STUDIES IN HUMAN INHERITANCE XIII A TABLE TO DETERMINE THE EXPECTED PROPORTION OF FEMALES SHOWING A SEX-INFLUENCED CHARACTER CORRESPONDING TO ANY GIVEN PROPORTION OF MALES SHOWING THE CHARACTER

STUDIES I N HUMAN INHERITANCE

XI11

A TABLE TO DETERMINE THE EXPECTED PROPORTION OF

FEMALES SHOWING A SEX-INFLUENCED CHARACTER

CORRESPONDING TO ANY GIVEN PROPORTION

OF MALES SHOWING

THE CHARACTER

LAURENCE H. SNYDER AND CHARLES W. COTTERMAN

Genetics Laboratory, Ohio State University, Columbus, Ohio

Received September 1, 1935

N A PREVIOUS paper (Snyder,

L.

H.

and Yingling,

H., 1935) the

I

gene frequency method was applied to sex-influenced factors, and a

formula for testing the applicability of the hypothesis of sex-influenced fac-

tors to human data was derived. Since any character dependent upon sex-

influenced factors will usually be relatively frequent in males, but rela-

tively rare in females, another method of attack presents itself. Knowing

the proportion of males showing a character suspected of being due to a

sex-influenced factor, what proportion of females may be expected to show

the character, assuming random mating? As a practical example, if 40%

of males are bald, what proportion of females may be expected to be bald,

assuming that baldness is due to a sex-influenced factor and that random

mating occurs in regard to this character?

Assume a pair of allelomorphs

B

and

b,

such that

B

is dominant in

males, but recessive in females. Let p =frequency of

B ,

and q =frequency

of

b.

Then p + q

=

1. Here p and q may be separately derived, as follows:

e=proportion

of

BB 9 9

in

general population,

-=proportion

of bb 3 3

in

general population.

2

2

q2

Since the proportion of

B B

9

9

in the general population is equal to

half of their proportion among females alone, and the proportion of

bb

8 8

in the general population

is

equal to half of their proportion among males

alone, we may write as follows:

=proportion among females of females who show the character

represented by

B ,

let ==proportion

among males of males who show

the character represented by

B ,

and let

8 b

=proportion among males of

males who show the character represented by

b.

Then

Let

-

~

q = d / a b .

(2)

dTTi+dZ=l.

( 3 )

d?m+dX=

1

So that

-

__

From equation

( 3 )

it is possible to derive a value of

9

B in terms of 8 B .

- -

dZ=l-dZ=l-dl-

3B

(4)

- (

Q B =

i-di-3x)2.

From equation

(4) a table may be constructed, giving, for any propor-

tion of males showing a dominant sex-influenced character, the correspond-

ing proportion of females who may be expected to show the character.

Employing the maximum likelihood method of

R.

A.

FISHER

(1930),

the

probable error formula for equation

(4)

may be derived as follows:

The distribution of dominant and recessive individuals in a sample of

N males follows the terms of the expansion of the binomial, [(p2+2pq)

+q2IN.

More exactly, the chance P

of

getting n dominants and N-n re-

cessives is

where

P = K(p2+2pq)”(q2)N-”= K ( l -q2)>”(q2)N-”

( 5 )

N!

n

!(N-

n) !

K =

The value to be estimated is p2,

so

that setting

we obtain

e=p2

q=l-V%.

Putting this value of q in equation

( S ) ,

we get

P

=

K(

248-

e).(

1

-

4)

2N-2n.

Taking natural logarithms,

L=log

P=log

K + n log (248-8)+(2N-2n)

log ( l - d @

aL

n ( 1 - 4 )

N-n

ae

e(2-dijj

&-e

d2L

( N - n ) ( l - j 4 )

de2

2dif(&-

e 1 2

2 e 4 @ 2

-

de12

e2(2

-

4 8 )

*

-__-

-=

n

n(1-

dF)

_______-

-=

Substituting

6

= p 2 and n =Np(2 -p), we have

N

--

a x

_ -

d(P2I2

P3(2-P)

STUDIES IN HUMAN INHERITANCE XI11

The variance of p2,

Substituting

p

=

1

-41

-=,

we get

The complete equation involved is thus

-

2

7

9

B

= 1-41

-2%)

i

.6745( 1

- 4 m )

(E

N

.

81

(9)

From table 1, for any proportion

of

males showing a dominant sex-influ-

enced character, the proportion of females who may be expected to show

the character may be directly read.

TABLE 1

Table of values of

~ = ( l - d l - ~ ) *

The values of to two decimal places are given in the left-hand column; the third decimar

-

place for each value is given in the top row. T h w for

a=

.310, the proportion of- is .0287; for

$ B = .316 it is .0299.

d B .OOO .001 .002 .003 .004 .005 .006 .007 .008 .009

-

.ooo

.0000 .OoOo

.oooo

.m

.oooo

.m

.o000

.m

.oooo

.oOOo .010 .0000

.oooo

.m

.oooo .oooo

.o001 .om1 .OoO1 .0001

.om1

,020 .WO1 ,0001 .o001 .o001 .OOol .0002 .0002 .WO2 ,0002 .0002 .030 .0002 .OoO2 ,0003 .0003 .0003 .0003 .0003 .WO3 .ON4 .0004 .040 .0004 .OW4 .0005 .OoO5 .OoO5 .o005 .o005 .WO6 .0006 .0006 .OS0 .OW6 .0007 .0007 .0007 .WO7 .o008 .WO8 .OW8 .WO9 .WO9 .060 .0009 .0010 .0010 .0010 .0011 .0011 .0011 .0012 .0012 .0012 .070 .0013 .0013 .0013 .0014 .0014 .0015 .0015 .0015 .0016 .0016

.080 .0017 .0017 ,0018 .0018 .0018 .0019 .0019 .0020 .0020 .0021

.090 .0021 .0022 .0022 .0023 .0023 .0024 .0024 .0025 .0025 ,0026 .lo0 .0026 .0027 .0027 .0028 .0029 .0029 .0030 .0030 .0031 .0031

.

110 .0032 .0033 .0033 .0034 .0034 .0035 .0036 .0036 .0037 .0038 .120 ,0038 .0039 .0040 .0040 .0041 .0042 .0042 . a 3 .0044 .0045 .130 .0045 .0046 .0047 .0047 .0048 .0049 3 0 5 0 .0050 .0051 .0052 .140 .0053 .0054 .0054 .0055 .0056 .0057 .0058 .0058 .0059 .0060

.150 .0061 .0062 .0063 .0063 .0064 .0065 .0066 .0067 .0065 .0069 .160 .0070 .0071 .0072 .0072 .0073 .0074 .0075 .0076 .0077 .0078 .170 .0079 .0080 .0081 .0082 .0083 .0084 .0085 .0086 .0087 ,0038

TABLE 1. (Continued)

Table of Values of FB=(l--l/l--dB)2

.OOO .001 -002 .003 .004 .005 .006 .007 .008 .009

.200 .210 .220 .230 .240

.2SO .260 .270 .280 .290

.300

.310

.320 .330 .340

.3SO .360 .370 .380 .390

.400

.410 .420 .430

.440

.4SO .460 .470

. a 0

.490

.so0 .510 .S20 .S30

.540

.5SO .560 .570

.580 .590

.0111 .0113 .0114 .0115 .0116 .0117 .0119 .0120 .0121 .0122

.0124 .0125 .0126 .0127 .0129 .0130 .0131 .0133 .0134 .0135 .0136 .0138 .0139 .0140 .0142 .0143 .0145 .0146 .0147 .0149 .0150 .0151 .0153 .0154 .0156 .0157 .0159 .0160 .0161 .0163 .0164 .0166 .0167 .0169 .0170 .0172 .0173 .0175 .0176 .0178

.0179 .0181 .OH3 .ON4 .ON6 .OH7 .OH9 .0191 .0192 .0194 .0195 .0197 .0199 .0200 .0202 .0204 .0205 .0207 .0209 .0210 .0212 .0214 .0215 .0217 .0219 .0221 .0222 .0224 .0226 .0228 .0229 .0231 .0233 .0235 .0237 .0238 .0240 .0242 .0244 .0246 .0248 .0250 .0251 .0253 .0255 .0257 .0259 ,0261 .0263 .0265

.0267 .0269 .0271 .0273 .0275 ,0277 .0279 .0281 .0283 .0285 .0287 .OB9 .0291 .0293 .0295 .0297 .0299 .0301 .0303 .0305 .0308 .0310 .0312 .0314 .0316 .0318 .0320 .0323 .0325 .0327 .0329 .0332 .0334 .0336 .0338 .0340 .0343 .0345 .0347 .0350 .0352 .0354 .0357 .0359 .0361 .0364 .0366 .0365 .0371 .0373

.0375 .0378 .0380 .0383 .0385 .0388 .0390 .0393 .0395 .0398 .0400 .0403 .0405 .OM8 .0410 .0413 .0415 .0418 .0420 .0423 .0425 .0428 .0431 .0433 .0436 .0439 .0441 .0444 .0447 .0449 .0452 .0455 .0457 .0460 .0463 .0466 .0468 .0471 .0474 .0476 .0479 .0482 .0485 .0488 .0491 .0494 .o497 .0499 .OS02 .OSOS

.0508 .OS11 .OS14

.OS17

.OS20 .OS23 .OS26 .OS29 .0532 .OS35 .OS38 .OS41 .OS44 .OS47 .OS50 .OS53 .OS56 .OS59 .OS62 .OS65 .OS68 .OS72 .OS75 .OS78 .OS81 .OS84 .0587 .OS91 .OS94 .OS97

.Om0 .0604

.Om7

.0610 .0613 .0617 .0620 .0623 .0627 .0630 .0633 .0637 . O M .0644 .0647 .0650 .0654 .0657 .0661 .0664 .0668 .0671 .0675 .0678 .0682 .0685 .0689 .0692 .0696 .0699 .0703 .0707 .0710 .0714 .0718 .0721 .0725 .0729 .0732 .0736 .0740 .0744 .0747 .0751 .0755 .0758 .0762 .0766 .0770 .0774 .0778 .0782 .0786 .0789 .0793 .0797 .0801 .0805 .0809 .OS13

.OS17 .0821 .0825 .0829 ,0832 .0837 .0841 .OS46 .0550 .0854

.0858 .0862 .0866 .0870 .0875 .0879 .0883 .0887 .0891 .0896

.o900 .0904 .0909 .0913 .0917 .0922 .0926 .0930 .0935 .0939

,0944 .o948 .0952 .0957 .0961 .0966 .0970 .0975 .o980 .0984 .0989 .0993 .0998 .lo03 .lo07 .lo12 .lo16 .lo21 .lo26 .lo31 .lo35 .lo40 .lo45 .lo50 .lo54 .lo59 .lo64 .lo69 .lo74 .lo79

.lo84 .lo89 .lo93 .lo98 .1103 .1108 .1113 .1118 .1123 .1128 .1134 .1139 .1144 .1149 .1154 .1159 .1164 .1169 .1175 .1180 .1185 .1190 .1196 .1201 .1206 .1212 .1217 .1222 .1228 .1233

STUDIES IN HUMAN INHERITANCE XI11

TABLE

1. (Continued)

Table of Values of ~ = ( ( I - d l - ~ ) '

83

.600 .610 .620 .630 .640 .650 .660 .670 .a0 .690 .000 .1351 .1410 .1471 .1534 .1600 .1668 .1738 .1811 .1886 .1964 .001 .1357 .1416 .1477 .1541 .1607 .1675 .1745 .1818 .1894 .1972

.002 .003 .004 .005 .006 .1363 .1368 .1374 .1380 .1386 .1422 .1428 .1434 .1440 .1446 .1484 .1490 .1496 .1502 .1509 .1547 .1554 .1560 .1567 .1574 .1613 .1620 .1627 .1634 .1640 .1682 .1689 .1696 .1703 .1710 .1752 .1760 .1767 .1774 .1781 .1826 .1833 .1841 .1848 .1856 .1902 .1909 .1917 .1925 .1933 .1980 .1988 .1997 .2005 .2013

.007 .1392 .1453 .1515 .1580 .1647 .1717 .1789 .1863 .1941 .2021 .008 .1398 .1459 .152! .1587 .1654 .1724 .1796 .1871 .1949 .2029 .009 .1404 .1465 .1528 .1593 .1661 .1731 .1803 .1879 .1957 .2037 .700 .2046 .2054 .2062 .2070 .2079 .2087 .2096 .2104 .2113 .2121 .710 .2130 .2138 .2147 .2156 .2164 .2173 .2182 .2190 .2199 .2208 .720 .2217 .2226 .2235 .2244 .2253 .2262 .2271 ,2280 .2289 .2298 .730 .2308 .2317 .2326 .2336 .2345 .2354 .2364 ,2373 .2383 .2392 .740 .2402 .2412 .2421 .2431 .2441 .2450 .2460 .2470 .2480 .2490 .750 .2500 .2510 .2520 .2530 .2540 .2551 .2561 .2571 .2581 .2592 .760 .2602 .2612 .2623 .2633 .2644 ,2655 .2665 ,2676 .2637 .2698 .770 .2708 .2719 .2730 .2741 .2752 .2763 .2774 .2785 .2797 .2808 .780 .2819 .2831 .2842 .2853 .2865 .2876 .2888 .2900 .2911 .2923 .790 .2935 ,2947 .2959 .2971 .2983 .2995 .3007 .3019 .3031 .3043

.800 .3056 .3068 .3081 .3093 .3106 .3118 .3131 .3144 .3156 .3169 .810 .3182 .3195 .3208 .3221 .3234 .3248 .3261 .3274 .3288 .3301 .820 .3315 .3328 .3342 .3356 .3370 .3383 .3397 .3411 .3425 .3440 .830 .3454 .3468 .3482 .3497 .3511 .3526 .3540 .3555 .3570 ,3585 .840 .3600 .3615 .3630 .3645 .3661 .3676 .3691 .3707 .3723 ,3738 .850 .3754 ,3770 .3786 .3802 .3818 .3834 .3851 .3867 .3883 .3900 ,860 .3917 .3933 .3950 .3967 .3984 .4002 .4019 .4036 .4054 .4071 .870 .4089 .4107 .4125 .4143 .4161 .4179 .4197 .4216 .4234 .4253 .880 .4272 .4291 .4310 .4329 .4348 .4368 .4387 .4407 .4427 .4447 .890 .4467 .4487 .4507 .4528 .4548 .4569 .4590 .4611 .4633 .4654 .900 .4675 .4697 .4719 .4741 .4763 .4786 .4808 . a 3 1 .4854 .4877 .910 .4900 .4923 .4947 .4971 .4995 .SO19 .SO43 .SO68 .SO93 .5118 .920 .5143 .5169 .5194 .5220 S246 .5273 .5299 .5326 .5353 .5381 .930 .5408 .5436 .5465 .5493 .5522 .5551 .5580 .5610 5640 .5670 .940 .5701 ,5732 .5763 .5795 .5827 .5860 .5892 S926 .5959 .5993 .950 .6028 .6063 .6098 .6134 .6170 .6207 .6245 .6283 .6321 .6360 .960 .6400 .6440 .6481 .6523 .6565 .6608 .6652 .6697 .6742 .6789 .970 .6836 .6884 .6933 .6984 .7035 .7088 .7142 .7197 .7254 .7312 .980 .7372 .7433 .7497 .7562 .7630 .7701 .7774 .7850 .7929 .8012

.990 .8100 .8193 .8291 .8397 .SS11 .8636 .8775 .8935 .9126 .9378

LITERATURE CITED

FISHER, R. A., 1930 Statistical methods for research workers. Edinburgh, Oliver and Boyd.