Analyzing Complex Survey Data (Quantitative Applications in the Social Sciences)

Series/Number 07–071

ANALYZING COMPLEX SURVEY DATA Second Edition

Eun Sul Lee Division of Biostatistics, School of Public Health, University of Texas Health Science Center—Houston Ronald N. Forthofer

SAGE PUBLICATIONS International Educational and Professional Publisher Thousand Oaks London New Delhi

Copyright Ó 2006 by Sage Publications, Inc. All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. For information: Sage Publications, Inc. 2455 Teller Road Thousand Oaks, California 91320 E-mail: [email protected] Sage Publications Ltd. 1 Oliver’s Yard 55 City Road London EC1Y 1SP United Kingdom Sage Publications India Pvt. Ltd. B-42, Panchsheel Enclave Post Box 4109 New Delhi 110 017 India Printed in the United States of America Library of Congress Cataloging-in-Publication Data Lee, Eun Sul. Analyzing complex survey data / Eun Sul Lee, Ronald N. Forthofer.—2nd ed. p. cm.—(Quantitative applications in the social sciences ; vol. 71) Includes bibliographical references and index. ISBN 0-7619-3038-8 (pbk. : alk. paper) 1. Mathematical statistics. 2. Social surveys—Statistical methods. I. Forthofer, Ron N., 1944-II. Title. III. Series: Sage university papers series. Quantitative applications in the social sciences ; no. 07–71. QA276.L3394 2006 2005009612 001.4
22—dc22 This book is printed on acid-free paper. 05 06 07 08 09 10 9 Acquisitions Editor: Editorial Assistant: Production Editor: Copy Editor: Typesetter:

8 7 6 5

Lisa Cuevas Shaw Karen Gia Wong Melanie Birdsall A. J. Sobczak C&M Digitals (P) Ltd

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CONTENTS Series Editor’s Introduction Acknowledgments

v vii

1.

Introduction

1

2.

Sample Design and Survey Data

3

Types of Sampling The Nature of Survey Data A Different View of Survey Data*

4 7 9

3.

4.

5.

6.

Complexity of Analyzing Survey Data

11

Adjusting for Differential Representation: The Weight Developing the Weight by Poststratification Adjusting the Weight in a Follow-Up Survey Assessing the Loss or Gain in Precision: The Design Effect The Use of Sample Weights for Survey Data Analysis*

11 14 17 18 20

Strategies for Variance Estimation

22

Replicated Sampling: A General Approach Balanced Repeated Replication Jackknife Repeated Replication The Bootstrap Method The Taylor Series Method (Linearization)

23 26 29 35 36

Preparing for Survey Data Analysis

39

Data Requirements for Survey Analysis Importance of Preliminary Analysis Choices of Method for Variance Estimation Available Computing Resources Creating Replicate Weights Searching for Appropriate Models for Survey Data Analysis*

39 41 43 44 47 49

Conducting Survey Data Analysis

49

A Strategy for Conducting Preliminary Analysis Conducting Descriptive Analysis Conducting Linear Regression Analysis

50 52 57

7.

Conducting Contingency Table Analysis Conducting Logistic Regression Analysis Other Logistic Regression Models Design-Based and Model-Based Analyses*

61 65 69 74

Concluding Remarks

78

Notes

80

References

83

Index

88

About the Authors

91

SERIES EDITOR’S INTRODUCTION When George Gallup correctly predicted Franklin D. Roosevelt as the 1936 presidential election winner, public opinion surveys entered the age of scientific sampling, and the method used then was quota sampling, a type of nonprobability sampling representative of the target population. The same method, however, incorrectly predicted Thomas Dewey as the 1948 winner though Harry S. Truman actually won. The method failed because quota sampling is nonprobabilistic and because Gallup’s quota frames were based on the 1940 census, overlooking the urban migration during World War II. Today’s survey sampling has advanced much since the early days, now relying on sophisticated probabilistic sampling designs. A key feature is stratification: The target population is divided into subpopulations of strata, and the sample sizes in the strata are controlled by the sampler and are often proportional to the strata population sizes. Another feature is cluster and multistage sampling: Groups are sampled as clusters in a hierarchy of clusters selected at various stages until the final stage, when individual elements are sampled within the final-stage clusters. The General Social Survey, for example, uses a stratified multistage cluster sampling design. (Kalton [1983] gives a nice introduction to survey sampling.) When the survey design is of this complex nature, statistical analysis of the data is no longer a simple matter of running a regression (or any other modeling) analysis. Surveys today all come with sampling weights to assist with correct statistical inference. Most texts on statistical analysis, by assuming simple random sampling, do not include treatment of sampling weights, an omission that may have important implications for making inferences. During the last two to three decades, statistical methods for data analysis have also made huge strides. These must have been the reason my predecessor, Michael Lewis-Beck, who saw through the early stages of the editorial work in this volume, chose to have a second edition of Analyzing Complex Survey Data. Lee and Forthofer’s second edition of the book brings us up to date in uniting survey sampling designs and survey data analysis. The authors begin by reviewing common types of survey sample designs and demystifying sampling weights by explaining what they are, and how they are developed and adjusted. They then carefully discuss the major issues of variance estimation and of preliminary as well as multivariate analysis of complex cross-sectional survey data when sampling weights are taken into account. They focus on the design-based approach that directly engages sample designs in the analysis (although they also discuss the model-based perspective, which v

vi

can augment a design-based approach in some analyses), and they illustrate the approach with popular software examples. Students of survey analysis will find the text of great use in their efforts in making sample-based statistical inferences. —Tim Futing Liao Series Editor

vii

ACKNOWLEDGMENTS We sadly acknowledge that Dr. Ronald J. Lorimor, who had collaborated on the writing of the first edition of this manuscript, died in 1999. His insights into survey data analysis remain in this second edition. We are grateful to Tom W. Smith for answering questions about the sample design of the General Social Survey and to Barry L. Graubard, Lu Ann Aday, and Michael S. Lewis-Beck for their thoughtful suggestions for the first edition. Thanks also are due to anonymous reviewers for their helpful comments for both editions and to Tim F. Liao for his thoughtful advice for the contents of the second edition. Special thanks go to many students in our classes at the University of Texas School of Public Health who participated in discussions of many topics contained in this book.

ANALYZING COMPLEX SURVEY DATA, SECOND EDITION Eun Sul Lee Division of Biostatistics, School of Public Health, University of Texas Health Science Center—Houston Ronald N. Forthofer

1. INTRODUCTION Survey analysis often is conducted as if all sample observations were independently selected with equal probabilities. This analysis is correct if simple random sampling (SRS) is used in data collection; however, in practice the sample selection is more complex than SRS. Some sample observations may be selected with higher probabilities than others, and some are included in the sample by virtue of their membership in a certain group (e.g., household) rather than being selected independently. Can we simply ignore these departures from SRS in the analysis of survey data? Is it appropriate to use the standard techniques in statistics books for survey data analysis? Or are there special methods and computer programs available for a more appropriate analysis of complex survey data? These questions are addressed in the following chapters. The typical social survey today reflects a combination of statistical theory and knowledge about social phenomena, and its evolution has been shaped by experience gained from the conduct of many different surveys during the last 70 years. Social surveys were conducted to meet the need for information to address social, political, and public health issues. Survey agencies were established within and outside the government in response to this need for information. In the early attempts to provide the required information, however, the survey groups were mostly concerned with the practical issues in the fieldwork—such as sampling frame construction, staff training/supervision, and cost reduction—and theoretical sampling issues received only secondary emphasis (Stephan, 1948). As these practical matters were resolved, modern sampling practice had developed far beyond SRS. Complex sample designs had come to the fore, and with them, a number of analytic problems. Because the early surveys generally needed only descriptive statistics, there was little interest in analytic problems. More recently, demands for 1

2

analytic studies by social and policy scientists have increased, and a variety of current issues are being examined, using available social survey data, by researchers who were not involved with the data collection process. This tradition is known as secondary analysis (Kendall & Lazarsfeld, 1950). Often, the researcher fails to pay due attention to the development of complex sample designs and assumes that these designs have little bearing on the analytic procedures to be used. The increased use of statistical techniques in secondary analysis and the recent use of log-linear models, logistic regression, and other multivariate techniques (Aldrich & Nelson, 1984; Goodman, 1972; Swafford, 1980) have done little to bring design and analysis into closer alignment. These techniques are predicated on the use of simple random sampling with replacement (SRSWR); however, this assumption is rarely met in social surveys that employ stratification and clustering of observational units along with unequal probabilities of selection. As a result, the analysis of social surveys using the SRSWR assumption can lead to biased and misleading results. Kiecolt and Nathan (1985), for example, acknowledged this problem in their Sage book on secondary analysis, but they provide little guidance on how to incorporate the sample weights and other design features into the analysis. A recent review of literature in public health and epidemiology shows that the use of design-based survey analysis methods is gradually increasing but remains at a low level (Levy & Stolte, 2000). Any survey that puts restrictions on the sampling beyond those of SRSWR is complex in design and requires special analytic considerations. This book reviews the analytic issues raised by the complex sample survey, provides an introduction to analytic strategies, and presents illustrations using some of the available software. Our discussion is centered on the use of the sample weights to correct for differential representations and the effect of sample designs on estimation of sampling variance with some discussion of weight development and adjustment procedures. Many other important issues of dealing with nonsampling errors and handling missing data are not fully addressed in this book. The basic approach presented in this book is the traditional way of analyzing complex survey data. This approach is now known as design-based (or randomization-based) analysis. A different approach to analyzing complex survey data is the so-called model-based analysis. As in other areas of statistics, the model-based statistical inference has gained more attention in survey data analysis in recent years. The modeling approaches are introduced in various steps of survey data analysis in defining the parameters, defining estimators, and estimating variances; however, there are no generally accepted rules for model selection or validating a specified model. Nevertheless, some understanding of the model-based approach is essential

3

for survey data analysts to augment the design-based approach. In some cases, both approaches produce the same results; but different results occur in other cases. The model-based approach may not be useful in descriptive data analysis but can be useful in inferential analysis. We will introduce the model-based perspective where appropriate and provide references for further treatment of the topics. Proper conduct of model-based analysis would require knowledge of general statistical models and perhaps some consultation from survey statisticians. Sections of the book relevant to this alternative approach and related topics are marked with asterisks (*). Since the publication of the first edition of this book, the software situation for the analysis of complex survey data has improved considerably. Userfriendly programs are now readily available, and many commonly used statistical methods are now incorporated in the packages, including logistic regression and survival analysis. These programs will be introduced with illustrations in this edition. These programs are perhaps more open to misuse than other standard software. The topics and issues discussed in this book will provide some guidelines for avoiding pitfalls in survey data analysis. In our presentation, we assume some familiarity with such sampling designs as simple random sampling, systematic sampling, stratified random sampling, and simple two-stage cluster sampling. A good presentation of these designs may be found in Kalton (1983) and Lohr (1999). We also assume general understanding of standard statistical methods and one of the standard statistical program packages, such as SAS or Stata.

2. SAMPLE DESIGN AND SURVEY DATA Our consideration of survey data focuses on sample designs that satisfy two basic requirements. First, we are concerned only with probability sampling in which each element of a population has a known (nonzero) probability of being included in the sample. This is the basis for applying statistical theory in the derivation of the properties of the survey estimators for a given design. Second, if a sample is to be drawn from a population, it is necessary to be able to construct a sampling frame that lists suitable sampling units that encompass all elements of the population. If it is not feasible or is impractical to list all population elements, some clusters of elements can be used as sampling units. For example, it is impractical to construct a list of all households in the United States, but we can select the sample in several stages. In the first stage, counties are randomly sampled; in the second stage, census tracts within the selected counties are sampled; in the third stage, street blocks are sampled within the selected tracts. Then, in the final stage of selection, a list of households is needed only for the selected

4

blocks. This multistage design satisfies the requirement that all population elements have a known nonzero probability of being selected. Types of Sampling The simplest sample design is simple random sampling, which requires that each element have an equal probability of being included in the sample and that the list of all population elements be available. Selection of a sample element can be carried out with or without replacement. Simple random sampling with replacement (SRSWR) is of special interest because it simplifies statistical inference by eliminating any relation (covariance) between the selected elements through the replacement process. In this scheme, however, an element can appear more than once in the sample. In practice, simple random sampling is carried out without replacement (SRSWOR), because there is no need to collect the information more than once from an element. Additionally, SRSWOR gives a smaller sampling variance than SRSWR. However, these two sampling methods are practically the same in a large survey in which a small fraction of population elements is sampled. We will use the term SRS for SRSWOR throughout this book unless otherwise specified. The SRS design is modified further to accommodate other theoretical and practical considerations. The common practical designs include systematic sampling, stratified random sampling, multistage cluster sampling, PPS sampling (probability proportional to size), and other controlled selection procedures. These more practical designs deviate from SRS in two important ways. First, the inclusion probabilities for the elements (also the joint inclusion probabilities for sets for the elements) may be unequal. Second, the sampling unit can be different from the population element of interest. These departures complicate the usual methods of estimation and variance calculation and, if proper methods of analysis are not used, can lead to a bias in estimation and statistical tests. We will consider these departures in detail, using several specific sampling designs, and examine their implications for survey analysis. Systematic sampling is commonly used as an alternative to SRS because of its simplicity. It selects every k-th element after a random start (between 1 and k). Its procedural tasks are simple, and the process can easily be checked, whereas it is difficult to verify SRS by examining the results. It is often used in the final stage of multistage sampling when the fieldworker is instructed to select a predetermined proportion of units from the listing of dwellings in a street block. The systematic sampling procedure assigns each element in a population the same probability of being selected. This ensures that the sample mean will be an unbiased estimate of the population mean

5

when the number of elements in the population (N) is equal to k times the number of elements in the sample (n). If N is not exactly nk, then the equal probability is not guaranteed, although this problem can be ignored when N is large. In that case, we can use the circular systematic sampling scheme. In this scheme, the random starting point is selected between 1 and N (any element can be the starting point), and every k-th element is selected assuming that the frame is circular (the end of the list is connected to the beginning of the list). Systematic sampling can give an unrealistic estimate, however, when the elements in the frame are listed in a cyclical manner with respect to survey variables and the selection interval coincides with the listing cycle. For example, if one selects every 40th patient coming to a clinic and the average daily patient load is about 40, then the resulting systematic sample would contain only those who came to the clinic at a particular time of the day. Such a sample may not be representative of the clinic patients. Moreover, even when the listing is randomly ordered, unlike SRS, different sets of elements may have unequal inclusion probabilities. For example, the probability of including both the i-th and the (i + k)-th element is 1/k in a systematic sample, whereas the probability of including both the i-th and the (i + k + 1)-th is zero. This complicates the variance calculation. Another way of viewing systematic sampling is that it is equivalent to selecting one cluster from k systematically formed clusters of n elements each. The sampling variance (between clusters) cannot be estimated from the one selected cluster. Thus, variance estimation from a systematic sample requires special strategies. A modification to overcome these problems with systematic sampling is the so-called repeated systematic sampling (Levy & Lemeshow, 1999, pp. 101–110). Instead of taking a systematic sample in one pass through the list, several smaller systematic samples are selected, going down the list several times with a new starting point in each pass. This procedure not only guards against possible periodicity in the frame but also allows variance estimation directly from the data. The variance of an estimate from all subsamples can be estimated from the variability of the separate estimates from each subsample. This idea of replicated sampling offers a strategy for estimating variance for complex surveys, which will be discussed further in Chapter 4. Stratified random sampling classifies the population elements into strata and samples separately from each stratum. It is used for several reasons: (a) The sampling variance can be reduced if strata are internally homogeneous, (b) separate estimates can be obtained for strata, (c) administration of fieldwork can be organized using strata, and (d) different sampling needs can be accommodated in separate strata. Allocation of the sample across the strata is proportionate when the sampling fraction is uniform across the

6

strata or disproportionate when, for instance, a higher sampling fraction is applied to a smaller stratum to select a sufficient number of subjects for comparative studies. In general, the estimation process for a stratified random sample is more complicated than in SRS. It is generally described as a two-step process. The first step is the calculation of the statistics—for example, the mean and its variance—separately within each stratum. These estimates are then combined based on weights reflecting the proportion of the population in each stratum. As will be discussed later, it also can be described as a one-step process using weighted statistics. The estimation simplifies in the case of proportionate stratified sampling, but the strata must be taken into account in the variance estimation. The formulation of the strata requires that information on the stratification variable(s) be available in the sampling frame. When such information is not available, stratification cannot be incorporated in the design. But stratification can be done after data are collected to improve the precision of the estimates. The so-called poststratification is used to make the sample more representative of the population by adjusting the demographic compositions of the sample to the known population compositions. Typically, such demographic variables as age, sex, race, and education are used in poststratification in order to take advantage of the population census data. This adjustment requires the use of weights and different strategies for variance estimation because the stratum sample size is a random variable in the poststratified design (determined after the data are collected). Cluster sampling is often a practical approach to surveys because it samples by groups (clusters) of elements rather than by individual elements. It simplifies the task of constructing sampling frames, and it reduces the survey costs. Often, a hierarchy of geographical clusters is used, as described earlier. In multistage cluster sampling, the sampling units are groups of elements except for the last stage of sampling. When the numbers of elements in the clusters are equal, the estimation process is equivalent to SRS. However, simple random sampling of unequal-sized clusters leads to the elements in the smaller clusters being more likely to be in the sample than those in the larger clusters. Additionally, the clusters are often stratified to accomplish certain survey objectives and field procedures, for instance, the oversampling of predominantly minority population clusters. The use of disproportionate stratification and unequal-sized clusters complicates the estimation process. One method to draw a self-weighting sample of elements in one-stage cluster sampling of unequal-sized clusters is to sample clusters with probability proportional to the size of clusters (PPS sampling). However, this requires that the true size of clusters be known. Because the true sizes usually are unknown at the time of the survey, the selection probability is

7

instead made proportional to the estimated size (PPES sampling). For example, the number of beds can be used as a measure of size in a survey of hospital discharges with hospitals as the clusters. One important consequence of PPES sampling is that the expected sample size will vary from one primary sampling unit (PSU) to another. In other words, the sample size is not fixed but varies from sample to sample. Therefore, the sample size, the denominator in the calculation of a sample mean, is a random variable, and, hence, the sample mean becomes a ratio of two random variables. This type of variable, a ratio variable, requires special strategies for variance estimation. The Nature of Survey Data If we are to infer from sample to population, the sample selection process is an integral part of the inference process, and the survey data must contain information on important dimensions of the selection process. Considering the departures from SRS in most social surveys, we need to view the survey data not only as records of measurements, but also as having different representation and structural arrangements. Sample weights are used to reflect the differing probabilities of selection of the sample elements. The development of sample weights requires keeping track of selection probabilities separately in each stratum and at each stage of sampling. In addition, it can involve correcting for differential response rates within classes of the sample and adjusting the sample distribution by demographic variables to known population distributions (poststratification adjustment). Moreover, different sample weights may be needed for different units of analysis. For instance, in a community survey it may be necessary to develop person weights for an analysis of individual data and household weights for an analysis of household data. We may feel secure in the exclusion of the weights when one of the following self-weighting designs is used. True PPS sampling in a one-stage cluster sampling will produce a self-weighting sample of elements, as in the SRS design. The self-weighting can also be accomplished in a two-stage design when true PPS sampling is used in the first stage and a fixed number of elements is selected within each selected PSU. The same result will follow if simple random sampling is used in the first stage and a fixed proportion of the elements is selected in the second stage (see Kalton, 1983, chaps. 5 and 6). In practice, however, the self-weighting feature is destroyed by nonresponse and possible errors in the sampling frame(s). This unintended selfselection process can introduce bias, but it is seldom possible to assess the bias from an examination of the sample data. Two methods employed in an attempt to reduce the bias are poststratification and nonresponse adjustments. Poststratification involves assigning weights to bring the sample proportion

8

in demographic subgroups into agreement with the population proportion in the subgroups. Nonresponse adjustment inflates the weights for those who participate in the survey to account for the nonrespondents with similar characteristics. Because of the nonresponse and poststratification adjustments by weighting, the use of weights is almost unavoidable even when a self-weighting design is used. The sample design affects the estimation of standard errors and, hence, must also be incorporated into the analysis. A close examination of the familiar formulas for standard errors found in statistics textbooks and incorporated into most computer program packages shows that they are based on the SRSWR design. These formulas are relatively simple because the covariance between elements is zero, as a result of the assumed independent selection of elements. It is not immediately evident how the formulas should be modified to adjust for other complex sampling designs. To better understand the need for adjustment to the variance formulas, let us examine the variance formula for several sample designs. We first consider variance for a sample mean from the SRSWOR design. The familiar variance formula for a sample mean, y
(selecting a sample of n elements from a population of N elements by SRSWR where the population mean P
2 /N.
in elementary statistics textbooks is
2/n, where
2 = (Yi − Y) is Y), This formula needs to be modified for the SRSWOR design because the selection of an element is no longer independent of the selection of another element. Because of the condition of not allowing duplicate selection, there is a negative covariance [−
2/(N − 1)] between i-th and j-th sample elements. Incorporating n(n − 1) times the covariance, the variance of the sam−n ple mean for SRSWOR is
n (N N − 1), which is smaller than that from SRSWR by the factor of (N − n)/(N − 1). Substituting the unbiased estimator of
2 of [(N − 1)s2 /N], the estimator for the variance of the sample mean from SRSWOR is 2

s2 V^(
y) = (1 − f ), n

(2:1)


(xi − x
)2 /(n − 1) and f = n/N. Both (N − n)/(N − 1) and where s2 = (1 − f ) are called the finite population correction (FPC) factor. In a large population, the covariance will be very small because the sampling fraction is small. Therefore, SRSWR and SRSWOR designs will produce practically the same variance, and these two procedures can be considered equivalent for all practical purposes. Stratified sampling is often presented as a more efficient design because it gives, if used appropriately, a smaller variance than that given by a comparable SRS. Because the covariances between strata are zero, the variance of the

9

sample estimate is derived from the within-stratum variances, which are combined based on the stratum sample sizes and the stratum weights. The value of a stratified sample variance depends on the distribution of the strata sample sizes. An optimal (or Neyman) allocation produces a sampling variance less than or equal to that based on SRS except in extremely rare situations. For other disproportionate allocations, the sampling variance may turn out to be larger than that based on SRS when the finite population correction factor (FPC) within strata cannot be ignored. Therefore, it cannot be assumed that stratification will always reduce sampling variance compared to SRS. The cluster sampling design usually leads to a larger sampling variance than that from SRS. This is because the elements within naturally formed clusters are often similar, which then yield a positive covariance between elements within the cluster. The homogeneity within clusters is measured by the intraclass correlation coefficient (ICC)—the correlation between all possible pairs of elements within clusters. If clusters were randomly formed (i.e., if each cluster were a random sample of elements), the ICC would be zero. In many natural clusters, the ICC is positive and, hence, the sampling variance will be larger than that for the SRS design. It is difficult to generalize regarding the relative size of the sampling variance in a complex design because the combined effects of stratification and clustering, as well as that of the sample weights, must be assessed. Therefore, all observations in survey data must be viewed as products of a specific sample design that contains sample weights and structural arrangements. In addition to the sample weights, strata and cluster identification (at least PSUs) should be included in sample survey data. Reasons for these requirements will become clearer later. One complication in the variance calculation for a complex survey stems from the use of weights. Because the sum of weights in the denominator of any weighted estimator is not fixed but varies from sample to sample, the estimator becomes a ratio of two random variables. In general, a ratio estimator is biased, but the bias is negligible if the variation in the weights is relatively small or the sample size is large (Cochran, 1977, chap. 6). Thus, the problem of bias in the ratio estimator is not an issue in large social surveys. Because of this bias, however, it is appropriate to use the mean square error—the sum of the variance plus the square of the bias—rather than the variance. However, because the bias often is negligible, we will use the term ‘‘variance’’ even if we are referring to the mean square error in this book. A Different View of Survey Data* So far, the nature of survey data is described from the design-based perspective—that is, sample data are observations sampled from a finite

10

population using a particular sample selection design. The sampling design specifies the probability of selection of each potential sample, and a proper estimator is chosen to reflect the design. As mentioned in the introduction, the model-based perspective offers an alternative view of sample survey data. Observations in the finite population are viewed as realizations of a random variable generated from some model (a random variable that followed some probability distribution). The assumed probability model supplies the link between units in the sample and units not in the sample. In the model-based approach, the sample data are used to predict the unobserved values, and thus inferences may be thought of as prediction problems (Royall, 1970, 1973). These two points of view may not make a difference in SRS, where we can reasonably assume that sample observations were independent and identically distributed from a normal distribution with mean µ and variance
. From the model point of view, the population total is the sum of observations in thePsample and Pthe sum of observations that are not in the sample; that is, Y = i∈S yi + i∈  S yi . Based on the assumption of common mean, the estimate of population total can be made as Y^ = n
y + ðN − nÞ
y = N y
, where y
is the best unbiased predictor of the unobserved observations under the model. It turns out to be the same as the expansion estimator in the designP based approach, namely, Y^ = (N/n) ni= 1 yi = N y
; where (N/n) is the sample weight (inverse of selection probability in SRS). Both approaches lead to the same variance estimate (Lohr, 1999, sec. 2.8). If a different model were adopted, however, the variance estimates might differ. For example, in the case of ratio1 and regression estimation under SRS, the assumed model is Yi = βxi + εi , where Yi is for a random variable and xi is an auxiliary variable for which the population total is known. Under this model, the linear estimate of the population total will be P P P y + β^ i∈ Y^ = i∈S yi + i∈  S yi = n
 S xi . The first part is from the sample, and the second part is the prediction for the unobserved units based on ^ as the sample ratio of y
/
the assumed model. If we take β x, then we have P y
P y
y
^ Y = n
y + x
i∈ x + i∈  S xi = x
ðn
 S xi Þ = x
X, where X is the population total of xi . This is simply the ratio estimate of Y. If we take β^ as the estimated regression coefficient, then we have a regression estimation. Although the ratio estimate is known to be slightly biased from the design-based viewpoint, it is unbiased from the model-based reasoning if the model is correct. But the estimate of variance by the model-based approach is slightly different from the estimate by the design-based approach. The design-based estimate of variance of the estimated population total

11


P ½yi −ð
y=
xÞxi
2 ^ = ð1 − n Þ N 2 is V^D ðYÞ . The model-based estimator is N n n−1 pffiffiffiffi 2
2 P ½{yi −ð
y=
xÞ}= xi
^ = ð1 − x Þ X V^M ðYÞ , where x is the sample total and X X x n−1 is the population total of the auxiliary variable (see Lohr, 1999, sec. 3.4). The ratio estimate model is valid when (a) the relation between yi and xi is a straight line through the origin and (b) the variance of yi about this line is proportional to xi . It is known that the ratio estimate is inferior to the expansion estimate (without the auxiliary variable) when the correlation between yi and xi is less than one-half the ratio of coefficient of variation of xi over the coefficient of variation of yi (Cochran, 1977, chap. 6). Therefore, the use of ratio estimation in survey analysis would require checking the model assumptions. In practice, when the data set includes a large number of variables, ratio estimation would be cumbersome to select different auxiliary variables for different estimates. To apply the model-based approach to a real problem, we must first be able to produce an adequate model. If the model is wrong, the model-based estimators will be biased. When using model-based inference in sampling, one needs to check the assumptions of the model by examining the data carefully. Checking the assumptions may be difficult in many circumstances. The adequacy of a model is to some extent a matter of judgment, and a model adequate for one analysis may not be adequate for another analysis or another survey.

3. COMPLEXITY OF ANALYZING SURVEY DATA Two essential aspects of survey data analysis are adjusting for the differential representation of sample observations and assessing the loss or gain in precision resulting from the complexity of the sample selection design. This chapter introduces the concept of weight and discusses the effect of sample selection design on variance estimation. To illustrate the versatility of weighting in survey analysis, we present two examples of developing and adjusting sample weights. Adjusting for Differential Representation: The Weight Two types of sample weights are commonly encountered in the analysis of survey data: (a) the expansion weight, which is the reciprocal of the selection probability, and (b) the relative weight, which is obtained by scaling down the expansion weight to reflect the sample size. This section reviews these two types of weights in detail for several sample designs.

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Consider the following SRS situation: A list of N = 4,000 elements in a population is numbered from 1 to 4,000. A table of random numbers is used to draw a fixed number of elements (for example, n = 200) from the population, not allowing duplicate selection (without replacement). The selection probability or sampling fraction is f = n/N = .05. The expansion weight is the reciprocal of the selection probability, wi = 1/f = N/n = 20 (i = 1, . . . , n), which indicates the number of elements represented by a sample observation in the population. These weights for the n elements selected sum to N: An estimator of the population total of variable Y based on the sample elements is   yi = N y
: (3:1) Y^ = wi yi = (N/n) Equation 3.1 shows the use of the expansion weight in the weighted sum of sample observations. Because the weight is the same for each element in SRS, the estimator can be simplified to N times of the sample mean (the last quantity in Equation 3.1). Similarly, the estimator of the

wi , which is the weighted population mean is defined as Y^¯ = wi yi /
sample mean. In SRS, this simplifies to (N/n) yi /N = y
, showing that the sample mean is an estimator for the population mean. However, even if the weights are not the same (in unequal probability designs), the estimators are still a weighted sum for the population total and a weighted average for the population mean. Although the expansion weight appears appropriate for the estimator of the population total, it may play havoc with the sample mean and other statistical measures. For example, using the sum of expansion weights in contingency tables in place of relative frequencies based on sample size may lead to unduly large confidence in the data. To deal with this, the expansion weight can be scaled down to produce the relative weight, (rw)i , which is defined to be the expansion weightP divided by the mean of the expansion
where w
= wi /n. These relative weights for all weights, that is, wi /w, elements in the sample add up to n: For the SRS design, (rw)i is 1 for each element. The estimator for the population total weighted by the relative weights is  
yi = N y
: (3:2) Y^ = w (rw)i yi = (N/n) Note in Equation 3.2 that the relative weighted sum is multiplied by the average expansion weight, which yields the same simplified estimator for the case of SRS as in Equation 3.1. Hence, the expansion weight is simpler to use than the relative weight in estimating the population total. The relative weight is appropriate in analytic studies, but it is inappropriate in estimating totals and computing finite population corrections.

13

Most public-use survey data from government agencies and survey organizations use the expansion weight, and it can easily be converted to the relative weight. Such conversion is not necessary, however, because the user-friendly statistical programs for survey analysis automatically perform the conversion internally when appropriate. Let us consider expansion weights in a stratified random sampling design. In this design, the population of N elements is grouped into L strata based on a certain variable with N1 , N2 , . . . , NL elements respectively, from which nh (h = l, 2, . . . , L) elements are independently selected from the h-th stratum. A stratified design retains a self-weighting quality when the sampling fraction in each stratum is the same. If a total of 200 elements are proportionately selected from two strata of N1 = 600 and N2 = 3,400 elements, then f = 200/4,000 = .05. A proportionate selection (a 5% sample from each stratum) yields n1 = 30 and n2 = 170 because f1 = 30/600 = .05 and f2 = 170/3,400 = .05. The weighting scheme is then exactly the same as in SRS design. The situation is slightly different with a disproportionate stratified sample design. For example, if the total sample of 200 were split equally between the two strata, f1 (¼100/600) and f2 (¼100/3,400) could have different values and the expansion weights would be unequal for the elements in the two strata, with w1i = 6 and w2i = 34. The expansion weights sum to 600 in the first stratum and to 3,400 in the second with their total being 4,000, the population
= (100 × 6 + 100 × 34)/200 = 20, size. The mean expansion weight, w and, hence, the relative weights sum to 30 in the first stratum and to 170 in the second. The sums of relative weights in both strata add up to the total sample size. Note that the use of either type of weight is equivalent to weighting stratum yh ) using the population distribution across the strata P means (
yh , the standard procedure]. Both the expansion and relative [i.e., (Nh /N)
weights in stratum 1 sum to 15% of their respective total sums, and the first stratum also contains 15% of the population elements. Although we have used SRS and stratified sample designs to introduce the sample weights, the same concept extends easily to more complex designs. In summary, the sample weight is the inverse of the selection probability, although it often is further modified by poststratification and nonresponse adjustments. The assignment of the sample weight to each sample element facilitates a general estimation procedure for all sample designs. As a general rule, all estimates take the form of weighted statistics in survey data analysis. The scale of these weights does not matter in estimating parameters and standard errors except when estimating totals and computing finite population corrections. Next, we present two examples of developing/adjusting the weight. The first example shows how sample weights are modified by the poststratification

14

procedure in order to make sample compositions conform to population compositions. The same procedure can be used to adjust for differential response rates in demographic subgroups. The poststratification approach works well when only a few demographic variables are involved. The second example demonstrates that differential attrition rates in a follow-up survey can be adjusted for a large number of variables using a logistic regression model. Developing the Weight by Poststratification To demonstrate the development of sample weights, we shall work with the 1984 General Social Survey (GSS). This is a complex sample survey conducted by the National Opinion Research Center (NORC) to obtain general social information from the civilian noninstitutionalized adult population (18 years of age and older) of the United States. A multistage selection design was used to produce a self-weighting sample at the household level. One adult was then randomly selected from each sampled household (Davis & Smith, 1985). There were 1,473 observations available for analysis in the data file. For the purpose of illustration, the expansion weight for these data at the household level could be calculated by dividing the number of households in the United States by 1,473. The expansion weight within the sampled household is the number of adults in the household. The product of these two weights gives the expansion weight for sample individuals. For an analysis of the GSS data, we need to focus only on the weight within the household, because each household has the same probability of selection. The relative weight for the individual can be derived by dividing the number of adults in the household by the average number of adults (2,852/1,473 = 1.94) per household. This weight reflects the probability of selection of an individual in the sample while preserving the sample size. We further modified this weight by a poststratification adjustment in an attempt to make the sample composition the same as the population composition. This would improve the precision of estimates and could possibly reduce nonresponse and sample selection bias to the extent that it is related to the demographic composition.2 As shown in Table 3.1, the adjustment factor is derived to cause the distribution of individuals in the sample to match the 1984 U.S. population by age, race, and sex. Column 1 is the 1984 population distribution by race, sex, and age, based on the Census Bureau’s estimates. Column 2 shows the weighted number of adults in the sampled households by the demographic subgroups, and the proportional distribution is in Column 3. The adjustment factor is the ratio of Column 1 to Column 3. The adjusted weight is found by multiplying the adjustment factor by the relative weight, and the distribution of adjusted weights is then

15 TABLE 3.1

Derivation of Poststratification Adjustment Factor: General Social Survey, 1984 Demographic Subgroups

Population Distribution (1)

Weighted Number of Adults (2)

Sample Distribution (3)

Adjustment Factor (1)/(3)

White, male 18–24 years 25–34 35–44 45–54 55–64 65 and over

.0719660 .1028236 .0708987 .0557924 .0544026 .0574872

211 193 277 135 144 138

.0739832 .0676718 .0795933 .0473352 .0504909 .0483871

0.9727346 1.5194460 0.8907624 1.1786660 1.0774730 1.1880687

White, female 18–24 years 25–34 35–44 45–54 55–64 65 and over

.0705058 .1007594 .0777364 .0582026 .0610057 .0823047

198 324 267 196 186 216

.0694250 .1136045 .0936185 .0682737 .0652174 .0757363

1.0155668 0.8869317 0.8303528 0.8469074 0.9354210 1.0867272

Nonwhite, male 18–24 years 25–34 35–44 45–54 55–64 65 and over

.0138044 .0172057 .0109779 .0077643 .0064683 .0062688

34 30 30 37 12 18

.0119215 .0105189 .0105189 .0129734 .0042076 .0063113

1.1579480 1.6356880 1.0436290 0.5984774 1.5372900 0.9932661

Nonwhite, female 18–24 years 25–34 35–44 45–54 55–64 65 and over

.0145081 .0196276 .0130655 .0094590 .0079636 .0090016

42 86 38 33 30 27

.0145081 .0301543 .0133240 .0115708 .0105189 .0094670

.9851716 .6509067 .9806026 .8174890 .7570769 .9508398

1.0000000

2,852

1.0000000

Total

SOURCE: U.S. Bureau of the Census, Estimates of the population of the United States, by age, sex, and race, 1980 to 1985 (Current Population Reports, Series P-25, No. 985), April 1986. Noninstitutional population estimates are derived from the estimated total population of 1984 (Table1), adjusted by applying the ratio of noninstitutional to total population (Table Al).

the same as the population distribution. The adjustment factors indicate that without the adjustment, the GSS sample underrepresents males 25–34 years of age and overrepresents nonwhite males 45–54 years of age and nonwhite females 25–34 years of age.

16 TABLE 3.2

Comparison of Weighted and Unweighted Estimates in Two Surveys Surveys and Variables

Weighted Estimates

Unweighted Estimates

60.0

59.4

I. General Social Survey (Percentage approving ‘‘hitting’’) Overall By sex Male Female By education Some college High school Others

63.5 56.8

63.2 56.8

68.7 63.3 46.8

68.6 63.2 45.2

II. Epidemiologic Catchment Areas Survey (Prevalence rate of mental disorders) Any disorders Anxiety disorders

14.8 6.5

8.8 18.5

SOURCE: Data for the Epidemiologic Catchment Areas Survey are from E. S. Lee, Forthofer, and Lorimor (1986), Table 1.

The adjusted relative weights are then used in the analysis of the data—for example, to estimate the proportion of adults responding positively to the question, ‘‘Are there any situations that you can imagine in which you would approve of a man punching an adult male stranger?’’ As shown in the upper section of Table 3.2, the weighted overall proportion is 60.0%, slightly larger than the unweighted estimate of 59.4%. The difference between the weighted and unweighted estimates is also very small for the subgroup estimates shown. This may be due primarily to the self-weighting feature reflected in the fact that most households have two adults and, to a lesser extent, to the fact that the ‘‘approval of hitting’’ is not correlated with the number of adults in a household. The situation is different in the National Institute of Mental Health-Sponsored Epidemiologic Catchment Area (ECA) Survey. In this survey, the weighted estimates of the prevalence of any disorders and of anxiety disorders are, respectively, 20% and 26% lower than the unweighted estimates, as shown in Table 3.2. Finally, the adjusted weights should be examined to see whether there are any extremely large values. Extreme variation in the adjusted weights may imply that the sample sizes in some poststrata are too small to be reliable. In such a case, some small poststrata need to be collapsed, or some raking procedure must be used to smooth out the rough edges (Little & Rubin, 1987, pp. 59–60).

17

Adjusting the Weight in a Follow-Up Survey Follow-up surveys are used quite often in social science research. Unfortunately, it is not possible to follow up with all initial respondents. Some may have died, moved, or refused to participate in the follow-up survey. These events are not randomly distributed, and the differential attrition may introduce selection bias in the follow-up survey. We could make an adjustment in the same manner as in the poststratification, based on a few demographic variables, but we can take advantage of a large number of potential predictor variables available in the initial survey for the attrition adjustment in the follow-up survey. The stratification strategy may not be well suited for a large number of predictors. The logistic regression model, however, provides a way to include several predictors. Based on the initial survey data, we can develop a logistic regression model for predicting the attrition (binary variable) by a set of well-selected predictor variables from the initial survey, in addition to usual demographic variables. The predicted logit based on characteristics of respondents in the original sample can then be used to adjust the initial weight for each respondent in the follow-up survey. This procedure, in essence, can make up for those lost to the follow-up by differentially inflating the weights of those successfully contacted and, hence, removing the selection bias to the extent that it is related to the variables in the model. As more appropriate variables are used in the model, the attrition adjustment can be considered more effective than the nonresponse adjustment in the initial survey. Poststratification used to correct for nonresponse resulting from ‘‘lost to follow-up’’ (or for other types of nonresponse) does not guarantee an improved estimate. However, if ‘‘lost to follow-up’’ is related to the variables used in the adjustment process, the adjusted estimate should be an improvement over the unadjusted estimate. Let us look at an example from a community mental health survey (one site in the ECA survey). Only 74.3% of initial respondents were successfully surveyed in the first annual follow-up survey. This magnitude of attrition is too large to be ignored. We therefore conducted a logistic regression analysis of attrition (1 = lost; 0 = interviewed) on selected predictors shown in Table 3.3. The chi-square value suggests that the variables in the model are significantly associated with attrition. The effect coding requires that one level of each variable be omitted, as shown in the table. Based on the estimated beta coefficients, ^i , i = 1, 2, . . . , n) is calculated for each respondent. It is the predicted logit (λ ^ ^i = 1/ð1 + e −λi ). then converted to the predicted proportion of attrition by p The initial weight for a successfully interviewed respondent is inflated by ^i Þ and setting the weight for a respondent lost to followdividing it by ð1 − p up to zero. As shown in Table 3.3, the adjusted weight for the retained panel members added up to 300,172, only 59 more than the sum of the weights in the

18 TABLE 3.3

Logistic Regression Model for Attrition Adjustment in a Follow-Up Survey Factors Intercept Age Sampling

Logistic Regression Model_______________ Category Variable Beta Coefficient

18-24 yrs

25-34 yrs 35-44 yrs 45-54 yrs 55 and over Marital status: Sep./divorced (74.3%) Other Gender: Male Female Race White Black and Other estimates Socioeconomic 1st quartile status 2nd quartile Diff. 3rd and 4th Family size: One -4.6% 2-4 members -3.1 5 or more 0.5 Diagnosis: Cog. Impair. -0.4 Schizophrenia -4.2 Antisocial -0.4 Anorexia No disorder

Survey-Related Information

AGE1

-0.737* 0.196

Initial survey Design:

AGE2 AGE3 AGE4 — MAR

0.052 -0.338* -0.016

Sample size: Weighted sum:

SEX — RACE1 RACE2

SES1 SES2

0.051

0.084 -0.668* -0.865*

Multistage 4,967 300,113

Follow-up survey Sample size: Sum of attritionadjusted weights: Adjusted sum:

3,690

300,172 300,113

Comparison of attrition-adjusted attrition-unadjusted

0.534* 0.389*

Disorders

SIZE1

0.033

Any disorder

39.2%

43.8%

SIZE2

-0.003

Major depre.

15.0

18.1

Cog. Impair.

1.8

1.3

Phobias

8.4

7.6

Alcohol abuse

13.9

13.7

Schizophrenia

0.6

0.6

DX1 DX2

0.472* -0.049

DX3

0.412*

DX4 —

2.283*

Unadjusteda Adjustedb

Likelihood ratio chi-square(16) = 198.0, p < 0.00001.

a. Based on the initial weights. b. Based on attrition-adjusted weights. * p < .05.

initial survey, suggesting that the procedure worked reasonably well. (If there were a large discrepancy between the sum of the adjusted weights and the sum of the weights in the initial survey, there would be a concern about the adjustment process.) The adjusted weights are readjusted to align to the initial survey. To show the effect of using attrition-adjusted weights, the prevalence rates of six selected mental disorders were estimated with and without the attrition-adjusted weights, as shown in Table 3.3. Using the adjusted weights, we see that the prevalence of any disorders (DSM-III defined disorders) is nearly 5 percentage points higher than the unadjusted prevalence. Assessing the Loss or Gain in Precision: The Design Effect As shown in the previous chapter, the variance of an SRSWOR sample mean is the variance of an SRSWR sample mean times the finite population

19

correction factor (1 − f ). The ratio of the sampling variance of SRSWOR to the sampling variance of SRSWR is then (1 − f ), which reflects the effect of using SRSWOR compared to using SRSWR. This ratio comparing the variance of some statistic from any particular design to that of SRSWR is called the design effect for that statistic. It is used to assess the loss or gain in precision of sample estimates from the design used, compared to a SRSWR design. A design effect less than one indicates that fewer observations are needed to achieve the same precision as SRSWR, whereas a design effect greater than one indicates that more observations are needed to yield the same precision. In the case of SRSWOR, the design effect is less than one, but it is close to one when the sampling fraction is very small. Because SRSWOR is customarily used in place of SRSWR, researchers have tended to base the design effect calculation on SRSWOR instead of SRSWR. In addition, in complex surveys the design effect is usually calculated based on the variance of the weighted statistic under the SRSWOR design. We shall do that throughout the rest of the book. Relating this notion of the design effect to the sample size, the effective sample size can be defined to be the actual sample size divided by the design effect. If the design effect is greater than one for a sample design, then, in effect the sample size would be reduced for a statistical analysis, thus leading to a larger sampling error. In other words, when the design effect is greater than one, the effective sample size is smaller than the actual size. Let us examine the design effect in more complex sample designs. The properties and estimation of the sampling error for stratified random sampling are well known, and so are the conditions under which stratification will produce a smaller variance than SRS. However, stratification often is used in conjunction with other design features, such as cluster sampling in several stages within the strata. As discussed earlier, clustering tends to increase the sampling error. The effect of stratification can be diluted by the effect of clustering in many practical designs. Unfortunately, the assessment of the design effect cannot be determined theoretically from properties of stratification and clustering separately, but instead must be approximated numerically to account for their combined effects. The following example demonstrates the determination of the design effect in a relatively simple situation. Consider the case of single-stage cluster sampling in which all clusters are of the same size. Suppose that there are N English classes (clusters) in a high school, with M students in each class. From the N classes, n classes are selected by SRS, and all the students in the chosen clusters are asked to report the number of books they have read since the beginning of the year. The number of students is NM in the population and nM in the sample. The sampling fraction is f = nM/NM = n/N.

20

Because the class sizes are equal, the average number of books read per student (population mean) is the mean of the N class means. The n sample classes can be viewed as a random sample of n means from a population of N means. Therefore, the sample mean (y
) is unbiased for the population mean (Y
), and its variance, applying Equation 2.1, is given by V^ðy
Þ =

s2b ð1 − f Þ, n

(3:3)


where s2b = ðy
i − y
Þ2 /ðn − 1Þ, the estimated variance of the cluster means. Alternately, Equation 3.3 can be expressed in terms of estimated ICC (^ ρ) as follows (Cochran, 1977, chap. 9): s2 [1 + (M − 1)^ ρ] V^(y
) = (1 − f ), nM

(3:4)



(yij − y
)2 /ðnM − 1Þ, the variance of elements in the samwhere s2 = ple. If this is divided by the variance of the mean from an SRSWOR sample s2 ð1 − f Þ, applying Equation 2.1], then the design of size nM, [V^ðy
Þ = nM effect of the cluster sample is 1 + ðM − 1Þ^ ρ. When ICC = 0, the design effect will be one; when ICC > 0, the design effect will be greater than one. If the clusters were formed at random, then ICC = 0; when all the elements within each cluster have the same value, ICC = 1. Most clusters used in community surveys consist of houses in the same area, and these yield positive ICCs for many survey variables. The ICC is usually larger for socioeconomic variables than for the demographic variables, such as age and sex. The assessment of the design effect for a more complex sample design is not a routine task that can be performed using the formulas in statistics textbooks; rather, it requires special techniques that utilize unfamiliar strategies. The next chapter reviews several strategies for estimating the sampling variance for statistics from complex surveys and examines the design effect from several surveys. The Use of Sample Weights for Survey Data Analysis* As discussed above, sample weights are used when computing point estimates. All point estimates take the form of weighted statistics. This use of sample weights makes sense from the above discussion on development and adjustment of sample weights, especially for a descriptive analysis. However, the use of sample weights in analytical studies is not as clear as in descriptive analysis. As discussed in the last section of Chapter 2, there are two different points of view regarding survey data. From the design-based

21

position, the use of sample weights is essential in both descriptive and analytical studies. Inferences are based on repeated sampling for the finite population, and the probability structure used for inference is that defined by the random variables indicating inclusion in the sample. From the model-based perspective, however, it can be argued that the sample selection scheme is irrelevant when making inferences under a specified model. If the observations in the population really follow the model, then the sample design should have no effect as long as the probability of selection depends on the model-specified dependent variable only through the independent variables included in the model. Conditions under which the sampling scheme is ignorable for inference have been explored extensively (Nordberg, 1989; Sugden & Smith, 1984). Many survey statisticians have debated these two points of view since the early 1970s (Brewer, 1999; Brewer & Mellor, 1973; Graubard & Korn, 1996, 2002; Hansen, Madow, & Tepping, 1983; Korn & Graubard, 1995a; Pfeffermann, 1996; Royall, 1970; Sarndal, 1978; T. M. F. Smith, 1976, 1983). A good analysis of survey data would require general understanding of both points of view as well as consideration of some practical issues, especially in social surveys. The model-based approach is consistent with the increasing use of model-based inference in other areas of statistical analysis, and it provides some theoretical advantages. Model-based estimates can be used with relatively small samples and even with nonprobability samples. In addition, model-based analysis can be done using standard statistical software such as SAS and SPSS without relying on survey packages such as SUDAAN and others that are reviewed in this book. The modelbased approach, however, assumes that the model correctly describes the true state of nature. If the model is misspecified, then the analysis would be biased and lead to misinterpretation of the data. Unfortunately, theoretically derived models for all observations in the population seldom exist in social survey situations. In addition, omission of relevant variables in the model would be a real concern in secondary analysis of survey data because not all relevant variables are available to the analysts. Thus, the major challenge to model-based inference is specifying a correct model for the purpose of the analysis. It has been recognized that a weighted analysis is heavily influenced by observations with extremely large weights (that may often be a result of nonresponse and poststratification adjustments rather than the selection probabilities). Another recognized limitation of weighting is the increase in variance. In general, the increase is high when the variability of the weights is large. There is something to lose in using weighted analysis when it is actually unnecessary for bias reduction, and the weighted analysis will be inefficient as compared with the unweighted analysis. Korn and Graubard

22

(1999, chap. 4) discuss various issues dealing with weighted and unweighted estimates of population parameters and offer a measure of the inefficiency of using weighted estimates. They recommend using the weighted analysis if the inefficiency is not unacceptably large, to avoid the bias in the unweighted analysis. If the inefficiency is unacceptably large, they recommend using the unweighted analysis, augmenting the model with survey design variables, including the weight, to reduce the bias. Incorporation of design variables into the model is often problematic, however, because the inclusion of the design variables as additional covariates in the model may contradict the scientific purpose of the analysis. For example, when the objective of the analysis is to examine associations between health measures and risk factors, conditioning on the design variables may interfere with the relational pathway. In complex large-scale surveys, it is often not possible to include in the model all the design information, especially when the sample weights are modified for nonresponse and poststratification adjustments (Alexander, 1987). Another practical problem in incorporating the design variables into the model is the lack of relevant information in the data set. Not all designrelated variables are available to the analysts. Most public-use survey data provide only PSU (the primary sampling unit), leaving out secondary cluster units (census enumeration districts or telephone exchanges). Often, provision of secondary units may not be possible because of confidentiality issues. In a model-based analysis, one must guard against possible misspecification of the model and possible omission of covariates. The use of sample weights (design-based analysis) can provide protection against model misspecification (DuMouchel & Duncan, 1983; Pfeffermann & Homes, 1985). Kott (1991) points out that the sampling weights need to be used in linear regression because the choice of covariates in survey data is limited in most secondary analyses. The merits and demerits of using sample weights will be further discussed in the last section of Chapter 6.

4. STRATEGIES FOR VARIANCE ESTIMATION The estimation of the variance of a survey statistic is complicated not only by the complexity of the sample design, as seen in the previous chapters, but also by the form of the statistic. Even with an SRS design, the variance estimation of some statistics requires nonstandard estimating techniques. For example, the variance of the median is conspicuously absent in the standard texts, and the sampling error of a ratio estimator (refer again to Note 1) is complicated because both the numerator and denominator are random variables.

23

Certain variance-estimating techniques not found in standard textbooks are sufficiently flexible to accommodate both the complexities of the sample design and the various forms of statistics. These general techniques for variance estimation, to be reviewed in this chapter, include replicated sampling, balanced repeated replication (BRR), jackknife-repeated replication (JRR), the bootstrap method, and the Taylor series method.

Replicated Sampling: A General Approach The essence of this strategy is to facilitate the variance calculation by selecting a set of replicated subsamples instead of a single sample. It requires that each subsample be drawn independently using an identical sample selection design. Then an estimate is made in each subsample by the identical process, and the sampling variance of the overall estimate (based on all subsamples) can be estimated from the variability of these independent subsample estimates. This is the same idea as the repeated systematic sampling mentioned in Chapter 2. The sampling variance of the mean (
u) of t replicate estimates u1 , u2 , . . . , ut of the parameter U can be estimated by the following simple variance estimator (Kalton, 1983, p. 51): v(
u) =



(ui − u
)2 /t(t − 1)

(4:1)

This estimator can be applied to any sample statistic obtained from independent replicates of any sample design. In applying this variance estimator, 10 replicates are recommended by Deming (1960), and a minimum of 4 by others (Sudman, 1976) for descriptive statistics. An approximate estimate of the standard error can be calculated by dividing the range in the replicate estimates by the number of replicates when the number of replicates is between 3 and 13 (Kish, 1965, p. 620). However, because this variance estimator with t replicates is based on (t − 1) degrees of freedom for statistical inference, a larger number of replicates may be needed for analytic studies, perhaps 20 to 30 (Kalton, 1983, p. 52). To understand the replicated design strategy, let us consider a simple example. Suppose we want to estimate the proportion of boys among 200 newly born babies. We will simulate this survey using the random digits from Cochran’s book (1977, p. 19), assuming the odd numbers represent boys. The sample is selected in 10 replicate samples of n = 20 from the first 10 columns of the table. The numbers of boys in the replicates are as follows:

24

Replicate:

9

8

13

12

14

8

10

7

10

8

Total = 99

Proportion of Boys:

.45

.40

.65

.60

.70

.40

.50

.35

.50

.40

Proportion = .495

The overall percentage of boys is 49.5%, and its standard error is 3.54% pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (¼ 49:5∗ 50:5/200). The standard error estimated from the 10 replicate estimates using Equation 4.1 is 3.58%. It is easy to get an approximate estimate of 3.50% by taking one tenth of the range (70%–35%). The chief advantage of replication is ease in estimation of the standard errors. In practice, the fundamental principle of selecting independent replicates is somewhat relaxed. For one thing, replicates are selected using sampling without replacement instead of with replacement. For unequal probability designs, the calculation of basic weights and the adjustment for nonresponse and poststratification usually are performed only once for the full sample, rather than separately within each replicate. In cluster sampling, the replicates often are formed by systematically assigning the clusters to the t replicates in the same order that the clusters were first selected, to take advantage of stratification effects. In applying Equation 4.1, the sample mean from the full sample generally is used for the mean of the replicate means. These deviations from fundamental principles can affect the variance estimation, but the bias is thought to be insignificant in large-scale surveys (Wolter, 1985, pp. 83–85). The community mental health survey conducted in New Haven, Connecticut, in 1984 as part of the ECA Survey of the National Institute of Mental Health (E. S. Lee, Forthofer, Holzer, & Taube, 1986) provides an example of replicated sampling. The sampling frame for this survey was a geographically ordered list of residential electric hookups. A systematic sample was drawn by taking two housing units as a cluster, with an interval of 61 houses, using a starting point chosen at random. A string of clusters in the sample was then sequentially allocated to 12 subsamples. These subsamples were created to facilitate the scheduling and interim analysis of data during a long period of screening and interviewing. Ten of the subsamples were used for the community survey, with the remaining two reserved for another study. The 10 replicates are used to illustrate the variance estimation procedure. These subsamples did not strictly adhere to a fundamental principle of independent replicated sampling because the starting points were systematically selected, except for the first random starting point. However, the systematic allocation of clusters to subsamples in this case introduced an approximate stratification leading to more stable variance estimation and,

25 TABLE 4.1

Estimation of Standard Errors From Replicates: ECA Survey in New Haven, 1984 (n = 3,058) Regression Coefficientsa Replicate Full Sample 1 2 3 4 5 6 7 8 9 10 Range

Prevalence Rateb

Odds Ratioc

Intercept

Gender

Color

Age

17.17 12.81 17.37 17.87 17.64 16.65 18.17 14.69 17.93 17.86 18.91

0.990 0.826 0.844 1.057 0.638 0.728 1.027 1.598 1.300 0.923 1.111

0.2237 0.2114 0.2581 0.2426 0.1894 0.1499 0.2078 0.3528 0.3736 0.2328 0.3008

−0.0081 0.0228 0.0220 −0.0005 0.0600 0.0448 −0.0024 −0.0487 −0.0333 −0.0038 −0.0007

0.0185 0.0155 0.0113 0.0393 0.2842 −0.0242 −0.0030 −0.0860 −0.0629 0.0751 0.0660

−0.0020 −0.0020 −0.0027 −0.0015 −0.0029 −0.0012 −0.0005 −0.0028 −0.0032 −0.0015 −0.0043

6.10

0.960

0.2237

0.1087

0.3702

0.0038

0.090 0.097

0.0234 0.0228

0.0104 0.0141

0.0324 0.0263

0.0004 0.0004

Standard error based on: Replicates SRS

0.59 0.68

SOURCE: Adapted from ‘‘Complex Survey Data Analysis: Estimation of Standard Errors Using PseudoStrata,’’ E. S. Lee, Forthofer, Holzer, and Taube, Journal of Economic and Social Measurement, Ó copyright 1986 by the Journal of Economic and Social Measurement. Adapted with permission. a. The dependent variable (coded as 1 = condition present and 0 = condition absent) is regressed on sex (1 = male, 0 = female), color (1 = black, 0 = nonblack), and age (continuous variable). This analysis is used for demonstration only. b. Percentage with any mental disorders during the last 6 months. c. Sex difference in the 6-month prevalence rate.

therefore, may be preferable to a random selection of a starting point for this relatively small number of replicates. Therefore, we considered these subsamples as replicates and applied the variance estimator with replicated sampling, Equation 4.1. Because one adult was randomly selected from each sampled household using the Kish selection table (Kish, 1949), the number of adults in each household became the sample case weight for each observation. This weight was then adjusted for nonresponse and poststratification. Sample weights were developed for the full sample, not separately within each subsample, and these were the weights used in the analysis. Table 4.1 shows three types of statistics calculated for the full sample as well as for each of the replicates. The estimated variance of the prevalence

26

rate, in percent (p), can be calculated from the replicate estimates (pi ) using Equation 4.1:
(pi − 17:17)2 = 0:3474, v(p) = 10(10 − 1) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi and the standard error is 0:3474 = 0:59. The overall prevalence rate of 17.17% is slightly different from the mean of the 10 replicate estimates because of the differences in response rates. Note that one tenth of the range in the replicate estimates (0.61) approximates the standard error obtained by Equation 4.1. Similarly, standard errors can be estimated for the odds ratio and regression coefficients. The estimated standard errors have approximately the same values as those calculated by assuming simple random sampling (using appropriate formulas from textbooks). This indicates that design effects are fairly small for these statistics from this survey. Although the replicated sampling design provides a variance estimator that is simple to calculate, a sufficient number of replicates are required to obtain acceptable precision for statistical inference. But if there is a large number of replicates and each replicate is relatively small, it severely limits the use of stratification in each replicate. Most important, it is impractical to implement replicated sampling in complex sample designs. For these reasons, a replicated design is seldom used in large-scale, analytic surveys. Instead, the replicated sampling idea has been applied to estimate variance in the dataanalysis stage. This attempt gave rise to pseudo-replication methods for variance estimation. The next two techniques are based on this idea of pseudo-replication. Balanced Repeated Replication The balanced repeated replication (BRR) method is based on the application of the replicated sampling idea to a paired selection design in which two PSUs are sampled from each stratum. The paired selection design represents the maximum use of stratification and yet allows the calculation of variance. In this case, the variance between two units is one half of the squared difference between them. To apply the replicated sampling idea, we first divide the sample into random groups to form pseudo-replicates. If it is a stratified design, it requires all the strata to be represented in each pseudoreplicate. In a stratified, paired selection design, we can form only two pseudo-replicates: one containing one of the two units from each stratum and the other containing the remaining unit from each stratum (complement replicate). Each pseudo-replicate then includes approximately half of the total

27

sample. Applying Equation 4.1 with t = 2, we can estimate the sampling variance of the mean of the two replicate estimates, u , u , by v(
u) = [(u − u
)2 + (u − u
)2 ]/2:

(4:2)

As seen above, the mean of replicate estimates is often replaced by an overall estimate obtained from the full sample. However, this estimator is too unstable to have any practical value because it is based on only two pseudo-replicates. The BRR method solves this problem by repeating the process of forming half-sample replicates, selecting different units from different strata. The pseudo-replicated half samples then contain some common units, and this introduces dependence between replicates, which complicates the estimation. One solution, which leads to unbiased estimates of variance for linear statistics, is to balance the formation of pseudo-replicates by using an orthogonal matrix (Plackett & Burman, 1946). The full balancing requires that the size of the matrix be a multiple of four and the number of replicates be greater than or equal to the number of strata. Then the sampling variance of a sample statistic can be estimated by taking the average of variance estimates by Equation 4.2 over t pseudo-replicates:   (ui − ui )2 /4t: (4:3) v(
u) = [(ui − u
)2 + (ui − u
)2 ]/2t = It is possible to reduce computation by dropping the complement half-sample replicates:  u) = (ui − u
)2 /t: (4:4) v (
This is the estimator originally proposed by McCarthy (1966). This balancing was shown by McCarthy to yield unbiased estimates of variance for linear estimators. For nonlinear estimators, there is a bias in the estimates of variance, but numerical studies suggest that it is small. For a large number of strata, the computation can be further simplified by using a smaller set of partially balanced replicates (K. H. Lee, 1972; Wolter, 1985, pp. 125–130). As in replicated sampling, BRR assumes that the PSUs are sampled with replacement within strata, although in practice sampling without replacement generally is used. Theoretically, this leads to an overestimation of variance when applied to a sample selected without replacement, but the overestimation is negligible in practice because the chance of selecting the same unit more than once under sampling without replacement is low when the sampling fraction is small. The sampling fraction in a paired selection design (assumed in the BRR method) usually is small because only two PSUs are selected from each stratum.

28

When used with a multistage selection design, BRR usually is applied only to PSUs and disregards the subsampling within the PSUs. Such a practice is predicated on the fact that the sampling variance can be approximated adequately from the variation between PSU totals when the first-stage sampling fraction is small. This is known as the ultimate cluster approximation. As shown in Kalton (1983, chap. 5), the unbiased variance estimator for a simple two-stage selection design consists of a component from each of the two stages, but the term for the second-stage component is multiplied by the first-stage sampling fraction. Therefore, the second-stage contribution becomes negligible as the first-stage sampling fraction decreases. This shortcut procedure based only on PSUs is especially convenient in the preparation of complex data files for public use as well as in the analysis of such data, because detailed information on complex design features is not required except for the first-stage sampling. If the BRR technique is to be applied to other than the paired selection designs, it is necessary to modify the data structure to conform to the technique. In many multistage surveys, stratification is carried out to a maximum and only one PSU is selected from each stratum. In such case, PSUs can be paired to form collapsed strata to apply the BRR method. This procedure generally leads to some overestimation of the variance because some of the between-strata variability is now included in the within-stratum calculation. The problem is not serious for the case of linear statistics if the collapsing is carried out judiciously; however, the collapsing generally is not recommended for estimating the variance of nonlinear statistics (see Wolter, 1985, p. 48). The Taylor series approximation method discussed later may be used for the nonlinear statistics. Although it is not used widely, there is a method of constructing orthogonal balancing for three PSUs per stratum (Gurney & Jewett, 1975). Now let us apply the BRR technique to the 1984 GSS. As introduced in the previous chapter, it used a multistage selection design. The first-stage sampling consisted of selecting one PSU from each of 84 strata of counties or county groups. The first 16 strata were large metropolitan areas and designated as self-representing (or automatically included in the sample). To use the BRR technique, the 84 strata are collapsed into 42 pairs of pseudo-strata. Because the numbering of non–self-representing PSUs in the data file approximately followed the geographic ordering of strata, pairing was done sequentially, based on the PSU code. Thus, the 16 self-representing strata were collapsed into 8 pseudo-strata, and the remaining 68 non–selfrepresenting strata into 34 pseudo-strata. This pairing of the self-representing strata, however, improperly includes variability among them. To exclude this and include only the variability within each of the self-representing strata, the combined observations within each self-representing pseudo-stratum were randomly grouped into two pseudo-PSUs.

29

To balance the half-sample replicates to be generated from the 42 pseudo-strata, an orthogonal matrix of order 44 (see Table 4.2) was used. This matrix is filled with zeros and ones. To match with the 42 strata, the first two columns were dropped (i.e., 44 rows for replicates and 42 columns for pseudo-strata). A zero indicates the inclusion of the first PSU from the strata, and a one denotes the inclusion of the second PSU. The rows are the replicates, and the columns represent the strata. For example, the first replicate contains the second PSU from each of the 42 pseudo-strata (because all the elements in the first row are ones). Using the rows of the orthogonal matrix, 44 replicates and 44 complement replicates were created. To estimate the variance of a statistic from the full sample, we needed first to calculate the statistic of interest from each of the 44 replicates and complement replicates. In calculating the replicate estimates, the adjusted sample weights were used. Table 4.3 shows the estimates of the proportion of adults approving the ‘‘hitting’’ for the 44 replicates and their complement replicates. The overall proportion was 60.0%. The sampling variance of the overall proportion, estimated by Equation 4.3, is 0.000231. Comparing this with the sampling variance of the proportion under the SRS design [pq/(n − 1) = 0.000163, ignoring FPC], we get the design effect of 1.42 (= 0.000231/0.000163). The design effect indicates that the variance of the estimated proportion from the GSS is 42% larger than the variance calculated from an SRS of the same size. The variance by Equation 4.4 also gives similar estimates. In summary, the BRR technique uses a pseudo-replication procedure to estimate the sampling variance and is primarily designed for a paired selection design. It also can be applied to a complex survey, which selects one PSU per stratum by pairing strata, but the pairing must be performed judiciously, taking into account the actual sample selection procedure. In most applications of BRR in the available software packages, the sample weights of the observations in the selected PSUs for a replicate are doubled to make up for the half of PSUs not selected. There is also a variant of BRR, suggested by Fay (Judkins, 1990), in creating replicate weights, which uses 2 − k or k times the original weight, depending on whether the PSU is selected or not selected based on the orthogonal matrix (0 ≤ k < 1). This will be illustrated further in the next chapter. Jackknife Repeated Replication The idea of jackknifing was introduced by Quenouille (1949) as a nonparametric procedure to estimate the bias, and later Tukey (1958) suggested how that same procedure could be used to estimate variance. Durbin (1959) first used this method in his pioneering work on ratio estimation. Later,

30 TABLE 4.2

Orthogonal Matrix of Order 44 Rows 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

Columns (44) 11111111111111111111111111111111111111111111 10100101001110111110001011100000100011010110 10010010100111011111000101110000010001101011 11001001010011101111100010111000001000110101 11100100101001110111110001011100000100011010 10110010010100111011111000101110000010001101 11011001001010011101111100010111000001000110 10101100100101001110111110001011100000100011 11010110010010100111011111000101110000010001 11101011001001010011101111100010111000001000 10110101100100101001110111110001011100000100 10011010110010010100111011111000101110000010 10001101011001001010011101111100010111000001 11000110101100100101001110111110001011100000 10100011010110010010100111011111000101110000 10010001101011001001010011101111100010111000 10001000110101100100101001110111110001011100 10000100011010110010010100111011111000101110 10000010001101011001001010011101111100010111 11000001000110101100100101001110111110001011 11100000100011010110010010100111011111000101 11110000010001101011001001010011101111100010 10111000001000110101100100101001110111110001 11011100000100011010110010010100111011111000 10101110000010001101011001001010011101111100 10010111000001000110101100100101001110111110 10001011100000100011010110010010100111011111 11000101110000010001101011001001010011101111 11100010111000001000110101100100101001110111 11110001011100000100011010110010010100111011 11111000101110000010001101011001001010011101 11111100010111000001000110101100100101001110 10111110001011100000100011010110010010100111 11011111000101110000010001101011001001010011 11101111100010111000001000110101100100101001 11110111110001011100000100011010110010010100 10111011111000101110000010001101011001001010 10011101111100010111000001000110101100100101 11001110111110001011100000100011010110010010 10100111011111000101110000010001101011001001 11010011101111100010111000001000110101100100 10101001110111110001011100000100011010110010 10010100111011111000101110000010001101011001 11001010011101111100010111000001000110101100

SOURCE: Adapted from Wolter (1985, p. 328) with permission of the publisher.

31 TABLE 4.3

Estimated Proportions Approving One Adult Hitting Another in the BRR Replicates: General Social Survey, 1984 (n = 1,473) Estimate (%) Replicate Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Estimate (%)

Replicate

Complement

Replicate Number

Replicate

Complement

60.9 60.1 62.1 58.5 59.0 59.8 58.5 59.0 61.3 59.2 61.7 60.2 62.1 59.7 58.1 56.0 59.8 58.6 58.9 60.8 63.4 58.3

59.2 59.9 57.9 61.7 61.0 60.2 61.5 61.0 58.8 60.8 58.3 59.8 58.7 60.4 62.0 64.2 60.3 61.3 61.1 59.3 56.5 61.7

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

61.4 57.7 60.4 61.7 59.3 62.4 61.0 61.2 60.9 61.6 61.8 60.6 58.6 59.4 59.8 62.0 58.1 59.6 58.8 59.2 58.7 60.5

58.6 62.4 59.6 58.2 60.6 57.6 58.9 58.7 59.1 58.5 58.2 59.4 61.5 60.7 60.3 58.1 61.9 60.5 61.2 60.8 61.4 59.5

Overall estimate = 60.0 Variance estimates

Variance

Standard Error

Design Effect

By Equation 4.3 By Equation 4.4

0.000231 0.000227

0.0152 0.0151

1.42 1.40

it was applied to computation of variance in complex surveys by Frankel (1971) in the same manner as the BRR method and was named the jackknife repeated replication (JRR). As is BRR, the JRR technique generally is applied to PSUs within strata. The basic principle of jackknifing can be illustrated by estimating sampling variance of the sample mean from a simple random sample. Suppose n = 5 and sample values of y are 3, 5, 2, 1, and 4. The sample mean then is y
= 3, and its sampling variance, ignoring the FPC, is
(yi − y
)2 = 0:5: (4:5) v(
y) = n(n − 1)

32

The jackknife variance of the mean is obtained as follows. 1. Compute a pseudo sample mean deleting the first sample value, which results in y
(1) = (5 + 2 + 1 + 4)/4 = 12/4: Now, by deleting the second sample value instead, we obtain the second pseudo-mean y
(2) = 10/4; likewise y
(3) = 13/4, y
(4) = 14/4, and y
(5) = 11/4: P 2. Compute the mean of the five pseudo-values; y
= y
(i) /n = (60/4)/5 = 3, which is the same as the sample mean. 3. The variance can then be estimated from the variability among the five pseudo-means, each of which contains four observations, v(y
) =

(n − 1)




(
y(i) − y
)2 = 0:5, n

(4:6)

which gives the same result as Equation 4.5. The replication-based procedures have a distinct advantage: They can be applied to estimators that are not expressible in terms of formulas, such as the sample median as well as to formula-based estimators. No formula is available for the sampling variance of the median, but the jackknife procedure can offer an estimate. With the same example as used above, the sample median is 3 and the five pseudo-medians are 3, 2.5, 3.5, 3.5, and 2.5 (the mean of these pseudo-medians is 3). The variance of the median is estimated as 0.8, using Equation 4.6. In the same manner, the jackknife procedure also can be applied to the replicated sampling. We can remove replicates one at a time and compute pseudo-values to estimate the jackknife variance, although this does not offer any computational advantage in this case. But it also can be applied to any random groups that are formed from any probability sample. For instance, a systematic sample can be divided into random or systematic subgroups for the jackknife procedure. For other sample designs, random groups can be formed following the practical rules suggested by Wolter (1985, pp. 31–33). The basic idea is to form random groups in such a way that each random group has the same sample design as the parent sample. This requires detailed information on the actual sample design, but unfortunately such information usually is not available in most public-use survey data files. The jackknife procedure is, therefore, usually applied to PSUs rather than to random groups. For a paired selection design, the replicate is formed removing one PSU from a stratum and weighting the remaining PSU to retain the stratum’s proportion in the total sample. The complement replicate is formed in the same manner by exchanging the removed and retained PSU in the stratum.

33

A pseudo-value is estimated from each replicate. For a weighted sample, the sample weights in the retained PSU need to be inflated to account for the observations in the removed PSU. The inflated weight is obtained by dividing the sum of the weights in the retained PSU by a factor (1 − wd /wt ), where wd is the sum of weights in the deleted PSU and wt is the sum of weights in all the PSUs in that stratum. The factor represents the complement of the deleted PSU’s proportion of the total stratum weight. Then the variance of a sample statistic in a paired selection design calculated from the full sample can be estimated from pseudo-values uh and complement pseudo-values uh in stratum h by   v(
u) = [(uh − u
)2 + (uh − u
)2 ]/2 = (uh − uh )2 /4: (4:7) This estimator has the same form as Equation 4.3 and can be modified to include one replicate, without averaging with the complement, from each stratum, as in Equation 4.4 for the BRR method, which gives  u) = (uh − u
)2 : (4:8) v (
The JRR is not restricted to a paired selection design but is applicable to any number of PSUs per stratum. If we let uhi be the estimate of U from the h-th stratum and i-th replicate, nh be the number of sampled PSUs in the h-th stratum, and rh be the number of replicates formed in stratum h, then the variance is estimated by vð u¯ Þ =

 rh L   nh − 1  h

rh

(uhi − u) ¯ 2:

(4:9)

i

If each of the PSUs in stratum h is removed to form a replicate, rh = nh in each stratum, but the formation of nh replicates in h-th stratum is not required. When the number of strata is large and nh is two or more, the computation can be reduced by using only one replicate in each stratum. However, a sufficient number of replicates must be used in analytic studies to ensure adequate degrees of freedom. Table 4.4 shows the results of applying the JRR technique to the collapsed paired design of the 1984 GSS used in the BRR computation. Estimated proportions of adults approving ‘‘the hitting of other adults’’ are shown for the 42 jackknife replicates and their complements. Applying Equation 4.7, we obtain a variance estimate of 0.000238 with a design effect of 1.46, and these are about the same as those obtained by the BRR technique. Using only the 42 replicates and excluding the complements (Equation 4.8), we obtain a variance estimate of 0.000275 with a design effect of 1.68.

34 TABLE 4.4

Estimated Proportions Approving One Adult Hitting Another in the JRR Replicates: General Social Survey, 1984 Estimate (%) Replicate Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Estimate (%)

Replicate

Complement

Replicate Number

Replicate

Complement

60.2 60.2 60.0 60.3 60.0 59.9 60.0 60.0 59.9 60.1 59.8 59.9 59.8 60.0 59.6 60.4 59.9 59.8 59.8 59.9 60.0

59.8 59.8 60.0 59.8 60.1 60.1 60.0 60.0 60.2 60.0 60.2 60.1 60.2 60.1 60.5 59.6 60.0 60.2 60.2 60.1 60.0

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

60.3 60.0 60.4 60.1 59.8 59.9 60.1 59.5 59.9 59.6 60.5 60.1 60.3 60.1 60.2 60.0 59.6 59.9 60.5 60.4 60.7

60.0 60.0 59.6 59.8 60.3 60.1 60.0 60.3 60.1 60.2 59.6 59.9 59.8 59.8 59.8 60.0 60.4 60.1 59.6 59.8 59.4

Overall estimate = 60.0 Variance estimates

Variance

Standard Error

Design Effect

By Equation 4.7 By Equation 4.8

0.000238 0.000275

0.0152 0.0166

1.46 1.68

From a closer examination of data in Table 4.4, one may get an impression that there is less variation among the JRR replicate estimates than among the BRR replicate estimates in Table 4.3. We should note, however, that the JRR represents a different strategy that uses a different method to estimate the variance. Note that Equation 4.3 for the BRR includes the number of replicates (t) in the denominator, whereas Equation 4.7 for the JRR is not dependent on the number of replicates. The reason is that in the JRR, the replicate estimates themselves are dependent on the number of replicates formed. Because the replicate is formed deleting one unit, the replicate estimate would be closer to the overall estimate when a large number of units is available to form the replicates, compared to the situation where a small number of units is used.

35

Therefore, there is no reason to include the number of replicates in Equations 4.7 and 4.8. However, the number of replicates needs to be taken into account when the number of replicates used is smaller than the total number of PSUs, as in Equation 4.9. In summary, the JRR technique is based on a pseudo-replication method and can estimate sampling variances from complex sample surveys. No restrictions on the sample selection design are needed, but forming replicates requires considerable care and must take into account the original sample design. As noted, this detailed design information is seldom available to secondary data analysts. For instance, if more information on ultimate clusters had been available in the GSS data file, we could have formed more convincing random groups adhering more closely to actual sample design rather than applying the JRR technique to a collapsed paired design.

The Bootstrap Method Closely related to BRR and JRR is the bootstrap method popularized by Efron (1979). The basic idea is to create replicates of the same size and structure as in the design by repeatedly resampling the PSUs in the observed data. Applying the bootstrap method to 84 PSUs in 42 pseudo-strata in the GSS data, one will sample 84 PSUs (using a with-replacement sampling procedure), two from each stratum. In some strata, the same PSU may be selected twice. The sampling is repeated a large number of times, a minimum of 200 (referred to as B) times (Efron & Tibshirani, 1993, sec. 6.4). However, a much larger number of replications usually is required to get a less variable estimate (Korn & Graubard, 1999, p. 33). For each replicate created (ui ), the parameter estimate is calculated. Then the bootstrap estimate of the variance of the mean of all replicate estimates is given by vðu
Þ =

B 1 (u − u
)2 : B i=1 i

(4:10)

This estimator needs to be corrected for bias by multiplying it by (n − 1)/n: When n is small, the bias can be substantial. In our example, there are two PSUs in each stratum, and the estimated variance needs to be halved. An alternative approach to correct the bias is to resample (nh − 1) PSUs in stratum h and multiply the sample weights of the observations in the resampled PSUs by nh /(nh − 1) (Efron, 1982, pp. 62–63). In our example, this will produce half-sample replicates as in BRR. The bootstrap estimate based on at least 200 replicates will then be about the same as the BBR estimate based on 44 half-sample replicates. Because of the large

36

number of replications required in the bootstrap method, this method has not yet been used extensively for variance estimation in complex survey analysis. Various procedures of applying the bootstrap method for variance estimation and other purposes have been suggested (Kovar, Rao, & Wu, 1988; Rao & Wu, 1988; Sitter, 1992). Although the basic methodology is widely known, many different competing procedures have emerged in selecting bootstrap samples. For example, Chao and Lo (1985) suggested duplicating each observation in the host sample N/n times to create the bootstrap population for simple random sampling without replacement. For sampling plans with unequal probability of selection, the replication of the observations needs to be proportionate to the sample weight; that is, the bootstrap sample should be selected using the PPS procedure. These options and the possible effects of deviating from the fundamental assumption of independent and identically distributed samples have not been thoroughly investigated. Although it is promising for handling many statistical problems, the bootstrap method appears less practical than BRR and JRR for estimating the variance in complex surveys, because it requires such a large number of replicates. Although BRR and JRR will produce the same results when applied by different users, the bootstrap results may vary for different users and at different tries by the same user, because the replication procedure is likely to yield different results each time. As Tukey (1986, p. 72) put it, ‘‘For the moment, jackknifing seems the most nearly realistic approach to assessing many of the sources of uncertainty’’ when compared with bootstrapping and other simulation methods. The bootstrap method is not implemented in the available software packages for complex survey analysis at this time, although it is widely used in other areas of statistical computing.

The Taylor Series Method (Linearization) The Taylor series expansion has been used in a variety of situations in mathematics and statistics. One early application of the series expansion was designed to obtain an approximation to the value of functions that are hard to calculate: for example, the exponential ex or logarithmic [log(x)] function. This application was in the days before calculators had special function keys and when we did not have access to the appropriate tables. The Taylor series expansion for ex involves taking the first- and higher-order derivatives of ex with respect to x; evaluating the derivatives at some value, usually zero; and building up a series of terms based on the derivatives. The expansion for ex is 1+x+

x2 x3 x4 + + + ... 2! 3! 4!

37

This is a specific application of the following general formula expanded at a: f (x) = f (a) + f  (a)(x − a) +

f  (a)(x − a)2 f  (a)(x − a)3 + +... 2! 3!

In statistics, the Taylor series is used to obtain an approximation to some nonlinear function, and then the variance of the function is based on the Taylor series approximation to the function. Often, the approximation provides a reasonable estimate to the function, and sometimes the approximation is even a linear function. This idea of variance estimation has several names in the literature, including the linearization method, the delta method (Kalton, 1983, p. 44), and the propagation of variance (Kish, 1965, p. 583). In statistical applications, the expansion is evaluated at the mean or expected value of x, written as E(x). If we use E(x) for a in the above general expansion formula, we have f (x) = f [E(x)] + f  [E(x)][x − E(x)] + f  [E(x)][x − E(x)]2 /2 ! + . . . The variance of f(x) is V [f (x)] = E[f 2 (x)] − E2 [f (x)] by definition, and using the Taylor series expansion, we have V [f (x)] = {f  [E(x)]}2 V (x) + . . .

(4:11)

The same ideas carry over to functions of more than one random variable. In the case of a function of two variances, the Taylor series expansion yields    ∂f ∂f (4:12) Cov(x1 , x2 ) V [f (x1 , x2 )] ffi ∂x1 ∂x2 Applying Equation 4.12 to a ratio of two variables x and y—that is, r = y/x—we obtain the variance formula for a ratio estimator V (y) + r2 V (x) − 2r Cov(x, y) + ... x2   V ðxÞ 2 Covðx, yÞ 2 V ð yÞ =r + 2 − + ... xy y2 x

V (r) =

Extending Equation 4.12 to the case of c random variables, the approximate variance of θ = f (x1 , x2 , . . . , xc ) is   ∂f  ∂f  V (θ) ffi (4:13) Cov(xi , xj ): ∂xi ∂xj

38 TABLE 4.5

Standard Errors Estimated by Taylor Series Method for Percentage Approving One Adult Hitting Another: General Social Survey, 1984 (n = 1,473) Subgroup Overall

Estimate (%)

Standard Error (%)

Design Effect

60.0

1.52

1.41

Gender

Male Female

63.5 56.8

2.29 1.96

1.58 1.21

Race

White Non white

63.3 39.1

1.61 3.93

1.43 1.30

Education

Some college High school graduate All others

68.7 63.3 46.8

2.80 2.14 2.85

1.06 1.55 1.27

Applying Equation 4.13 to a weighted estimator,  f (Y) = Y^i = wi yij , j = 1, 2, . . . , c, involving c variables in a sample of n observations, Woodruff (1971) showed that     ∂f V (θ) ffi V wi yij : (4:14) ∂yj This alternative form of the linearized variance of a nonlinear estimator offers computational advantages because it bypasses the computation of the c × c covariance matrix in Equation 4.13. This convenience of converting a multistage estimation problem into a univariate problem is realized by a simple interchange of summations. This general computational procedure can be applied to a variety of nonlinear estimators, including regression coefficients (Fuller, 1975; Tepping, 1968). For a complex survey, this method of approximation is applied to PSU totals within the stratum. That is, the variance estimate is a weighted combination of the variation in Equation 4.14 across PSUs within the same stratum. These formulas are complex but can require much less computing time than the replication methods discussed above. This method can be applied to any statistic that is expressed mathematically—for example, the mean or the regression coefficient—but not to such nonfunctional statistics as the median and other percentiles. We now return to the GSS example of estimating the variance of sample proportions. Table 4.5 shows the results of applying the Taylor series method

39

to the proportion of adults approving the hitting of other adults, analyzed by gender, race, and level of education. The proportion is computed as a ratio of weighted sums of all positive responses to the sum of all the weights. Its standard error is computed applying Equation 4.14 modified to include the PSUs and strata. The design effect for the overall proportion is 1.41, which is about the same as those estimated by using the other two methods, whose results are shown in Tables 4.3 (BBR) and 4.4 (JRR). The estimated proportion varies by gender, race, and level of education. Because the subgroup sizes are small, the standard errors for the subgroups are larger than that for the overall estimate. In addition, the design effects for subgroup proportions are different from that for the overall estimate. In this chapter, we presented several methods of estimating variance for statistics from complex surveys (for further discussion, see Rust and Rao, [1996]). Examples from GSS and other surveys tend to show that the design effect is greater than one in most complex surveys. Additional examples can be found in E. S. Lee, Forthofer, and Lorimor (1986) and Eltinge, Parsons, and Jang (1997). Examples in Chapter 6 will demonstrate the importance of using one of the methods reviewed above in the analysis of complex survey data.

5. PREPARING FOR SURVEY DATA ANALYSIS The preceding chapters have concentrated on the complexity of survey designs and techniques for variance estimation for these designs. Before applying the sample weights and the methods for assessing the design effect, one must understand the survey design and the data requirements for the estimation of the statistics and the software intended to be used. These requirements are somewhat more stringent for complex survey data than for data from an SRS because of the weights and design features used in surveys. Data Requirements for Survey Analysis As discussed in Chapter 3, the weight and the design effect are basic ingredients needed for a proper analysis of survey data. In preparing for an analysis of survey data from a secondary source, it is necessary to include the weights and the identification of sampling units and strata in the working data file, in addition to the variables of interest. Because these designrelated data items are labeled differently in various survey data sources, it is important to read the documentation or consult with the source agency or person to understand the survey design and the data preparation procedures. The weights usually are available in major survey data sources. As noted earlier, the weights reflect the selection probabilities and the adjustments

40

for nonresponse and poststratification. The weights generally are expressed as expansion weights, which add up to the population size, and in certain analyses it may be more convenient to convert them to the relative weights, which add up to the sample size. In some survey data, several weight variables are included to facilitate proper use of the data, which may be segmented for different target populations or subsampled for certain data items. It is necessary to choose appropriate weights for different uses of the data, through a careful review of the documentation. For some surveys, the weight is not explicitly labeled as such, and it is necessary to study the sample design to realize the weight. As seen in Chapter 4, in the GSS the weight was derived from the number of adults in the household. It was also necessary to perform poststratification adjustments to make the demographic composition of the sample comparable to the population as in Table 3.1. If the weight is not available even after contact with the provider of the data, the user cannot assume a self-weighting sample. If the data are used without the weight, the user must take responsibility for clearly acknowledging this when reporting findings. It is hard to imagine analyzing survey data without the weight, even in model-based analysis (although used differently from the design-based analysis), when one recognizes the likelihood of unequal selection probability and differential response rates in subgroups. The calculation of the design effect usually requires information on the first-stage selection procedure, that is, the identification of strata and PSUs, although secondary sampling units and associated strata may be required for certain nested designs. If one PSU is selected from each stratum as in the GSS, the stratum identification is the same as the PSU identification. If stratification is not used or the stratum identification is not available from the data, one can perform the analysis assuming an unrestricted cluster sampling. If there is no information on the stratum and PSU, it is important to investigate whether treating the data as SRS is reasonable for the given sample design. When stratification is used and the stratum identification is available, we need to make sure that at least two PSUs are available in each stratum. Otherwise, it is not possible to estimate the variance. If only one cluster is selected from each stratum, it is necessary to pair the strata to form pseudostrata. Pairing the strata requires good understanding of the sample design. An example of a particular strategy for pairing, using the GSS, was presented in Chapter 3. In the absence of any useful information from the data document, a random pairing may be acceptable. Stanek and Lemeshow (1977) have investigated the effect of pairing based on the National Health Examination Survey and found that variance estimates for the weighted mean and combined ratio estimate were insensitive to different pairings of the strata, but this conclusion may not apply to all surveys.

41

Importance of Preliminary Analysis Survey data analysis begins with a preliminary exploration to see whether the data are suitable for a meaningful analysis. One important consideration in the preliminary examination of a secondary data source is to examine whether there is a sufficient number of observations available in the various subgroups to support the proposed analysis. Based on the unweighted tabulations, the analyst determines whether sample sizes are large enough and whether categories of the variables need to be collapsed. The unweighted tabulations also give the number of the observations with missing values and those with extreme values, which could indicate either measurement errors or errors of transcription. Although the unit nonresponse adjustment is handled by the data collection agency when developing the sample weight, the analyst must deal with the missing values (item nonresponse). With a small number of missing values, one can ignore the respondents with missing values for the analysis. Instead of completely excluding the observations with missing values, however, the design-based survey analysis requires use of the entire data set by setting the weights to zero for the observations with missing values. This is necessary to estimate the variance that is inherent in the sample design. Although the point estimate would be the same either excluding or setting the weights to zero for the observations with missing values, the estimated variance would be different. The exclusion tends to underestimate the variance. If the amount of item nonresponse is not trivial, then ignoring missing values would set too many weights to zero, and the original weighting scheme would be destroyed. This can lead to bias, and it will no longer be possible to refer accurately to the target population. One method of handling this problem is to inflate the weights of the observations without missing values to compensate for the ignored observations. When performing this type of adjustment, it generally is assumed that there is no systematic pattern among the subjects with missing values, but this assumption may not be valid. For example, if all the subjects with missing values were males or all fell into a limited age range, then it would be inappropriate simply to inflate the weight of the remaining observations. An alternative to the weight adjustment is to impute the missing values by some reasonable method, although it is not necessarily a better solution than the weight adjustment. Imputation is not a routine statistical task. There are many ways to impute missing values (Kalton & Kasprszky, 1986; Little & Rubin, 2002). It is essential to understand the mechanism that leads to missing values before choosing a particular method of imputation. In some situations, simple procedures can be used. For example, an extra category can be created for missing values for categorical variables. Another simple procedure would

42

be mean imputation for continuous variables, but this procedure will distort the shape of the distribution. To preserve the shape of the distribution, one can use hot deck imputation, regression imputation, or multiple imputation. Some simple illustrations of imputation will be presented in the next chapter without going into detailed discussion. If imputation is used, the variance estimators may need some adjustment (Korn & Graubard, 1999, sec. 5.5), but that topic is beyond the scope of this book. Prior to any substantive analysis, it is also necessary to examine whether each of the PSUs has a sufficient number of observations. It is possible that some PSUs may contain only a few observations, or even none, because of nonresponse and exclusion of missing values. A PSU with none of or only a few observations may be combined with an adjacent PSU within the same stratum. The stratum with a single PSU as a result of combining PSUs may then be combined with an adjacent stratum. However, collapsing too many PSUs and strata destroys the original sample design. The resulting data analysis may be of questionable value, because it is no longer possible to determine what population is represented by the sample resulting from the combined PSUs and strata. The number of observations that is needed in each PSU depends on the type of analysis planned. The required number will be larger for analytic studies than for estimation of descriptive statistics. A general guideline is that the number should be large enough to estimate the intra-PSU variance for the given estimate. To illustrate this point, we consider the GSS data. An unweighted tabulation by stratum and PSU has shown that the number of observations in the PSU ranges from 8 to 49, with most of the frequencies being larger than 13, indicating that the PSUs probably are large enough for estimating variances for means and proportions. For an analytic study, we may want to investigate the percentage of adults approving of hitting, by education and gender. For this analysis we need to determine if there are a number of PSUs without observations in a particular education-by-gender category. If there are many PSUs with no observation for some education-by-gender category, this calls into question the estimation of the variance–covariance matrix, which is based on the variation in the PSU totals within the strata. The education (3 levels) by gender (2 levels) tabulation by PSU showed that 42 of the 84 PSUs had at least one gender-by-education cell with no observations. Even after collapsing education into two categories, we will have to combine nearly half of the PSUs. Therefore, we should not attempt to investigate, simultaneously, the gender and education variables in relation to the question about hitting. However, it is possible to analyze gender or education alone in relation to hitting without combining many PSUs and strata. Subgroup analysis of complex survey data cannot be conducted by selecting out the observations in the analytic domain. Although the case selection

43

would not alter the basic weights, it might destroy the basic sample design. For example, selecting one small ethnic group may eliminate a portion of the PSUs and reduce the number of observations substantially in the remaining PSUs. As a result, it would be difficult to assess, from the subset, the design effect inherent in the basic design. Even though the basic design is not totally destroyed, selecting out observations from a complex survey sample may lead to an incorrect estimation of variance, as explained above in conjunction with the method of handling missing values. Correct estimation of variance requires keeping the entire data set in the analysis and assigning weights of zero to observations outside the analytic domain. The subpopulation analysis procedures available in software packages are designed to conduct a subgroup analysis without selecting out the observations in the analytic domain. The subpopulation analysis will be discussed further in Chapter 6. The first step in a preliminary analysis is to explore the basic distributions of key variables. The tabulations may point out the need for refining operational definitions of variables and for combining categories of certain variables. Based on summary statistics, one may learn about interesting patterns and distributions of certain variables in the sample. After analyzing the variables one at a time, we can next investigate the existence of relations to screen out variables that are clearly not related to one another or to some dependent variables. It may be possible to conduct a preliminary exploration using the standard graphic SRS-based statistical methods. Given the role of weights in survey data, however, any preliminary analysis ignoring the weights may not accomplish the goal of a preliminary analysis. One way to conduct a preliminary analysis taking weights into account is to select a subsample of manageable size, with selection probability proportional to the magnitude of the weights, and to explore the subsample using the standard statistical and graphic methods. This procedure will be illustrated in the first section of Chapter 6. Choices of Method for Variance Estimation Incorporating the design features into the analysis requires choosing the method of variance estimation. As discussed in Chapter 4, three methods of variance estimation (BRR, JRR, and Taylor series approximation) are used in practice. Several researchers (Bean, 1975; Frankel, 1971; Kish & Frankel, 1974; Lemeshow & Levy, 1979) have evaluated these three general methods empirically, and Krewski and Rao (1981) have performed some theoretical comparisons of these approaches. These evaluation studies tend to show that none of the three methods consistently performs better or worse, and that the choice may depend in most cases on the availability of and familiarity with the software. In a few cases, the choice may depend on the type of statistic to be estimated or the sample design used, as in the paired selection design.

44

The formula-based Taylor series approximation (linearization) is perhaps the most widely used method of variance estimation for complex surveys because it is found in most available software. It may be preferable to the replication-based methods (BRR and JRR) for practical reasons, but, as discussed in Chapter 4, it is not applicable for the median or other percentiles and nonparametric statistics. The replication-based methods are more general and can be applied with these statistics, but they require creating and handling the replicates. Another advantage of the replication approach is that it provides a simple way to incorporate adjustments for nonresponse and poststratification more appropriately. By separately computing the weighting adjustments for each replicate, it is possible to incorporate the effects of adjustments in variance estimation. For small surveys and small domain estimation, the JRR estimate may be more stable than the BRR estimate because every replicate in JRR includes most of the full sample, whereas only half of a sample is included in the BRR replicate. However, BRR is reported to be more reliable than JRR for the estimation of quartiles (Kovar et al., 1988; Rao, Wu, & Yue, 1992). A variation of BRR suggested by Fay (Judkins, 1990) is used to stabilize the variance estimator. In this variation, the replicate weights are to be inflated by factor of 2 − k or k instead of 2 or 0, and the variance estimator is to be modified by multiplying the right side of Equation 4.4 by 1/(1 − k)2 (k can take a value between 0 and 1). Korn and Graubard (1999, pp. 35–36) showed that Fay’s BRR with k = 0.3 produced about the same result as the standard BRR but produced a somewhat smaller variance when adjustments for nonresponse and poststratification were incorporated in the replicate weights. Fay’s method can be seen as a compromise between JRR and BRR. Judkins (1990) demonstrated that for estimation of quartiles and other statistics, Fay’s method with k = 0.3 performed better than either standard BRR or JRR in terms of bias and stability. The BRR method is designed for a paired selection design. When one PSU is selected from each stratum, the PSUs must be paired to create pseudo-strata in order to apply the BRR method. When more than two PSUs are selected from each stratum, it is difficult to create a paired design, and it is better to use the JRR or the Taylor series method. Procedurally, the Taylor series approximation method is the simplest, and the replication-based methods require extra steps to create replicate weights. Available Computing Resources Over the last three decades, several different programs were developed for complex survey data analysis. Early programs were developed for different purposes in government agencies, survey research organizations, and

45

universities. Some of these programs evolved into program packages for general users. Initial program packages were developed for mainframe computing applications. With the enhanced computing capability of PCs, more efforts were devoted to developing PC versions, and several new software packages emerged.3 Although some failed to implement upgrades, three program packages kept up with the current state of software standards and are user-friendly. These are SUDAAN, Stata, and WesVar. One set of programs that has been around for more than 20 years is the SUDAAN package, which is available from the Research Triangle Institute in North Carolina. It has two different versions: a stand-alone version and a SAS (Statistical Analysis System) version. The latter is especially convenient to use in conjunction with the SAS data step. As with SAS, the SUDAAN license needs to be renewed annually. The default method for variance estimation is the Taylor series method with options to use BRR and JRR. It can handle practically all types of sample designs including multiple-layered nesting and poststratified designs. It has the most comprehensive set of analytical features available for analysis of complex survey data, but it is more expensive to maintain than the other two packages. It supports a variety of statistical procedures, including CROSSTAB, DESCRIPT, RATIO, REGRESS, LOGISTIC, LOGLINK (log-linear regression), MULTILOG (multinomial and ordered logistic regression), SURVIVAL (Cox proportional hazards model), and others. Over the years, the designers have added new procedures and dropped some (e.g., CATAN for weighted least-square modeling). The SAS user may find these procedures easy to implement, but it may be difficult for new users to specify the design and to interpret the output without help from an experienced user. Consulting and technical assistance are available on the SAS Web site. Stata is a general-purpose statistical program package that includes a survey analysis module. It is available from Stata Corporation, College Station, Texas. Its survey analysis component supports a variety of analytical procedures including svymean, svytotal, svyprop (proportion), svyratio (ratio estimation), svytab (two-way tables), svyregress (regression), svylogit (logistic regression), svymlogit (multinomial logistic regression), svypois (Poisson regression), svyprobit (probit models), and others. It uses the Taylor series method for variance estimation using PSUs (ultimate cluster approximation). Although it does not support complicated designs such as multilayered nesting designs, it can be used for analyzing most of the survey designs used in practice. Most of its survey analysis procedures are parallel to its general (nonsurvey) statistical procedures, which means that many general features in its statistical analysis can be integrated easily with the survey analysis components. The output is relatively easy to understand, and new users may find Stata easier to learn than SUDAAN.

46

The WesVar program is developed and distributed by Westat, Inc., Rockville, Maryland. It is designed to compute descriptive statistics, linear regression modeling, and log-linear models using replication methods. Five different replication methods are available, including JK1 (delete-1 jackknife for unstratified designs), JK2 (jackknife for 2-per-stratum designs), JKn (delete-1 jackknife for stratified designs), BRR, and Fay (BRR using Fay’s method). It was developed at Westat, Inc. (Flyer & Mohadjer, 1988), and older versions of this package are available for users at no charge. It is now commercially available, and a student version is also available. Although data can be imported from other systems, it is designed to be a stand-alone package. In addition to the sample weights for the full sample, it requires that each record in the input data file contain the replicate weight. For simple designs, the program can create the replicate weights before running any procedure. The program documentation is adequate, but new users may find instructions for preparing the replicate weights somewhat difficult to understand without help from experienced users. Cohen (1997) evaluated early versions of these three software packages (Release 7 of SUDAAN, Release 5 of Stata, WesVarPC Version 2.02) with respect to programming effort, efficiency, accuracy, and programming capability. The evaluation showed that the WesVar procedure consistently required the fewest programming statements to derive the required survey estimates, but it required additional data preparation for the creation of replicate weights necessary for the derivation of variance estimates. Stata tended to require more program statements to obtain the same results, but it provided no undue burden to implement them. As far as computational efficiency is concerned, the SUDAAN procedure was consistently superior in generating the required estimates. It is difficult to recommend one software package over another. In choosing a software package, one should consider the method of variance estimation to be used, the cost of maintaining the software, and the strengths and limitations of the respective packages reviewed here in the context of one’s analytical requirements. Perhaps more important is the analyst’s familiarity with statistical packages. For example, for SAS users it is natural to choose SUDAAN. Stata users probably will choose to use Stata’s survey analysis component. Detailed illustrations presented in Chapter 6 utilized primarily Stata (Version 8) and SUDAAN (Release 8.0.1). These illustrations may suggest additional points to consider for choosing a software package. The availability of computing resources is getting better all the time, as more statistical packages incorporate complex survey data analysis procedures. SPSS 13.0 now provides an add-on module for survey data analysis, SPSS Complex Samples. It includes four procedures: CSDESCRIPTIVES,

47

CSTABULATE (contingency table analysis), CSGLM (regression, ANOVA, ANCOVA), and CSLOGISTIC. The SAS system also provides survey data analysis capabilities in its latest release, SAS 9.1. The SURVEYFREQ procedure produces one-way and multiway contingency table analyses with tests of association. The SURVEYLOGISTIC procedure performs logistic regression, and it can also fit other link functions. Survey data analysis often can now be done by many statistical packages currently in use without resorting to any special-purpose software. Creating Replicate Weights The BRR method requires replicate weights. These weights can be included in the data set or created prior to running any analysis. Usually H (a multiple of 4, larger than the number of strata) sets of replicate weights need to be created. For SUDAAN, the replicate weights are entered as data, and WesVar can generate them with proper specifications. For example, for a survey with 6 strata and 2 PSUs in each stratum, 8 sets of replicate weights need to be created, as shown in Table 5.1. The SUDAAN statements for BRR for this example are shown in the right side of the table. The input data consist of stratum, PSU, the number of hospital beds, the number of AIDS patients, the sample weight (wt), and 8 sets of replicate weights (w1 through w8). Note that the replicate weights are either twice the sample weights, if the units are to be selected, or zero, if not to be selected. The zero or twice the weight is arranged based on 6 rows of the 8 × 8 orthogonal matrix, similar to Table 4.2. The ratio estimate (refer again to Note 2) of the total AIDS patients is to be calculated applying the ratio of AIDS/beds to the total number of beds (2,501) in the target area. PROC RATIO with DESIGN = brr specifies the desired statistic and the method of estimating variance, and deff requests the design effect. The NEST (specifying strata and PSU) statement is not needed, because the BRR design is used. REPWGT designates the replicate weight variables. NUMER and DENOM specify the numerator and denominator of the ratio estimate. The BRR estimates by SUDAAN are shown in the lower left side of Table 5.1. The estimate is 1,191, and its standard error is 154.2 (the design effect is 2.10). For the same data, WesVar produced the same point estimate and the standard error of 155.0 by creating replicate weights based on different rows of the same orthogonal matrix. The BRR using the alternative replicate weights (1 or 0 time the sample weights) gave a standard error of 149.5. Fay’s variant of BRR uses replicate weights created by taking 2 − k or k(0 ≤ k < 1) times the sample weights, depending on whether an observation is in the selected unit or not. Using k = 0.3, the standard error was estimated to be 137.7.

48 TABLE 5.1

Creation of Replicate Weights for BRR and Jackknife Procedure (A) SUDAAN statements for BRR: data brr; input stratum psu beds aids wt w1-w8; aids2=aids*2501; datalines; 1 1 72 20 2 0 0 0 4 0 4 4 4 1 2 87 49 6 12 12 12 0 12 0 0 0 2 1 99 38 2 4 0 0 0 4 0 4 4 2 2 48 23 2 0 4 4 4 0 4 0 0 3 1 99 38 2 4 4 0 0 0 4 0 4 3 2 131 78 4 0 0 8 8 8 0 8 0 4 1 42 7 2 0 4 4 0 0 0 4 4 4 2 38 28 2 4 0 0 4 4 4 0 0 5 1 42 26 2 4 0 4 4 0 0 0 4 5 2 34 9 2 0 4 0 0 4 4 4 0 6 1 39 18 4 0 8 0 8 8 0 0 8 6 2 76 20 2 4 0 4 0 0 4 4 0 ; proc ratio design=brr deff; weight wt; repwgt w1-w8; numer aids2; denom beds; run;

(B) SUDAAN statements for jackknife method: Data jackknife; input stratum psu beds aids wt; aids2=aids*2501; datalines; 1 1 72 20 2 1 2 87 49 6 2 1 99 38 2 2 2 48 23 2 3 1 99 38 2 3 2 131 78 4 4 1 42 7 2 4 2 38 28 2 5 1 42 26 2 5 2 34 9 2 6 1 39 18 4 6 2 76 20 2 ; proc ratio design=jackknife deff; nest stratum; weight wt; numer aids2; denom beds; run;

SUDAAN output for BRR:

SUDAAN output for jackknife procedure:

The RATIO Procedure Variance Estimation Method: BRR by: Variable, One. ---------------------------------------------| Variable | | One | | | 1 | ---------------------------------------------| AIDS2/BEDS | Sample Size | 12 | | | Weighted Size | 32.00 | | | Weighted X-Sum | 2302.00 | | | Weighted Y-Sum | 2741096.00 | | | Ratio Est. | 1190.75 | | | SE Ratio | 154.15 | | | DEFF Ratio #4 | 2.10 | ----------------------------------------------

The RATIO Procedure Variance Estimation Method: Delete-1 Jackknife by: Variable, One. -----------------------------------------------| Variable | | One | | | 1 | -----------------------------------------------| AIDS2/BEDS | Sample Size | 12 | | | Weighted Size | 32.00 | | | Weighted X-Sum | 2302.00 | | | Weighted Y-Sum | 2741096.00 | | | Ratio Est. | 1190.75 | | | SE Ratio | 141.34 | | | DEFF Ratio #4 | 1.76 | ------------------------------------------------

SOURCE: Data are from Levy and Lemeshow (1999, p. 384).

The jackknife procedure does not require replicate weights. The program creates the replicates by deleting one PSU in each replicate. The SUDAAN statements for JRR and the results are shown in the right side of Table 5.1. The standard error estimated by the jackknife procedure is 141.3, which is smaller than the BRR estimate. The standard error calculated by the Taylor series method (assuming with-replacement sampling) was 137.6, slightly less than the jackknife estimate but similar to the estimate from Fay’s BRR. As discussed in Chapter 4, BRR and JRR assume with-replacement sampling. If we assume without-replacement sampling (the finite population correction is used), the standard error is estimated to be 97.3 for this example. The third National Health and Nutrition Examination Survey (NHANES III)4 from the National Center for Health Statistics (NCHS) contains the replicate weights for BRR. The replicate weights were created for Fay’s method with k = 0.3 incorporating nonresponse and poststratification adjustments at different stages of sampling. As Korn and Graubard (1999) suggested, a preferred approach is using Fay’s method of creating replicate weights

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incorporating adjustments for the nonresponse and poststratification. But such weights usually are not included in many survey data sets, nor is there appropriate information for creating such replicate weights. Searching for Appropriate Models for Survey Data Analysis* It has been said that many statistical analyses are carried out with no clear idea of the objective. Before analyzing the data, it is essential to think around the research question and formulate a clear analytic plan. As discussed in a previous section, a preliminary analysis and exploration of data are very important in survey analysis. In a model-based analysis, this task is much more formidable than in a design-based analysis. Problem formulation may involve asking questions or carrying out appropriate background research in order to get the necessary information for choosing an appropriate model. Survey analysts often are not involved in collecting the survey data, and it is often difficult to comprehend the data collection design. Asking questions about the initial design may not be sufficient, but it is necessary to ask questions about how the design was executed in the field. Often, relevant design-related information is neither documented nor included in the data set. Moreover, some surveys have overly ambitious objectives given the possible sample size. So-called general-purpose surveys cannot possibly include all the questions that are relevant to all future analysts. Building an appropriate model including all the relevant variables is a real challenge. There should also be a check on any prior knowledge, particularly when similar sets of data have been analyzed before. It is advisable not to fit a model from scratch but to see if the new data are compatible with earlier results. Unfortunately, it is not easy to find model-based analyses using complex survey data in social and health science research. Many articles dealing with the model-based analysis tend to concentrate on optimal procedures for analyzing survey data under somewhat idealized conditions. For example, most public-use survey data sets contain only strata and PSUs, and opportunities for defining additional target parameters for multilevel or hierarchical linear models (Bryk & Raudenbush, 1992; Goldstein & Silver, 1989; Korn & Graubard, 2003) are limited. The use of mixed linear models for complex survey data analysis would require further research and, we hope, stimulate survey designers to bring design and analysis into closer alignment.

6. CONDUCTING SURVEY DATA ANALYSIS This chapter presents various illustrations of survey data analysis. The emphasis is on the demonstration of the effects of incorporating the weights and the data structure on the analysis. We begin with a strategy for conducting

50

a preliminary analysis of a large-scale, complex survey. Data from Phase II of NHANES III (refer to Note 4) will be used to illustrate various analyses, including descriptive analysis, linear regression analysis, contingency table analysis, and logistic regression analyses. For each analysis, some theoretical and practical considerations required for the survey data will be discussed. The variables used in each analysis are selected to illustrate the methods rather than to present substantive findings. Finally, the model-based perspective is discussed as it relates to analytic examples presented in this chapter. A Strategy for Conducting Preliminary Analysis Sample weights can play havoc in the preliminary analysis of complex survey data, but exploring the data ignoring the weights is not a satisfactory solution. On the other hand, programs for survey data analysis are not well suited for basic data exploration. In particular, graphic methods were not designed with complex surveys in mind. In this section, we present a strategy for conducting preliminary analyses taking the weights into account. Prior to the advent of the computer, the weight was handled in various ways in data analysis. When IBM sorting machines were used for data tabulations, it was common practice to duplicate the data cards to match the weight value to obtain reasonable estimates. To expedite the tabulations of large-scale surveys, the PPS procedure was adopted in some surveys (Murthy & Sethi, 1965). Recognizing the difficulty of analyzing complex survey data, Hinkins, Oh, and Scheuren (1994) advocated an ‘‘inverse sampling design algorithm’’ that would generate a simple random subsample from the existing complex survey data, so that users could apply their conventional statistical methods directly to the subsample. These approaches are no longer attractive to survey data analysis because programs for survey analysis are now readily available. However, because there is no need to use the entire data file for preliminary analysis, the idea of subsampling by the PPS procedure is a very attractive solution for developing data for preliminary analysis. The PPS subsample can be explored by the regular descriptive and graphic methods, because the weights are already reflected in the selection of the subsample. For example, the scatterplot is one of the essential graphic methods for preliminary data exploration. One way to incorporate the weight in the scatterplot is the use of bubbles that represent the magnitude of the weight. Korn and Graubard (1998) examined alternative procedures to scatterplot bivariate data and showed advantages of using the PPS subsample. In fact, they found that ‘‘sampled scatterplots’’ are a preferred procedure to ‘‘bubble scatterplots.’’ For a preliminary analysis, we generated a PPS sample of 1,000 from the adult file of Phase II (1991–1994) of NHANES III (refer to Note 4), which

51 TABLE 6.1

Subsample and Total Sample Estimates for Selected Characteristics of U.S. Adult Population, NHANES III, Phase II

Total sample (n = 9,920)c Unweighted Weighted PPS subsample (n = 1,000) Unweighted

Mean Age

Vitamin Use

Hispanic Population

SBPa

Correlation Between Sample BMIb and SBP

46.9 years 43.6

38.4% 42.9

26.1% 5.4

125.9 mmHg 122.3

0.153 0.243

42.9

43.0

122.2

0.235

5.9

a. Systolic blood pressure b. Body mass index c. Adults 17 years of age and older

consisted of 9,920 adults. We first sorted the total sample by stratum and PSU and then selected a PPS subsample systematically using a skipping interval of 9.92 on the scale of cumulated relative weights. The sorting by stratum and PSU preserved in essence the integrity of the original sample design. Table 6.1 demonstrates the usefulness of a PPS subsample that can be analyzed with conventional statistical packages. In this demonstration, we selected several variables that are most affected by the weights. Because of oversampling of the elderly and ethnic minorities, the weighted estimates are different from the unweighted estimates for mean age and percentage of Hispanics. The weights also make a difference for vitamin use and systolic blood pressure because they are heavily influenced by the oversampled categories. The subsample estimates, although not weighted, are very close to the weighted estimates in the total sample, demonstrating the usefulness of a PPS subsample for preliminary analysis. A similar observation can be made based on the correlation between body mass index and systolic blood pressure. The PPS subsample is very useful in exploring the data without formally incorporating the weights, especially for the students in introductory courses. It is especially well suited for exploring the data by graphic methods such as scatterplot, side-by-side boxplot, and the median-trace plot. The real advantage is that the resampled data are approximately representative of the population and can be explored ignoring the weights. The point estimates from the resampled data are approximately the same as the weighted estimates in the whole data set. Any interesting patterns discovered from the resampled data are likely to be confirmed by a more complete analysis using Stata or SUDAAN, although the standard errors are likely to be different.

52

Conducting Descriptive Analysis For a descriptive analysis, we used the adult sample (17 years of age or older) from Phase II of NHANES III. It included 9,920 observations that were arranged in 23 pseudo-strata, with 2 pseudo-PSUs in each stratum. The identifications for the pseudo-strata (stra) and PSUs (psu) were included in our working data file. The expansion weights in the data file were converted to relative weights (wgt). To determine whether there were any problems in the distribution of the observations across the PSUs, an unweighted tabulation was performed. It showed that the numbers of observations available in the PSUs ranged from 82 to 286. These PSU sample sizes seem sufficiently large for further analysis. We chose to examine the body mass index (BMI), age, race, poverty index, education, systolic blood pressure, use of vitamin supplements, and smoking status. BMI was calculated by dividing the body weight (in kilograms) by the square of the height (in meters). Age was measured in years, education (educat) was measured as the number of years of schooling, the poverty index (pir) was calculated as a ratio of the family income to the poverty level, and systolic blood pressure (sbp) was measured in mmHg. In addition, the following binary variables were selected: Black (1 = black; 0 = nonblack), Hispanic (1 = Hispanic; 0 = non-Hispanic), use of vitamin supplements (vituse) (1 = yes; 0 = no), and smoking status (smoker) (1 = ever smoked; 0 = never smoked). We imputed missing values for the variables selected for this analysis to illustrate the steps of survey data analysis. Various imputation methods have been developed to compensate for missing survey data (Brick & Kalton, 1996; Heitjan, 1997; Horton & Lipsitz, 2001; Kalton & Kasprszky, 1986; Little & Rubin, 2002; Nielsen, 2003; Zhang, 2003). Several software packages are available (e.g., proc mi and proc mianalyze in SAS/STAT; SOLAS; MICE; S-Plus Missing Data Library). There are many ways to apply them to a specific data set. Choosing appropriate methods and their course of application ultimately depends on the number of missing values, the mechanism that led to missing values (ignorable or nonignorable), and on the pattern of missing values (monotone or general). It is tempting to apply sophisticated statistical procedures, but that may do more harm than good. It will be more helpful to look at concrete examples (Kalton & Kasprszky, 1986; Korn & Graubard, 1999, sec. 4.7 and chap. 9) rather than reading technical manuals. Detailed discussions of these issues are beyond the scope of this book. The following brief description is for illustrative purposes only. There were no missing values for age and ethnicity in our data. We first imputed values for variables with the fewest missing values. There were fewer than 10 missing values for vituse and smoker and about 1% of values

53

missing for educat and height. We used a hot deck5 procedure to impute values for these four variables by selecting donor observations randomly with probability proportional to the sample weights within 5-year age categories by gender. The same donor was used to impute values when there were missing values in one or more variables for an observation. Regression imputation was used for height (3.7% missing; 2.8% based on weight, age, gender, and ethnicity; and 0.9%, based on age, gender, and ethnicity), weight (2.8% missing, based on height, age, gender, and ethnicity), sbp (2.5% missing, based on height, weight, age, gender and ethnicity), and pir (10% missing, based on family size, educat, and ethnicity). About 0.5% of imputed pir values were negative, and these were set to 0.001 (the smallest pir value in the data). Parenthetically, we could have brought other anthropometric measures into the regression imputation, but our demonstration was based simply on the variables selected for this analysis. Finally, the bmi values (5.5% missing) were recalculated based on updated weight and height information. To demonstrate that the sample weight and design effect make a difference, the analysis was performed under three different options: (a) unweighted, ignoring the data structure; (b) weighted, ignoring the data structure; and (c) survey analysis, incorporating the weights and sampling features. The first option assumes simple random sampling, and the second recognizes the weight but ignores the design effect. The third option provides an appropriate analysis for the given sample design. First, we examined the weighted means and proportions and their standard errors with and without the imputed values. The imputation had inconsequential impact on point estimates and a slight reduction in estimated standard errors under the third analytic option. The weighted mean pir without imputed values was 3.198 (standard error = 0.114) compared with 3.168 (s:e: = 0.108) with imputed values. For bmi, the weighted mean was 25.948 (s:e: = 0.122) without imputation and 25.940 (s:e: = 0.118) with imputation. For other variables, the point estimates and their standard errors were identical to the third decimal point because there were so few missing values. The estimated descriptive statistics (using imputed values) are shown in Table 6.2. The calculation was performed using Stata. The unweighted statistics in the top panel were produced by the nonsurvey commands summarize for point estimates and ci for standard errors. The weighted analysis (second option) in the top panel was obtained by the same nonsurvey command with the use of [w
wgt]. The third analysis, incorporating the weights and the design features, is shown in the bottom panel. It was conducted using svyset [pweight
wgt], strata (stra), and psu (psu) for setting complex survey features and svymean for estimating the means or proportions of specified variables.

54 TABLE 6.2

Descriptive Statistics for the Variables Selected for Regression Analysis of Adults 17 Years and Older From NHANES III, Phase II (n = 9,920): An Analysis Using Stata (A) Weighted and unweighted statistics, ignoring the design features Unweighted Analysis Weighted Analysis Variable | Mean Std. Err. Mean Std. Err. Min Max ------------+---------------------------------------------------------------------bmi | 26.4465 .05392 25.9402 .05428 10.98 73.16 age | 46.9005 .20557 43.5572 .17865 17 90 black | .2982 .00459 .1124 .00317 0 1 hispanic | .2614 .00441 .0543 .00228 0 1 pir | 2.3698 .01878 3.1680 .02086 0 11.89 educat | 10.8590 .03876 12.3068 .03162 0 17 sbp | 125.8530 .20883 122.2634 .18397 81 244 vituse | .3844 .00488 .4295 .00497 0 1 smoker | .4624 .00501 .5114 .00502 0 1 ----------------------------------------------------------------------------------(B) Survey analysis, using the weights and design features . svyset [pweight=wgt], strata(stra) psu(psu) . svymean bmi age black hispanic pir educat sbp vituse smoker Survey mean estimation pweight: wgt Strata: stra PSU: psu

Number of obs(*) = 9920 Number of strata = 23 Number of PSUs = 46 Population size = 9920.06 -----------------------------------------------------------------------Mean | Estimate Std. Err. [95% Conf. Interval] Deff ---------+-------------------------------------------------------------bmi | 25.9402 .11772 25.6946 26.2013 4.9903 age | 43.5572 .57353 42.3708 44.7436 10.3067 black | .1124 .00973 .0923 .1326 9.4165 hispanic | .0543 .00708 .0397 .0690 9.6814 pir | 3.1680 .10779 2.9622 3.4328 25.6331 educat | 12.3068 .12312 12.0565 12.5671 15.0083 sbp | 122.2634 .38543 121.4010 122.980 4.1995 vituse | .4295 .01215 .4043 .4546 5.9847 smoker | .5114 .01155 .4874 .5352 5.2829 ------------------------------------------------------------------------

*Some variables contain missing values.

The statistics shown in Table 6.2 are the estimated means for the continuous variables, proportions for the binary variables, and standard errors. There are slight differences between the weighted and unweighted means/ proportions for a few variables, and the differences are considerable for some variables. The weighted proportion is more than 60% smaller than the weighted proportion for blacks and nearly 80% smaller for Hispanics, reflecting oversampling of these two ethnic groups. The weighted mean age is about 3.5 years less than the unweighted mean because the elderly also were oversampled. On the other hand, the weighted mean is considerably greater than the unweighted mean for the poverty index and for the number of years of schooling, suggesting that the oversampled minority groups are concentrated in the lower ranges of income and schooling. The weighted

55

estimate for vitamin use is also somewhat greater than the unweighted estimate. This lower estimate may reflect a lower use by minority groups. The bottom panel presents the survey estimates that reflect both the weights and design features. Although the estimated means and proportions are exactly the same as the weighed statistics in the top panel, the standard errors increase substantially for all variables. This difference is reflected in the design effect in the table (the square of the ratio of standard error in the bottom panel to that for the weighted statistic in the top panel). The large design effects for poverty index, education, and age partially reflect the residential homogeneity with respect to these characteristics. The design effects of these socioeconomic variables and age are larger than those for the proportion of blacks and Hispanics. The opposite was true in the NHANES II conducted in 1976–1980 (data presented in the first edition of this book), suggesting that residential areas are now increasingly becoming more homogeneous with respect to socioeconomic status than by ethnic status. The bottom panel also shows the 95% confidence intervals for the means and proportions. The t value used for the confidence limits is not the familiar value of 1.96 that might be expected from the sample of 9,920 (the sum of the relative weights). The reason for this is that in a multistage cluster sampling design, the degrees of freedom are based on the number of PSUs and strata, rather than the sample size, as in SRS. Typically, the degrees of freedom in complex surveys are determined as the number of PSUs sampled minus the number of strata used. In our example, the degrees of freedom are 23 (= 46 − 23) and t23, 0:975 = 2.0687; and this t value is used in all confidence intervals in Table 6.2. In certain circumstances, the degrees of freedom may be determined somewhat differently from the above general rule (see Korn & Graubard, 1999, sec. 5.2). In Table 6.3, we illustrate examples of conducting subgroup analysis. As mentioned in the previous chapter, any subgroup analysis using complex survey data should be done using the entire data set without selecting out the data in the analytic domain. There are two options for conducting proper subgroup analysis in Stata: the use of by or subpop. Examples of conducting a subgroup analysis for blacks are shown in Table 6.3. In the top panel, the mean BMI is estimated separately for nonblacks and blacks by using the by option. The mean BMI for blacks is greater than for nonblacks. Although the design effect of BMI among nonblacks (5.5) is similar to the overall design effect (5.0 in Table 6.2), it is only 1.1 among blacks. Stata also can be used to test linear combinations of parameters. The equality of the two population subgroup means can be tested using the lincom command ([bmi]1—[bmi]0, testing the hypothesis of the difference between the population mean BMI for black = 1 and the mean BMI for nonblack = 0), and the difference is statistically significant based on the

56 TABLE 6.3

Comparison of Mean Body Mass Index Between Black and Nonblack Adults 17 Years and Older, NHANES III, Phase II (n = 9,920): An Analysis Using Stata (A) . svyset [pweight=wgt], strata(stra) psu(psu) . svymean bmi, by (black) Survey mean estimation pweight: wgt Strata: stra PSU: psu

Number of obs = 9920 Number of strata = 23 Number of PSUs = 46 Population size = 9920.06 -----------------------------------------------------------------------------Mean Subpop. | Estimate Std. Err. [95% Conf. Interval] Deff ---------------+-------------------------------------------------------------bmi black==0 | 25.7738 .12925 25.5064 26.0412 5.512 black==1 | 27.2536 .17823 26.8849 27.6223 1.071 -----------------------------------------------------------------------------(B) . lincom [bmi]1-[bmi]0, deff ( 1) - [bmi]0 + [bmi]1 = 0.0 -----------------------------------------------------------------------------Mean | Estimate Std. Err. t P>|t| [95% Conf. Interval] Deff ------------+----------------------------------------------------------------(1) | 1.4799 .21867 6.77 0.000 1.0275 1.9322 1.462 ------------------------------------------------------------------------------(C) . svymean bmi, subpop(black) Survey mean estimation pweight: wgt Number of obs = 9920 Strata: stra Number of strata = 23 PSU: psu Number of PSUs = 46 Subpop.: black==1 Population size = 9920.06 -----------------------------------------------------------------------------Mean | Estimate Std. Err. [95% Conf. Interval] Deff ---------+-------------------------------------------------------------------bmi | 27.2536 .17823 26.8849 27.6223 1.071 -----------------------------------------------------------------------------(D) . svymean bmi if black==1 stratum with only one PSU detected (E) . replace (479 real . replace (485 real

stra=14 changes stra=16 changes

if stra==13 made) if stra==15 made)

. svymean bmi if black==1 Survey mean estimation pweight: wgt Strata: stra PSU: psu

Number of obs = 2958 Number of strata = 21 Number of PSUs = 42 Population size = 1115.244 -----------------------------------------------------------------------------Mean | Estimate Std. Err. [95% Conf. Interval] Deff ---------+-------------------------------------------------------------------bmi | 27.2536 .17645 26.8867 27.6206 2.782

t test. The design effect is 1.46, indicating that the t value for this test is reduced about 20% to compensate for the sample design features. Alternatively, the subpop option can be used to estimate the mean BMI for blacks, as shown in the bottom panel. This option uses the entire data set by setting the weights to zero for those outside the analytic domain. The mean, standard error, and design effect are the same as those calculated for

57

blacks using the by option in the top panel. Next, we selected out blacks by specifying the domain (if black = = 1) to estimate the mean BMI. This approach did not work because there were no blacks in some of the PSUs. The tabulation of blacks by stratum and PSU showed that only one PSU remained in the 13th and 15th strata. When these two strata are collapsed with adjacent strata, Stata produced a result. Although the point estimate is the same as before, the standard error and design effect are different. As a general rule, subgroup analysis with survey data should avoid selecting out a subset, unlike in the analysis of SRS data. Besides the svymean command for descriptive analysis, Stata supports the following descriptive analyses: svytotal (for the estimation of population total), svyratio (for the ratio estimation), and svyprop (for the estimation of proportions). In SUDAAN, these descriptive statistics can be estimated by the DESCRIPT procedure, and subdomain analysis can be accommodated by the use of the SUBPOPN statement. Conducting Linear Regression Analysis Both regression analysis and ANOVA examine the linear relation between a continuous dependent variable and a set of independent variables. To test hypotheses, it is assumed that the dependent variable follows a normal distribution. The following equation shows the type of relation being considered by these methods for i = 1, 2, . . . , n: Yi = β0 + β1 X1i + β2 X2i +    + βp Xpi + εi

(6:1)

This is a linear model in the sense that the dependent variable (Yi ) is represented by a linear combination of the βj ’s plus εi : The βj is the coefficient of the independent variable (Xj ) in the equation, and εi is the random error term in the model that is assumed to follow a normal distribution with a mean of 0 and a constant variance and to be independent of the other error terms. In regression analysis, the independent variables are either continuous or discrete variables, and the βj ’s are the corresponding coefficients. In the ANOVA, the independent variables (Xj ’s) are indicator variables (under effect coding, each category of a factor has a separate indicator variable coded 1 or 0) that show which effects are added to the model, and the βj ’s are the effects. Ordinary least squares (OLS) estimation is used to obtain estimates of the regression coefficients or the effects in the linear model when the data result from a SRS. However, several changes in the methodology are required to deal with data from a complex sample. The data now consist of the individual observations plus the sample weights and the design descriptors. As was discussed in Chapter 3, the subjects from a complex sample usually have

58

different probabilities of selection. In addition, in a complex survey the random-error terms often are no longer independent of one another because of features of the sample design. Because of these departures from SRS, the OLS estimates of the model parameters and their variances are biased. Thus, confidence intervals and tests of hypotheses may be misleading. A number of authors have addressed these issues (Binder, 1983; Fuller, 1975; Holt, Smith, & Winter, 1980; Konijn, 1962; Nathan & Holt, 1980; Pfeffermann & Nathan, 1981; Shah, Holt, & Folsom, 1977). They do not concur on a single approach to the analysis, but they all agree that the use of OLS as the estimation methodology can be inappropriate. Rather than providing a review of all these works, we focus here on an approach that covers the widest range of situations and that also has software available and widely disseminated. This approach to the estimation of the model parameters is the design-weighted least squares (DWLS), and its use is supported in SUDAAN, Stata, and other software for complex survey data analysis. The weight in the DWLS method is the sample weight discussed in Chapter 3. DWLS is slightly different from the weighted least squares (WLS) method for unequal variances, which derives the weight from an assumed covariance structure (see Lohr, 1999, chap. 12). To account for the complexities introduced by the sample design and other adjustments to the weights, one of the methods discussed in Chapter 4 may be used in the estimation of the variance–covariance matrix of the estimates of the model parameters. Because these methods use the PSU total rather than the individual value as the basis for the variance computation, the degrees of freedom for this design equal the number of PSUs minus the number of strata, instead of the sample size. The degrees of freedom associated with the sum of squares for error are then the number of PSUs minus the number of strata, minus the number of terms in the model. Table 6.4 presents the results of the multiple regression analysis of BMI on the selected independent variables under the three options of analysis. For independent variables, we used the same variables used for descriptive analysis. In addition, age squared is included to account for a possible nonlinear age effect on BMI. For simplicity, the interaction terms are not considered in this example, although their inclusion undoubtedly would have increased the R-squared, apart from a heightened multicollinearity problem. Imputed values were used in this analysis. The regression coefficients were almost the same as those obtained from the same analysis without using imputed values. The standard errors of the coefficients were also similar between the analyses with and without imputed values. The top panel shows the results of unweighted and weighted analyses ignoring the design features. The regress command is used for both the

59 TABLE 6.4

Summary of Multiple Regression Models for Body Mass Index on Selected Variables for U.S. Adults From NHANES III, Phase II (n = 9,920): An Analysis Using Stata (A) Unweighted and weighted analysis, ignoring design features Unweighted analysis Weighted analysis ------------------------------------------- ------------------------------------Source | SS df MS | SS df MS ----------+------------------------------+-----------------------------Model | 33934.57 9 3770.48 | 37811.46 9 4201.27 Residual | 252106.39 9910 25.44 | 236212.35 9910 23.84 ----------+------------------------------+-----------------------------Total | 286040.68 9919 28.84 | 274023.81 9919 27.63 -----------------------------------------------------------------------F( 9, 9910) = 148.21 F( 9, 9910) = 176.26 Prob > F = 0.0000 Prob > F = 0.0000 R-squared = 0.1186 R-squared = 0.1380 Adj R-squared = 0.1178 Adj R-squared = 0.1372 Root MSE = 5.0438 Root MSE = 4.8822 ------------------------------------------------ ------------------------------------bmi | Coef. Std. Err. t P>|t| Coef. Std. Err. t P>|t| -----------+------------------------------------ ------------------------------------age | .38422 .01462 26.27 0.000 .39778 .01528 26.03 0.000 agesq | -.00391 .00014 -27.61 0.000 -.00421 .00016 -27.06 0.000 black | 1.15938 .13178 8.80 0.000 .96291 .16108 5.98 0.000 hispanic | .70375 .14604 4.82 0.000 .64825 .22761 2.85 0.004 pir | -.14829 .03271 -4.53 0.000 -.12751 .02758 -4.62 0.000 educat | -.00913 .01680 -0.54 0.587 -.11120 .01865 -5.96 0.000 sbp | .05066 .00313 16.18 0.000 .07892 .00338 23.35 0.000 vituse | -.72097 .10752 -6.71 0.000 -.64256 .10176 -6.31 0.000 smoker | -.47851 .10456 -4.58 0.000 -.34981 .10033 -3.49 0.001 _cons | 12.70443 .49020 25.92 0.000 10.36452 .52213 19.85 0.000 ----------------------------------------------------------------------------------(B) Survey analysis using the data features . svyset [pweight=wgt], strata(stra) psu(psu) . svyregress bmi age agesq black hispanic pir educat sbp vituse smoker, deff Survey linear regression pweight: wgt Strata: stra PSU: psu

Number of obs = 9920 Number of strata = 23 Number of PSUs = 46 Population size = 9920.06 F( 9, 15) = 71.84 Prob > F = 0.0000 R-squared = 0.1380 --------------------------------------------------------------------------------------bmi | Coef. Std. Err. t P>|t| [95% Conf. Interval] Deff -----------+--------------------------------------------------------------------------age | .39778 .02110 18.85 0.000 .35412 .44143 2.0539 agesq | -.00421 .00023 -18.02 0.000 -.00469 -.00373 2.3647 black | .96291 .22418 4.30 0.000 .49916 1.42666 1.5778 hispanic | .64825 .20430 3.17 0.004 .22562 1.07087 .8897 pir | -.12751 .05624 -2.27 0.033 -.24855 -.01117 4.5323 educat | -.11203 .02703 -4.11 0.000 -.16712 -.05529 2.1457 sbp | .07892 .00514 15.35 0.000 .06828 .08956 1.8798 vituse | -.64256 .17793 -3.61 0.001 -1.01063 -.27449 3.0546 smoker | -.34982 .20405 -1.71 0.100 -.77192 .07229 4.0343 _cons | 10.36452 .80124 12.94 0.000 8.70704 12.02201 2.3041 ---------------------------------------------------------------------------------------

unweighted and weighted analyses and the weight is specified by [w
wgt] in the weighted analysis. First, our attention is called to the disappointingly low R-squared values, 0.12 in the unweighted analysis and 0.14 in the weighted analysis. It shows that most of the variation in BMI is

60

unaccounted for by the model. Other important variables are not included in this model. Perhaps the satisfactory specification of a model for predicting BMI may not be possible within the scope of NHANES III data. Both the unweighted and weighted analyses indicate that age is positively related, and age squared is negatively related, to BMI. This indicates that the age effect is curvilinear, with a dampening trend for older ages, as one might expect. The poverty index and education are negatively associated with BMI. Examining the regression coefficients for the binary variables, both blacks and Hispanics have positive coefficients, indicating that these two ethnic groups have greater BMI than their counterparts. The systolic blood pressure is positively related to BMI, and the vitamin users, who may be more concerned about their health, have a lower BMI than the nonusers. Those who have ever smoked have BMIs less than half a point lower than those who never smoked. There is a small difference between the unweighted and weighted analyses. Although the education effect is small (beta coefficient ¼ −0.009) in the unweighted analysis, it increases considerably in absolute value (beta coefficient = −0.111) in the weighted analysis. If a preliminary analysis were conducted without using the sample weights, one could have overlooked education as an important predictor. This example clearly points to the advantage of using a PPS subsample for a preliminary analysis that was discussed at the beginning of this chapter. The negative coefficient for smoking status dampens slightly, suggesting that the negative effect of smoking on BMI is more pronounced for the oversampled groups than for their counterparts. Again, the importance of sample weights is demonstrated here. The analysis also points to the advantage of using a PPS subsample for preliminary analysis rather than using unweighted analysis. The analytical results taking into account the weights and design features are shown in the bottom panel. This analysis was done using the svyregress command. The estimated regression coefficients and R2 are the same as those shown in the weighted analysis because the same formula is used in the estimation. However, the standard errors of the coefficients and the t statistics are considerably different from those in the weighted analysis. The design effects of the estimated regression coefficients ranged from 0.89 for Hispanics to 4.53 for poverty-to-income ratio. Again we see that a complex survey design may result in a larger variance for some variables than for their SRS counterparts, but not necessarily for all the variables. In this particular example, the general analytic conclusions that were drawn in the preliminary analysis also were true in the final analysis, although the standard errors for regression coefficients increased for all but one variable. Comparing the design effects in Tables 6.2 and 6.4, one finds that the design effects for regression coefficients are somewhat smaller than for

61

the means and proportions. So, applying the design effect estimated from the means and totals to regression coefficients (when the clustering information is not available from the data) would lead to conclusions that are too conservative. Smaller design effects may be possible in a regression analysis if the regression model controls for some of the cluster-to-cluster variability. For example, if part of the reason for people in the same cluster having similar BMI is similar age and education, then one would expect that adjusting for age and education in the regression model might account for some of clusterto-cluster variability. The clustering effect would then have less impact on the residuals from the model. Regression analysis can also be conducted by using the REGRESS procedure in SUDAAN as follows: PROC REGRESS DESIGN = wr; NEST stra psu; WEIGHT wgt; MODEL = bmi age agesq black hispanic pir educat sbs vituse smoker; RUN;

Conducting Contingency Table Analysis The simplest form of studying the association of two discrete variables is the two-way table. If data came from an SRS, we could use the Pearson chisquare statistic to test the null hypothesis of independence. For the analysis of a two-way table based on complex survey data, the test procedure needs to be changed to account for the survey design. Several different test statistics have been proposed. Koch, Freeman, and Freedman (1975) proposed using the Wald statistic,6 and it has been used widely. The Wald statistic usually is converted to an F statistic to determine the p value. In the F statistic, the numerator degrees of freedom are tied to the dimension of the table, and the denominator degrees of freedom reflect the survey design. Later, Rao and Scott (1984) proposed correction procedures for the log-likelihood statistic, using an F statistic with non-integer degrees of freedom. Based on a simulation study (Sribney, 1998), Stata implemented the Rao-Scott corrected statistic as the default procedure, but the Wald chi-square and the log-linear Wald statistic are still available as an option. On the other hand, SUDAAN uses the Wald statistic in its CROSSTAB procedure. In most situations, these two statistics lead to the same conclusion. Table 6.5 presents an illustration of two-way table analysis using Stata. In this analysis, the association between vitamin use (vituse) and years of education (edu) coded in three categories (1 = less than 12 years of

62 TABLE 6.5

Comparison of Vitamin Use by Level of Education Among U.S. Adults, NHANES III, Phase II (n = 9,920): An Analysis Using Stata (A) . tab vituse edu, column chi -------------------------------------------------------| edu vituse | 1 2 3 | Total -----------+---------------------------------+---------0 | 2840 1895 1372 | 6107 | 68.43 61.89 50.66 | 61.56 -----------+---------------------------------+---------1 | 1310 1167 1336 | 3813 | 31.57 38.11 49.34 | 38.44 -----------+---------------------------------+---------Total | 4150 3062 2708 | 9920 | 100.00 100.00 100.00 | 100.00 Pearson chi2(2) = 218.8510 Pr = 0.000 -------------------------------------------------------(B) . svyset [pweight=wgt], strata(stra) psu(psu) . svytab vituse edu, column ci pearson wald pweight: Strata: PSU:

wgt stra psu

Number of obs = 9920 Number of strata = 23 Number of PSUs = 46 Population size = 9920.06 ---------------------------------------------------------------------| edu vituse | 1 2 3 Total ----------+----------------------------------------------------------0 | .6659 .6018 .4834 .5705 | [.6307,.6993] [.5646,.6379] [.4432,.5237] [.5452,.5955] 1 | .3341 .3982 .5166 .4295 | [.3007,.3693] [.3621,.4354] [.4763,.5568] [.4045,.4548] Total | 1 1 1 1 ---------------------------------------------------------------------Key: column proportions [95% confidence intervals for column proportions] Pearson: Uncorrected chi2(2) = 234.0988 Design-based F(1.63, 37.46) = 30.2841 P = 0.0000 Wald (Pearson): Unadjusted chi2(2) = 51.9947 Adjusted F(2, 22) = 24.8670 P = 0.0000 ---------------------------------------------------------------------(C) . svyset [pweight=wgt], strata(stra) psu(psu) . svytab vituse edu, subpop(hispanic) column ci wald pweight: Strata: PSU:

wgt stra psu

Number of obs = 9920 Number of strata = 23 Number of PSUs = 46 Population size = 9920.06 Subpop.: hispanic==1 Subpop. no. of obs = 2593 Subpop. size = 539.043 ---------------------------------------------------------------------| edu vituse | 1 2 3 Total ----------+----------------------------------------------------------0 | .7382 .6728 .5593 .6915 | [.6928,.7791] [.6309,.7122] [.4852,.6309] [.6509,.7293] 1 | .2618 .3272 .4407 .3085 | [.2209,.3072] [.2878,.3691] [.3691,.5148] [.2707,.3491] Total | 1 1 1 1 ---------------------------------------------------------------------Key: column proportions [95% confidence intervals for column proportions] Wald (Pearson): Unadjusted chi2(2) = 47.1625 Adjusted F(2, 22) = 22.5560 P = 0.0000 ----------------------------------------------------------------------

education; 2 = 12 years; 3 = more than 12 years). In Panel A, the ordinary chi-square analysis is performed ignoring the weights and the data structure. There is a statistically significant relation between education and use

63

of vitamins, with those having a higher education being more inclined to use vitamins. The percentage of vitamin users varies from 32% in the lowest level of education to 49% in the highest level. Panel B shows the analysis of the same data taking the survey design into account. The weighted percentage of vitamin users by the level of education varies slightly more than in the unweighted percentages, ranging from 33% in the first level of education to 52% in the third level of education. Note that with the request of ci, Stata can compute confidence intervals for the cell proportions. In this analysis, both Pearson and Wald chi-square statistics are requested. The uncorrected Pearson chi-square, based on the weighed frequencies, is slightly larger than the chi-square value in Panel A, reflecting the slightly greater variation in the weighted percentages. However, a proper p value reflecting the complex design cannot be evaluated based on the uncorrected Pearson chi-square statistic. A proper p value can be evaluated from the design-based F statistic of 30.28 with 1.63 and 37.46 degrees of freedom, which is based on the test procedure as a result of the Rao-Scott correction. The unadjusted Wald chi-square test statistic is 51.99, but a proper p value must be determined based on the adjusted F statistic. The denominator degrees of freedom in both F statistics reflect the number of PSUs and strata in the sample design. The adjusted F statistic is only slightly smaller than the Rao-Scott F statistic. Either one of these test statistics would lead to the same conclusion. In Panel C, the subpopulation analysis is performed for the Hispanic population. Note that the entire data file is used in this analysis. The analysis is based on 2,593 observations, but it represents only 539 people when the sample weights are considered. The proportion of vitamin users among Hispanics (31%) is considerably lower than the overall proportion of vitamin users (43%). Again, there is a statistically significant relation between education and use of vitamins among Hispanics, as the adjusted F statistic indicates. Let us now look at a three-way table. Using the NHANES III, Phase II adult sample data, we will examine gender difference in vitamin use across the levels of education. This will be a 2 × 2 × 3 table, and we can perform a two-way table analysis at each level of education. Table 6.6 shows the analysis of three 2 × 2 tables using SAS and SUDAAN. The analysis ignoring the survey design is shown in the top panel of the table. At the lowest level of education, the percentage of vitamin use for males is lower than for females, and the chi-square statistic suggests the difference is statistically significant. Another way of examining the association in a 2 × 2 table is the calculation of the odds ratio. In this table, the odds of using vitamins for males is 0.358 [¼ 0.2634/ (1 − 0.2634)], and for females it is 0.567 [¼ 0.3617/(1 − 0.3617)]. The ratio of male odds over female odds is 0.63 (¼ 0.358/0.567), indicating that

64 TABLE 6.6

Analysis of Gender Difference in Vitamin Use by Level of Education Among U.S. Adults, NHANES III, Phase II (n = 9,920): An Analysis Using SAS and SUDAAN (A) Unweighted analysis by SAS: proc freq; tables edu*sex*vituse / nopercent nocol chisq measures cmh; run; [Output summarized below] Level of education: Vitamin use status: Gender-

Male: Female:

Chi-square: P-value: Odds ratio: 95% CI:

Less than H.S. (n) User (1944) (2206)

26.34% 36.17

H.S. graduate (n) User (1197) (1865)

46.29 <.0001 0.63 (0.56, 0.72)

31.91% 42.09

32.02 <.0001 0.64 (0.56, 0.75)

Some college (n) User (1208) (1500)

43.54% 54.00

29.27 <.0001 0.66 (0.56, 0.76)

CMH chi-square: 107.26 (p<.0001) CMH common odds ratio: 0.64 95%CI: (0.59, 0.70) -------------------------------------------------------(B) Survey analysis by SUDAAN: proc crosstab design=wr; nest stra psu; weight wgt; subgroup edu sex vituse; levels 3 2 2; tables edu*sex*vituse; print nsum wsum rowper cor upcor lowcor chisq chisqp cmh cmhpval; run; [Output summarized below] Level of education: Vitamin use status: Gender-

∗

Male: Female:

Less than H.S. (n)* User (1274.9) (1299.7)

28.04% 38.62

H.S. graduate (n)* User (1432.2) (1879.4)

32.66% 45.43

Some college (n)* User (2031.1) (2002.7)

45.62% 57.37

Chi-square:

19.02

38.01

10.99

P-value: Odds ratio: 95% CI:

.0002 0.62 (0.50, 0.77)

<.0001 0.58 (0.49, 0.69)

.0030 0.62 (0.48, 0.80)

CMH chi-square:

42.55

(p<.0001)

Weighted sum

the males’ odds of taking vitamins are 63% of the females’ odds. The 95% confidence interval does not include 1, suggesting that the difference is statistically significant. The odds ratios are consistent across three levels of education. Because the ratios are consistent, we can combine 2 × 2 tables across the three levels of education. We can then calculate the CochranMantel-Haenszel (CMH) chi-square (df = 1) and the CMH common odds ratio. The education-adjusted odds ratio is 0.64, and its 95% confidence interval does not include 1. The lower panel of Table 6.6 shows the results of using the CROSSTAB procedure in SUDAAN to perform the same analysis, taking the survey design into account. On the PROC statement, DESIGN = wr designates with-replacement sampling, meaning that the finite population correction is

65

not used. The NEST statement designates the stratum and PSU variables. The WEIGHT statement gives the weight variable. The SUBGROUP statement declares three discrete variables, and the LEVELS statement specifies the number of levels in each discrete variable. The TABLES statement defines the form of contingency table. The PRINT statement requests nsum (frequencies), wsum (weighted frequencies), rowper (row percent), cor (crude odds ratio), upcor (upper limit of cor), lowcor (lower limit of cor), chisq (chi-square statistic), chisqp (p value for chi-square statistic), cmh (CMH statistic), and cmhpval (p value for CMH). The weighted percentages of vitamin use are slightly different from the unweighted percentages. The Wald chi-square values in three separate analyses are smaller than the Pearson chi-square values in the upper panel except for the middle level of education. Although the odds ratio remained almost the same at the lower level of education, it decreased somewhat at the middle and higher levels of education. The CROSSTAB procedure in SUDAAN did not compute the common odds ratio, but it can be obtained from a logistic regression analysis to be discussed in the next section. Conducting Logistic Regression Analysis The linear regression analysis presented earlier may not be useful to many social scientists because many of the variables in social science research generally are measured in categories (nominal or ordinal). A number of statistical methods are available for analyzing categorical data, ranging from basic cross-tabulation analysis, shown in the previous section, to generalized linear models with various link functions. As Knoke and Burke (1980) observed, the modeling approach revolutionized contingency table analysis in the social sciences, casting aside most of the older methods for determining relationships among variables measured at discrete levels. Two approaches have been widely used by social scientists: log-linear models using the maximum likelihood approach (Knoke & Burke, 1980; Swafford, 1980) and the weighted least square approach (Forthofer & Lehnen, 1981; Grizzle, Starmer, & Koch, 1969). The use of these two methods for analyzing complex survey data was illustrated in the first edition of this book. In these models, the cell proportions or functions of them (e.g., natural logarithm in the log-linear model) are expressed as a linear combination of effects that make up the contingency table. Because these methods are restricted to contingency tables, continuous independent variables cannot be included in the analysis. In the past decade, social scientists have begun to use logistic regression analysis more frequently because of its ability to incorporate a larger

66

number of explanatory variables, including continuous variables (Aldrich & Nelson, 1984; DeMaris, 1992; Hosmer & Lemeshow, 1989; Liao, 1994). Logistic regression and other generalized linear models with different link functions are now implemented in software packages for complex survey data analysis. Survey analysts can choose the most appropriate model from an array of models. The application of the logit model is illustrated below, using Stata and SUDAAN. The ordinary linear regression analysis represented by Equation 6.1 examines the relationship between a continuous dependent variable and one or more independent variables. The logistic regression is a method for examining the association of a categorical outcome with a number of independent variables. The following equation shows the type of modeling a binary outcome variable Y for i = 1, 2, . . . , n: log[πi /(1 − πi )] = β0 + β1 x1i + β2 x2i +    + βp−1 xp−1, i :

(6:2)

In Equation 6.2, πi is the probability that yi = 1. This is a generalized linear model with a link function of the log odds or logit. Instead of least square estimation, the maximum likelihood approach (see Eliason, 1993) is used to estimate the parameters. Because the simultaneous equations to be solved are nonlinear with respect to the parameters, iterative techniques are used. Maximum likelihood theory also offers an estimator of the covariance matrix of the estimated β’s, assuming individual observations are random and independent. Just as in the analysis of variance model, if a variable has l levels, we only use l − 1 levels in the model. We shall measure the effects of the l − 1 levels from the effect of the omitted or reference level of the variable. The esti^ is the difference in logit between the level in the model and the mated β(β) omitted level, that is, the natural log of odds ratio of the level in the model ^

over the level omitted. Thus, taking eβ gives the odds ratio adjusted for other variables in the model. The results of logistic regression usually are summarized and interpreted as odds ratios (see Liao, 1994, chap. 3). With complex survey data, the maximum likelihood estimation needs to be modified, because each observation has a sample weight. The maximum likelihood solution incorporating the weights is generally known as pseudo or weighted maximum likelihood estimation (Chambless & Boyle, 1985; Roberts, Rao, & Kumar, 1987). Whereas the point estimates are calculated by the pseudo likelihood procedure, the covariance matrix of the estimated ^ is calculated by one of the methods discussed in Chapter 4. As discussed β’s earlier, the approximate degrees of freedom associated with this covariance matrix are the number of PSUs minus the number of strata. Therefore, the

67

standard likelihood-ratio test for model fit should not be used with the survey logistic regression analysis. Instead of the likelihood-ratio test, the adjusted Wald test statistic is used. The selection and inclusion of appropriate predictor variables for a logistic regression model can be done similarly to the process for linear regression. When analyzing a large survey data set, the preliminary analysis strategy described in the earlier section is very useful in preparing for a logistic regression analysis. To illustrate logistic regression analysis, the same data used in Table 6.6 are analyzed using Stata. The analytical results are shown in Table 6.7. The Stata output is edited somewhat to fit into a table. The outcome variable is vitamin use (vituse), and explanatory variables are gender (1 = male; 0 = female) and level of education (edu). The interaction term is not included in this model, based on the CMH statistic shown in Table 6.6. First, we performed standard logistic regression analysis, ignoring the weight and design features. The results are shown in Panel A. Stata automatically performs the effect (or dummy) coding for discrete variables with the use of the xi option preceding the logit statement and adding i. in front of the variable name. The output shows the omitted level of each discrete variable. In this case, the level ‘‘male’’ is in the model, and its effect is measured from the effect of ‘‘female,’’ the reference level. For education, being less than a high school graduate is the reference level. The likelihood-ratio chi-square value is 325.63 (df = 3) with p value of < 0.00001, and we reject the hypothesis that gender and education together have no effect on vitamin usage, suggesting that there is a significant effect. However, the pseudo R2 suggests that most of the variation in vitamin use is unaccounted for by these two variables. The parameter estimates for gender and education and their estimated standard errors are shown as well as the corresponding test statistics. All factors are significant. Including or in the model statement produces the odds ratios instead of beta coefficients. The estimated odds ratio for males is 0.64, meaning that the odds of taking vitamins for a male is 64% of the odds that a female uses vitamins after adjusting for education. This odds ratio is the same as the CMH common odds ratio shown in Table 6.6. The significance of the odds ratio can be tested using either a z test or a confidence interval. The odds ratio for the third level of education suggests that persons with some college education are twice likely to take vitamins than those with less than 12 years of education for the same gender. None of the confidence intervals includes 1, suggesting that all effects are significant. Panel B of Table 6.7 shows the goodness-of-fit statistic (chi-square with df = 2). The large p value suggests that the main effects model fits the data (not significantly different from the saturated model). In this simple situation,

68 TABLE 6.7

Logistic Regression Analysis of Vitamin Use on Gender and Level of Education Among U.S. Adults, NHANES III, Phase II (n = 9,920): An Analysis Using Stata (A) Standard logistic regression (unweighted, ignoring sample design): . xi: logit vituse i.male i.edu i.male i.edu Iteration Iteration Iteration Iteration Logit estimates

(naturally coded; _Imale_0 omitted) (naturally coded; _Iedu_1 omitted) -6608.3602 -6445.8981 -6445.544 -6445.544 Number of obs = 9920 LR chi2(3) = 325.63 Prob > chi2 = 0.0000 Log likelihood = -6445.544 Pseudo R2 = 0.0246 --------------------------------------------------------------------------------------------vituse | Coef. Std. Err. z P>|z| [95% Conf.Int.] Odds Ratio [95% Conf.Int.] -----------+-------------------------------------------------------------------------------_Imale_1 | -.4418 .0427 -10.34 0.000 -.5256 -.3580 .6429 .5912 .6990 _Iedu_2 | .2580 .0503 5.12 0.000 .1593 .3566 1.2943 1.1773 1.4285 _Iedu_3 | .7459 .0512 14.56 0.000 .6455 .8462 2.1082 1.9069 2.3308 _cons | -.5759 .0382 -15.07 0.000 -.6508 -.5010 --------------------------------------------------------------------------------------------0: 1: 2: 3:

log log log log

_Imale_0-1 _Iedu_1-3 likelihood likelihood likelihood likelihood

= = = =

(B) Testing goodness-of-fit: . lfit Logistic model for vit, goodness-of-fit test number of observations = 9920 number of covariate patterns = 6 Pearson chi2(2) = 0.16 Prob > chi2 = 0.9246 (C) Survey logistic regression (incorporating the weights and design features): . svyset [pweight=wgt], strata (stra) psu(psu) . xi: svylogit vituse i.male i.edu i.male i.edu

_Imale_0-1 _Iedu_1-3

(naturally coded; _Imale_0 omitted) (naturally coded; _Iedu_1 omitted)

Survey logistic regression pweight: wgt Strata: stra PSU: psu

Number of obs = 9920 Number of strata = 23 Number of PSUs = 46 Population size = 9920.06 F( 3, 21) = 63.61 Prob > F = 0.0000 ------------------------------------------------------------------------------------------------vituse | Coef. Std. Err. t P>|t| [95% Conf. Int.] Deff Odds Rat.[95% Conf. Int.] ------------+-----------------------------------------------------------------------------------_Imale_1 | -.4998 .0584 -8.56 0.000 -.6206 -.3791 1.9655 .6066 .5376 .6845 _Iedu_2 | .2497 .0864 2.89 0.008 .0710 .4283 2.4531 1.2836 1.0736 1.5347 _Iedu_3 | .7724 .0888 8.69 0.000 .5885 .9562 2.8431 2.1649 1.8013 2.6019 _cons | -.4527 .0773 -5.86 0.000 -.6126 -.2929 2.8257 ------------------------------------------------------------------------------------------------(D) Testing linear combination of coefficients: . lincom _Imale_1+_Iedu_3, or ( 1) _Imale_1 + _Iedu_3 = 0.0 -----------------------------------------------------------------------------vituse | Odds Ratio Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------(1) | 1.3132 .1518 2.36 0.027 1.0340 1.6681 ------------------------------------------------------------------------------

the two degrees of freedom associated with the goodness of fit of the model can also be interpreted as the two degrees of freedom associated with the gender-by-education interaction. Hence, there is no interaction of gender and education in relation to the proportion using vitamin supplements, confirming the CMH analysis shown in Table 6.6.

69

Panel C of Table 6.7 shows the results of logistic regression analysis for the same data, with the survey design taken into account. The log likelihood is not shown because the pseudo likelihood is used. Instead of a likelihoodratio statistic, the F statistic is used. Again, the p value suggests that the main effects model is a significant improvement over the null model. The estimated parameters and odds ratios changed slightly because of the sample weights, and the estimated standard errors of beta coefficients increased as reflected in the design effects. Despite the increased standard errors, the beta coefficients for gender and education levels are significantly different from 0. The odds ratio for males adjusted for education decreased to 0.61 from 0.64. Although the odds ratio remained about the same for the second level of education, its p value increased considerably, to 0.008 from < 0.0001, because of the design taken into account. After the logistic regression model was run, the effect of linear combination of parameters was tested as shown in Panel D. We wanted to test the hypothesis that the sum of parameters for male and the third level of education is zero. Because there is no interaction effect, the resulting odds ratio of 1.3 can be interpreted as indicating that the odds of taking vitamin for males with some college education are 30% higher than the odds for the reference (females with less than 12 years of education). SUDAAN also can be used to perform a logistic regression analysis, using its LOGISTIC procedure in the stand-alone version or the RLOGIST procedure in the SAS callable version (a different name used to distinguish it from the standard logistic procedure in SAS). Finally, the logistic regression model also can be used to build a prediction model for a synthetic estimation. Because most health surveys are designed to estimate the national statistics, it is difficult to estimate health characteristics for small areas. One approach to obtain estimates for small areas is the synthetic estimation utilizing the national health survey and demographic information of local areas. LaVange, Lafata, Koch, and Shah (1996) estimated the prevalence of activity limitation among the elderly for U.S. states and counties using a logistic regression model fit to the National Health Interview Survey (NHIS) and Area Resource File (ARF). Because the NHIS is based on a complex survey design, they used SUDAAN to fit a logistic regression model to activity limitation indicators on the NHIS, supplemented with county-level variables from ARF. The model-based predicted probabilities were then extrapolated to calculate estimates of activity limitation for small areas. Other Logistic Regression Models The binary logistic regression model discussed above can be extended to deal with more than two response categories. Some such response categories are ordinal, as in perceived health status: excellent, good, fair, and poor.

70

Other response categories may be nominal, as in religious preferences. These ordinal and nominal outcomes can be examined as functions of a set of discrete and continuous independent variables. Such modeling can be applied to complex survey data, using Stata or SUDAAN. In this section, we present two examples of such analyses without detailed discussion and interpretation. For details of the models and their interpretation, see Liao (1994). To illustrate the ordered logistic regression model, we examined obesity categories based on BMI. Public health nutritionists use the following criteria to categorize BMI for levels of obesity: obese (BMI ≥ 30), overweight (25 ≤ BMI < 30), normal (18.5 ≤ BMI < 25), and underweight (BMI < 18.5). Based on NHANES III, Phase II data, 18% of U.S. adults are obese, 34% overweight, 45% normal, and 3% underweight. We want to examine the relationship between four levels of obesity (bmi2: 1 = obese, 2 = overweight, 3 = normal, and 4 = underweight) and a set of explanatory variables including age (continuous), education (edu), black, and Hispanic. For the four ordered categories of obesity, the following three sets of probabilities are modeled as functions of explanatory variables: Prfobeseg versus Prfall other levelsg Prfobese plus overweightg versus Prfnormal plus underweightg Prfobese plus overweight plus normalg versus Prfunderweightg Then three binary logistic regression models could be used to fit a separate model to each of three comparisons. Recognizing the natural ordering of obesity categories, however, we could estimate the ‘‘average’’ effect of explanatory variables by considering the three binary models simultaneously, based on the proportional odds assumption. What is assumed here is that the regression lines for the different outcome levels are parallel to each other and that they are allowed to have different intercepts (this assumption needs to be tested using the chi-square statistic; the test result is not shown in the table). The following represents the model for j = 1, 2, . . . , c − 1 (c is the number of categories in the dependent variable): ! p X Pr(category ≤j ) β i xi (6:3) = αj + log Pr(category ≥ ðj+1Þ ) i=1 ^ From this model, we estimate (c − 1) intercepts and a set of β’s. Table 6.8 shows the result of the above analysis using SUDAAN. The SUDAAN statements are shown at the top. The first statement, PROC MULTILOG, specifies the procedure. DESIGN, NEST, and WEIGHT specifications are the same as in Table 6.6. REFLEVEL declares the first level of education as the reference (the last level is used as the reference if not

71 TABLE 6.8

Ordered Logistic Regression Analysis of Obesity Levels on Education, Age, and Ethnicity Among U.S. Adults, NHANES III, Phase II (n = 9,920): An Analysis Using SUDAAN proc multilog design=wr; nest stra psu; weight wgt; reflevel edu=1; subgroup bmi2 edu; levels 4 3; model bmi2=age edu black hispanic/ cumlogit; setenv decwidth=5; run; Independence parameters have converged in 4 iterations -2*Normalized Log-Likelihood with Intercepts Only: 21125.58 -2*Normalized Log-Likelihood Full Model : 20791.73 Approximate Chi-Square (-2*Log-L Ratio) : 333.86 Degrees of Freedom : 5 Variance Estimation Method: Taylor Series (WR) SE Method: Robust (Binder, 1983) Working Correlations: Independent Link Function: Cumulative Logit Response variable: BMI2 ---------------------------------------------------------------------BMI2 (cum-logit), Independent P-value Variables and Beta T-Test Effects Coeff. SE Beta T-Test B=0 B=0 ---------------------------------------------------------------------BMI2 (cum-logit) Intercept 1 -2.27467 0.11649 -19.52721 0.00000 Intercept 2 -0.62169 0.10851 -5.72914 0.00001 Intercept 3 2.85489 0.11598 24.61634 0.00000 AGE 0.01500 0.00150 9.98780 0.00000 EDU 1 0.00000 0.00000 . . 2 0.15904 0.10206 1.55836 0.13280 3 -0.20020 0.09437 -2.12143 0.04488 BLACK 0.49696 0.08333 5.96393 0.00000 HISPANIC 0.55709 0.06771 8.22744 0.00000 ---------------------------------------------------------------------------------------------------------------------------Contrast Degrees of P-value Freedom Wald F Wald F ------------------------------------------------------OVERALL MODEL 8.00000 377.97992 0.00000 MODEL MINUS INTERCEPT 5.00000 36.82064 0.00000 AGE 1.00000 99.75615 0.00000 EDU 2.00000 11.13045 0.00042 BLACK 1.00000 35.56845 0.00000 HISPANIC 1.00000 67.69069 0.00000 ----------------------------------------------------------------------------------------------------------------BMI2 (cum-logit), Independent Variables and Lower 95% Upper 95% Effects Odds Ratio Limit OR Limit OR ----------------------------------------------------------AGE 1.01511 1.01196 1.01827 EDU 1 1.00000 1.00000 1.00000 2 1.17239 0.94925 1.44798 3 0.81857 0.67340 0.99503 BLACK 1.64372 1.38346 1.95295 HISPANIC 1.74559 1.51743 2.00805 -----------------------------------------------------------

72

specified). The categorical variables are listed on the SUBGROUP statement, and the number of categories of each of these variables is listed on the LEVELS statement. The MODEL statement specifies the dependent variable, followed by the list of independent variables. The keyword CUMLOGIT on the model statement fits a proportional odds model. Without this keyword, SUDAAN fits the multinomial logistic regression model that will be discussed in the next section. Finally, SETENV statement requests five decimal points in printing the output. The output shows three estimates of intercepts and one set of beta coefficients for independent variables. The statistics in the second box indicate that main effects are all significant. The odds ratios in the third box can be interpreted in the same manner as in the binary logistic regression. Hispanics have 1.7 times higher odds of being obese than non-Hispanics, controlling for the other independent variables. Before interpreting these results, we must check whether the proportional odds assumption is met, but the output does not give any statistic for checking this assumption. To check this assumption, we ran three ordinary logistic regression analyses (obese vs. all other, obese plus overweight vs. normal plus underweight, and obese plus overweight plus normal vs. underweight). The three odds ratios for age were 1.005, 1.012, and 1.002, respectively, and they are similar to the value of 1.015 shown in the bottom section of Table 6.8. The odds ratios for other independent variables also were reasonably similar, and we have concluded that the proportional odds assumption seems to be acceptable. Stata also can be used to fit a proportional odds model using its svyolog procedure, but Stata fits a slightly different model. Whereas the set of βi xi ’s is added to the intercept in Equation 6.3, it is subtracted in the Stata model. Thus, the estimated beta coefficients from Stata carry the sign opposite from those of SUDAAN, while the absolute values are the same. This means that the odds ratios from Stata are the reciprocal of odds ratios estimated from SUDAAN. The two programs give identical intercept estimates. Stata uses the term cut instead of intercept. For nominal outcome categories, a multinomial logistic regression model can be used. Using this model, we can examine the relationship between a multilevel nominal outcome variable (no ordering is recognized) and a set of explanatory variables. The model designates one level of the outcome as the base category and estimates the log of the ratio of the probability being in the j-th category relative to the base category. This ratio is called the relative risk, and the log of this ratio is known as the generalized logit. We used the same obesity categories used above. Although we recognized the ordering of obesity levels previously, we considered it as a nominal variable this time because we were interested in comparing the levels of obesity to the normal category. Accordingly, we coded the obesity levels differently

73

[bmi3: 1 = obese, 2 = overweight, 3 = underweight, and 4 = normal (the base)]. We used three predictor variables including age (continuous variable), sex [1 = male (reference); 2 = female], and current smoking status [csmok: 1 = current smoker; 2 = never smoked (reference); 3 = previous smoker]. The following equations represent the model:   Pr(obese) = β0,1 + β1,1 (age) + β2,1 (male) log Pr(normal) + β3,1 (p:smo ker ) + β4,1 (p:smo ker )  log

 Pr(overweight) = β0,2 + β1,2 (age) + β2,2 (male) Pr(normal) + β3,2 (c:smo ker ) + β4,2 (p:smo ker )

  Pr(underweight) = β0,3 + β1,3 (age) + β2,3 (male) log Pr(normal) + β3,3 (c:smo ker )

(6:4)

+ β4,3 (p:smo ker ) We used SUDAAN to fit the above model, and the results are shown in Table 6.9 (the output is slightly edited to fit into a single table). The SUDAAN statements are similar to the previous statements for the proportional odds model except for omitting CUMLOGIT on the MODEL statement. The svymlogit procedure in Stata can also fit the multinomial regression model. Table 6.9 shows both beta coefficients and relative risk ratios (labeled as odds ratios). Standard errors and the p values for testing β = 0 also are shown. Age is a significant factor in comparing obese versus normal and overweight versus normal, but not in comparing underweight versus normal. Although gender makes no difference in comparing obese and normal, it makes a difference in the other two comparisons. Looking at the table of odds ratios, the relative risk ratio of being overweight to normal for males is more than 2 times as great as for females, provided age and smoking status are the same. The relative risk of being obese to normal for current smokers is only 0.68% of those who never smoked, holding age and gender constant. Available software also supports other statistical models that can be used to analyze complex survey data. For example, SUDAAN supports Cox’s regression model (proportional hazard model) for a survival analysis, although cross-sectional surveys seldom provide longitudinal data. Other generalized linear models defined by different link functions also can be applied to complex survey data, using the procedures supported by SUDAAN, Stata, and other programs.

74 TABLE 6.9

Multinomial Logistic Regression Analysis of Obesity on Gender and Smoking Status Among U.S. Adults, NHANES III, Phase II (n = 9,920): An Analysis Using SUDAAN proc multilog design=wr; nest stra psu; weight wgt; reflevel csmok=2 sex=2; subgroup bmi2 csmok sex; levels 4 3 2; model bmi3=age sex csmok; setenv decwidth=5; run; Independence parameters have converged in 6 iterations Approximate ChiSquare (-2*Log-L Ration) : 587.42 Degrees of Freedom : 12 Variance Estimation Method: Taylor Series (WR) SE Method: Robust (Binder, 1983) Working Correlations: Independent Link Function: Generalized Logit Response variable: BMI3 ------------------------------------------------------------------------------------------------| BMI3 log-odds)| | Independent Variables and Effects | | | | Intercept | AGE | SEX = 1 | CSMOK = 1 | CSMOK = 3 | ------------------------------------------------------------------------------------------------| 1 vs 4 | Beta Coeff. | -1.33334 | 0.01380 | 0.08788 | -0.39015 | -0.27372 | | | SE Beta | 0.14439 | 0.00214 | 0.12509 | 0.07203 | 0.13206 | | | T-Test B=0 | -9.23436 | 6.43935 | 0.70251 | -5.41617 | 2.07277 | | | P-value | 0.00000 | 0.00000 | 0.48941 | 0.00002 | 0.04958 | ------------------------------------------------------------------------------------------------| 2 vs 4 | Beta Coeff. | -1.25883 | 0.01527 | 0.76668 | -0.24271 | -0.02006 | | | SE Beta | 0.13437 | 0.00200 | 0.08275 | 0.11067 | 0.09403 | | | T-Test B=0 | -9.36835 | 7.64830 | 9.26512 | -2.19307 | -0.21335 | | | P-value | 0.00000 | 0.00000 | 0.00000 | 0.03868 | 0.83293 | ------------------------------------------------------------------------------------------------| 3 vs 4 | Beta Coeff. | -2.07305 | -0.01090 | -1.16777 | 0.33434 | 0.04694 | | | SE Beta | 0.48136 | 0.00742 | 0.25280 | 0.30495 | 0.26168 | | | T-Test B=0 | -4.30663 | -1.46824 | -4.61937 | 1.09637 | 0.17936 | | | P-value | 0.00026 | 0.15558 | 0.00012 | 0.28427 | 0.85923| ------------------------------------------------------------------------------------------------------------------------------------------------------Contrast Degrees of P-value Freedom Wald F Wald F ------------------------------------------------------OVERALL MODEL 15.00000 191.94379 0.00000 MODEL MINUS INTERCEP 12.00000 68.70758 0.00000 INTERCEPT . . . AGE 3.00000 22.97518 0.00000 SEX 3.00000 64.83438 0.00000 CSMOK 6.00000 6.08630 0.00063 ----------------------------------------------------------------------------------------------------------------------------------------------------| BMI3(log-odds)| | Independent Variables and Effects | | | | Intercept | AGE | SEX = 1 | CSMOK = 1 | CSMOK = 3 | ----------------------------------------------------------------------------------------------| 1 vs 4 | Odds Ratio | 0.26360 | 1.01390 | 1.09186 | 0.67695 | 0.76054 | | | Lower 95% Limit | 0.19553 | 1.00941 | 0.84291 | 0.58323 | 0.57874 | | | Upper 95% Limit | 0.35535 | 1.01840 | 1.41433 | 0.78573 | 0.99946 | ----------------------------------------------------------------------------------------------| 2 vs 4 | Odds Ratio | 0.28399 | 1.01539 | 2.15260 | 0.78450 | 0.98014 | | | Lower 95% Limit | 0.21507 | 1.01120 | 1.81393 | 0.62397 | 0.80689 | | | Upper 95% Limit | 0.37499 | 1.01959 | 2.55450 | 0.98633 | 1.19059 | ----------------------------------------------------------------------------------------------| 3 vs 4 | Odds Ratio | 0.12580 | 0.98916 | 0.31106 | 1.39702 | 1.04805 | | | Lower 95% Limit | 0.04648 | 0.97408 | 0.18439 | 0.74341 | 0.60994 | | | Upper 95% Limit | 0.34052 | 1.00447 | 0.52476 | 2.62526 | 1.80086 | -----------------------------------------------------------------------------------------------

Design-Based and Model-Based Analyses* All the analyses presented so far relied on the design-based approach, as sample weights and design features were incorporated in the analysis.

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Before relating these analyses to the model-based approach, let us briefly consider the survey data used for these analyses. For NHANES III, 2,812 PSUs were formed, covering the United States. These PSUs consisted of individual counties, but sometimes they included two or more adjacent counties. These are administrative units, and the survey was not designed to produce separate estimates for these units. From these units, 81 PSUs were sampled, with selection probability proportional to the sizes of PSUs—13 from certainty strata and 2 from each of 34 strata formed according to these demographic characteristics rather than geographic location. Again, strata are not designed to define population parameters. The second stage of sampling involved area segments consisting of city or suburban blocks or other contiguous geographic areas. Segments with larger minority populations were sampled with a higher probability. The third stage of sampling involved the listing of all the households within the sampled area segments and then sampling them at a rate that depended on the segment characteristics. The fourth stage of sampling was to sample individuals within sampled households to be interviewed. These secondary units were used to facilitate sampling rather than to define population parameters. The public-use data file included only strata and PSUs; identification of seconding sampling units was not included. Sample weights were calculated based on the selections probabilities of interviewed persons with weighting adjustments for nonresponse and poststratification. Many of the analytic issues discussed in the previous chapters arise because of the way large-scale social and health surveys are conducted. Available data are not prepared to support the use of hierarchical linear models for incorporating the multistage selection design. Because of the unequal selection probabilities of interviewed individuals, coupled with adjustments for nonresponse and poststratification, it is unconvincing to ignore the sample weights in descriptive analysis of NHANES III data. The rationale for weighting in descriptive analysis has been quite clear. As shown in Table 6.2, the bias in the unweighted estimate is quite high for age and race-related variables. The standard error of the weighted estimate is quite similar to that of the unweighted estimate for all variables, suggesting that the variability in sample weights does not increase the variance. However, the standard error for the weighted estimates taking into account PSUs and strata is quite high, as reflected by the design effects. One way to reduce the variance is the use of a model using auxiliary information. Classical examples are the familiar ratio and regression estimators (Cochran, 1977, chap. 6). In these estimators, the auxiliary information is in the form of known population means of concomitant variables that are related to the target variables. The use of auxiliary information can be extended to the estimation of distribution functions (Rao, Kovar, & Mantel, 1990);

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however, the use of auxiliary information in routine descriptive analysis is limited because finding suitable auxiliary information is difficult. The use of a model in regression analysis is quite obvious. The unweighted estimates in Table 6.4 are based strictly on a model-based approach. It ignored the sample weights and the design features, but much of the relevant design information is included among the independent variables: for example, age (oversampling of elderly persons) and black and Hispanic (oversampling of minority populations). Indeed, the model-based estimates of coefficients are similar to the weighted estimates, suggesting that the model-based analysis is quite reasonable in this case. The one notable exception is the coefficient of education, which is very different under the two estimation procedures. Education is highly insignificant in the model-based analysis but highly significant in the weighted analysis. The education effect could not be detected by the model-based analysis because of the diminishing education effect for older ages. Age is included in the model, but the interaction effect between age and education is not included in the model. This example suggests that the use of the sample weights protects against misspecification of the model. Korn and Graubard (1995b) further illustrate the advantage of using sample weights in a regression analysis, using data from the 1988 National Maternal and Infant Health Survey. This survey oversampled lowbirthweight infants. The estimated regression lines of gestational age on birthweight from the unweighted and weighted analyses turn out to be very different. Although the unweighted fitting reflects sample observations equally and does not describe the population, the weighted fitting pulls the regression line to where the population is estimated to be. The relationship between the two variables actually is curvilinear. If a quadratic regression were fit instead, then the unweighted and weighted regressions would show greater agreement. As discussed above concerning the analytic results in Table 6.4, a careful examination of the differences between the weighted and unweighted regressions can sometimes identify important variables or interactions that should be added to the model. The differences between the unweighted and weighted estimates suggest that incorporating the sample design provides protection against the possible misspecification of the population model. Several statistics for testing the differences between the weighted and unweighted estimates have been proposed in the literature (DuMouchel & Duncan, 1983; Fuller, 1984; Nordberg, 1989). Korn and Graubard (1995a) apply these test statistics to the NHANES I and II data using design-based variances. They recommend the design-based analysis when the inefficiency is small. Otherwise, additional modeling assumptions can be incorporated into the analysis. They have noted that secondary sampling units are not

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available to the public and point to the need to increase the number of PSUs in the design of large health surveys. These tests are limited to point estimation, and therefore their conclusions may not apply to all circumstances. More detailed discussion of these and related issues is provided by Pfeffermann (1993, 1996). The fact that the design-based analysis provides protection against possible misspecification of the model suggests that the analysis illustrated using SUDAAN, Stata, and other software for complex survey analysis is appropriate for NHANES data. Even in the design-based analysis, a regression model is used to specify the parameters of interest, but inference takes the sample design into account. The design-based analysis in this case may be called a model-assisted approach (Sarndal, Swensson, & Wretman, 1992). The design-based theory relies on large sample sizes to make inferences about the parameters. The model-based analysis may be a better option for a small sample. When probability sampling is not used in data collection, there is no basis for applying the design-based inference. The model-based approach would make more sense where substantive theory and previous empirical investigations support the proposed model. The idea of model-based analysis is less obvious in a contingency table analysis than in a regression analysis. The rationale for design-based analysis taking into account the sampling scheme already has been discussed. As in the regression analysis, it is wise to pay attention to the differences between the weighted proportions and the unweighted proportions. If there is a substantial difference, one should explore why they differ. In Table 6.6, the unweighted and weighted proportions are similar, but the weighted odds ratios for vitamin use and gender are slightly lower than the unweighted odds ratios for high school graduates and those with some college education, while the weighted and unweighted odds ratios are about the same for those with less than high school graduation. The small difference for the two higher levels of education may be due to race or some other factor. If the difference between the unweighted and weighted odds ratios is much larger and it is due to race, one should examine the association separately for different racial groups. The consideration of additional factors in the contingency table analysis can be done using a logistic regression model. The uses of a model and associated issues in a logistic regression are exactly the same as in a linear regression. A careful examination of the weighted and unweighted analysis provides useful information. In Table 6.7, the weighted and unweighted estimates of coefficients are similar. It appears that the weighting affects the intercept more than the coefficients. The analysis shown in Table 6.7 is a simple demonstration of analyzing data using logistic regression, and no careful consideration is given to choosing an

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appropriate model. Comparable model-based analyses without using the weights and sample design were not performed for the ordered logistic regression model in Table 6.8 and multinomial logistic regression model in Table 6.9, because we feel that an appropriate model including all relevant independent variables was not specified. In summary, analysis of complex survey data would require both the model-based and design-based analysis. Design-based methods yield approximately unbiased estimators or associations, but standard errors can be ineffective. Model-based methods require assumptions in choosing the model, and wrong assumptions can lead to biased estimators of associations and standard errors.

7. CONCLUDING REMARKS In this book, we have discussed the problematic aspects of survey data analysis and methods for dealing with the problems caused by the use of complex sample designs. The focus has been on understanding the problems and the logic of the methods, rather than on providing a technical manual. We also have presented a practical guide for preparing for an analysis of complex survey data and demonstrated the use of some of the software available for performing various analyses. Software for complex survey analysis is now readily available, and with the increasing computing power of personal computers, many sophisticated analytical methods can be implemented easily. Nevertheless, data analysts need to specify the design, to create replicate weights for certain analysis, and to choose appropriate test statistics for survey analysis. Therefore, the user should have a good understanding of the sample design and related analytical issues. Although the material presented on these issues has been addressed mainly to survey data analysts, we hope that this introduction also stimulates survey designers and data producers to pay more attention to the needs of the users of survey data. As more analytic uses are made of the survey data that were initially collected for enumerative purposes, the survey designers must consider including certain design-related information that allows more appropriate analysis to be performed as well as easing the user’s burden. The data producers should develop the sample weights appropriate to the design, and even replicate weights incorporating adjustments for nonresponse and poststratification in addition to codes for strata and sampling units. Finally, we must point out that we have been taking the position of design-based statistical inference with some introduction to an alternative approach known as model-based inference. Each has its own strengths.

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Briefly, model-based inference assumes that a sample is a convenience set of observations from a conceptual superpopulation. The population parameters under the specified model are of primary interest, and the sample selection scheme is considered secondary to the inference. Consequently, the role of the sample design is deemphasized here, and statistical estimation uses the prediction approach under the specified model. Naturally, estimates are subject to bias if the model is incorrectly specified, and the bias can be substantial even in large samples. The design-based inference requires taking the sample design into account, and it is the traditional approach. The finite population is of primary interest, and the analysis aims at finding estimates that are design-unbiased in repeated sampling. We believe that the sample design does matter when inference is made from sample data, especially in the description of social phenomena, in comparison to more predictable physical phenomena. At the same time, the appropriateness of a model needs to be assessed, and the role of the analytical model must be recognized in any data analysis. Any inference using both the design and the model is likely to be more successful than that using either one alone. We further believe that these two approaches tend to be complementary and that there is something to be gained by using them in combination. Similar views are expressed by Brewer (1995, 1999) and Sundberg (1994).7 We cannot lose sight of either the many practical issues that prevail in social survey designs or lingering problems of nonresponse and other sources of nonsampling errors. These practical issues and the current state of substantive theory in social science tend to force us to rely more on the traditional approach for the time being. The model-based approach should provide an added perspective in bridging the gap between survey design and analysis. There is a considerable body of theoretical and practical literature on complex survey data analysis. In addition to the references cited, more advanced treatment of the topics discussed in this book and other related issues are available in single volumes (Korn & Graubard, 1999; Lehtonen & Pahkinen, 1995; Skinner, Holt, & Smith, 1989). For a secondary use of complex survey data, the analyst should recognize that the analysis may be seriously marred by the limitations or even mistakes made in the sample design. It will be wise for an analyst with limited sampling knowledge to consult with an experienced practicing survey statistician.

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NOTES 1. The method of ratio estimation is used in estimating the population ratio of two variables (for example, the ratio of the weight of fruits to the amount of juice produced). It is also used in obtaining a more accurate estimate of a variable (e.g., current income, y) by forming a ratio to another closely related variable (e.g., previous income at the time of the last census, x). The sample ratio (y/x, or change in income) is then applied to the previous census income to obtain the current estimate of income, which is more accurate than that estimated without using an auxiliary variable. For details, see Cochran (1977, chap. 6). 2. Holt and Smith (1979) characterized poststratification as a robust technique for estimation. Based on the conditional distribution, they showed that the self-weighted sample mean is generally biased and poststratification offers protection against extreme sample configurations. They suggested that poststratification should be more strongly considered for use in sample surveys than appears to be the case at present. In addition, poststratification may also provide some protection against any anomalies introduced by nonresponse and other problems in sample selection. 3. SUPER CARP and PC CARP were available from the Statistical Laboratory of Iowa State University and were useful to the statistically inclined users. CPLX was made available by Dr. Fay of the U.S. Bureau of the Census and was useful for conducting discrete multivariate analysis using modified BRR and the jackknife method (Fay, 1985). CENVAR and VPLX programs are now available from the U.S. Bureau of the Census. The Epi Info system for epidemiological and statistical analysis, developed by CDC (U.S. Centers for Disease Control and Prevention), includes the CSAMPLE procedure for complex survey data analysis. The OSIRIS Statistical Software System from the Institute for Social Research, University of Michigan, included some procedures for descriptive statistics and regression analysis (available only for mainframe computers). There were also other programs written for some special survey projects such as the World Fertility Survey (CLUSTERS). There are two programs written as a series of SAS macros for survey data analysis, including GES, available from Statistics Canada, and CLAN, available from Statistics Sweden. 4. The National Health and Nutrition Examination Survey (NHANES) is a continuing series of surveys carried out by the National Center for Health Statistics (NCHS) to assess the health and nutritional status of the U.S. population. There have been several rounds of NHANES. NHANES I was conducted in 1971–1973, NHANES II in 1976–1980, and NHANES III in 1988–1994. A special survey of the Hispanic population (Hispanic HANES)

81 was conducted in 1982–1984. NHANES became a continuing survey, and data are now released every 2 years (NHANES 1999–2000, 2001–2002, and 2003–2004). NHANES collects information on a variety of health-related subjects from a large number of individuals through personal interviews and medical examinations, including diagnostic tests and other procedures used in clinical practice (S. S. Smith, 1996). NHANES designs are complex to accommodate the practical constraints of cost and survey requirements, resulting in a stratified, multistage, probability cluster sample of eligible persons in households (NCHS, 1994). The PSUs are counties or small groups of contiguous counties, and the subsequent hierarchical sampling units include census enumeration districts, clusters of households, households, and eligible persons. Preschool children, the aged, and the poor are oversampled to provide sufficient numbers of persons in these subgroups. The sample weight contained in the public-use micro data files is the expansion weight (inverse of selection probability adjusted for nonresponse and poststratification). NHANES III was conducted in two phases. The multistage sampling design resulted in 89 sample areas, and these were randomly divided into two sets: 44 sites were surveyed in 1988–1991, and the remaining 45 sites in 1991–1994. Each phase sample can be considered as an independent sample, and the combined sample can be used for a large-scale analysis. 5. The hot deck method of imputation borrows values from other observations in the data set. There are many ways to select the donor observations. Usually, imputation cells are established by sorting the data by selected demographic variables and other variables such as stratum and PSU. The donor is then selected from the same cell as the observation with the missing value. By imputing individual values rather than the mean values, this method avoids the underestimation of variances to a large degree. This method is widely used by the U.S. Bureau of the Census and other survey organizations. For further details, see Levy and Lemeshow (1999, pp. 409–411). 6. A Wald statistic with one degree of freedom is basically the square of a normal variable with a mean of zero divided by its standard deviation. For hypotheses involving more than one degree of freedom, the Wald statistic is the matrix extension of the square of the normal variable. 7. The difference between design-based and model-based approaches is well illustrated by Brewer and Mellor (1973). The combined use of different approaches is well articulated by Brewer (1995) and illustrated further based on the case of stratified vs. stratified balanced sampling (Brewer, 1999). Sundberg (1994) expressed similar views in conjunction with variance estimation. Advantages and disadvantages of moving from designbased to model-based sample designs and sampling inference are reviewed and illustrated with examples by Graubard and Korn (2002). They point out

82 that with stratified sampling, it is not sufficient to drop FPC factors from standard design-based variance formulas to obtain appropriate variance formulas for model-based inference. With cluster sampling, standard design-based variance formulas can dramatically underestimate modelbased variability, even with a small sampling fraction of the final units. They conclude that design-based inference is an efficient and reasonably model-free approach to infer about finite population parameters but suggest simple modifications of design-based variance estimators to make inferences with a few model assumptions for superpopulation parameters, which frequently are the ones of primary scientific interest.

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INDEX ANOVA, 57 Balanced repeated replication (BRR), 26–29, 44 jackknife repeated replication (JRR) and, 33–34 replicate weights and, 47 Binary logistic regression analysis, 65–69 Bootstrap method, 35–36 Cluster sampling, 6, 9 Cochran–Mantel–Haenszel (CMH) chi–square, 64–65 Computer software, 44–47, 51, 56, 61–63, 72–75, 77 Contingency table analysis, 61–65, 77 Data imputation, 41–42, 53, 81n5 missing, 41–42 preliminary analysis of, 41–43, 50–51 requirements for survey analysis, 39–40 Descriptive analysis, 52–57 Design–based analysis, 2–3, 75–78, 79, 81–82n7 Design effects, 18–20, 40, 60–61 Design–weighted least squares (DWLS), 58 Errors, estimation of sample design and, 8 Taylor series method for, 38–39 Expansion weights, 11–14 Follow–up surveys, 17–18 General Social Survey (GSS), 14–16 Goodness–of–fit statistic, 67–69

Imputation, 41–42, 53, 81n5 Inference, model–based, 79, 82n7 Intraclass correlation coefficient (ICC), 9 Jackknife repeated replication (JRR), 29–35, 44 replicate weights and, 48 Linearization, 36–39 Linear regression analysis, 57–61, 66, 78 Logistic regression analysis binary, 65–69 nominal, 72–75 ordered, 70–72 Maximum likelihood estimations, 66–67 Missing data, 41–42 Model–based analysis, 2–3, 9–11, 75–78, 81–82n7 Model–based inference, 79, 82n7 Multistage sample design, 3–4 National Opinion Research Center (NORC), 14–16 Nature of survey data, 7–9 Nominal logistic regression analysis, 72–75 Ordered logistic regression analysis, 70–72 Ordinary least squares (OLS), 57–58 Pearson chi–square statistic, 61, 63 Poststratification, 14–16, 80n1 Precision, assessing loss or gain in, 18–20 Prediction models, 69 Preliminary analysis of data, 41–43, 50–51

89 Preparation for survey data analysis data requirements and, 39–40 importance of preliminary analysis in, 41–43 Primary sampling units (PSUs), 7, 9 balanced repeated replication (BRR) and, 26–29 bootstrap method and, 35–36 jackknife repeated replication (JRR) and, 29–35 preliminary analysis of, 42–43 Probability proportional sampling (PPS sampling), 6–7, 7, 50–51 Problem formulation, 49 Proportional to estimated size (PPES) sampling, 7 Ratio estimation, 80n1 Regression analysis, logistic, 65–75 contingency table analysis compared to, 77 imputation, 53 linear, 57–61, 66, 78 use of models in, 76 Relative weights, 11–14 Repeated systematic sampling, 5 Replication balanced repeated (BRR), 26–29, 33–34, 44 jackknife repeated (JRR), 29–35, 44 sampling, 23–26 weights, 47–49 Sample design design effect and, 18–20 multistage, 3–4 standard error estimation and, 8 types of sampling in, 4–7 variance estimation and, 22–39, 43–44 Sampling balanced repeated replication (BRR), 26–29, 44, 47 cluster, 6, 9

expansion weights in, 11–14 follow–up, 17–18 jackknife repeated replication (JRR), 29–35, 44, 48 probability proportional (PPS), 6–7, 7, 50–51 proportional to estimated size (PPES), 7 repeated systematic, 5 replicated, 23–26 simple random (SRS), 1, 4 stratified random, 5–6, 8–9 systematic, 4–5 types of, 4–7 units, primary (PSUs), 7, 9, 26–29 weights, 7–8, 11–14, 20–22, 79 Simple random sampling (SRS), 1, 4 with replacement (SRSWR), 2 without replacement (SRSWOR), 4 Software programs, 44–47, 51, 56, 61–63, 72–75, 77 SPSS 13.0, 46–47 Stata software package, 45, 51, 56, 61–63, 72 Stratified random sampling, 5–6, 8–9, 40 Subgroup analysis, 42–43, 55–57 SUDAAN software package, 45, 46, 51, 61, 72 Survey data analysis, 1–3 complexity of, 11–22 computer software for, 44–47, 51, 56, 61–63, 72–75, 77 contingency table analysis and, 61–65, 77 data requirements for, 39–40 descriptive analysis and, 52–57 design–based, 2–3, 75–78, 81–82n7 importance of preliminary analysis in, 41–43 linear regression analysis and, 57–61, 66, 78 logistic regression analysis and, 65–69 model–based, 9–11, 75–78, 81–82n7

90 nature of, 7–9 preliminary, 50–51 preparing for, 39–49 problem formulation before, 49 sample design and, 3–11 searching for appropriate models for, 49 subgroup analysis in, 42–43, 55–57 use of sample weights for, 20–22 variance estimation in, 22–39 Systematic sampling, 4–5 Taylor series method, 36–39, 44, 45 Unweighted versus weighted analysis, 58–60 Variance estimation balanced repeated replication (BRR), 26–29, 33–34, 47 bootstrap method, 35–36

choosing method for, 43–44 jackknife repeated replication (JRR), 29–35, 44, 48 replicated sampling, 23–26 Taylor series method, 36–39, 44, 45 Wald statistic, 61, 63, 81n6 Weights adjusting in follow–up surveys, 17–18 creating replicate, 47–49 design effect and, 18–20 development by poststratification, 14–16 expansion, 11–14 sample, 7–8, 79 for survey data analysis, 20–22, 39–40 and unweighted versus weighted analysis, 58–60 WesVar software package, 46

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ABOUT THE AUTHORS Eun Sul Lee is Professor of Biostatistics at the School of Public Health, the University of Texas Health Science Center at Houston, where he teaches sampling techniques for health surveys and intermediate biostatistics methods. He received his undergraduate education at Seoul National University in Korea. His PhD, from North Carolina State University, is in experimental statistics and sociology. His current research interests involve sample survey design, analysis of health-related survey data, and the application of life table and survival analysis techniques in demography and public health. He is junior author of Introduction to Biostatistics: A Guide to Design, Analysis and Discovery (with Ronald Forthofer, 1995). Ronald N. Forthofer is retired as a professor of biostatistics at the School of Public Health, the University of Texas Health Science Center at Houston. He now lives in Boulder County, Colorado. His PhD, from the University of North Carolina at Chapel Hill, is in biostatistics. His past research has involved the application of linear models and categorical data analysis techniques in health research. He is senior author of Public Program Analysis: A New Categorical Data Approach (with Robert Lehnen, 1981) and also senior author of Introduction to Biostatistics: A Guide to Design, Analysis and Discovery (with Eun Sul Lee, 1995).