Skew-Growth Curves

16

STA TISTICS: PEARL AND REED

PROC. N. A. S.

7-day test in tension were 210 pounds the predicted value for the 28-day test would be 275 pounds or 50 per cent of the 28-day tests shoild be above 275 pounds. Seventy-five per cent of them should be above 275 pounds minus 29 pounds or above 246 pounds. In like manner the limit necessary to include 90 per cent or 99 per cent of the tests may be found. Thus in the illustrated case 90 per cent would be above 275 pounds minus 55 pounds or 230 pounds, 99 per cent of the 28-day tests should be above 275 - 100 pounds or 175 pounds. The results for tensile ratios, compression and compressive ratios, are similarly obtained. TABLE III

28-DAY TEST PREDICTIONS NECESSARY TO HAVE THE GIVEN PERCENTAGE oF ACTUAL 28-DAY TEsT RXSULTS ABOVE THE PREDICTED VALUE, THE PREDICTION BEING MADE PROM TH 7-DAY TEST RESULTS Percentage of actual 28-day tests above predicted value KIND OP TEST

Tension (pounds) Tensile ratio

Compression (pounds) Compressive ratio

90

75

50

Predicted value Predicted value Predicted value Predicted value

-29

pounds

-11.7% 416 pounds

-13.5%

-55

pounds

-22.6% 801

99

-100

-40.8%

pounds -1447

-26.1%

pounds

pounds

-47.1%

In way of conclusion it may be said that the results of this study indicate that the factors determining the strength of mortar at the end of the 7-day period of curing compared with those determining strength at the end of the 28-day period of curing are sufficiently alike to make possible the prediction of the 28-day breaking strength from the 7-day breaking strength. The error of such a prediction is on the whole not excessive.

SKEW-GROWTH CURVES' By RAYMOND PZARL AND LOWZLL J. RiUUD SCHOOL or HYGIENE AND PUBLIC HEALTH, JOHNS HOPKINS UNIVERSITY Read before the Academy November 12, 1924

In an earlier paper2 we have pointed out that if, in the general growth curve k 1 +mealx + a2X + ax3 + ......±+a

(1)

the exponent of e be cut off at the cubic term, appropriate values of the constants will give a single cycle growth curve which is unsymmetrical or skew about the point of inflection. We gave by way of illustration a theoretical case of such a skew-growth curve, but have not hitherto published any ex-

voi. 1 1, 1925

17

STATISTICS: PEARL AND REED

ample of the actualfitting of this curveto observational data on growthasymmetrically distributed about the point of inflection. The wide applicability of the general curve (1) to symmetrical population growth has been demonstrated.3 It seems desirable now to show the ability of the curve to deal with phenomena of skew growth. Hitherto observational growth series which were plainly skew have been graduated (if graduated at all) either by a process of fitting two or more curves to different parts of the observed TABLE I GROWTH oF MALE ALBINO RAT. (DONALDSON'S DATA) CALCULATED AGE IN DAYS

OBSERVED WEIGHT IN GRAMS

10 11 12 13 14 15 17 19 21 23 25 27 29 31 34 37 40 43 46 49 52 55 58 61 64

67

13.5 13.3 14 8 15.3 15.2 16.5 17.8 19.5 21.2 22.9 25.3 27.4 29.5 31.8 34.9

37.8 42.2 46.3 50.5 56.7 62.5 68.5 73.9

81.7 89.1 99.3

WEIGHT BY OUR EQUATION I

14.1 14.5 15.0 15.5 16.1 16.7

17.9 19.3 20.8 22.4 24.2 26.1 28.2 30.5 33.2 38.3 42.7 48.6 52.8 58.3 64.2 70.4 76.8 83.4 90.1 97.0

AGE IN DAYS

OBSERVXD WEIGHT IN GRAMS

70 73 76 79 82 85 88 92

97 102 107 112 117 124 131

13.8

143 150 157

164 171 178 185

216 256 365 Root mean square

de-

106.3 113.8 121.3 128.2 135.0 143.8 148.4 152.3 160.0 168.8 177.6 183.8 191.4 197.3 202.5 209.7 218.3 225.4 227.0 231.4 235.8 239.4 239.8 252.9 265.4 279.0

CALCULATED WIGHT BY OUR EQUATION

I

103.8 110.7 117.6 124.3 130.9 137.4 143.7 151.7 161.2 170.0 178.1 185.5 192.2 200.6 208.1 214.5 218.6 223.7 228.2 232.1 235.7 238.9 241.9 252.7 264.4 279.6 4.96

viation

series and then welding the separate curves together where they overlap, or by attempting to make a symmetrical curve fit obviously skew material. One or the other of these plans has been followed by Hatai on Donaldson's data,4 and by Robertson5 variously. Aside from the mathematical inelegance of such procedures, in the particular case of skew-growth data the fits obtained have in some cases been grotesquely poor.

18

STATISTICS: PEARL AND REED

PROC. N. A. S.

We present here four examples of skew-growth curves fitted with the cubic form of (1). These are: (a) The growth in weight of male albino rats from 10 days of age on (data from Donaldson, loc. cit., pp. 108-109); (b) the growth of female albino rats from 10 days of age on (data from TABLE II GROWTH oF 1sMALms ALBINO RAT.

(DONALDSON'S DATA)

CAI,CUJEATED AGE IN DAYS _ __

10 11 12 13 14 15 17 19 21 23 25 27 29 31 34 37 40 43 46 49 52 55 58 61 64 67

OBSERVED WRIGHT IN GRAMS

WRIGHT BY OUR RQUATION II

13.0 12.8 15.1 15.1 15.6 17.7 19.2 20.6 22.6 24.9 27.4 30.0 31.4 32.9 35.7 39.5 43.7 47.9 52.0 57.7 62.9 68.4 74.6 78.4 85.8 96.0

14.0 14.5 15.0 15.5 16.1 16.7 18.0 19.4 21.0 22.7 24.5 26.5 28.7 31.0 34.8 38.9 43.4 48.3 53.4 58.8 64.4 70.1 76.0 82.0 88.0 93.9

AGE IN DAYS

70 73 76 79 82 85 88 92 97 102 107 112 117 124 131 138 143 150 157 164 171 178 185 192 365 Root mean square deviation

OBSERVED WRIGHT IN

CALCULATED WXIGHT BY OUR EQUA-

GRAMS

TION

99.8 105.6 110.4 118.8 124.7 131.5 136.0 139.8 146.3 153.1 155.8 161.4 168.0

99.8 105.6 111.2 116.7 122.0 127.0 131.8 137.9 144.8 151.2 156.9 162.0 166.6 172.2 177.0 181.1 183.8 187.0 189.9 192.4 194.7

172.6 181.0 185.0 186.6 188.2 188.0 189.5 192.2 197.0 200.0 202.2 226.4

...

II

196.7 198.6 200.5 226.9 2.022

Donaldson, loc. cit., pp. 110-111); (c) the growth in weight of Cucurbita pepo (Anderson's data from Robertson, loc. cit., p. 74), (d) the growth in length of the regenerating tail of the tadpole (data from Durbin6) The equations in the several cases are: 2>73 I. Male Rats. y = 7 + 1 +e4.3204-7.2196x+30.0878X2-0.5291Xa where y = weight in grams, and x = age in hundred day units.

VOL. 11, 1925

STATISTICS: PEARL AND REED

II. Female Rats.

y

=

19

220 22 +3.5065x -0.62o2x. 7+1+ y1 + e4.426 - 7.6OIx y and x in same units as above.

5190 III. Cucurbita pepo. y= 174 + 1 + 51O.3l48i6.3399x + where y = weight in grams, and x = age in ten day units. 9.60 IV. Tadpole tails. = 1 + e4.4715-7.2829x+3.3833x*-0.6370xs where y = length of regenerated tail in number and x = age in ten day units. The observed values and those calculated from the equations above set forth are exhibited in tables I to IV inclusive. TABI<E III GROWTH oF CUCURBITA PEPO. (DATA PROM ROBIRTSON) GRAMS

CALCULATED WEIGHT BY OUR SQUATION nI

267 443 658 961 1498 2200 2920 3366 3758 4092 4488

267 399 645 1044 1586 2210 2829 3378 3829 4186 4464

OBSERVED DAYS

5 6 7 8 9 10 11 12 13 14 15

WEIGHT IN

GRAMS

CALCUILATED WEIGHT BY OUR EQUATION m

4720 4864 4980 5114 5176 5242 5298 5352 5360 5366

4680 4850 4984 5089 5172 5236 5282 5315 5337 5350

OBSERVED

DAYS

16 17 18

19 20 21 22 23 24 25 Root mean square deviation

WEIGHT IN

46.1

The observations and fitted curves are shown graphically in figures 1 to 4 inclusive. To give a concrete idea of the degree of skewness in these curves table V is presented. The figures in the last column of the table are the significant ones. If the curves were all symmetrical their values would all be 50 per cent. In fact the highest is 38.4 and the lowest 32.5. They indicate that the de-parture of these curves from the symmetrical condition is substantial. The results exhibited in the tables and graphs require little comment. All of the curves are obviously skew. The graduations are all excellent. They demonstrate the ability of our generalized growth curve to deal

20

STA TISTICS: PEA RL AND REED

PRoc. N. A. S.

150

VA

120

60

30

0

30

g

90

J0

/50

/v0

21

2 0

7'0

300

3,0

360

3lo04

Days FIGURE 1

Growth of male albino rats (Donaldson's data). The circles give the observations in this and the following diagrams. The smooth curve is the graph of our equation I. The x in this and the following curves denotes the point of inflection. 300 270 _ 240

_Asymptote

2270

~10

0

7

30

tO

O /0i20

A

8t Age

210

240

270

300

is0

3

420

in Days FIGURE 2

Growth of female albino rats (Donaldson's data). The smooth our equation II.

curve

is the graph of

21

STA TISTICS: PEARL AND REED

VOL. 1 1, 1925

5400

4200

Cucuxrbilta DeDo / 3600

3000

2400 1800

600 v

2

4

0

a

10

10

/v

XV

Xz

24

Z6,

Days3

FIGURE

Growth of Cucurbita pepo (data from Robertson). The smooth curve is the graph of our equation III.

k. ll. -

IZ lz.,)

-

4

Days FIGURE 4

Regeneration of tadpole tail (Durbin's data). The smooth curve is the graph of our equation IV.

22

PRoc. N. A. S_

STATISTICS: PEARL AND REED

adequately with unsymmetrical growth phenomena as well as with symmetrical. This is all that we desire to demonstrate in this paper. TABLE IV REGENERATION OF TADPOLE TAIL. DAYS

0 1 2 3

OBSERVED REGENERATION IN NUMBER

6 7 8 9 10 11 12

DAYS

0.11

13 14 15 17

... 0.68 ...

1.09

5.00 .

.

CALCULATED REGENERATION FROM OUR EQUA-TION IV

6.37 6.71 7.00 7.48 7.88 8.58 9.17 9.38 9.50 9.57 9.60 9.60 9.60

... 6.60

7.60

19 23 27 29 31 33 37

*... 9.68 9.20 9.30 9.60 9.40

41 51 Root mean square deviation

5.97

..

TION IN

NUMBER

1.62 2.25 2.95 3.66 4.34 4.96 5.50

2.95

*.

OBSERVED

REGENERA-

OUR EQUATION IV

0.22 0.40 0.68

4 5

(DURBIN'S DATA)

CALCULATED REGENERATION FROM

0.196

TABLE V

SEKwNzss OF GROWTH CuRVas POSITION OP CURVE

ON X AXIS ____

Male rats Female rats III. Cucurbita pepo IV. Regeneration I. II.

___

70.0 62.3 9.9 7.3

___

POINT

OP

___

___

___

103.5 84.5

2168.0 3.12

INPLECTION PER CENT OP DISTANCE FROM I,OWER TO UPPXER

ON Y AXIS ___

ASYMPTOTE

35.3 35.2 38.4 32.5

1 Papers from the Department of Biometry and Vital Statistics, School of Hygiene andc Public Health, Johns Hopkins University, No. 111. 2 Pearl, R., and Reed, L. J. Oqr the mathematical theory of population growthMetron, 3, pp. 6-19, 1923. ' Pearl, R. Studies in Human Biology. Baltimore (Williams & Wilkins Co.), 1924. 4 Donaldson, H. H. The Rat. Philadelphia (Wistar Institute), 1915. ' Robertson, T. B. The Chemical Basis of Growth and Senescence. Philadelphia (J. B. Lippincott Co.), 1923. 6 Durbin, M. L. An analysis of the rate of regeneration throughout the regenerative: process. J. Expt. Zool. 7, pp. 397-420, 1909.