2 vaccine group and 0 for the placebo group (see the SAS code for Example 21.1). With PROC PHREG, the MODEL statement is used with the time variable and censoring indicator on the left side of the equal sign (=) and the covariates on the right side of the equal sign (=) . Recall from Example 21.1 that the indicator variable CENS takes a value of 1 if the observation is censored, 0 if it is not censored. The left side of the MODEL equality is specified in the same format as the TIME statement in PROC LIFETEST for identifying censored values. You specify the vaccine group factor, TRT, and the HSV-2 lesion frequency during the prior year, X, as the two covariates. The TIES option is used in the event that more than one observation has tied event times. This is discussed further in Section 22.4.9. The results are shown in Output 22.1. The estimate of the parameter associated with the covariate X is 0.176 , which is significant (p = 0.0073) based on the chi-square value of 7.1896. This means that prior lesional frequency is an important consideration when analyzing the event times. The test for the group effect (TRT) is also significant (p = 0.0160) and indicates a difference in the distributions of event times between vaccine groups after adjusting for the covariate, X. The hazard ratio, 0.404 , indicates that the covariate-adjusted ‘hazard’ of HSV-2 recurrence for the active (gD2) group is only about 40% of that for the placebo group. The log-rank test using the same data set (ignoring the covariate, X) resulted in a significant finding with a p-value close to 0.05 (Example 21.1). The Cox proportional hazards model resulted in an even greater significance (p = 0.016) after adjusting for the important covariate, X.

Chapter 22: The Cox Proportional Hazards Model 369

SAS Code for Example 22.1 . . . (continued from Example 21.1) PROC PHREG DATA = HSV; MODEL WKS*CENS(1) = TRT X / TIES = EXACT; TITLE1 'Cox Proportional Hazards Model'; TITLE2 “Example 22.1: HSV-2 Episodes with gD2 Vaccine cont'd.” ; RUN;

OUTPUT 22.1. SAS Output for Example 22.1 Cox Proportional Hazards Model Example 22.1: HSV-2 Episodes with gD2 Vaccine - cont'd. The PHREG Procedure Model Information Data Set Dependent Variable Censoring Variable Censoring Value(s) Ties Handling

WORK.HSV WKS CENS 1 EXACT

Summary of the Number of Event and Censored Values

Total

Event

Censored

Percent Censored

48

31

17

35.42

Convergence Status Convergence criterion (GCONV=1E-8) satisfied. Model Fit Statistics

Criterion -2 LOG L AIC SBC

Without Covariates

With Covariates

183.628 183.628 183.628

172.792 176.792 179.660

Testing Global Null Hypothesis: BETA=0 Test Likelihood Ratio Score Wald

Chi-Square 10.8364 11.0159 10.7236

DF 2 2 2

Pr > ChiSq 0.0044 0.0041 0.0047

Analysis of Maximum Likelihood Estimates

Variable TRT X

DF 1 1

Parameter Standard Estimate Error Chi-Square Pr > ChiSq -0.90578 0.37600 5.8033 0.0160 0.17627 0.06574 7.1896 0.0073

Hazard Ratio 0.404 1.193

370

Common Statistical Methods for Clinical Research with SAS Examples

The next example illustrates the use of stratification in Cox regression and introduces the concept of time-dependent covariates. p Example 22.2 -- Hyalurise in Vitreous Hemorrhage

A new treatment, Hyalurise, is being tested to facilitate the clearance of vitreous hemorrhage, a condition that results in severe vision impairment. Patients were randomly assigned to receive a single injection of either Hyalurise (n =83) or saline (n = 60) in the affected eye and were followed for 1 year. Time, in weeks, to ‘response’ is shown in Table 22.1. (‘Response’ is defined as sufficient clearance of the hemorrhage to permit diagnosis of the underlying cause and appropriate intervention.) The time for patients who discontinued or completed the trial before achieving ‘response’ is considered censored (†). Covariates include a measure of baseline hemorrhage density (Grade 3 or 4), study center (A, B, or C), and whether the patient developed certain infectious complications (ocular infection, inflammation, or herpetic lesion) during the study (based on ‘Infect. Time’ in Table 22.1). Does the time to response differ between the patients treated with Hyalurise and the control group treated with saline?

Solution The response variable is time, in weeks, from randomization to ‘response’, some of whose values are censored. Because this analysis clearly calls for ‘survival’ methodology that incorporates covariates, you use PROC PHREG in SAS to fit a Cox regression model under the assumption of proportional hazards. In addition to the treatment group effect, you want to include as covariates: baseline hemorrhage density, study center, and the occurrence of infectious complications. Notice that the last covariate is a bit unusual. The occurrence of complications can take the values ‘yes’ or ‘no’, which can be coded 1 and 0, respectively. However, these values can change during the study and depend on the time of the response. Clearly, the degree to which complications affect ‘response’ is only relevant if the complication develops before ‘response’ occurs. For this reason, this is called a ‘time-dependent’ covariate. Such time-dependent covariates can be included in the Cox proportional hazards model, as illustrated in the SAS analysis that follows.

SAS Analysis of Example 22.2 In the SAS program for Example 22.2, the variables are coded as follows: RSPTIM TRT DENS CENTER INFTIM

= time (weeks) from randomization to ‘response’ = treatment group (0=saline, 1=Hyalurise) = baseline hemorrhage density (3 or 4) = Study Center (A, B, or C) = time (weeks) from randomization to onset of complication.

Chapter 22: The Cox Proportional Hazards Model 371

TABLE 22.1. Raw Data for Example 22.2 -------------------------------- Hyalurise Injection --------------------------------Patient Number

Density

Infect. Time

Resp. Time

Patient Number

Density

Infect. Time

Resp. Time

A

101 102 105 106 108 110 112 114 116 117 119 121 123 125 126 128 130

3 4 4 3 4 3 4 3 3 4 3 4 3 4 3 3 4

. 10 52 . . . . 36 . . . . 28 . . . .

32 20 24 41 32 10 † 32 45 6 52 † 13 6 44 23 30 2 26

132 133 134 136 139 140 141 144 146 148 150 151 153 155 156 159 160

4 4 3 3 3 4 3 3 4 4 4 3 3 3 3 4 4

18 . . . . 20 12 . . 24 . 4 . . . . .

46 21 14 24 † 19 24 31 52 † 40 7 16 36 28 38 † 9 30 6

B

202 203 206 207 208 210 212 214 215 217 219

3 4 3 4 3 4 3 4 4 3 3

. . . . . . . 30 . 33 .

4† 10 11 52 † 12 46 10 31 4 20 2

220 223 224 225 226 228 230 231 233 234 237

4 4 3 3 3 3 4 4 3 4 3

. . 24 . . . . . 10 . .

25 20 9 22 23 13 8 37 24 52 † 27

C

301 303 304 306 309 311 312 313 315 317 318 320 321 323

3 4 4 4 3 3 3 4 3 4 3 3 3 3

. . 20 18 . . . . . . . . 16 .

10 32 14 14 7 5 9 12 10 52 † 10 5 34 6

324 327 329 330 332 334 337 338 340 341 342 344 346

4 3 3 3 3 4 3 3 3 4 4 4 3

. 30 . 30 . . . 20 . 18 . . .

22 15 20 52 † 10 2† 16 33 12 27 2 20 † 3

Center

† indicates censored observation

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TABLE 22.1. Data Set for Example 22.2 (continued) ----------------------------------- Saline Injection ------------------------------------Patient Number

Density

Infect. Time

Resp. Time

Patient Number

Density

Infect. Time

A

103 104 107 109 111 113 115 118 120 122 124 127 129

3 4 4 3 3 4 4 4 4 4 3 3 3

. . 12 20 . 38 . 28 . . . . 6

52 † 4 32 25 8 52 † 14 † 9 36 42 † 52 † 16 21

131 135 137 138 142 143 145 147 149 152 154 157 158

3 3 3 3 4 4 3 3 4 3 3 3 4

. . . . . 34 . . . . 42 . .

8 52 † 22 † 35 † 28 52 † 35 4 2† 12 48 11 20

B

201 204 205 209 211 213 216 218

3 4 3 3 4 3 3 3

. 9 . . 44 . . .

15 † 10 8 20 32 42 24 † 16

221 222 227 229 232 235 236

4 4 3 4 4 4 4

. 40 . . 26 . .

11 52 † 11 7† 32 20 34

C

302 305 307 308 310 314 316 319 322 325

4 3 3 3 4 3 4 4 4 4

8 . . . . . 36 12 . 12

10 † 24 36 52 † 6 21 26 27 14 45

326 328 331 333 335 336 339 343 345

3 4 4 3 4 4 3 4 3

. 22 . 15 . . 11 . .

16 33 5 16 17 16 24 6† 14

Center

† indicates censored observation

In this example, the variable CENS is created to identify censored observations associated with a value of 0 . Thus, the left side in the MODEL statement in PROC PHREG is ‘RSPTIM*CENS(0) =’ . The baseline hemorrhage density (DENS) is easily included in the MODEL statement as a dichotomous numeric covariate (Grade 4 indicates the most severe condition). Notice that Study Center is a nominal categorical factor that cannot be incorporated directly as a covariate because PROC PHREG can only use numeric valued covariates. You can create two new 0 and 1 dummy variables as described in Chapter 20. However, because differences among centers are not of direct interest, it is easier to include CENTER in a STRATA statement . Essentially,

Resp. Time

Chapter 22: The Cox Proportional Hazards Model 373

this tells SAS to treat the centers as three distinct clusters and combine the results across centers. CENTER effects are not estimated, and use of CENTER in the STRATA statement does not require the proportional hazards assumption with respect to Study Center. To include the possible effect of occurrence of complications in the analysis, you must create a new, time-dependent covariate (INFCTN) using a special feature in PROC PHREG that allows SAS DATA step type programming statements to be used in the procedure. This ensures that the covariate takes the correct values (0 or 1) at the appropriate time points. INFCTN is defined as 1 if a complication occurs prior to response, and 0, if a complication occurs after response or not at all . In Output 22.2, you see the analysis of the maximum likelihood estimates of the model parameters, including p-values based on the Wald chi-square tests. The pvalue for treatment effect (TRT), which controls for the covariates, is 0.0780 11 . The hazard ratio of 1.412 12 indicates that the ‘hazard’ of responding at any arbitrary time point is 41.2% higher with Hyalurise injection compared with saline. Neither the numeric covariate, baseline hemorrhage density (DENS), nor the timedependent covariate (INFCTN) is significant. Notice that no tests are performed for the Study Center effect because Study Center is a categorical factor included in the STRATA statement. SAS Code for Example 22.2 DATA VITCLEAR; INPUT PAT RSPTIM TRT CENTER $ DENS INFTIM @@; CENS = (RSPTIM GE 0); RSPTIM = ABS(RSPTIM); /* RSPTIM = time (wks) from randomization to response (RSPTIM is censored if negative) */ /* TRT = 1 for Hyalurise, TRT = 0 for Saline */ /* CENTER = study center (A, B, or C) */ /* DENS = 3, 4 for Grade 3 or 4, respectively */ /* INFTIM = time (wks) from randomization to onset of infection or other complications */ DATALINES; 101 32 1 A 3 . 102 20 1 A 4 10 103 -52 0 A 3 . 104 4 0 A 4 . 105 24 1 A 4 52 106 41 1 A 3 . 107 32 0 A 4 12 108 32 1 A 4 . 109 25 0 A 3 . 110 -10 1 A 3 . 111 8 0 A 3 . 112 32 1 A 4 . 113 -52 0 A 4 38 114 45 1 A 3 36 115 -14 0 A 4 . 116 6 1 A 3 . 117 -52 1 A 4 . 118 9 0 A 4 28 119 13 1 A 3 . 120 36 0 A 4 . 121 6 1 A 4 . 122 -42 0 A 4 . 123 44 1 A 3 28 124 -52 0 A 3 . 125 23 1 A 4 . 126 30 1 A 3 . 127 16 0 A 3 . 128 2 1 A 3 . 129 21 0 A 3 6 130 26 1 A 4 . 131 8 0 A 3 . 132 46 1 A 4 18 133 21 1 A 4 .

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Common Statistical Methods for Clinical Research with SAS Examples

SAS Code for Example 22.2 (continued) 134 137 140 143 146 149 152 155 158 201 204 207 210 213 216 219 222 225 228 231 234 237 303 306 309 312 315 318 321 324 327 330 333 336 339 342 345 ;

14 -22 24 -52 40 -2 12 -38 20 -15 10 -52 46 42 -24 2 -52 22 13 37 -52 27 32 14 7 9 10 10 34 22 15 -52 16 16 24 2 14

1 0 1 0 1 0 0 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0

A A A A A A A A A B B B B B B B B B B B B B C C C C C C C C C C C C C C C

3 3 4 4 4 4 3 3 4 3 4 4 4 3 3 3 4 3 3 4 4 3 4 4 3 3 3 3 3 4 3 3 3 4 3 4 3

. . 20 34 . . . . . . . . . . . . . . . . . . . . . . . . 16 . 30 30 . . 11 . .

135 -52 0 A 3 . 138 -35 0 A 3 . 141 31 1 A 3 12 144 -52 1 A 3 . 147 4 0 A 3 . 150 16 1 A 4 . 153 28 1 A 3 . 156 9 1 A 3 . 159 30 1 A 4 . 202 -4 1 B 3 . 205 8 0 B 3 . 208 12 1 B 3 . 211 32 0 B 4 44 214 31 1 B 4 20 217 20 1 B 3 33 220 25 1 B 4 . 223 20 1 B 4 . 226 23 1 B 3 . 229 -7 0 B 4 . 232 32 0 B 4 26 235 20 0 B 4 . 301 10 1 C 3 . 304 14 1 C 4 20 307 36 0 C 3 . 310 6 0 C 4 . 313 12 1 C 4 . 316 26 0 C 4 36 319 27 0 C 4 12 322 14 0 C 4 . 325 45 0 C 4 12 328 33 0 C 4 22 331 5 0 C 4 . 334 -2 1 C 4 . 337 16 1 C 3 . 340 12 1 C 3 . 343 -6 0 C 4 . 346 3 1 C 3 .

136 139 142 145 148 151 154 157 160 203 206 209 212 215 218 221 224 227 230 233 236 302 305 308 311 314 317 320 323 326 329 332 335 338 341 344

-24 19 28 35 7 36 48 11 6 10 11 20 10 4 16 11 9 11 8 24 34 -10 24 -52 5 21 -52 5 6 16 20 10 17 33 27 -20

1 1 0 0 1 1 0 0 1 1 1 0 1 1 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 1 1 0 1 1 1

A A A A A A A A A B B B B B B B B B B B B C C C C C C C C C C C C C C C

3 3 4 3 4 3 3 3 4 4 3 3 3 4 3 4 3 3 4 3 4 4 3 3 3 3 4 3 3 3 3 3 4 3 4 4

. . . . 24 4 42 . . . . . . . . . . . . 10 . 8 . . . . . . . . . . . 20 18 .

PROC PHREG DATA = VITCLEAR; MODEL RSPTIM*CENS(0) = TRT DENS INFCTN / TIES = EXACT; IF INFTIM GT RSPTIM OR INFTIM = . THEN INFCTN = 0; ELSE INFCTN = 1; STRATA CENTER; TITLE1 'Cox Proportional Hazards Model'; TITLE2 'Example 22.2: Hyalurise in Vitreous Hemorrhage' ; RUN;

Chapter 22: The Cox Proportional Hazards Model 375

OUTPUT 22.2. SAS Output for Example 22.2 Cox Proportional Hazards Model Example 22.2: Hyalurise in Vitreous Hemorrhage The PHREG Procedure Model Information Data Set Dependent Variable Censoring Variable Censoring Value(s) Ties Handling

WORK.VITCLEAR RSPTIM CENS 0 EXACT

Summary of the Number of Event and Censored Values

Stratum

CENTER

Total

Event

Censored

Percent Censored

1 A 60 45 15 25.00 2 B 37 30 7 18.92 3 C 46 39 7 15.22 ------------------------------------------------------------------Total 143 114 29 20.28 Convergence Status Convergence criterion (GCONV=1E-8) satisfied. Model Fit Statistics

Criterion -2 LOG L AIC SBC

Without Covariates

With Covariates

641.460 641.460 641.460

636.479 642.479 650.688

Testing Global Null Hypothesis: BETA=0 Test Likelihood Ratio Score Wald

Chi-Square

DF

4.9809 4.9703 4.9251

3 3 3

Pr > ChiSq 13 0.1732 0.1740 0.1774

Analysis of Maximum Likelihood Estimates

Variable

DF

Parameter Estimate

Standard Error

Chi-Square

TRT DENS INFCTN

1 1 1

0.34500 -0.23884 0.12835

0.19577 0.19598 0.27797

3.1057 1.4852 0.2132

Pr>ChiSq 0.0780 11 0.2230 0.6443

Hazard Ratio 1.412 12 0.788 1.137

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22.4 Details & Notes 22.4.1. The Cox proportional hazards model is a ‘semi-parametric’ approach that does not require the assumption of any specific hazard function or distribution of survival times. In this sense, it takes the character of a nonparametric test. However, as in regression analysis, this method models the survival times as a function of the covariates with the goal of obtaining estimates of the model parameters (βi’s). Various types of functions can be used and different models can be specified for the distribution of the error variation. In this sense, the Cox proportional hazards model is parametric. 22.4.2. The hazard is a function of time, t. Formally, the hazard represents the probability that the event of interest occurs at a specified time (t) given that it has not occurred prior to time (t). If the hazard is constant over time, event times can be modeled with the exponential distribution. The natural log of the hazard function is ln h(t) = λ(t) + β1X1 + β2X2 +…+ βkXk If the event times have an exponential distribution, λ(t) is a constant, independent of t, such as α, and the model becomes ln h(t) = α + β1X1 + β2X2 +…+ βkXk which means the hazard is constant with respect to time and is a simple linear function of the covariates, Xi’s, on the log-scale. Section 21.4.5 describes how to test for exponential event times by checking whether a plot of the log of the survival probabilities is linear through the origin. This test is a direct result of the fact that the exponential distribution is associated with a constant hazard function (i.e., risk is independent of time). In general, if a model for the hazard function can be assumed, then the survival distributions can be specified. In fact, the cumulative survival distribution is a function of the hazard function integrated over time. Therefore, the survival distribution can be described by the hazard function, and, conversely, the survival distribution uniquely defines a hazard function. When the survival distribution is known or can be accurately assumed, a parametric analysis can be applied, often with greater efficiency than the nonparametric log-rank test or the ‘semi-parametric’ Cox proportional hazards model. The parametric approach with a known survival distribution can easily incorporate covariates into the analysis. In addition to exponential event times, widely used models include the log-normal, Weibull, gamma, and log-logistic distributions. The SAS procedure LIFEREG can be used to fit any of these parametric models with any number of covariates.

Chapter 22: The Cox Proportional Hazards Model 377

The main disadvantage of the parametric approach is that, in many cases, model assumptions might be incorrect and there might not be enough data to adequately test the assumptions or, perhaps, the actual hazard function is not well represented by any of those in common usage. Often, the results are highly dependent on the appropriate distributional assumptions. 22.4.3. The strength of the Cox proportional hazards model is its ability to incorporate background factors as covariates. When many covariates are being considered, this method can be used in an exploratory fashion to build a model that includes significant covariates and excludes the covariates that have little apparent effect on the event times. This stepwise procedure is demonstrated in the SAS OnlineDoc for PROC PHREG. 22.4.4. PROC PHREG prints the results of a global test that determines whether any of the covariates included in the model are significant after adjusting for all the other covariates. As shown in the output for Example 22.1 , three different methods are used: "Likelihood Ratio", "Score" and "Wald". Each of these is a goodness-of-fit test with an approximate chi-square distribution with 2 degrees of freedom under the global null hypothesis of ‘no covariate effects’. With the significant results of p < 0.005 for all three methods in the example, you conclude that fluctuations in event times are associated with either vaccine group (TRT) or prior lesional frequency (X) or both. A poor fit is indicated by the global tests in Example 22.2, all of which show non-significant chi-square tests with p-values of approximately 0.17 13 . 22.4.5. A positive parameter estimate indicates that the hazard increases with increasing values of the covariate. A negative value indicates that the hazard decreases with increasing values of the covariate. In Example 22.1, the negative parameter estimate for TRT (–0.906) indicates that the hazard of HSV-2 recurrence decreases with active treatment (TRT=1) compared with a placebo (TRT=0). The hazard increases with larger prior lesional frequencies (X), as seen by the positive parameter estimate for X (0.176). The term ‘hazard’ is often associated with the event of death or disease. However, in survival analysis, ‘hazard’ does not always connote a danger or an undesirable event. In Example 22.2, the event of interest is ‘response,’ which is a desirable result. Although it sounds awkward, an increasing ‘hazard’ in the desirable response with active treatment compared with saline injection is seen. Notice that the parameter estimate for TRT is positive, and the hazard ratio for the treatment effect is greater than 1 (1.412 12 ), just the opposite of Example 22.1. 22.4.6. The model for the hazard function using Cox proportional hazards with only one covariate (k=1) is

h(t) = λ(t) ⋅ e βX

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Common Statistical Methods for Clinical Research with SAS Examples

For two patients with different values of X, such as Xu and Xv, the ratio of hazards at time t is

λ(t) ⋅ eβ X u h u (t) = = e β( X u − X v ) λ(t) ⋅ e β X v h v (t) which is constant with respect to time, t. This is the proportional hazards assumption. If X is a dummy variable with Xu = 1 (the active group) and Xv = 0 (the β placebo group), Xu – Xv = 1 so that e represents the ratio of the hazard that’s associated with the active treatment to the hazard that’s associated with the placebo. This is the hazard ratio in the SAS printout for Example 22.1 and Example 22.2 12 . Note: In SAS releases prior to Version 7, the hazard ratio is labeled ‘Risk Ratio’ in SAS output. –0.906

= 0.404. This can be In Example 22.1, the hazard ratio for TRT is e interpreted as the hazard at any time t for the active group is only 40.4% of the placebo group’s hazard. The hazard ratio for X is 1.193. Because X is a numeric variable (rather than a numerically coded categorical variable), you interpret this as a 19.3% increase in the hazard of an HSV-2 recurrence for each additional prior lesion that was experienced. 22.4.7. Example 22.2 illustrates the inclusion of a dichotomous timedependent covariate. Continuous numeric covariates can also be timedependent. For example, time to a certain response in a nutritional study might depend on a subject’s protein intake, which might change during the course of the study. This type of continuous numeric time-dependent covariates can also be included in PROC PHREG. Details and an excellent discussion of timedependent covariates using PROC PHREG can be found in Survival Analysis Using the SAS System, by Paul Allison (1995), which was written under the SAS Books By Users program. 22.4.8. ‘Non-proportional hazards’ implies an increasing or decreasing hazard ratio over time. In PROC PHREG, this can be represented by a covariate-by-time ‘interaction’. The proportional hazards assumption for any covariate can be checked by including the interaction of that covariate with time as a new time-dependent covariate in the MODEL statement in PROC PHREG. For example, to check the proportional hazards assumption for the covariate DENS in Example 22.2, you can include the time-dependent covariate, TDENS, by specifying the following SAS statement: TDENS = DENS*RSPTIM;

after the MODEL statement. A significant test for the parameter that’s associated with TDENS would indicate that the hazard changes over time.

Chapter 22: The Cox Proportional Hazards Model 379

This test is also illustrated in the SAS Help and Documentation for PROC PHREG, which recommends using the natural logarithm of the response times decreased by the average of the log response times to obtain greater stability. Use of this stabilizing transformation in Example 22.2 would require the statement TDENS = DENS*(LOG(RSPTIM ) – 2.85);

in place of the preceding statement (2.85 is the mean of the ln(RSPTIM) over all observations). A visual aid in determining conformance with the assumption of proportional hazards is a plot of the log(–log(S(t))) versus time, where S(t) is the estimated survival distribution. This can be done in SAS using the PLOTS=(LLS) option in the LIFETEST procedure. The graphs of these functions for each group should be parallel under the proportional hazards assumption. Lack of parallelism suggests deviations from the proportional hazards assumption. When the proportional hazards assumption does not seem to hold for a specific covariate, the Cox proportional hazards model might still be used by performing the analysis within each of a series of sub-intervals of the range of the covariate’s values, then combining the results. This is done in SAS by using the STRATA statement, which is illustrated in the SAS code for Example 22.2. If the covariate has continuous numeric values, the strata can be defined by numeric intervals in the STRATA statement (for details, see the SAS Help and Documentation for PROC PHREG). 22.4.9. An adjustment might be necessary when ties occur in event times. SAS has 4 adjustment options for handling ties: BRESLOW, EFRON, EXACT, and DISCRETE. BRESLOW is the default, although it might not be the best option to use. With small data sets or a small number of tied values, the EXACT option is usually preferable. Because of computational resources, the EXACT option is not efficient for larger data sets. The EFRON option provides a good approximation and is computationally faster when large amounts of data are being analyzed.

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Common Statistical Methods for Clinical Research with SAS Examples

CHHAAPPTTEERR 23 Exercises 23.1 23.2 23.3 23.4

Introduction...................................................................................................... 381 Exercises ........................................................................................................... 381 Appropriateness of the Methods .................................................................... 383 Summary .......................................................................................................... 387

23.1 Introduction The data set, TRIAL, in Chapter 3, is based on a contrived, simplified clinical study that measured three types of response variables for 100 patients. The exercises in this section provide an opportunity to practice applying some of the methods discussed in this book using this data set. Section 23.2 lists the exercises and requests an analysis that uses a specific method referenced in an earlier chapter. The SAS code for performing these exercises is shown in Appendix G. Section 23.3 provides a critique of the methods used and a discussion of their appropriateness, based on the assumptions the analyst is willing to make.

23.2 Exercises Write a SAS program to perform the following analyses using the ‘TRIAL’ data set in Chapter 3. Analyses 1 to 11 use the continuous numeric response SCORE, Analyses 12 to 17 use the dichotomous response RESP, and Analyses 18 to 27 use the ordinal categorical response SEV. 23.2.1.

Using the SCORE response variable:

Analysis 1:

Compare mean responses between treatment groups by using the two-sample t-test (Chapter 5).

Analysis 2:

Apply the Wilcoxon rank-sum test to test for any shifts in the distribution between Treatment Groups A and B (Chapter 13).

Analysis 3:

Using a one-way ANOVA model (Chapter 6), test for a difference in mean response between treatment groups.

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Common Statistical Methods for Clinical Research with SAS Examples

Analysis 4:

Determine if there is an interaction between treatment group and study center by using the two-way ANOVA (Chapter 7).

Analysis 5:

Re-run the two-way ANOVA without the interaction effect to determine the significance of treatment group differences.

Analysis 6:

Compare treatment group responses by using ANOVA accounting for any effects due to Study Center and Sex. Include all interactions.

Analysis 7:

Check for a significant treatment group effect adjusted for Study Center and Sex by using a main effects ANOVA model.

Analysis 8:

Ignoring all group factors, determine if the response variable SCORE is linearly related to Age (Chapter 10). Plot the data.

Analysis 9:

Determine if the slope of the linear regression in Analysis 8 differs between treatment groups (Chapter 11).

Analysis 10:

Estimate the mean response for each treatment group using Age as a covariate. Assuming the answer to 9 is ‘no’, determine the pvalue for comparing these adjusted means (Chapter 11). Plot the data by treatment group.

Analysis 11:

Compare treatment group means adjusted for Age and Study Center (Chapter 11). Assume there are no interactions.

23.2.2.

Using the RESP response variable:

Analysis 12:

Compare response rates between treatment groups by using the chi-square test (Chapter 16).

Analysis 13:

Compare response rates between treatment groups by using Fisher’s exact test (Chapter 17).

Analysis 14:

Compare response rates between treatment groups controlling for Study Center by using the Cochran-Mantel-Haenszel test (Chapter 19).

Analysis 15:

Compare response rates between treatment groups by using logistic regression (Chapter 20).

Analysis 16:

Compare response rates between treatment groups adjusted for Age assuming a logistic model (Chapter 20).

Analysis 17:

Using a logistic regression model, test for a significant treatment group effect adjusted for Age, Sex, Study Center, and Treatmentby-Center interaction (Chapter 20).

Chapter 23: Exercises 383

23.2.3.

Using the SEV response variable:

Analysis 18:

Test for general association between severity of response and treatment group by using the chi-square test (Chapter 16).

Analysis 19:

Apply the generalized Fisher’s exact test to determine if the distributions of response by severity differ between treatment groups (Chapter 17).

Analysis 20:

Use the chi-square test to determine if the row mean scores differ between treatment groups (Chapter 16). Use ‘table’ scores for coding severity.

Analysis 21:

Repeat Analysis 20 using modified ridit scores in place of ‘table’ scores.

Analysis 22:

Test for a shift in the distribution of severity between treatment groups by using the Wilcoxon rank-sum test (Chapter 13).

Analysis 23:

Test for a difference in mean severity scores between treatment groups controlling for Study Center (Chapter 19). Use ‘table’ scores.

Analysis 24:

Repeat Analysis 23 using modified ridit scores in place of ‘table’ scores.

Analysis 25:

Determine the significance of the treatment group effect by using a logistic regression model under the assumption of proportional odds (Chapter 20).

Analysis 26:

Using a proportional odds model, compare response rates between treatment groups adjusted for Age (Chapter 20).

Analysis 27:

Adjusting for Age, Sex, and Study Center, determine the significance of the treatment group effect by using the proportional odds model (Chapter 20).

23.3 Appropriateness of the Methods SAS code for conducting the analyses in the previous section is given in Appendix G. Table 23.1 summarizes the p-values for the treatment group effect for each of these analyses. Although the exercises are assigned for practice in using SAS to perform commonly used statistical analyses, some of these methods would clearly be inappropriate to use in practice. Let’s take a look at the results.

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Common Statistical Methods for Clinical Research with SAS Examples

TABLE 23.1. Summary of Significance of Treatment Group Effects from Exercises

Analysis

Method

p-Value for Treatment Group effect

1 2

Analyses Using the SCORE Variable Two-sample t-test Wilcoxon rank-sum test

0.0422 0.0464

3 4 5 6 7 8 9 10 11

One-way ANOVA Two-way ANOVA with interaction Two-way ANOVA without interaction Three-way ANOVA with all interactions Three-way ANOVA main effects model Linear regression ANCOVA / unequal slopes ANCOVA / equal slopes Stratified ANCOVA / equal slopes

0.0422 0.0702 0.0378 0.1036 0.0493 -0.5663 0.0257 0.0252

12 13 14 15 16 17

Analyses Using the RESP Variable Chi-square test Fisher’s exact test Cochran-Mantel-Haenszel test Logistic regression, TRT effect only Logistic regression, TRT and AGE effects Logistic regression, TRT, AGE, SEX, CENTER, TRTxCENTER effects

0.4510 0.4794 0.4113 0.4523 0.3576 0.6984

18 19 20 21 22 23 24 25 26 27

Analyses Using the SEV Variable Chi-square test Generalized Fisher’s exact test Mantel-Haenszel, row mean scores differ / table scores Mantel-Haenszel, row mean scores differ / mod. ridit scores Wilcoxon rank-sum test CMH, row mean scores differ / table scores CMH, row mean scores differ / mod. ridit scores Proportional odds model, TRT effect only Proportional odds model, TRT and AGE effects Proportional odds model, TRT, AGE, SEX, CENTER

0.1216 0.1277 0.0514 0.0468 0.0472 0.0445 0.0471 0.0447 0.0244 0.0313

The SCORE variable is considered a continuous numeric response. As such, you can apply a variety of ANOVA, ANCOVA, and non-parametric methods to test for equality of treatment groups.

Chapter 23: Exercises 385

23.3.1. The two-sample t-test and one-way ANOVA produce the same pvalues (Analyses 1 and 3). This will always be the case when there are only two groups because the ANOVA F-test is functionally equivalent to the t-test in this situation. When using SAS, PROC TTEST would usually be preferable because it automatically provides a test for the assumption of equal variances and computes Satterthwaite’s adjustment for the unequal variance case. 23.3.2. The p-values differ considerably for the two-way ANOVA with (Analysis 4) and without the interaction (Analysis 5). In an unbalanced design, the inclusion of an interaction term in the model, when, in fact, no interaction exists, can produce conservative main effects comparisons based on the SAS Type III sums of squares in the ANOVA. For this reason, many analysts prefer to use the interaction model only as a preliminary test, rather than as a final model. 23.3.3. Notice that inclusion of the Study Center effect using the two-way main effects model (Analysis 5) provides sufficient blocking effect to bring out a slightly more significant treatment effect than the one-way ANOVA (Analysis 3), even though it is not a significant effect. When a third main effect (Sex) is added (Analysis 7), the treatment comparison actually becomes less significant than either the one-way or two-way main effects models. Although such a scenario can occur for a number of reasons, in this case, it appears to be an inconsequential difference based on the sample size imbalance, which affects the SAS Type III sums of squares. The treatment effect can be masked, as in Analysis 6, when exploring interaction effects. 23.3.4. ANCOVA (Analysis 10) reveals that Age might be an important covariate, and the adjusted Treatment Group differences appear more significant than those unadjusted for Age. This comes at the expense of having to assume equal slopes between the two groups, an assumption that is supported by Analysis 9. As you’ve seen previously, the Treatment Group effect is masked when testing for the interaction (equal slope assumption) as in Analysis 9. 23.3.5. Addition of the Study Center effect to the ANCOVA (Analysis 11) does not appreciably alter the significance of the Treatment effect noted in Analysis 10. Overall, it is clear that the inclusion of the Age effect in the model removes more background noise than the inclusion of either Study Center or Sex. 23.3.6. Although the t-test and ANOVA methods assume a normal distribution of the response data, these methods are fairly robust against departures from this assumption, especially with larger sample sizes. A histogram of the data (Output 3.5) does not support the normality assumption in this case. However, treatment comparisons based on ranked data (Analysis 2) result in similar p-values with this size sample (N=100). Therefore, any of the following analyses: 1, 2, 3, 5, 7, 10, or 11, would be

386

Common Statistical Methods for Clinical Research with SAS Examples

appropriate. The analyses intended to explore interactions (4, 6, and 9) would not be the best tests to use for treatment differences when interactions do not exist. In the presence of an interaction that involves Treatment Group, alternative approaches, such as subgroup analyses, might be more appropriate. Notice that the treatment group comparisons adjusted for Age result in the greatest significance. This might be due to Age differences for this sample, or it might be a repeatable effect. In any case, to select appropriate statistical methods for pivotal studies prior to examination of the data, you must rely on trends found from previous studies. If a specific covariate (such as Age) was consistently found to be an important factor in Phase II studies, for example, the pre-specified statistical model to be used for the Phase III program should include that covariate. Usually, the simplest models that contain all factors known to have an important effect on the treatment comparisons should be used in pivotal studies. The RESP variable is a dichotomous response. For this, you can apply multiple categorical methods to analyze differences between treatment groups. 23.3.7. Comparisons of response rates between treatment groups are not significant. Similar p-values for the treatment group comparisons, which range between 0.41 and 0.48, are found by using the chi-square test (Analysis 12), Fisher’s exact test (Analysis 13), the CMH test stratified by Study Center (Analysis 14), and a logistic regression model with no covariates (Analysis 15) (a finding that might be expected in studies of moderate sample sizes, such as this one). With very small sample sizes or response rates close to 0 or 1, Fisher’s exact test would be more appropriate than the other methods. Exact methods, useful in the case of small samples, are also available in logistic regression analysis. 23.3.8. The inclusion of Age as a covariate in the logistic regression model results in a smaller p-value for the treatment effect (Analysis 16). As shown in the ANCOVA (Analysis 10), Age appears to be an important factor when modeling treatment differences. 23.3.9. The inclusion of Study Center, Sex, and the Treatment-by-Center interaction in the logistic regression model results in even less significant treatment comparisons (Analysis 17), further supporting the recommendation to use parsimonious models. This can often be done by using logistic regression in a stepwise fashion to eliminate non-significant effects. Stepwise procedures are most often used in an exploratory role or in earlier phase studies. The SEV variable is an ordinal categorical response with 4 levels. Here, you can apply a rank test, a logistic regression analysis using the proportional odds model, or a number of other categorical procedures.

Chapter 23: Exercises 387

23.3.10. Both the chi-square test (Analysis 18) and the generalized Fisher’s exact test (Analysis 19) are tests for general association. Notice that the p-values for treatment comparisons are much larger than those of the other tests (Analyses 20 to 27) for the SEV response variable. As explained in Chapter 16, these methods are testing a less-specific hypothesis. 23.3.11. The Mantel-Haenszel chi-square test (Analysis 20) for ‘row mean scores differ’ is more appropriate for ordinal data than the test for ‘general association’. The Mantel-Haenszel chi-square tests for equality of mean scores between treatment groups, when the scores are assigned to the categorical levels in integer increments (‘table scores’). In this case, greater significance is found by using the modified ridit scores (Analysis 21) compared with the ‘table scores’ (Chapter 16). This will not always be the case, as seen in Analyses 23 and 24. In these analyses, the CMH test controlling for Study Center is used for comparing mean scores between treatment groups. The p-value using the ‘table scores’ (Analysis 23) is slightly smaller than that based on the modified ridit scores (Analysis 24). 23.3.12. The Wilcoxon rank-sum test (Analysis 22) is actually a special version of the Kruskal-Wallis test. Recall that the Mantel-Haenszel chi-square using modified ridit scores is identical to the Kruskal-Wallis test (Chapter 18). The difference in p-values (0.0472 in Analysis 22 vs. 0.0468 in Analysis 21) is due only to a correction factor. 23.3.13. Logistic regression modeling, similar to the ANCOVA method, shows the most significant treatment comparison (Analysis 26) when adjusting only for Age. Interestingly, this p-value (0.0244) is slightly less than that obtained by using the ANCOVA method (Analysis 10), perhaps due to the deviation from normality of the SCORE variable.

23.4 Summary When you can quantify degree of response, it is usually easier to discriminate between treatment groups because more powerful statistical methods are available. With only the knowledge of whether or not a patient responded, you see that treatment group differences are non-significant for all of the applicable methods (Analyses 12 to 17). When the response is categorized by severity, you obtain significant treatment group discrimination. Notably, when using a more precise response (continuous numeric representation of severity), the significance of treatment comparisons was not improved. This might be due, in part, to the nonnormality of the SCORE results, but it also points out the power of categorical analyses when using ordinal response categories relative to ANOVA methods. Comparison of these results points out the need to know how closely the underlying assumptions are satisfied before selecting an appropriate statistical method.

388

Common Statistical Methods for Clinical Research with SAS Examples

Appendix A Probability Tables A.1 Probabilities of the Standard Normal Distribution .............................................. 390 A.2 Critical Values of the Student t-Distribution.................................................... 391 A.3 Critical Values of the Chi-Square Distribution................................................ 392

390

Common Statistical Methods for Clinical Research with SAS Examples

P 0

z

APPENDIX A.1. Probabilities of the Standard Normal Distribution

z

P

z

P

z

P

z

P

z

P

z

P

0.00

0.5000

0.01 0.02 0.03 0.04 0.05

0.4960 0.4920 0.4880 0.4840 0.4801

0.51 0.52 0.53 0.54 0.55

0.3050 0.3015 0.2981 0.2946 0.2912

1.01 1.02 1.03 1.04 1.05

0.1562 0.1539 0.1515 0.1492 0.1469

1.51 1.52 1.53 1.54 1.55

0.0655 0.0643 0.0630 0.0618 0.0606

2.01 2.02 2.03 2.04 2.05

0.0222 0.0217 0.0212 0.0207 0.0202

2.51 2.52 2.53 2.54 2.55

0.0060 0.0059 0.0057 0.0055 0.0054

0.06 0.07 0.08 0.09 0.10

0.4761 0.4721 0.4681 0.4641 0.4602

0.56 0.57 0.58 0.59 0.60

0.2877 0.2843 0.2810 0.2776 0.2743

1.06 1.07 1.08 1.09 1.10

0.1446 0.1423 0.1401 0.1379 0.1357

1.56 1.57 1.58 1.59 1.60

0.0594 0.0582 0.0571 0.0559 0.0548

2.06 2.07 2.08 2.09 2.10

0.0197 0.0192 0.0188 0.0183 0.0179

2.56 2.57 2.58 2.59 2.60

0.0052 0.0051 0.0049 0.0048 0.0047

0.11 0.12 0.13 0.14 0.15

0.4562 0.4522 0.4483 0.4443 0.4404

0.61 0.62 0.63 0.64 0.65

0.2709 0.2676 0.2643 0.2611 0.2578

1.11 1.12 1.13 1.14 1.15

0.1335 0.1314 0.1292 0.1271 0.1251

1.61 1.62 1.63 1.64 1.65

0.0537 0.0526 0.0516 0.0505 0.0495

2.11 2.12 2.13 2.14 2.15

0.0174 0.0170 0.0166 0.0162 0.0158

2.61 2.62 2.63 2.64 2.65

0.0045 0.0044 0.0043 0.0041 0.0040

0.16 0.17 0.18 0.19 0.20

0.4364 0.4325 0.4286 0.4247 0.4207

0.66 0.67 0.68 0.69 0.70

0.2546 0.2514 0.2483 0.2451 0.2420

1.16 1.17 1.18 1.19 1.20

0.1230 0.1210 0.1190 0.1170 0.1151

1.66 1.67 1.68 1.69 1.70

0.0485 0.0475 0.0465 0.0455 0.0446

2.16 2.17 2.18 2.19 2.20

0.0154 0.0150 0.0146 0.0143 0.0139

2.66 2.67 2.68 2.69 2.70

0.0039 0.0038 0.0037 0.0036 0.0035

0.21 0.22 0.23 0.24 0.25

0.4168 0.4129 0.4090 0.4052 0.4013

0.71 0.72 0.73 0.74 0.75

0.2389 0.2358 0.2327 0.2296 0.2266

1.21 1.22 1.23 1.24 1.25

0.1131 0.1112 0.1093 0.1075 0.1056

1.71 1.72 1.73 1.74 1.75

0.0436 0.0427 0.0418 0.0409 0.0401

2.21 2.22 2.23 2.24 2.25

0.0136 0.0132 0.0129 0.0125 0.0122

2.71 2.72 2.73 2.74 2.75

0.0034 0.0033 0.0032 0.0031 0.0030

0.26 0.27 0.28 0.29 0.30

0.3974 0.3936 0.3897 0.3859 0.3821

0.76 0.77 0.78 0.79 0.80

0.2236 0.2206 0.2177 0.2148 0.2119

1.26 1.27 1.28 1.29 1.30

0.1038 0.1020 0.1003 0.0985 0.0968

1.76 1.77 1.78 1.79 1.80

0.0392 0.0384 0.0375 0.0367 0.0359

2.26 2.27 2.28 2.29 2.30

0.0119 0.0116 0.0113 0.0110 0.0107

2.76 2.77 2.78 2.79 2.80

0.0029 0.0028 0.0027 0.0026 0.0026

0.31 0.32 0.33 0.34 0.35

0.3783 0.3745 0.3707 0.3669 0.3632

0.81 0.82 0.83 0.84 0.85

0.2090 0.2061 0.2033 0.2005 0.1977

1.31 1.32 1.33 1.34 1.35

0.0951 0.0934 0.0918 0.0901 0.0885

1.81 1.82 1.83 1.84 1.85

0.0351 0.0344 0.0336 0.0329 0.0322

2.31 2.32 2.33 2.34 2.35

0.0104 0.0102 0.0099 0.0096 0.0094

2.81 2.82 2.83 2.84 2.85

0.0025 0.0024 0.0023 0.0023 0.0022

0.36 0.37 0.38 0.39 0.40

0.3594 0.3557 0.3520 0.3483 0.3446

0.86 0.87 0.88 0.89 0.90

0.1949 0.1922 0.1894 0.1867 0.1841

1.36 1.37 1.38 1.39 1.40

0.0869 0.0853 0.0838 0.0823 0.0808

1.86 1.87 1.88 1.89 1.90

0.0314 0.0307 0.0301 0.0294 0.0287

2.36 2.37 2.38 2.39 2.40

0.0091 0.0089 0.0087 0.0084 0.0082

2.86 2.87 2.88 2.89 2.90

0.0021 0.0021 0.0020 0.0019 0.0019

0.41 0.42 0.43 0.44 0.45

0.3409 0.3372 0.3336 0.3300 0.3264

0.91 0.92 0.93 0.94 0.95

0.1814 0.1788 0.1762 0.1736 0.1711

1.41 1.42 1.43 1.44 1.45

0.0793 0.0778 0.0764 0.0749 0.0735

1.91 1.92 1.93 1.94 1.95

0.0281 0.0274 0.0268 0.0262 0.0256

2.41 2.42 2.43 2.44 2.45

0.0080 0.0078 0.0075 0.0073 0.0071

2.91 2.92 2.93 2.94 2.95

0.0018 0.0018 0.0017 0.0016 0.0016

0.46 0.47 0.48 0.49 0.50

0.3228 0.3192 0.3156 0.3121 0.3085

0.96 0.97 0.98 0.99 1.00

0.1685 0.1660 0.1635 0.1611 0.1587

1.46 1.47 1.48 1.49 1.50

0.0721 0.0708 0.0694 0.0681 0.0668

1.96 1.97 1.98 1.99 2.00

0.0250 0.0244 0.0239 0.0233 0.0228

2.46 2.47 2.48 2.49 2.50

0.0069 0.0068 0.0066 0.0064 0.0062

2.96 2.97 2.98 2.99 3.00

0.0015 0.0015 0.0014 0.0014 0.0013

Appendix A: Probability Tables

p 0

t(p)

APPENDIX A.2. Critical Values of the Student t-Distribution

df

t(0.100)

t(0.050)

t(0.025)

t(0.010)

1 2 3 4 5

3.078 1.886 1.638 1.533 1.476

6.314 2.920 2.353 2.132 2.015

12.706 4.303 3.182 2.776 2.571

31.821 6.965 4.541 3.747 3.365

6 7 8 9 10

1.440 1.415 1.397 1.383 1.372

1.943 1.895 1.860 1.833 1.812

2.447 2.365 2.306 2.262 2.228

3.143 2.998 2.896 2.821 2.764

11 12 13 14 15

1.363 1.356 1.350 1.345 1.341

1.796 1.782 1.771 1.761 1.753

2.201 2.179 2.160 2.145 2.131

2.718 2.681 2.650 2.624 2.602

16 17 18 19 20

1.337 1.333 1.330 1.328 1.325

1.746 1.740 1.734 1.729 1.725

2.120 2.110 2.101 2.093 2.086

2.583 2.567 2.552 2.539 2.528

21 22 23 24 25

1.323 1.321 1.319 1.318 1.316

1.721 1.717 1.714 1.711 1.708

2.080 2.074 2.069 2.064 2.060

2.518 2.508 2.500 2.492 2.485

26 27 28 29 30

1.315 1.314 1.313 1.311 1.310

1.706 1.703 1.701 1.699 1.697

2.056 2.052 2.048 2.045 2.042

2.479 2.473 2.467 2.462 2.457

31 32 33 34 35

1.309 1.309 1.308 1.307 1.306

1.696 1.694 1.692 1.691 1.690

2.040 2.037 2.035 2.032 2.030

2.453 2.449 2.445 2.441 2.438

391

392

Common Statistical Methods for Clinical Research with SAS Examples

p 0

X(p)

APPENDIX A.3. Critical Values of the Chi-Square Distribution

df

X(0.100)

X(0.050)

X(0.025)

X(0.010)

1 2 3 4 5

2.706 4.605 6.251 7.779 9.236

3.841 5.991 7.815 9.488 11.070

5.024 7.378 9.348 11.143 12.832

6.635 9.210 11.345 13.277 15.086

6 7 8 9 10

10.645 12.017 13.362 14.684 15.987

12.592 14.067 15.507 16.919 18.307

14.449 16.013 17.535 19.023 20.483

16.812 18.475 20.090 21.666 23.209

11 12 13 14 15

17.275 18.549 19.812 21.064 22.307

19.675 21.026 22.362 23.685 24.996

21.920 23.337 24.736 26.119 27.488

24.725 26.217 27.688 29.141 30.578

16 17 18 19 20

23.542 24.769 25.989 27.204 28.412

26.296 27.587 28.869 30.144 31.410

28.845 30.191 31.526 32.852 34.170

32.000 33.409 34.805 36.191 37.566

21 22 23 24 25

29.615 30.813 32.007 33.196 34.382

32.671 33.924 35.172 36.415 37.652

35.479 36.781 38.076 39.364 40.646

38.932 40.289 41.638 42.980 44.314

26 27 28 29 30

35.563 36.741 37.916 39.087 40.256

38.885 40.113 41.337 42.557 43.773

41.923 43.195 44.461 45.722 46.979

45.642 46.963 48.278 49.588 50.892

31 32 33 34 35

41.422 42.585 43.745 44.903 46.059

44.985 46.194 47.400 48.602 49.802

48.232 49.480 50.725 51.966 53.203

52.191 53.486 54.775 56.061 57.342

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Appendix E Multiple Comparison Methods E.1 E.2 E.3

Multiple Comparisons of Means..................................................................... 421 Multiple Comparisons of Binomial Proportions ........................................... 432 Summary........................................................................................................... 436

E.1

Multiple Comparisons of Means The Treatment effect is of primary concern in comparative clinical trials. Many of the statistical methods described in this book are illustrated with examples comparing two treatment groups. When there are more than two treatment groups, say K (K > 2), an ‘omnibus’ approach, such as analysis of variance, can be applied to test the overall null hypothesis of ‘no difference among the treatment groups’. If rejected, the next step is to determine which pairs of groups differ, resulting in as many as K· (K–1) / 2 possible pairwise comparisons. First, consider the situation in which it is appropriate to compare mean responses among groups, such as ANOVA. This is one of the most common situations involving pairwise treatment comparisons. A number of approaches to multiple comparisons are shown using the following example.

422

Common Statistical Methods for Clinical Research with SAS Examples

p Example E.1

Consider a parallel study that includes a placebo group, a reference group, and 3 different dose levels of an experimental treatment. The response, Y, is measured for each patient in each of the 5 groups shown in Table E.1. TABLE E.1. Raw Data for Example E.1 A Lo-Dose

B Mid-Dose

21 13 25 18 17 23

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16 12

Treatment Group C D Hi-Dose Reference

19 18 24 27 29

E Placebo

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23 18 22 19 24 14

14 17 21 20 13 23

29 28 32

20 29

27 12 16

Treatment group summaries are shown in Table E.2, with a pooled standard deviation of 4.671 (see Chapter 6 for calculating formula). TABLE E.2. Summary Statistics by Treatment Group for Example E.1 A

B

C

D

E

Mean

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24.700

26.556

21.125

18.111

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4.473

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4.486

5.011

N

8

10

9

8

9

With 5 treatment groups, there are (5) · (4) / 2 = 10 possible pairwise comparisons. You can proceed by conducting a series of 10 two-sample t-tests (Chapter 5) to obtain the ‘raw’ (unadjusted) p-values for these comparisons. With the assumption of homogeneous variability across groups, the estimate of the standard deviation pooled over the 5 groups (4.671) is used in the calculation of the t-tests, as demonstrated in Chapter 5. Alternatively, you could use the one-way ANOVA, as shown next. In SAS, pairwise t-tests can be performed as part of the ANOVA by using PROC GLM and specifying the PDIFF option in the LSMEANS statement as shown in the SAS code and output that follow. You can omit the ADJUST=T option , which requests pairwise t-tests, because that is the default.

Appendix E: Multiple Comparison Methods

SAS Code for Example E.1 DATA MC; INPUT TRT $ Y @@; DATALINES; A 21 A 13 A 25 A 18 A 17 A 23 A 16 A 12 B B 31 B 24 B 20 B 19 B 18 B 24 B 27 B 29 C C 22 C 24 C 34 C 26 C 29 C 28 C 32 D 23 D D 19 D 24 D 14 D 20 D 29 E 14 E 17 E 21 E E 23 E 27 E 12 E 16 ; PROC GLM DATA = MC; CLASS TRT; MODEL Y = TRT / SS3; LSMEANS TRT / PDIFF ADJUST=T; TITLE1 'EXAMPLE E.1: Multiple Comparisons'; TITLE2 'All-Pairwise Comparisons of Means'; RUN;

27 25 18 20

B C D E

28 19 22 13

OUTPUT E.1. SAS Output with Pairwise t-Tests for Example E.1 EXAMPLE E.1: Multiple Comparisons All-Pairwise Comparisons of Means The GLM Procedure Class Level Information Class Levels Values TRT 5 A B C D E Number of observations 44 Dependent Variable: Y DF

Sum of Squares

Model 4 Error 39 Corrected Tota1 43

521.470707 850.961111 1372.431818

Source

Source TRT

Mean Square

F Value

130.367677 21.819516

5.97

Pr > F 0.0008

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Coeff Var

Root MSE

Y Mean

0.379961

21.34268

4.671136

21.88636

DF

Type III SS

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F Value

Pr > F

4

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130.3676768

5.97

0.0008

TRT A B C D E

Least Squares Means LSMEAN Y LSMEAN Number

18.1250000 24.7000000 26.5555556 21.1250000 18.1111111

1 2 3 4 5

423

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Common Statistical Methods for Clinical Research with SAS Examples

OUTPUT E.1. (continued) Least Squares Means for effect TRT Pr > |t| for H0: LSMean(i)=LSMean(j) Dependent Variable: Y i/j 1 2 3 4 5

1 0.0051 0.0006 0.2066 0.9951

2 0.0051 0.3926 0.1147 0.0039

3 0.0006 0.3926 0.0216 0.0004

4 0.2066 0.1147 0.0216

5 0.9951 0.0039 0.0004 0.1919

0.1919

NOTE: To ensure overall protection level, only probabilities associated with pre-planned comparisons should be used.

The p-values associated with the 10 pairwise comparisons are summarized in Table E.3 based on the results shown in Output E.1. (Note: The LS-means are the same as the arithmetic means for the one-way ANOVA). The values that are flagged with an asterisk (*) indicate a value less than 0.05, the nominal significance level of each test. Because the ‘raw’ p-values are not adjusted for multiple testing, the overall ‘experiment-wise’ error rate is not controlled. To maintain an overall significance level at the 0.05 level, you can use one of the methods described in the sections that follow Table E.3. TABLE E.3. p-Values from t-Tests for Pairwise Treatment Group Comparisons in Example E.1 Comparison

|Mean Difference|

‘Raw’ p-value

C vs. E A vs. C B vs. E A vs. B C vs. D B vs. D D vs. E A vs. D B vs. C A vs. E

8.444 8.431 6.589 6.575 5.431 3.575 3.014 3.000 1.856 0.014

0.0004* 0.0006* 0.0039* 0.0051* 0.0216* 0.1147 0.1919 0.2066 0.3926 0.9951

The Tukey-Kramer Method Tukey developed a test statistic for simultaneous comparisons of means, which is related to the maximum mean difference over all pairwise comparisons. This statistic has the so-called ‘studentized range’ distribution when sample sizes are the same for all groups. Probabilities associated with this distribution can be found by using numerical techniques and are tabulated in many statistical texts.

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The Tukey-Kramer method is based on an approximation to the studentized range distribution when sample sizes differ among groups, as is usually the case in clinical trials. The SAS function PROBMC with the “RANGE” distribution can be used to obtain the adjusted p-values for all pairwise comparisons associated with the TukeyKramer approach. Let Tij represent the two-sample t-statistic for comparing the means of Groups i and j.

Tij =

xi − x j s⋅

1 1 + ni n j

where s is the square root of the ANOVA mean square error (MSE). The TukeyKramer p-value can be found by using a SAS statement in the form Pij = 1 – PROBMC(“RANGE”, ABS(SQRT(2)*Tij), . , df, K) where df is the number of ANOVA error degrees of freedom, and K is the number of treatment groups. For example, to compute the Tukey-Kramer adjusted p-value for comparing Groups A (Lo-Dose) and B (Mid-Dose) in Example E.1, find TAB = (18.125 – 24.700) / 4.671((1/8) + (1/10))½ = –2.9675 then use SAS to obtain the adjusted p-value PAB, P_AB = 1 – PROBMC(“RANGE”, SQRT(2)*2.9675, . , 39, 5);

for which SAS returns a value of 0.0386. These probabilities can also be obtained directly by using PROC GLM for all comparisons. Simply specify the ADJUST=TUKEY option in the LSMEANS statement, as follows. LSMEANS TRT / PDIFF ADJUST=TUKEY;

The results from this statement are shown in Output E.2.

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OUTPUT E.2. Tukey-Kramer Results for Example E.1 EXAMPLE E.1: Multiple Comparisons All-Pairwise Comparisons of Means The GLM Procedure Least Squares Means Adjustment for Multiple Comparisons: Tukey-Kramer

TRT A B C D E

Y LSMEAN

LSMEAN Number

18.1250000 24.7000000 26.5555556 21.1250000 18.1111111

1 2 3 4 5

Least Squares Means for effect TRT Pr > |t| for H0: LSMean(i)=LSMean(j) Dependent Variable: Y i/j

1

1 2 3 4 5

0.0386 0.0054 0.7021 1.0000

2

3

4

0.0386

0.0054 0.9080

0.7021 0.4980 0.1389

0.9080 0.4980 0.0300

0.1389 0.0039

5 1.0000 0.0300 0.0039 0.6759

0.6759

Dunnett’s Test Although the Tukey-Kramer method has wide appeal for all pairwise comparisons, Dunnett’s test is the preferred method if the goal is to maintain the overall significance level when performing multiple tests to compare a set of treatment means with a control group. With K groups, you now have only K–1 comparisons. Similar to the Tukey-Kramer p-values, the adjusted p-values based on a two-sided Dunnett’s test can be obtained by using the SAS function PROBMC with the distribution “DUNNETT2”, as shown below. Let Ti represent the two-sample t-statistic (Chapter 5) for comparing the mean of Group i with the control Group mean.

xi − x0

Ti = s⋅

1 1 + ni n0

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427

where s is the square root of the ANOVA mean square error (MSE) and the subscript 0 refers to the control group. The Dunnett p-value can be found by using a SAS statement of the form Pi = 1 – PROBMC(“DUNNETT2”, ABS(Ti), . , df, K–1, L1, L2,…, LK–1) where df is the number of ANOVA error degrees of freedom, K is the number of treatment groups (including the control group), and the Li’s are distributional parameters used to adjust for unequal sample sizes, defined as

Li =

ni ni + n0

For example, to compute Dunnett’s adjusted p-value for comparing Group B (Middle-Dose) with Group E (Placebo) in Example E.1, find TB = (24.700 – 18.111) / 4.671((1/10) + (1/9))½ = 3.0701 and L1 = (8 / (8+9))½ = 0.6860 L2 = (10/ (10+9))½ = 0.7255 L3 = (9/ (9+9))½

= 0.7071

L4 = (8 / (8+9))½

= 0.6860

then, evaluate P_B=1–PROBMC(“DUNNETT2”,3.0701, . ,39,4,0.6860,0.7255,0.7071,0.6860);

to obtain the adjusted p-value, PB, for which SAS returns a value of 0.0138. These probabilities can be easily obtained for all comparisons by using PROC GLM. You must use the CONTROL option to identify Group E as the control group, and specify the ADJUST=DUNNETT option in the LSMEANS statement, as follows: LSMEANS TRT / PDIFF=CONTROL(‘E’) ADJUST=DUNNETT;

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The results from this statement are shown in Output E.3. OUTPUT E.3. Dunnett’s Results for Example E.1 EXAMPLE E.1: Multiple Comparisons Treatment Comparisons vs. Control Group (TRT=E) The GLM Procedure Least Squares Means Adjustment for Multiple Comparisons: Dunnett

TRT A B C D E

Y LSMEAN 18.1250000 24.7000000 26.5555556 21.1250000 18.1111111

H0:LSMean= Control Pr > |t| 1.0000 0.0138 0.0017 0.4901

p-Value Adjustment Methods Both the Tukey-Kramer and Dunnett methods rely on numerical integration to evaluate complex probability distributions and require the assumption of normality. They are robust statistical methods that have good power and enjoy widespread usage in statistical programs, including SAS. Let’s look at the Bonferroni, the Sidak, the step-down Bonferroni (Holm), and the step-up Bonferroni (Hochberg) methods, which are useful in multiple comparisons. These methods attempt to control the overall significance level by adjusting p-values without requiring the evaluation of complex distributions. They can be used in making pairwise comparisons of means but also have a much wider application. Although versatile and very easy to compute, these methods are generally very conservative and might fail to detect important differences. The adjusted p-values for the 10 pairwise comparisons of Example E.1 are shown in Table E.4 based on the methods listed in the preceding paragraph, in addition to the results previously obtained by using the Tukey-Kramer and Dunnett methods.

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TABLE E.4. Adjusted p-Values for Example E.1

(i) Comparison

t-Test (‘Raw’)

TukeyKramer

Dunnett

Bonferroni

Sidak

Step-Down Bonferroni

Step-Up Bonferroni

pRAW

pTUK

pDUN

pBON

pSID

pSTPDWN

pSTPUP

0.0017*

0.004*

0.004*

0.004*

0.004*

0.006*

0.006*

0.005*

0.005*

0.039*

0.038*

0.031*

0.031*

(1)

C v. E

0.0004*

0.0039*

(2)

A v. C

0.0006*

0.0054*

(3)

B v. E

0.0039*

0.0300*

(4)

A v. B

0.0051*

0.0386*

0.051

0.050*

0.036*

0.036*

(5)

C v. D

0.0216*

0.1389

0.216

0.196

0.130

0.130

(6)

B v. D

0.1147

0.4980

1.000

0.704

0.574

0.574

(7)

D v. E

0.1919

0.6759

1.000

0.881

0.768

0.620

(8)

A v. D

0.2066

0.7021

1.000

0.901

0.768

0.620

(9)

B v. C

0.3926

0.9080

1.000

0.993

0.785

0.785

(10) A v. E

0.9951

1.0000

1.000

1.000

0.995

0.995

0.0138*

0.4901

1.0000

The adjusted p-values using the Bonferroni and Sidak methods, are found from the raw p-values by using the formulas,

pBON = m · pRAW and

pSID = 1 – (1 – pRAW)m where m is the number of comparisons and pRAW is the raw (unadjusted) p-value. Values greater than 1.0 are truncated to 1.0. For example, adjusted p-values for Comparison (5) in Table E.4 are 10

pBON = 10(0.0216) = 0.216 and pSID = 1– (1–0.0216)

= 0.1962

The Bonferroni step-down method adjusts the raw p-values in a stepwise fashion starting with the smallest. Comparison (1) is adjusted as pSTPDWN(1) = m·pRAW(1) or 10(0.0004) = 0.004. The next smallest raw p-value, Comparison (2), is adjusted as pSTPDWN(2) = (m–1)pRAW(2) = 9(0.0006) = 0.0054. Comparison (3) is adjusted as pSTPDWN(3) = (m–2)pRAW = 8(0.0039) = 0.0312, and so on. Any adjusted p-value calculated to be greater than 1 is set to 1.0. Also, all comparisons that follow the first non-significant comparison must be nonsignificant, and an adjusted p-value is set equal to its predecessor if its calculated value is less. Therefore for Comparison (8), the adjusted p-value is calculated to be 3(0.2066) = 0.6198, which is less than the preceding adjusted p-value (0.7676), so the Comparison 8 p-value is set to 0.7676. The Bonferroni step-down method is sometimes referred to as Holm’s method.

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The Bonferroni step-up-method adjusts the raw p-values in a stepwise fashion, but it starts with the largest value. For Comparison (10) in Table E.4., you compute pSTPUP(10) = 1 · pRAW(10) = 0.9951. The next largest value, Comparison (9), is adjusted as pSTPUP(9) = 2·pRAW(9) = 2(0.3926) = 0.7852. Comparison (8) is adjusted as pSTPUP(8) = 3·pRAW(8) = 3(0.2066) = 0.6198, and so on. Each adjusted p-value must be no greater than its predecessor. Those values that are greater are set equal to the preceding adjusted p-value. The Bonferroni step-up method is also known as Hochberg’s method. Adjusted p-values using the Bonferroni, Sidak, and Bonferroni stepwise methods can be found by using PROC MULTTEST in SAS as shown below with the MC data set of Example E.1. The BONFERRONI, SIDAK, HOLM, and HOC options are specified in the PROC MULTTEST statement, and the results are written to the data set NEWMC . The treatment group (TRT) is specified as the grouping variable in a CLASS statement , and the MEAN option in the TEST statement identifies the response variable (Y) for which comparisons of means are to be performed. One CONTRAST statement is used for each pairwise comparison. You see very similar results between the Bonferroni step-down and step-up methods for Example E.1 (stpbon_p and hoc_p, respectively, in Output E.4). However, the step-up method is actually more powerful and, in many cases, will find more significant comparisons, but it must be used with caution because the step-up method assumes that the raw p-values are mutually independent. Notice also that the Bonferroni step-down and step-up methods produce results that are very comparable to the Tukey-Kramer method for Example E.1. Although the Bonferroni and Sidak methods are very easy to use, they are not recommended for the primary analysis in confirmatory or pivotal clinical trials because of their overly conservative nature. SAS Code for Example E.1 (continued) PROC MULTTEST BONFERRONI SIDAK HOLM HOC NOPRINT OUT=NEWMC DATA=MC; CLASS TRT; TEST MEAN(Y); CONTRAST 'A v B' -1 1 0 0 0; CONTRAST 'A v C' -1 0 1 0 0; CONTRAST 'A v D' -1 0 0 1 0; CONTRAST 'A v E' -1 0 0 0 1; CONTRAST 'B v C' 0 -1 1 0 0; CONTRAST 'B v D' 0 -1 0 1 0; CONTRAST 'B v E' 0 -1 0 0 1; CONTRAST 'C v D' 0 0 -1 1 0; CONTRAST 'C v E' 0 0 -1 0 1; CONTRAST 'D v E' 0 0 0 -1 1; RUN; PROC PRINT DATA=NEWMC; TITLE1 'EXAMPLE E.1: Multiple Comparisons'; TITLE2 'p-Value Adjustment Procedures'; RUN;

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OUTPUT E.4. p-Values for Pairwise Treatment Comparisons EXAMPLE E.1: Multiple Comparisons p-Value Adjustment Procedures

O b s

_ t e s t _

_ v a r _

1 2 3 4 5 6 7 8 9 10

MEAN MEAN MEAN MEAN MEAN MEAN MEAN MEAN MEAN MEAN

Y Y Y Y Y Y Y Y Y Y

_ c o n t r a s t _ A A A A B B B C C D

v v v v v v v v v v

_ v a l u e _ B C D E C D E D E E

_ s e _

_ n v a l _

r a w _ p

b o n _ p

s t p b o n _ p

s i d _ p

h o c _ p

289.300 97.491 39 0.00511 0.05109 0.03577 0.04993 0.03577 370.944 99.870 39 0.00064 0.00637 0.00573 0.00635 0.00573 132.000 102.765 39 0.20655 1.00000 0.76780 0.90110 0.61965 -0.611 99.870 39 0.99515 1.00000 0.99515 1.00000 0.99515 81.644 94.435 39 0.39257 1.00000 0.78513 0.99316 0.78513 -157.300 97.491 39 0.11470 1.00000 0.57352 0.70428 0.57352 -289.911 94.435 39 0.00389 0.03887 0.03109 0.03820 0.03109 -238.944 99.870 39 0.02165 0.21645 0.12987 0.19654 0.12987 -371.556 96.888 39 0.00045 0.00447 0.00447 0.00446 0.00447 -132.611 99.870 39 0.19195 1.00000 0.76780 0.88132 0.61965

The Dunnett’s p-values are not directly comparable to the other p-values in Table E.4 because they are testing different hypotheses. The Dunnett p-values are included to show how the values change when the alternative hypothesis is changed from “at least one pair of means differs” to “at least one treatment mean differs from the control mean”. Adjusted p-values can be found for the latter hypothesis by using the Bonferroni and Sidak methods in the same way that is described for the all-pairwise comparisons method. For example, when restricting attention only to the treatment versus control comparisons, the Sidak adjusted p-value for the B v. E comparison would be 1–(1–0.0039)4 = 0.0155.

‘Closed’ Testing Multiple testing of individual null hypotheses that control for the overall significance level can be performed by using a principle called ‘closure’. This concept is discussed in depth with good examples in Chi (1998) and Bauer (1991). Here, the concept is introduced for the case of K=3 (3 groups) and m=3 pairwise comparisons. Under the principle of closure, each of the 3 pairwise comparisons can be conducted at the full α (0.05) level with no p-value adjustments, but only if the omnibus hypothesis (i.e., ‘all groups equal’) is first rejected at the α level. This method has great utility in the analysis of clinical trials data because many studies include 3 treatment groups. Pairwise comparisons of the mean responses among the 3 groups can be performed with t-tests for example, each at the 0.05 level of significance following a significant F-test when analysis of variance is used. If the Treatment effect is not significant based on the ANOVA F-test, each pairwise comparison must be declared non-significant under closed testing in order to maintain the experimentwise error rate at 0.05.

432

E.2

Common Statistical Methods for Clinical Research with SAS Examples

Multiple Comparisons of Binomial Proportions Because the p-value adjustment methods discussed in the previous section are based only on the raw p-values and not the underlying distribution of response data, you can use these same methods for a wide range of applications, including those with non-normal data (e.g., categorical responses and time-to-event data). When responses are binomial, you can use a set of raw p-values from a series of chisquare tests, Fisher’s exact tests, Cochran-Mantel-Haenszel tests, and many other methods to adjust for multiple comparisons of group proportions, such as response rates among treatments. In fact, PROC MULTTEST in SAS can be executed from an input list of raw p-values, as shown next. The comparison identification must be in the variable named TEST, and the raw (input) p-values must be values of the variable named RAW_P. The output shows the same adjusted p-values as previously obtained. Notice that in this program, no TEST or CLASS statements are used with PROC MULTTEST. Also, these adjusted p-values are not as accurate as those shown in Output E.4 because the raw p-values are entered using only 4-decimal place accuracy. DATA ADJST; INPUT TEST RAW_P @@; DATALINES; 1 0.0004 2 0.0006 3 0.0039 6 0.1147 7 0.1919 8 0.2066 ;

4 0.0051 5 0.0216 9 0.3926 10 0.9951

PROC MULTTEST BONFERRONI SIDAK HOLM HOC PDATA = ADJST; TITLE 'DATA SET ADJST'; RUN;

OUTPUT E.5. p-Value Adjustments Using PROC MULTTEST for Example E.1 DATA SET ADJST The Multtest Procedure p-Values

Test

Raw

Bonferroni

Stepdown Bonferroni

Sidak

Hochberg

1 2 3 4 5 6 7 8 9 10

0.0004 0.0006 0.0039 0.0051 0.0216 0.1147 0.1919 0.2066 0.3926 0.9951

0.0040 0.0060 0.0390 0.0510 0.2160 1.0000 1.0000 1.0000 1.0000 1.0000

0.0040 0.0054 0.0312 0.0357 0.1296 0.5735 0.7676 0.7676 0.7852 0.9951

0.0040 0.0060 0.0383 0.0498 0.1962 0.7043 0.8812 0.9012 0.9932 1.0000

0.0040 0.0054 0.0312 0.0357 0.1296 0.5735 0.6198 0.6198 0.7852 0.9951

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433

Resampling Methods ‘Resampling’ is another method that can be used for performing multiple comparisons with binomial data. This method uses the same randomization principle on which Fisher’s Exact test (Chapter 17) is based and requires the use of a computer program. In SAS, you can use PROC MULTTEST to perform ‘permutation’ resampling, as shown in Example E.2. In essence, the algorithm randomly selects, with replacement, a sample of all possible permutations of outcomes assuming the null hypothesis is true. The adjusted p-value is the proportion of those samples that have outcomes more extreme than the observed outcome. p Example E.2

Suppose a binomial response (‘response’ or ‘non-response’) is measured on a new sample of 168 patients randomized to the 5-group parallel study given in Example E.1, with summary results as follows. TABLE E.5. Response Rates for Example E.2 A

B

C

D

E

Lo-Dose

Mid-Dose

Hi-Dose

Reference

Placebo

N Responders

33 9

31 14

34 23

35 10

35 5

Non-Responders

24

17

11

25

30

Response Rate

27.3%

45.2%

67.6%

28.6%

14.3%

Use PROC MULTTEST to perform permutation resampling and compare the results with other p-value adjustment methods. SAS Code for Example E.2 DATA RR; INPUT TRT $ RESP COUNT @@; /* RESP=0 is “non-response”, RESP=1 is “response” */ DATALINES; A 0 24 A 1 9 B 0 17 B 1 14 C 0 11 C 1 23 D 0 25 D 1 10 E 0 30 E 1 5 ;

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Common Statistical Methods for Clinical Research with SAS Examples

ODS SELECT Multtest.pValues; PROC MULTTEST ORDER = DATA PERMUTATION NSAMPLE=20000 SEED=28375 DATA = RR; CLASS TRT; TEST FISHER(RESP); FREQ COUNT; CONTRAST 'A vs B' -1 1 0 0 0; CONTRAST 'A vs C' -1 0 1 0 0; CONTRAST 'A vs D' -1 0 0 1 0; CONTRAST 'A vs E' -1 0 0 0 1; CONTRAST 'B vs C' 0 -1 1 0 0; CONTRAST 'B vs D' 0 -1 0 1 0; CONTRAST 'B vs E' 0 -1 0 0 1; CONTRAST 'C vs D' 0 0 -1 1 0; CONTRAST 'C vs E' 0 0 -1 0 1; CONTRAST 'D vs E' 0 0 0 -1 1; TITLE1 'EXAMPLE E.2: Multiple Comparisons of Proportions'; TITLE2 'Permutation Resampling Method'; RUN; /* (prespecify the seed value so results can be duplicated) */;

OUTPUT E.6. Resampling Method Using PROC MULTTEST for Example E.2 EXAMPLE E.2: Multiple Comparisons of Proportions Permutation Resampling Method The Multtest Procedure p-Values Variable

Contrast

RESP RESP RESP RESP RESP RESP RESP RESP RESP RESP

A A A A B B B C C D

vs vs vs vs vs vs vs vs vs vs

B C D E C D E D E E

Raw

Permutation

0.1932 0.0014 1.0000 0.2365 0.0831 0.2038 0.0072 0.0017 <.0001 0.2436

0.5530 0.0080 1.0000 0.7095 0.3666 0.6039 0.0397 0.0093 <.0001 0.7095

The ‘raw’ p-values are the values that result from unadjusted pairwise comparisons using two-tailed Fisher’s exact tests (Chapter 17). The values are found directly by using PROC MULTTEST with the FISHER option in the TEST statement , which eliminates the need to run PROC FREQ. A resampling size of 20,000 is requested, and a random seeding value is provided . If no value is specified for SEED, SAS randomly selects a value, which results in slightly different adjusted pvalues each time the program is run.

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For comparison, compute adjusted p-values by using the Bonferroni, Sidak, stepdown Bonferroni and step-up Bonferroni methods from the raw p-values based on Fisher’s exact test (Output E.6).

DATA BINP; INPUT TEST RAW_P @@; DATALINES; 1 0.1932 2 0.0014 3 1.0000 6 0.2038 7 0.0072 8 0.0017 ;

4 0.2365 5 0.0831 9 0.0001 10 0.2436

PROC MULTTEST BONFERRONI SIDAK HOLM HOC PDATA = BINP; RUN;

The p-values from the SAS output are reproduced in Table E.6, which compares the adjusted p-values for the permutation resampling method with the methods previously described. Notice that permutation resampling results in 4 significant comparisons (p < 0.05), while each of the other methods find only 3 that are significant. Resampling techniques are preferred, when possible, because they generally have greater power than the other methods previously discussed. In the past, application of resampling methods has been limited by the huge computing resources required, but they are gaining in use with the increasing power of desktop computing. One disadvantage of resampling is that the adjusted p-values will differ slightly with each implementation. When using PROC MULTTEST, results can be duplicated by specifying the same value for SEED each time you run the program. TABLE E.6. Adjusted p-Values for Example E.2

Comparison

Fisher’s (‘Raw’)

Permutation Resampling

Bonferroni

Sidak

Stepdown Bonferroni

Stepup Bonferroni

C v. E

0.0001*

0.0001*

0.001*

0.001*

0.001*

0.001*

A v. C

0.0014*

0.0080*

0.014*

0.014*

0.013*

0.013*

C v. D

0.0017*

0.0093*

0.017*

0.0170*

0.014*

0.014*

B v. E

0.0072*

0.0397*

0.072

0.070

0.050

0.050

B v. C

0.0831

0.3666

0.831

0.580

0.499

0.487

A v. B

0.1932

0.5530

1.000

0.883

0.966

0.487

B v. D

0.2038

0.6039

1.000

0.898

0.966

0.487

A v. E

0.2365

0.7095

1.000

0.933

0.966

0.487

D v. E

0.2436

0.7095

1.000

0.939

0.966

0.487

A v. D

1.0000

1.0000

1.000

1.000

1.000

1.000

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The adjusted p-values found by PROC MULTTEST are based on the entire set of CONTRAST statements used. To compare each treatment group with the control, use the same SAS statements as shown in the SAS Code for Example 4.2, and replace the ten CONTRAST statements with the following set of four statements: CONTRAST CONTRAST CONTRAST CONTRAST

E.3

'A 'B 'C 'D

vs vs vs vs

E' -1 0 0 0 E' 0 -1 0 0 E' 0 0 -1 0 E' 0 0 0 -1

1; 1; 1; 1;

Summary This introduction to multiple comparisons illustrates just some of the many methods available for conducting multiple comparisons for hypothesis testing. Many of the same methods can be used when addressing the multiplicity problem caused by simultaneous analysis of multiple response variables, such as that which occurs with multiple primary endpoints or study objectives. Adjustments based on methods discussed here can also be implemented for simultaneous confidence intervals. The Tukey-Kramer and Dunnett’s methods presented in this section using one-way ANOVA (Chapter 6) are widely used when response data have a normal distribution. These methods can also be used with more complex models, such as two-way ANOVA (Chapter 7), repeated measures ANOVA (Chapter 8), and analysis of covariance (Chapter 11), although these methods can be somewhat more conservative with these models. You have also seen how to adjust the raw pairwise p-values to maintain a prespecified, overall α-level without requiring ANOVA assumptions. The p-value adjustment methods discussed here can be used for pairwise treatment comparisons when response data do not follow the normal distribution. This approach can be applied to most of the methods discussed in this book that involve comparisons of more than two treatment groups. In addition, resampling techniques and simulation methods are gaining greater attention with the increasing power of low-cost computing. The field of multiple comparisons is a broad and rapidly evolving field. Entire books have been written and professional conferences are held, about this one subject. Standard tests of the past, such as Duncan’s test, have been replaced by more powerful methods. ‘Stepwise’ methods can be used with the Tukey, Dunnett, and Sidak methods, re-sampling methods, and tests under the closure principle. Improvement in power often results by implementing stepwise versions of the methods discussed in this chapter. Note: For a more detailed discussion of multiple comparisons with SAS examples, refer to the excellent book, “Multiple Comparisons and Multiple Tests Using the SAS System,” by Westfall, Tobias, Rom, and Wolfinger, which was written under the SAS Books By Users program.

Appendix F Data Transformations F.1 F.2

Introduction...................................................................................................... 437 The Log Transformation................................................................................. 438

F.1

Introduction Many statistical tests, including the two-sample t-test (Chapter 5) and ANOVA (Chapters 6, 7, and 8) are performed under the assumption of normally distributed data with variance homogeneity among groups. Although such tests are known to be robust against mild deviations to these assumptions, especially with larger sample sizes, clinical data are often encountered that appear to have skewed distributions or larger variability within certain groups. The underlying distribution in such tests might actually be non-normal, or there might be a small proportion of the data called ‘outliers’ that occur outside the range of usual expectation under the normality assumption, for known or unknown reasons. Data transformations can often be used effectively in normalizing the data and/or stabilizing the variability. The logarithmic, square root, arcsine, and rank transformations are among the most frequently used data transformations in applied statistics. The square root and arcsine transformations are infrequently used in clinical statistics, however. Converting the observations to their ranks before analysis is one of the most common methods of transforming the data (see Chapters 12, 13, and 14). The logarithmic (‘log’) transformation is also frequently used, and its application is discussed in the next section. Log transformations are commonly used in analyzing many types of clinical research data. One example is in vaccine and immunogenicity studies in which antibody titer values are measured. These values (xi’s) are usually modeled with the log-normal distribution, and results are summarized in terms of the geometric mean titer and the geometric mean ratio (or ‘n-fold increase’). Another common application of the log-transformation is in the analysis of area-under-the-curve (AUC) data based on blood levels from bioavailability and pharmacokinetic studies.

438

F.2

Common Statistical Methods for Clinical Research with SAS Examples

The Log Transformation The Log-Normal Distribution If the logarithm of a response, Y, denoted as log(Y), has a normal distribution, the random variable Y is said to come from a ‘log-normal’ distribution. The lognormal distribution looks like a normal distribution (see Chapter 1 or Figure B.1 in Appendix B) with its left tail compressed and its right tail stretched out due to a small number of very large values. (This is also called a ‘skewed’ distribution). It is appropriate to perform the log transformation on clinical data prior to analysis if the data come from a log-normal distribution. More generally, the log transformation is often used to ‘improve’ the normality of data sets that have any type of skewed distribution as described above. What Is a Logarithm? A logarithm has an associated base, b. The base b logarithm of X is expressed as logbX. A logarithm is the inverse function of exponentiation. That is, if Y = logbX, then X = bY. When not specified, b is usually 10. Most often, b is selected to be (“Euler’s constant”) e. In fact, this selection is so common, the base e logarithm has a special name, the ‘natural logarithm’, which is abbreviated as ‘ln’. Thus, ln(X) = loge(X), which implies X = eY (also denoted, X = exp(Y)). Only the natural logs are referred to in the discussion that follows. Analyzing Data Using the Log Transformation Following transformation, analyses are carried out using an ANOVA on the transformed data, and summary statistics are computed in the usual way. These statistics can be expressed in terms of the original data by using the relationships discussed next. Observe the response measures x1, x2, …, xn. Apply the log transformation as yi = ln(xi), and conduct analyses as usual on the yi’s. Let Y and sY denote the mean and standard deviation of the yi’s, and let LY and UY represent the lower and upper limits of the 100(1–α)% confidence interval, found by Y ± tα/2 ּ s y

that is,

LY = Y – tα/2 ּ s y and LU = Y + tα/2 ּ s y A 100(1–α)% confidence interval based on the original data (xi’s) is found by exponentiating LY and UY. L

U

(eY –e

Y

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Common Statistical Methods for Clinical Research with SAS Examples

$SSHQGL[* 6$6&RGHIRU([HUFLVHVLQ &KDSWHU 6$6&RGHIRU([HUFLVHVVHH&KDSWHU LIBNAME EXAMP ’c:\bookfiles\examples\sas’; OPTIONS NODATE NONUMBER;

*======================================================* | STATISTICAL ANALYSES for Response Variable = "SCORE"| | (continuous numeric response variable) | *=======================================================* ; /* ANALYSIS #1: Two-Sample t-Test */ PROC TTEST DATA = EXAMP.TRIAL; CLASS TRT; VAR SCORE; TITLE ’ANALYSIS #1: Two-Sample t-Test’; RUN;

/* ANALYSIS #2: Wilcoxon Rank-Sum Test */ PROC NPAR1WAY WILCOXON DATA = EXAMP.TRIAL; CLASS TRT; VAR SCORE; TITLE ’ANALYSIS #2: Wilcoxon Rank-Sum Test’; RUN;

/* ANALYSIS #3: One-Way ANOVA */ PROC GLM DATA = EXAMP.TRIAL; CLASS TRT; MODEL SCORE = TRT; MEANS TRT; TITLE ’ANALYSIS #3: One-Way ANOVA’; RUN;

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/* ANALYSIS #4: Two-Way ANOVA with Interaction */ PROC GLM DATA = EXAMP.TRIAL; CLASS TRT CENTER; MODEL SCORE = TRT CENTER TRT*CENTER; LSMEANS TRT / STDERR PDIFF; TITLE ’ANALYSIS #4: Two-Way ANOVA With Interaction’; RUN;

/* ANALYSIS #5: Two-Way ANOVA without Interaction */ PROC GLM DATA = EXAMP.TRIAL; CLASS TRT CENTER; MODEL SCORE = TRT CENTER; TITLE ’ANALYSIS #5: Two-Way ANOVA Omitting Interaction’; RUN;

/* ANALYSIS #6: Three-Way ANOVA, with All Interactions */ PROC GLM DATA = EXAMP.TRIAL; CLASS TRT CENTER SEX; MODEL SCORE = TRT | CENTER | SEX; LSMEANS TRT / STDERR PDIFF; TITLE ’ANALYSIS #6: Three-Way ANOVA - all Interactions’; RUN;

/* ANALYSIS #7: Three-Way Main Effects ANOVA */ PROC GLM DATA = EXAMP.TRIAL; CLASS TRT CENTER SEX; MODEL SCORE = TRT CENTER SEX; TITLE ’ANALYSIS #7: Three-Way Main Effects ANOVA’; RUN;

/* ANALYSIS #8: Regression of Score on Age */ PROC GLM DATA = EXAMP.TRIAL; MODEL SCORE = AGE / SOLUTION; TITLE ’Linear Regression of Score on Age’; RUN; PROC PLOT DATA = EXAMP.TRIAL; PLOT SCORE*AGE; RUN;

/* ANALYSIS #9: Analysis of Covariance - Test for Equal Slopes */ PROC GLM DATA = EXAMP.TRIAL; CLASS TRT; MODEL SCORE = TRT AGE TRT*AGE; /* use interaction to check for equal slopes */ TITLE1 ’ANALYSIS #9: ANCOVA Using Age as Covariate,’; TITLE2 ’Test for Equal Slopes’; RUN;

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/* ANALYSIS #10: Analysis of Covariance - Assuming Equal Slopes */ PROC GLM DATA = EXAMP.TRIAL; CLASS TRT; MODEL SCORE = TRT AGE / SOLUTION; LSMEANS TRT / STDERR PDIFF; TITLE ’ANALYSIS #10: ANCOVA Using Age as Covariate’; RUN; PROC PLOT DATA = EXAMP.TRIAL; PLOT SCORE*AGE=TRT; RUN;

/* ANALYSIS #11: Stratified ANCOVA */ PROC GLM DATA = EXAMP.TRIAL; CLASS TRT CENTER; MODEL SCORE = TRT CENTER AGE / SOLUTION; TITLE1 ’ANALYSIS #11: ANCOVA -- using Age as covariate,’; TITLE2 ’Stratified by Center’; RUN;

*=======================================================* | STATISTICAL ANALYSES for Response Variable = "RESP" | | (dichotomous response variable) | *=======================================================*

;

PROC FORMAT; VALUE RSPFMT 0 = ’NO ’ 1 = ’YES’; RUN;

/* ANALYSES #12 and #13: Chi-Square and Fishers Exact Test -- Dichotomous Response */ PROC FREQ DATA = EXAMP.TRIAL; TABLES TRT*RESP / CHISQ NOCOL NOPCT; FORMAT RESP RSPFMT.; TITLE1 ’ANALYSIS #12: Chi-Square Test’; TITLE2 ’ANALYSIS #13: Fishers Exact Test’; RUN;

/* ANALYSIS #14: Stratified CMH Test for Response Rate Analysis */ PROC FREQ DATA = EXAMP.TRIAL; TABLES CENTER*TRT*RESP / CMH NOCOL NOPCT; FORMAT RESP RSPFMT.; TITLE ’ANALYSIS #14: CMH Test Controlling for Center’; RUN;

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/* ANALYSIS #15: Logistic Regression - Dichotomous Response */ PROC LOGISTIC DATA = EXAMP.TRIAL; CLASS TRT / PARAM = REF; MODEL RESP = TRT; TITLE1 ’ANALYSIS #15: Logistic Regression Analysis for’; TITLE2 ’Treatment Group Differences Using Dichotomized Response’; RUN;

/* ANALYSIS #16: Logistic Regression - Dichotomous Response, Adjusted for Age */ PROC LOGISTIC DATA = EXAMP.TRIAL; CLASS TRT / PARAM = REF; MODEL RESP = TRT AGE; TITLE1 ’ANALYSIS #16: Logistic Regression Analysis for’; TITLE2 ’Treatment Group Differences Using Dichotomized Response’; TITLE3 ’Adjusted for Age’; RUN;

/* ANALYSIS #17: Logistic Regression - Dichotomous Response, Adjusted for Age, Sex, Center */ PROC LOGISTIC DATA = EXAMP.TRIAL; CLASS TRT SEX CENTER / PARAM = REF; MODEL RESP = TRT AGE SEX CENTER TRT*CENTER; TITLE1 ’ANALYSIS #17: Logistic Regression Analysis for’; TITLE2 ’Treatment Group Differences Using Dichotomized Response’; TITLE3 ’Adjusted for Age, Sex, Center & Interactions’; RUN;

*=======================================================* | STATISTICAL ANALYSES for Response Variable = "SEV" | | (ordinal categorical response variable) | *=======================================================*

;

\* Obtain Severity Distributions based on the Response = "SEV" *\ PROC FORMAT; VALUE SEV1FMT

0 = ’0 = None’ 2 = ’2 = Mod ’

1 = ’1 = Mild’ 3 = ’3 = Sev ’

VALUE SEV2FMT

0 = ’0 = None ’ 2 = ’2.185 = Mod ’

1 = ’1.036 = Mild’ 3 = ’3 = Sev ’ ;

VALUE SEV3FMT

0 = ’None’ 2 = ’Mod.’

1 = ’Mild’ 3 = ’Sev.’

;

;

RUN;

/* ANALYSES #18 and #19: Chi-Square / Fishers Exact Test -Multinomial Response */ PROC FREQ DATA = EXAMP.TRIAL; TABLES TRT*SEV / CHISQ EXACT NOCOL NOPCT; FORMAT SEV SEV1FMT.;

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TITLE1 ’ANALYSIS #18: Chi-Square Test’; TITLE2 ’ANALYSIS #19: Generalized Fishers Exact Test’; RUN;

/* ANALYSIS #20: CMH Test for Comparing Severity Distributions Using Table Scores */ PROC FREQ DATA = EXAMP.TRIAL; TABLES TRT*SEV / CMH NOCOL NOPCT; FORMAT SEV SEV1FMT.; TITLE1 ’ANALYSIS #20: Mantel-Haenszel Test on Severity’; TITLE2 ’(Using Table Scores)’; RUN;

/* ANALYSIS #21: CMH Test for Comparing Severity Distributions - Using Modified Ridit Scores */ PROC FREQ DATA = EXAMP.TRIAL; TABLES TRT*SEV / CMH SCORES = MODRIDIT NOCOL NOPCT; FORMAT SEV SEV2FMT.; TITLE1 ’ANALYSIS #21: Mantel-Haenszel Test on Severity’; TITLE2 ’(Using Modified Ridit Scores)’; RUN;

/* ANALYSES #22: Wilcoxon Rank-Sum Test on Response = "SEV" */ PROC NPAR1WAY WILCOXON DATA = EXAMP.TRIAL; CLASS TRT; VAR SEV; TITLE ’ANALYSIS #22: Wilcoxon Rank-Sum Test on "SEV" Response’; RUN;

/* ANALYSIS #23: Stratified CMH Test for Comparing Severity Distributions - Using Table Scores */ PROC FREQ DATA = EXAMP.TRIAL; TABLES CENTER*TRT*SEV / CMH NOCOL NOPCT; FORMAT SEV SEV3FMT.; TITLE1 ’ANALYSIS #23: CMH Test on Severity Distributions’; TITLE2 ’(Using Table Scores), Controlling for Center’; RUN;

/* ANALYSIS #24: Stratified CMH Test for Comparing Severity Distributions - Using Modified Ridit Scores */ PROC FREQ DATA = EXAMP.TRIAL; TABLES CENTER*TRT*SEV / CMH SCORES = MODRIDIT NOCOL NOPCT; FORMAT SEV SEV3FMT.; TITLE1 ’ANALYSIS #24: CMH Test on Severity Distributions’; TITLE2 ’(Using Modified Ridit Scores), Controlling for Center’; RUN;

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/* ANALYSIS #25: Logistic Regression: Proportional Odds Model */ PROC LOGISTIC DATA = EXAMP.TRIAL; CLASS TRT / PARAM = REF; MODEL SEV = TRT; TITLE1 ’ANALYSIS #25: Proportional Odds Model for Treatment’; TITLE2 ’Differences Using Ordinal Response Categories’; RUN;

/* ANALYSIS #26: Logistic Regression: Proportional Odds Model, Adjusted for Age */ PROC LOGISTIC DATA = EXAMP.TRIAL; CLASS TRT / PARAM = REF; MODEL SEV = TRT AGE; TITLE1 ’ANALYSIS #26: Proportional Odds Model for Treatment’; TITLE2 ’Differences Using Ordinal Response Categories’; TITLE3 ’Adjusted for Age’; RUN;

/* ANALYSIS #27: Logistic Regression: Proportional Odds Model, Adjusted for Age, Sex, Center */ PROC LOGISTIC DATA = EXAMP.TRIAL; CLASS TRT SEX CENTER; MODEL SEV = TRT AGE SEX CENTER; TITLE1 ’ANALYSIS #27: Proportional Odds Model for Treatment’; TITLE2 ’Group Differences Using Ordinal Response Categories’; TITLE3 ’Adjusted for Age, Sex & Center’; RUN;

REFERENCES and Additional Reading

Allison, P.D. (1995). Survival Analysis Using the SAS System: A Practical Guide. Cary, NC: SAS Institute Inc. Allison, P.D. (1999). Logistic Regression Using the SAS System: Theory and Application. Cary, NC: SAS Institute Inc. Altman, D.G. (1991). Practical Statistics for Medical Research. New York: Chapman & Hall. Anderson, S., Auquier, A., Hauck, W.W., Oakes, D., Vandaele, W., and Weisberg, H.I. (1980). Statistical Methods for Comparative Studies. New York: John Wiley & Sons. Armitage, P. and Berry, G. (1987). Statistical Methods in Medical Research. 2d ed. Palo Alto, CA: Blackwell Scientific Publications. Bancroft, T.A. (1968). Topics in Intermediate Statistical Methods, Volume One. Ames, IA: The Iowa University Press. Bauer, P. (1991). Multiple Testing in Clinical Trials. Statistics in Medicine 10: 871-890. Cantor, A. (1997). Extending SAS Survival Analysis Techniques for Medical Research. Cary, NC: SAS Institute Inc. Chi, G.Y.H. (1998). Multiple Testings: Multiple Comparisons and Multiple Endpoints. Drug Information Journal 32: 1347S-1362S. Cochran, W.G. and Cox, G.M. (1957). Experimental Designs. 2d ed. New York: John Wiley & Sons. Conover, W.J. and Iman, R.L. (1981). Rank Transformations as a Bridge between Parametric and Nonparametric Statistics. The American Statistician 35: 124-129.

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Common Statistical Methods for Clinical Research with SAS Examples

Cox, D.R. and Oakes, D. (1984). Analysis of Survival Data. New York: Chapman & Hall. D’Agostino, R.B., Massaro, J., Kwan, H., and Cabral, H. (1993). “Strategies for Dealing with Multiple Treatment Comparisons in Confirmatory Clinical Trials.” Drug Information Journal 27: 625641. Dawson-Saunders, B. and Trapp, R.G. (1990). Basic and Clinical Biostatistics. San Mateo, CA: Appleton & Lange. Delwiche, L.D. and Slaughter, S.J. (1998). The Little SAS Book: A Primer. 2d ed. Cary, NC: SAS Institute Inc. DeMets, D.L. and Lan, K.K.G. (1994). “Interim Analysis: The Alpha Spending Function Approach.” Statistics in Medicine 13: 13411352. Elashoff, J.D. and Reedy, T.J. (1984). “Two-Stage Clinical Trial Stopping Rules.” Biometrics 40: 791-795. Elliott, R.J. (2000). Learning SAS in the Computer Lab. 2d Ed. Pacific Grove, CA: Brooks/Cole. Fleiss, J.L. (1981). Statistical Methods for Rates and Proportions. 2d ed. New York: John Wiley & Sons. Freund, R.J. and Littell, R.C. (1991). SAS System for Regression. 2d ed. Cary, NC: SAS Institute Inc. Geller, N.L. and Pocock, S.J. (1987). “The Consultant’s Forum: Interim Analyses in Randomized Clinical Trials: Ramifications and Guidelines for Practitioners.” Biometrics 43: 213-223. Gillings, D. and Koch, G. (1991). “The Application of the Principle of Intention-to-Treat to the Analysis of Clinical Trials.” Drug Information Journal 25: 411-424. Hand, D.J. and Taylor, C.C. (1987). Multivariate Analysis of Variance and Repeated-Measures: A Practical Approach to Behavioural Scientists. New York: Chapman & Hall. Hatcher, L. and Stepanski, E.J. (1994). A Step-by-Step Approach to Using the SAS System for Univariate and Multivariate Statistics. Cary, NC: SAS Institute Inc.

References and Additional Reading

Hollander, M. and Wolfe, D.A. (1973). Nonparametric Statistical Methods. New York: John Wiley & Sons. Hosmer, D.W. and Lemeshow, S. (1989). Applied Logistic Regression. New York: John Wiley & Sons. Johnson, M.F. (1990). “Issues in Planning Interim Analyses.” Drug Information Journal 24: 361-370. Jones, B. and Kenward, M.G. (1989). Design and Analysis of Crossover Trials. New York: Chapman & Hall. Kalbfleisch, J.D. and Prentice, R.L. (1980). The Statistical Analysis of Failure Time Data. New York: John Wiley & Sons. Koch, G.G. and Gansky, S.A. (1996). “Statistical Considerations for Multiplicity in Confirmatory Protocols.” Drug Information Journal 30: 523-534. Lachin, J.M. (1981). “Introduction to Sample Size Determination and Power Analysis for Clinical Trials.” Controlled Clinical Trials 2: 93-113. Lachin, J.M. (2000). “Statistical Considerations in the Intent-to-Treat Principle.” Controlled Clinical Trials 21: 167-189. Lan, K.K.G., and DeMets, D.L. (1983). “Discrete Sequential Boundaries for Clinical Trials.” Biometrika 70: 659-663. Lee, J. (1981). “Covariance Adjustment of Rates Based on the Multiple Logistic Regression Model.” Journal of Chronic Diseases 34: 415426. Lehman, E.L. (1988). Theory of Point Estimation. New York: John Wiley & Sons. Littell, R.C., Freund, R.J. and Spector, P.C. (1991). SAS System for Linear Models. 3rd ed. Cary, NC: SAS Institute Inc. Littell, R.C., Milliken, G.A., Stroup, W.W., and Wolfinger, R.D. (1996). SAS System for Mixed Models. Cary, NC: SAS Institute Inc. Mendenhall, W. (1975). Introduction to Probability and Statistics. 4th ed. North Scituate, MA: Duxbury Press.

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Common Statistical Methods for Clinical Research with SAS Examples

Munro, B.H. and Page, E.B. (1993). Statistical Methods for Health Care Research. 2d ed. Philadelphia: JB Lippincott Co. O'Brien, P.C. and Fleming, T.R. (1979). “A Multiple Testing Procedure for Clinical Trials.” Biometrics 35: 549-556. PMA Biostatistics and Medical Ad Hoc Committee on Interim Analysis. (1993). “Interim Analysis in the Pharmaceutical Industry.” Controlled Clinical Trials 14: 160-173. Pocock, S.J. (1977). Group Sequential Methods in the Design and Analysis of Clinical Trials.” Biometrika 64, 191-199. Pocock, S.J., Geller, N.L., and Tsiatis, A.A. (1987). “The Analysis of Multiple Endpoints in Clinical Trials.” Biometrics 43: 487-498. Reboussin, D.M., DeMets, D.L., Kim, K.M., and Lan, K.K.G. (2000). “Computations for Group Sequential Boundaries Using the LanDeMets Spending Function Method. Controlled Clinical Trials 21:190-207. Sankoh, A.J. (1999). “Interim Analyses: An Update of an FDA Reviewer’s Experience and Perspective.” Drug Information Journal 33: 165-176. SAS OnlineDoc, Version 8. 1999. CD-ROM. SAS Institute Inc., Cary, NC. SAS Institute Inc. (1996). SAS/STAT Software: Changes and Enhancements through Release 6.11. Cary, NC: SAS Institute Inc. SAS Institute Inc. (1998). SAS/STAT User’s Guide, Version 6, 4th ed. 2 Vols. Cary, NC: SAS Institute Inc. Stokes, M.E., Davis, C.S., and Koch, G.G. (2000). Categorical Data Analysis Using the SAS System. 2d ed. Cary, NC: SAS Institute Inc. Tibshirani, R. (1982). “A Plain Man's Guide to the Proportional Hazards Model.” Clinical and Investigative Medicine 5: 63-68. Walker, G.A. (1999). Considerations in the Development of Statistical Analysis Plans for Clinical Studies. San Diego, CA: Collins-Wellesley Publishing.

References and Additional Reading

Westfall, P.H., Tobias, R.D., Rom, D., Wolfinger, R.D., and Hochberg, Y. (1999). Multiple Comparisons and Multiple Tests Using the SAS System. Cary, NC: SAS Institute Inc. Winer, B.J. (1971). Statistical Principles in Experimental Designs, 2d ed. New York: McGraw-Hill. Zeger, S.L., Liang, K.Y., and Albert, P.S. (1988). “Models for Longitudinal Data: A Generalized Estimating Equation Approach.” Biometrics 44: 1049-1060.

451

452

Common Statistical Methods for Clinical Research with SAS Examples

Index A a-spending function 36–37 active vs. placebo comparisons of degree of response (example) 277–278 ADJUST= option, LSMEANS statement (GLM) 425, 427 ADR frequency with antibiotics (example) 267–269 aerosol inhalation, antibiotic blood levels following (example) 165–170 AGGREGATE option, MODEL statement (LOGISTIC) 341 AGREE option, TABLES statement (FREQ) 297, 304 Akaike’s Information Criterion (AIC) 147, 340 Alzheimer’s progression (example) 137–148 AML remission, relapse rate adjusted for (example) 322–329 analysis of covariance See ANCOVA (analysis of covariance) analysis of variance, one-way See ANOVA (analysis of variance), one-way analysis of variance, three-way See ANOVA (analysis of variance), three-way analysis of variance, two-way See ANOVA (analysis of variance), two-way ANCOVA (analysis of covariance) 199–226 coefficient of determination 223 confidence intervals 222 GLM procedure 215, 224 GLM procedure (example), 209–220 least squares criterion 200, 216, 217 mean square 202, 205 mean square error 206, 216 MEANS procedure (example) 209–213, 215–220 multi-center hypertension study (example) 214–220 sum of squares 201, 202, 205 sum of squares for error 201, 205 sum of squares for group 201 triglyceride changes for glycemic control (example) 203–213 Type I sum of squares 216, 224 Type III sum of squares 216, 224 anemia, hemoglobin changes in (example) 94–102

ANOVA (analysis of variance), See also repeated measures analysis basic concepts of 399–407 four-way crossover (example) 165–170 random effects in 109–110 two-way crossover 155, 157–173 two-way crossover (example) 160–164 ANOVA (analysis of variance), one-way 77–90 confidence intervals 89–90 Dunnett’s test 88 F-distribution 78, 86, 90 F-statistic 78, 86, 90 F-test 78, 86, 87, 89–90 F-values 86, 89 GLM procedure (example) 82–86 HAM-A scores in GAD (example) 80–86 Kolmogorov-Smirnov test (example) 87 Kruskal-Wallis test 87, 250, 254 least squares mean 405–407 mean square error 78, 79, 81, 90 mean square group 78, 79, 81 MEANS procedure (example) 82–86 multiple comparison of treatment effects 87, 89 noise reduction by blocking 401–405 non-parametric analog (Kruskal-Wallis test) 247–256 on ranked data 254 pairwise t-tests 87–88 p-value (example) 90 Shapiro-Wilk test 87 sum of squares for error 79, 81 sum of squares for group 79, 81 two-sample t-test 87, 90, 237–245 Types I-IV sum of squares 90 within- vs. between-group variation 399–401 ANOVA (analysis of variance), repeated measures See repeated measures analysis ANOVA (analysis of variance), summary table ANCOVA (analysis of covariance) 200, 205 crossover design 159, 162 for two-way ANOVA 92 multiple regression analysis 193 one-way ANOVA (example) 86 repeated measures analysis 113 repeated measures analysis, univariate approach 118 ANOVA (analysis of variance), three-way 109

454

Index

ANOVA (analysis of variance), two-way 91–110 F-test 93, 96–97, 110 F-statistic 96–97 F-value 92, 97, 106 Fixed effects 109–110 Friedman’s test 256 GLM procedure 97, 98, 108 handling empty cells in SS calculations 416 hemoglobin changes in anemia (example) 94–102 interaction effects 102, 103 mean square 92 mean square error 92–93, 107, 108, 109 MEANS procedure (example) 97–99 memory function (example) 103–107 noise reduction by blocking 401–405 non-parametric analog (Friedman’s test) 256 pairwise t-tests 97 random effects 109–110 sum of squares 92–93, 95–96 sum of squares for error 96, 109 sums of squares, SAS types 97, 104, 108, 109 within- vs. between-group variation 399–401 anti-anginal response vs. disease history (example) 178–184 antibiotic blood levels following aerosol inhalation (example) 165–170 applied statistics, defined 1 arcsine transformation 197, 437 arthritic discomfort (example) multivariate approach to repeated measures analysis 122–127 univariate approach to repeated measures analysis 115–121 autocorrelation in time series analysis 197

B balanced residuals 172 Bartlett’s test 87 between- vs. within-group variation 399–401 bias in clinical research studies 10–12 bilirubin abnormalities (example) 294–299 binomial distribution 5 binomial data, multiple comparisons of 432–436 binomial response for more than two groups 274–276 binomial test 257–263 genital wart cure rate (example) 259–261 McNemar’s test 293–304

one-tailed hypothesis tests 263 two-tailed hypothesis tests 263 blinded randomization 11–12 blocking for noise reduction 401–405 BMI (body-mass index), example 56–60 Bonferroni adjustment 33, 428–431, 435 BONFERRONI option, MULTTEST procedure 430 Bowker’s test 304 BRESLOW option 379 Breslow-Day test 315

C CABG, CHF incidence in (example) 286–289 cardiac medication, diaphoresis following (example) 160–164 carryover effect See crossover design case report forms (CRFs), examples 40–41 CATMOD procedure logistic regression analysis 344 repeated measures ANOVA 155, 304 Central Limit Theorem 6–9, 235, 261 central tendency 14 characterizing populations 3 CHART procedure, TRIAL data set 46, 47, 50, 53 CHF incidence in CABG after ARA (example) 286–289 chi-square distribution critical values (table) 392 notation 393 shape of 397, 398 chi-square test 265–284 ADR frequency with antibiotics (example) 267–269 binomial response for more than two groups 274–276 confidence intervals 280–281 continuity correction 280 for contingency table analysis 270–273 for non-zero correlation 281 FREQ procedure 268, 275, 288 Kruskal-Wallis test 284 log-rank test 360 Mantel-Haenszel 275, 282 multinomial responses 276–278 one-tailed hypothesis tests 281 sample size determination 29 two-tailed hypothesis tests 281

Index Z-test statistic 279 CHISQ option, TABLES statement (FREQ) 268, 275, 288 CI= option, TTEST procedure 70 CINV function 294 CLASS statement GLM procedure 97 MIXED procedure 143 CLM option, MODEL statement (GLM) 182, 194 closed testing 431 clustered binomial data in logistic regression analysis 336–340, 344 CMH option, TABLES statement (FREQ) 272–278, 316 Cochran-Armitage test 31, 281–282 Cochran-Mantel-Haenszel test 305–316 chi-square test vs. 313 correlated binary responses and 336 Dermotel response in diabetic users (example) 307–312 FREQ procedure 272–278, 316 log-rank test 360 relapse rate adjusted for AML remission (example) 324 coefficient of determination ANCOVA (analysis of covariance) 223 linear regression analysis 187 multiple regression analysis 194 collinearity 188, 197, 224 COLLINOINT option, MODEL statement (REG) 197 compound symmetry 112, 122, 136 concatenating data 345 confidence intervals 16–17 ANCOVA (analysis of covariance) 222–223 chi-square test 280–281 odds ratios 341–342 one-way ANOVA (example) 89–90 confidence limits 70 contingency table analysis 270–272 continuity correction chi-square test 280 McNemar’s test 301 continuous probability distributions 5 CONTRAST statement GLM procedure 88 MULTTEST procedure 434, 436 CONTRAST(1) option, REPEATED statement (GLM) 130, 131 controlled studies 10

455

CORR option, REG procedure 187 correlated binary responses 336 covariate effect, linear regression analysis 193 Cox proportional hazards model 353, 365–379 HSV-2 episodes (example) 367–369 Hyalurise in vitreous hemorrhage (example) 370–375 log-rank test vs 368 Cox regression See Cox proportional hazards model CRFs (case report forms), examples 40–41 crossover design 155, 157–173 See also repeated measures analysis antibiotic blood levels following aerosol inhalation (example) 165–170 CLASS statement, GLM procedure 163 diaphoresis following cardiac medication (example) 160–164 GLM procedure (example) 163–164, 165–170 mean square 159 repeated measures ANOVA 155, 159 sums of squares 159, 161–162, 172 sum of squares for error 162 2-period 171–172 3-period 171 4-period 165–170

D data collection forms, examples 40–41 DATA step 45 data transformations 437–439 dehydration, symptomatic recovery in (example) 185–192 Dermotel response in diabetic ulcers (example) 307–312 DESCENDING option, LOGISTIC procedure 325, 332 descriptive statistics 3, 13–15 designing the study plan 9–12 diabetes Dermotel response in diabetic ulcers (example) 307–312 gastroparesis symptom relief (example) 331–335 Hyalurise in vitreous hemorrhage (example) 370–375 triglyceride changes for glycemic control (example) 203–214 diaphoresis following cardiac medication (example) 160–164

456

Index

DISCRETE option 379 discrete probability distribution 5 dispersion 14 DIST option, MODEL statement (GENMOD) 147 distributions used in statistical inference 393–398 double-blind studies 11 dropout rates for four dose groups (example) 274–277 DUNNETT option, MEANS statement (GLM) 88 Dunnett’s test 82, 88, 426–428 Durbin-Watson test 197 DW option, MODEL statement (REG) 197

FREQ procedure See also TABLES statement, FREQ procedure Bowker’s test 304 chi-square test 268, 275, 288 Cochran-Armitage test 282 Cochran-Mantel-Haenszel test 272, 274, 277, 316 EXACT statement 256 Fisher’s exact test 288, 291 McNemar’s test 297 TRIAL data set 46, 47, 51–52 WEIGHT statement 268, 288 Friedman’s test 256

E EDF option, NPAR1WAY procedure 245 EFRON option 379 empty cells 416 erectile dysfunction after surgery (example) 336–340 estimable functions SAS Type I 413 SAS Type II 414 SAS Type III 415 events, defined 5 EXACT option 279, 291 EXACT statement FREQ procedure 256 NPAR1WAY procedure 253 exploratory analyses, defined 3–4

F F-distribution notation 393 shape of 397, 398 F-tests 73–75 See also crossover design one-way ANOVA 78, 86, 87, 89–90 repeated measures analysis, univariate approach (example) 117 TEST statement, GLM procedure 110 two-way ANOVA 93, 96–97 factorial, defined 258 FEV1 changes (example) 68–73 FINV function 74 FISHER option, TABLES statement (FREQ) 288, 291 Fisher’s exact test 280, 285–291 fixed effects in ANOVA models 109–110 Freeman-Halton test 291

G GAD (general anxiety disorder), HAM-A scores (example) 80–86 gastroparesis symptom relief (example) 331–335 GEE analysis 147–148, 154, 155 Gehan test 361 generalized estimating equations (GEE analysis) 147–148, 154, 155 generalized Fisher’s exact text 291 generalized Wilcoxon test 361 genital wart cure rate (example) 259–261 GENMOD procedure 147–148, 155, 344 MODEL statement 147 repeated measures ANOVA, multivariate approach 147–148 TYPE3 option 153 WALD option 153 GLM procedure See also LSMEANS statement, GLM procedure See also MODEL statement, GLM procedure See also REPEATED statement, GLM procedure ANCOVA (analysis of covariance), examples 209–221 CLASS statement 97 CONTRAST statement 88 crossover design (example) 163–164, 165–170 linear regression analysis 181 one-way ANOVA (example) 82–83, 85–86 RANDOM statement 110, 149 repeated measures analysis, missing data (example) 140–143 repeated measures analysis, multivariate approach (example) 122–136 repeated measures analysis, univariate approach (example) 119–121

Index TEST statement 110 three-way ANOVA 109 two-way ANOVA (example) 97 global evaluations of Seroxatene (example) 239-243 glycemic control, triglyceride changes for (example) 203–214 Greenhouse-Geiser values 141, 150 group sequential methods 33 group variance, homogeneity of 78

H HAM-A scores in GAD (example) 80–86 hazard functions See Cox proportional hazards model hemoglobin changes in anemia (example) 94–102 histogram 13 HOC option, MULTTEST procedure 429 HOLM option, MULTTEST procedure 429 homogeneity of group variance 78 Hotelling-Lawley Trace test 130, 140, 150 HSV-2 episodes (example) 351–359, 367–370 Hyalurise in vitreous hemorrhage (example) 370–375 hypertension study (example) 214–220 hypothesis testing 18–21, 23–38 See also one-tailed hypothesis tests See also two-tailed hypothesis tests multiple hypotheses testing 30–37 p-value 27 sample size 27–29 significance levels 21, 23–24 test power 25–26

I inferences 2–3, 9–12 inferential methods 3 inferential statistics 16–21 interaction effects, two-way ANOVA 102 intercourse success rates after surgery (example) 336–340 interim analyses 33–37

J Jonckheere-Terpstra test 31, 255–256 JT option, TABLES statement (FREQ) 256

457

K Kaplan-Meier estimates 361–362 Kolmogorov-Smirnov test 76, 87, 245 Kruskal-Wallis test 247–256 chi-square test vs. 284 one-way ANOVA 87 psoriasis evaluation (example) 249–253

L LACKFIT option, MODEL statement (LOGISTIC) 341 Lan-DeMets a-spending function 36–37 last-observation-carried-forward (LOCF) 152 least squares criterion 177, 199, 217 least squares mean 405–407 LIFEREG procedure 376 LIFETEST procedure 355–359 PHREG procedure vs. 363 PLOTS= option 361, 362, 379 STRATA statement 355, 363 TEST statement 363 likelihood ratio test 361 limiting conditions for populations 2 linear contrasts 88 linear regression analysis 175–198 anti-anginal response vs. disease history (example) 178–184 coefficient of determination 187, 194 correlation coefficient 193–195 covariate effect 193 GLM procedure 181 mean square 193 sum of squares 193, 195 sum of squares for error 180, 195 zero regression slope 193 LOCF (last-observation-carried-forward) 152 log-rank test 349–363 chi-square test vs. 360, 361 Cochran-Mantel-Haenszel test vs. 360 Cox proportional hazards model vs. 368 generalized Wilcoxon test vs. 361 HSV-2 episodes (example) 351–359 likelihood ratio test vs. 361 Wilcoxon rank-sum test vs. 349–350 log transformations 197, 437–439 LOGISTIC procedure 322 See also MODEL statement, LOGISTIC procedure clustered binomial data 336–340

458

Index

LOGISTIC procedure (continued) DESCENDING option 325, 332 gastroparesis symptom relief (example) 331–335 OUTPUT statement 345–346, 347 PARAM= option 332 relapse rate adjusted for AML remission (example) 322–329 logistic regression 317–348 CATMOD procedure for 344 clustered binomial data 336–340 gastroparesis symptom relief (example) 331–335 intercourse success rates after surgery (example) 336–340 logit model, multiple covariates 321–322 logit model, one covariate 317–319 model estimation 320 multinomial responses 330–335 odds ratio 319 relapse rate adjusted for AML remission (example) 322–329 weighted-least squares 344 with nominal covariates 343 logit model multiple covariates 321–322 one covariate 317–319 longitudinal data 111 LS-mean 405–407 LSMEANS statement, GLM procedure ADJUST= option 425, 427 ANCOVA (analysis of covariance) 209 PDIFF option 98, 108 two-way ANOVA 98, 108

M Mann-Whitney U-test 244 Mantel-Haenszel chi-square test 275, 282 MAR (missing-at-random) values 152 marginal homogeneity (Stuart-Maxwell test) 298–300, 303 masked randomization 11 Mauchley’s Criterion 122–123 maximum likelihood method 320, 366 MCAR (missing values completely at random) 152 McNemar’s test 293–304 binomial test vs. 301–302 continuity correction 301 FREQ procedure 297–298 sign test and 302 trinomial response 298–300

mean square error See MSE mean square for group effect (MSG) 79 repeated measures analysis 113 means, multiple comparisons of 421–431 MEANS procedure one-sample t-test 59, 62 TRIAL data set 45, 47 two-sample t-test 70 MEANS statement (GLM) DUNNETT option 88 one-way ANOVA 82 two-way ANOVA 97 memory function (example) 103–107 methodology bias 12 mid-ranks 283 missing-at-random (MAR) values 152 missing data, repeated measures analysis with 136–147, 151–153 GENMOD procedure 147–148 GLM procedure 140–142 MIXED procedure 136, 143–147, 152, 153–154 sphericity test 122–123, 140, 150 missing values completely at random (MCAR) 152 MIXED procedure 152, 153–154 ANCOVA (analysis of covariance), example 224 CLASS statement 143 Handling missing data 136, 152 MODEL statement 143 repeated measures analysis (examples) 137–140, 143–147 REPEATED statement 143, 153 RCORR option 153 SUBJECT option 143 TYPE= option 143, 153 unbalanced mixed models 110 model parameters 14 MODEL statement, GENMOD procedure 147 MODEL statement, GLM procedure 181–182 ANCOVA (analysis of covariance), example 209–213, 215–220, 224 CLM option 182, 196 crossover design 163, 165–170, 171–172 NOUNI option 129 one-way ANOVA 82 P option 182, 196 repeated measures analysis, multivariate approach 122, 129 repeated measures analysis, univariate approach 119

Index SOLUTION option 209, 216 two-way ANOVA 97 MODEL statement, LOGISTIC procedure AGGREGATE option 341 LACKFIT option 341 RISKLIMITS option 342 SCALE option 338, 340 MODEL statement, MIXED procedure 143 MODEL statement, PHREG procedure 368, 378 MODEL statement, REG procedure 185, 187 COLLINOINT option 197 DW option 197 SS1 and SS2 options 195 VIF option 197–198 modified ridit scores 282–283, 383–384, 387 MS (mean square) ANCOVA (analysis of covariance) 202, 206 crossover design 159 two-way ANOVA 92 MSE (mean square error) ANCOVA (analysis of covariance), 206, 216 one-way ANOVA 79, 81, 85 repeated measures analysis 114, 151 repeated measures analysis, univariate approach 119 two-way ANOVA 92–93, 107, 108, 109 MSG (mean square for group effect) one-way ANOVA 79 repeated measures analysis 113 multi-center hypertension study (example) 214–220 multinomial responses chi-square test 276–278 logistic regression analysis 330–335 multiple comparisons 31–32, 421–436 one-way ANOVA (example) 87–89 pairwise t-tests for 97 multiple hypotheses testing 30–37 interim analyses 33–37 multiple linear regression 184–192, 193, 195, 197 coefficient of determination 194 REG procedure 185, 187 sum of squares for error 195 symptomatic recovery in pediatric dehydration (example) 185–192 Types I, II, III sum of squares 194 multiple response variables 32–33 multivariate approach, repeated measures ANOVA 122–136 Alzheimer’s progression (example) 140–143

459

missing data 136–147, 151–153 missing data, GENMOD procedure 147–148 missing data, GLM procedure 140–143 missing data, MIXED procedure 144–147 treadmill walking distance (example) 127–136 MULTTEST procedure BONFERRONI option 430 HOC option 430 HOLM option 430 p-value adjustment methods 430 permutation resampling 433–436 SIDAK option 430

N NOFREQ option, TABLES statement (FREQ) 309 noise reduction by blocking 401–405 nominal variables 270 non-parametric analog Friedman’s test 256 Kruskal-Wallis test 247–256 Wilcoxon rank-sum test 237–245 Wilcoxon signed-rank test 227–235 non-zero correlation, chi-square test for 281 non-zero slope 221 NOPERCENT option, TABLES statement (FREQ) 309 normal approximation 260–263 normal distribution 6 notation 393 probabilities of (table) 390 shape of 397, 398 NORMAL option, UNIVARIATE procedure 76, 231 normalizing data 437–439 NOROW option, TABLES statement (FREQ) 309 NOUNI option, MODEL statement (GLM) 129 NPAR1WAY procedure EDF option 245 EXACT statement 253 WILCOXON option 242, 245, 252

O O’Brien-Fleming method 35 odds ratio (OR) for covariate 319, 341 confidence intervals 341 logistic regression analysis 319 relapse rate adjusted for AML remission (example) 333 ODS statement 155–156

460

Index

one-sample t-test 55–65 body-mass index (example) 56–60 MEANS procedure 59, 62 non-parametric analog (Wilcoxon signed-rank test) 227–235 paired-difference in weight loss (example) 60–63 sample size determination 28 TTEST procedure 58–60, 62–63 one-sample Z-test 28 one-tailed tests 26 binomial test 263 chi-square test 281 two-tailed tests vs. 64 one-way ANOVA 77–90 confidence intervals 89–90 Dunnett’s test 88 F-distribution 78, 79, 81, 86, 90 F-test 82, 87, 89, 400 GLM procedure (example) 82–86 HAM-A scores in GAD (example) 80–86 Kolmogorov-Smirnov test 87 Kruskal-Wallis test 87, 247–256 mean square error (MSE) 78, 79 mean square group (MSG) 78, 79 MEANS procedure 82 MEANS statement (GLM) 82 MODEL statement (GLM) 82 multiple comparison of treatment effects 87, 89 noise reduction by blocking 401–405 non-parametric analog (Kruskal-Wallis test) 247–256 on ranked data 254 p-value 90 SAS Types I-IV sum of squares 90 Shapiro-Wilk test 87 within- vs. between-group variation 399–401 ordinal variables 270 outliers 47 OUTPUT statement, LOGISTIC procedure 345–346 overdispersion, correcting for 336–340

P P option, MODEL statement (GLM) 182, 196 p-values 27 adjustment methods 428–431 one-way ANOVA 90 paired-difference in weight loss (example) 60–63 paired-difference t-test 56

weight loss example 60–63 Wilcoxon signed-rank test vs. 229 PAIRED statement, TTEST procedure 62 pairwise t-tests 87–88, 422–424 for multiple comparisons 97–98 two-way ANOVA 99 parallel-group study 10 PARAM= option, LOGISTIC procedure 332 partial correlation coefficients 195 PDIFF option, LSMEANS statement (GLM) 98, 108 Pearson-product-moment correlation coefficient 193 pediatric dehydration, symptomatic recovery in (example) 185–192 permutation resampling (example) 433–436 PHREG procedure 366, 377, 378–379 HSV-2 episodes (example) 367–369 Hyalurise in Vitreons Hemorrhage (example) 370–375 LIFETEST procedure vs. 363 MODEL statement 368, 378 STRATA statement 372–373, 379 Pillai’s Trace test 130, 140, 150 placebo effect 11 placebo vs. active comparisons of degree of response (example) 277–278 PLOTS= option, LIFETEST procedure 361–362, 379 Pocock’s method 34 point estimates 17 pooled variance 68, 69 populations characterizing 3 defined 2 limiting conditions for 2 power (Type II error) 25–26 power curve, for Z-test 26 principal component analysis 198 PRINTE option, REPEATED statement 122, 129, 140 probability distributions 4–9, 390 probability tables 390–392 PROBBNML function 258, 263 PROBCHI function 249, 252, 268 PROBF function 74 probit transformation 342 PROBMC function 425 PROBNORM function 263

Index PROFILE option, REPEATED statement (GLM) 150 proportional hazards See Cox proportional hazards model proportional odds assumption 330, 344 PRT option, MEANS procedure 59, 62, 70 psoriasis evaluation (example) 249–253

R random effects in ANOVA models 109–110 RANDOM statement, GLM procedure 110, 149 random variables 4–5 notation 393 properties 394 results 395–396 randomization 10–12 randomized block design 91–92 See also two-way ANOVA RANK procedure 244 ranked data one-way ANOVA on 254 two-sample t-test on 244 RCORR option, REPEATED statement (MIXED) 153 REG procedure 185–192, 198 CORR option 187 MODEL statement 185, 187, 196, 197 multiple linear regression analysis 185 regression analysis See linear regression analysis See multiple linear regression relapse rate adjusted for AML remission (example) 322–329 relative risk ratio 341 repeated measures analysis 111–156 arthritic discomfort (example) 115–127 CATMOD procedure 304 crossover design 155 disease progression in Alzheimer’s (example) 137–148 GEE analysis 147–148, 155 mean square error 151 mean square for group effect 113 missing data 136–147, 151–153 multivariate approach 122–136, 150, 152 sum of squares for error 149 treadmill walking distance (example) 127–136 univariate approach 112–121, 150, 151 weighted-least-squares 155

461

within- vs. between-group variation 399–401 repeated measures analysis, with missing data 136–147, 151–153 GENMOD procedure 147–148 GLM procedure 136, 140–142 MIXED procedure 136–139, 143–147 sphericity test 140 REPEATED statement, GLM procedure CONTRAST(1) option 131, 150 PROFILE option 150 repeated measures analysis, multivariate approach 122 repeated measures analysis, missing data 140 SUMMARY option 150, 151 REPEATED statement, MIXED procedure RCORR option 153 SUBJECT option 143 repeated measures, missing data (example) 143–147 resampling methods 433–436 residuals, balanced 172 response variables, multiple 32–33 ridit See modified ridit scores RISKLIMITS option, MODEL statement (LOGISTIC) 342 risk ratio 341, 378 Rose-Bengal staining in KCS (example) 228–233 Roy’s Greatest Root test 130, 150

S samples characterizing populations from 3 defined 2 resampling methods 433–436 sample size 27–29 sampling theory 3 stratified sampling 3 Satterthwaite adjustment 74–75 SC (Schwartz Criterion) 340 SCALE= option, MODEL statement (LOGISTIC) 338, 340 Schwartz Criterion (SC) 340 SCORES= option, TABLES statement (FREQ) 283 Seroxatene evaluations (example) 239–243 Shapiro-Wilk test 76, 231 one-way ANOVA 87 Sidak method 428–431

462

Index

SIDAK option, MULTTEST procedure 430, 432 sign test 263, 302 significance levels 21, 23–24 simple linear regression analysis See linear regression analysis simple random sample, defined 2 single-blind studies 11 SOLUTION option, MODEL statement (GLM) 209, 216 sphericity test 122–123, 129, 150 repeated measures analysis 140 square root transformation 197, 347 SS (sum of squares) ANCOVA (analysis of covariance) 201, 205 crossover design 159, 161–162, 172 empty cells in SS calculations 416, 419 linear regression analysis 195 repeated measures analysis, univariate approach 116–117 SAS Types I, II, III, IV for ANOVA 97, 409–420 two-way ANOVA 92–93, 96 SS1 and SS2 options, MODEL statement (REG) 195 SS3 option, MODEL statement (GLM) crossover design 163 repeated measures analysis, univariate approach (example) 119 two-way ANOVA 97 SSE (sum of squares for error) 79, 177 ANCOVA (analysis of covariance) 200, 205 crossover design 162 linear regression analysis 180 multiple regression analysis 195 repeated measures analysis 117, 149 two-way ANOVA 96, 109 SSG (sum of squares for groups) 79 standard normal distribution, probabilities (table) 390 statistical design consideration 9–12 statistical inferences 10–12 distributions used in 393–398 statistical methods, selecting 12 statistics, defined 1 step-down Bonferroni method 428, 429 step-up Bonferroni method 428, 430 STRATA statement LIFETEST procedure 355 PHREG procedure 372–373, 379 stratified sampling 3

Stuart-Maxwell test 298–300, 303 See also McNemar’s test Student t-distribution critical values (table) 391 one-sample t-test 55–56 notation 393 shape of 398 two-sample t-test 67 study plan design 9–12 SUBJECT option, REPEATED statement (MIXED) 143 sum of squares See SS sum of squares for error See SSE sum of squares for groups (SSG) 79 SUMMARY option, REPEATED statement (GLM) 150 survival analysis 36, 350, 366 Cox proportional hazards model 365 log-rank test 349 symptom frequency, before and after treatment (example) 300 symptom relief, in gastroparesis (example) 331–335 symptomatic recovery in pediatric dehydration (example) 185–192

T T option, MEANS procedure 59, 62, 70 T option, MEANS statement (GLM) 88, 97 t-tests multiple response variables 32 paired difference 56, 62 pairwise 87–88, 97, 422–424 sample size determination 28–29 Satterthwaite adjustment 74–75 Wilcoxon signed-rank test vs. 229, 234, 235 Z-test vs. 64 t-tests, one-sample 55–65 body-mass index (example) 56–60 MEANS procedure 59 non-parametric analog (Wilcoxon signed-rank test) 227–235 paired-difference in weight loss (example) 60–63 sample size determination 28 Student t-distribution 55–56 TTEST procedure 58–60, 62 t-tests, two-sample 67–76

Index FEV1 changes (example) 68–73 MEANS procedure 70 non-parametric analog (Wilcoxon rank-sum test) 237–245 on ranked data 244 one-way ANOVA 87–88 sample size determination 28–29 Student t-distritbution 67 TTEST procedure 70, 244 Wilcoxon rank-sum test vs. 244–245 within-group tests vs. 76 TABLES statement, FREQ procedure AGREE option 297, 304 CHISQ option 268, 275, 288 CMH option 272, 274, 277, 316 EXACT option 291 FISHER option 288, 291 JT option 256 NOFREQ option 309 NOPERCENT option 309 NOROW option 309 SCORES option 278, 283 TREND option 282 test criterion 20–21 test power 25–26 TEST statement GLM procedure 110 LIFETEST procedure 362–363 test statistic 20–21 three-way ANOVA 109 time series analysis, autocorrelation in 197 treadmill walking distance (example) 127–136 treatment, symptom frequency before and after (example) 300 treatment effects See multiple comparisons TREND option, TABLES statement (FREQ) 282 TRIAL data set 39–54 statistical summary 45–53 triglyceride changes for glycemic control (example) 203–213 trinomial response 298–299 triple-blind studies 12 TTEST procedure 58–63 CI= option 70 one-sample t-test 58–60, 62, 70 PAIRED statement 62 two-sample t-test 70 two-sample t-test on ranked data 244 using with SAS 6.12 and earlier 59, 62

463

VAR statement 62 Tukey-Kramer method 424–426 two-sample comparison of proportions 29 two-sample t-test See t-test, two-sample two-sample Z-tests 28–29 two-tailed Z- tests 27 two-tailed tests 26–27, 235, 262 chi-square test 281 one-tailed tests vs. 64 two-tailed Z-test 27 two-way ANOVA 91–110 F-test 93, 96–97 F-values 92, 97 fixed effects 109–110 GLM procedure 97, 98, 109 handling empty cells 416 hemoglobin changes in anemia (example) 94–102 interaction effects (graphs) 102 mean square 92 mean square error 92–93, 107, 108, 109, 403 MEANS procedure 97 memory function (example) 103–107 noise reduction by blocking 403 non-parametric analog (Friedman’s test) 256 random effects 109–110 SAS Types I, II, III, IV sum of squares 97, 104, 108 sum of squares 93, 96 sum of squares for error 96, 109 within- vs. between-group variation 399–401 two-way crossover design 155, 157–173 See also repeated measures analysis antibiotic blood levels following aerosol inhalation (example) 165–170 diaphoresis following cardiac medication (example) 160–164 Type I error 23 Type I sum of squares 409–420 ANCOVA (analysis of covariance), 216, 224 estimable functions 412–413 multiple regression analysis 195 one-way ANOVA 90 two-way ANOVA 97, 108 Type II error 25 Type II sum of squares 409–420 estimable functions 413–414 one-way ANOVA 90 two-way ANOVA 97, 108

464

Index

Type III sum of squares 409–420 ANCOVA (analysis of covariance) 216, 224, 225 estimable functions 414–415, 418 handling empty cells 416, 419 more than two treatment groups 417–419 multiple regression analysis 195 one-way ANOVA 90 repeated measures analysis, 121 two-way ANOVA 97, 104, 108 Type IV sum of squares 409–420 estimable functions 415 handling empty cells 416, 419 more than two treatment groups 419 one-way ANOVA 90 two-way ANOVA 97 TYPE= option, MIXED procedure 143, 153 TYPE3 option, MODEL statement (GENMOD) 147, 153 REPEATED statement 143

U unbalanced layout (example) 103–107 unbiased comparisons 10–12 uniform distribution 7–8 univariate approach to repeated measures analysis 112–121 arthritic discomfort (example) 115–121 UNIVARIATE procedure NORMAL option 76, 231 sign test 263 test for normality 64 TRIAL data set 45, 48–49 Wilcoxon signed-rank test 231

V VAR statement, TTEST procedure 62 variables multiple response variables 32–33 nominal variables 270 ordinal variables 270–271 random variables 4, 5 variance homogeneity 78 variance inflation factor (VIF) 197–198 VIF option, MODEL statement (REG) 197 vitreous hemorrhage, Hyalurise in (example) 370–375

W WALD chi-square test 373 WALD option, GENMOD procedure 153 walking distance on treadmill (example) 127–136 weight loss, paired-difference in (example) 60–63 WEIGHT statement, FREQ procedure 268, 288 weighted-least-squares (WLS) 155, 344 WILCOXON option, NPAR1WAY procedure 242, 252 Wilcoxon rank-sum test 237–245 log-rank test vs. 349–350 Wilcoxon signed-rank test 64, 227–235 paired-difference t-test 229, 231 Rose-Bengal staining in KCS (example) 228–233 Seroxatene evaluations (example) 239–243 UNIVARIATE procedure 231 Z-test vs. 234 Wilcoxon test, generalized 361 Wilks’ Lambda test 130, 140, 150 Williams’ method 338 within-group tests instead of two-sample t-tests 76 within- vs. between-group variation 399–401 within-patient control study 10 WLS (weighted-least-squares) 155, 344

Z Z-test 24 comparing binomial proportions 279, 280 multiple response variables 32 one-tailed hypothesis test 26, 27 one-sample 28 power curve for 26 sample size determination 27–29 t-test vs. 64 two-sample 28–29 two-tailed hypothesis test 27 Wilcoxon signed-rank test vs. 234 zero regression slope 193

Numbers 2-period crossover design 171–172 3-period crossover design 171 4-period crossover design 165–170

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Annotate: Simply the Basics by Art Carpenter .......................................Order No. A57320

Applied Multivariate Statistics with SAS® Software, Second Edition by Ravindra Khattree and Dayanand N. Naik..............................Order No. A56903

Applied Statistics and the SAS Programming Language, Fourth Edition ®

by Ronald P. Cody and Jeffrey K. Smith.................................Order No. A55984

An Array of Challenges — Test Your SAS ® Skills by Robert Virgile.......................................Order No. A55625

Beyond the Obvious with SAS Screen Control Language ®

by Don Stanley .........................................Order No. A55073

Carpenter’s Complete Guide to the SAS® Macro Language

A Handbook of Statistical Analyses Using SAS®, Second Edition by B.S. Everitt and G. Der .................................................Order No. A58679

Health Care Data and the SAS® System by Marge Scerbo, Craig Dickstein, and Alan Wilson .......................................Order No. A57638

The How-To Book for SAS/GRAPH ® Software by Thomas Miron .....................................Order No. A55203

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by Art Carpenter .......................................Order No. A56100

Learning SAS ® in the Computer Lab, Second Edition The Cartoon Guide to Statistics by Larry Gonick and Woollcott Smith.................................Order No. A55153

Categorical Data Analysis Using the SAS ® System, Second Edition by Maura E. Stokes, Charles S. Davis, and Gary G. Koch .....................................Order No. A57998

Client/Server Survival Guide, Third Edition by Robert Orfali, Dan Harkey, and Jeri Edwards......................................Order No. A58099

Cody’s Data Cleaning Techniques Using SAS® Software by Ron Cody........................................Order No. A57198 Common Statistical Methods for Clinical Research with SAS ® Examples, Second Edition by Glenn A. Walker...................................Order No. A58086

Concepts and Case Studies in Data Management by William S. Calvert and J. Meimei Ma......................................Order No. A55220

Debugging SAS ® Programs: A Handbook of Tools and Techniques by Michele M. Burlew ...............................Order No. A57743

Efficiency: Improving the Performance of Your SAS ® Applications by Robert Virgile.......................................Order No. A55960

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Multivariate Data Reduction and Discrimination with SAS ® Software by Ravindra Khattree and Dayanand N. Naik..............................Order No. A56902

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The Next Step: Integrating the Software Life Cycle with SAS ® Programming

SAS ® Software Roadmaps: Your Guide to Discovering the SAS ® System

by Paul Gill ...............................................Order No. A55697

by Laurie Burch and SherriJoyce King ..............................Order No. A56195

Output Delivery System: The Basics by Lauren E. Haworth ..............................Order No. A58087

SAS ® Software Solutions: Basic Data Processing by Thomas Miron......................................Order No. A56196

Painless Windows: A Handbook for SAS ® Users by Jodie Gilmore ......................................Order No. A55769 (for Windows NT and Windows 95)

Painless Windows: A Handbook for SAS ® Users, Second Edition by Jodie Gilmore ......................................Order No. A56647 (updated to include Version 7 features)

PROC TABULATE by Example by Lauren E. Haworth ..............................Order No. A56514

Professional SAS ® Programmers Pocket Reference, Second Edition by Rick Aster ............................................Order No. A56646

Professional SAS ® Programmers Pocket Reference, Third Edition by Rick Aster ............................................Order No. A58128

Programming Techniques for Object-Based Statistical Analysis with SAS® Software

SAS ® System for Elementary Statistical Analysis, Second Edition by Sandra D. Schlotzhauer and Ramon C. Littell.................................Order No. A55172

SAS ® System for Forecasting Time Series, 1986 Edition by John C. Brocklebank and David A. Dickey ...................................Order No. A5612

SAS ® System for Linear Models, Third Edition by Ramon C. Littell, Rudolf J. Freund, and Philip C. Spector ...............................Order No. A56140

SAS ® System for Mixed Models by Ramon C. Littell, George A. Milliken, Walter W. Stroup, and Russell D. Wolfinger .........................Order No. A55235

SAS ® System for Regression, Third Edition by Rudolf J. Freund and Ramon C. Littell.................................Order No. A57313

by Tanya Kolosova and Samuel Berestizhevsky ....................Order No. A55869

SAS ® System for Statistical Graphics, First Edition

Quick Results with SAS/GRAPH ® Software

The SAS ® Workbook and Solutions Set (books in this set also sold separately)

by Arthur L. Carpenter and Charles E. Shipp ...............................Order No. A55127

Quick Start to Data Analysis with SAS ® by Frank C. Dilorio and Kenneth A. Hardy..............................Order No. A55550

Regression and ANOVA: An Integrated Approach Using SAS ® Software

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by Ron Cody .............................................Order No. A55594

Selecting Statistical Techniques for Social Science Data: A Guide for SAS® Users by Frank M. Andrews, Laura Klem, Patrick M. O’Malley, Willard L. Rodgers, Kathleen B. Welch, and Terrence N. Davidson .......................Order No. A55854

by Keith E. Muller and Bethel A. Fetterman ..........................Order No. A57559

Solutions for Your GUI Applications Development Using SAS/AF ® FRAME Technology

Reporting from the Field: SAS ® Software Experts Present Real-World Report-Writing

Statistical Quality Control Using the SAS ® System

Applications .............................................Order No. A55135

by Dennis W. King....................................Order No. A55232

SAS ®Applications Programming: A Gentle Introduction

A Step-by-Step Approach to Using the SAS ® System for Factor Analysis and Structural Equation Modeling

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by Don Stanley .........................................Order No. A55811

by Larry Hatcher.......................................Order No. A55129

SAS ® for Linear Models, Fourth Edition by Ramon C. Littell, Walter W. Stroup, and Rudolf J. Freund ...............................Order No. A56655

A Step-by-Step Approach to Using the SAS ® System for Univariate and Multivariate Statistics

SAS ® Macro Programming Made Easy

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by Michele M. Burlew ...............................Order No. A56516

SAS ® Programming by Example by Ron Cody and Ray Pass ............................................Order No. A55126

SAS ® Programming for Researchers and Social Scientists, Second Edition by Paul E. Spector....................................Order No. A58784

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Survival Analysis Using the SAS ® System: A Practical Guide by Paul D. Allison .....................................Order No. A55233

Table-Driven Strategies for Rapid SAS ® Applications Development by Tanya Kolosova and Samuel Berestizhevsky ....................Order No. A55198

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Using SAS ® in Financial Research by Ekkehart Boehmer, John Paul Broussard, and Juha-Pekka Kallunki .........................Order No. A57601

Using the SAS ® Windowing Environment: A Quick Tutorial by Larry Hatcher.......................................Order No. A57201

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Your Guide to Survey Research Using the SAS® System by Archer Gravely ....................................Order No. A55688

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Business Analysis Using Regression: A Casebook by Dean P. Foster, Robert A. Stine, and Richard P. Waterman........................Order No. A56818

JMP® Start Statistics, Version 3 by John Sall, Ann Lehman, and Leigh Creighton ................................Order No. A55626

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