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
I
N A PREVIOUS paper (Snyder, L. H. and Yingling, H., 1935) the
gene frequency method was applied to sex-influenced factors, and a
formula for testing the applicability of the hypothesis of sex-influenced factors to human data was derived. Since any character dependent upon sexinfluenced factors will usually be relatively frequent in males, but relatively 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,
2
q2
-=proportion
2
of bb 3 3 in general population.
Since the proportion of BB 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:
Let
=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
GENETICS
21: 79 M I 1936
80
LAURENCE H. SNYDER AND CHARLES W. COTTERMAN
~
(2)
q=d/ab.
So that
dTTi+dZ=l.
-
(3)
__
From equation ( 3 ) it is possible to derive a value of 9 B in terms of 8 B .
- -
d?m+dX=1
dZ=l-dZ=l-dl-3
B
-Q (B = i-di-3x)2.
(4)
From equation (4) a table may be constructed, giving, for any proportion of males showing a dominant sex-influenced character, the corresponding proportion of females who may be expected to show the character.
(1930), the
Employing the maximum likelihood method of R. A. FISHER
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 recessives is
P = K(p2+2pq)”(q2)N-”= K ( l -q2)>”(q2)N-”
(5)
where
N!
K=
n !(N- n) !
The value to be estimated is p2, so that setting
e=p2
we obtain
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)
aL
n(1-4)
ae
e(2-dijj
-=
N-n
-__-
d2L ( N - n ) ( l - j 4 )
2dif(&-
(7)
&-e
-=
de2
log ( l - d @
n
_______-
e12
2 e 4 @ 2 - de12
Substituting 6 =p2 and n =Np(2 -p), we have
a x _-d(P2I2
N
P3(2-P)
n(1-
dF)
e2(2 - 4 8 ) *
81
STUDIES IN HUMAN INHERITANCE XI11
The variance of p2,
(9)
Substituting p = 1-41
-=,
we get
The complete equation involved is thus
7
9 B = 1-41
-
2
-2%) i.6745( 1- 4 m )(E
N .
From table 1, for any proportion of males showing a dominant sex-influenced character, the proportion of females who may be expected to show
the character may be directly read.
TABLE1
Table of values of
~=(l-dl-~)*
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
.310, the proportion ofis .0287; for
$ B = .316 it is .0299.
a=
-
.003
.005
.OOO
.001
.ooo
.0000
.0000
.WO1
.0002
.0004
.OoOo
.010
,020
.030
.040
,0001
.OoO2
.OW4
.o001
,0003
.0005
.o001
.0003
.OoO5
.OOol
.0003
.OS0
.060
.070
.080
.090
.OW6
.0009
.0013
.0017
.0021
.0007
.0010
.0013
.0017
.0022
.0007
.0010
.0013
,0018
.0022
.0007
.0010
.0014
.0018
.0023
.WO7
.0011
.0014
.0018
.0023
.o008
.0011
.110
.lo0
.0026
.0032
.0045
.0053
.0027
.0033
.0040
.0047
.0054
.0028
.0034
.120
.130
.140
.0027
.0033
.0039
.0046
.0054
.150
.160
.170
.180
.190
.0061
.0070
.0079
.0089
.0100
.0062
.0071
.0080
.0090
.0101
.0063
.0072
.0081
.0091
.0102
,0038
.002
.004
dB
.007
.o000
.om1
.0002
.0003
.o005
.m .oooo
.OoO1
.WO2
.WO3
.WO6
.0001
,0002
.ON4
.0006
.0006
.0019
.0024
.WO8
.0011
.0015
.0019
.0024
.OW8
.0012
.0015
.0020
.0025
.WO9
.0012
.0016
.0020
.0025
.WO9
.0012
.0016
.0021
,0026
.0029
.0034
.0041
.0048
.0056
.0029
.0035
.0042
.0049
.0057
.0030
.0036
.0042
3050
.0058
.0030
.0036
. a 3
.0050
.0058
.0031
.0037
.0044
.0051
.0059
.0031
.0038
.0045
.0052
.0060
.0064
.0073
.0083
.0093
.0104
.0065
.0074
.0084
.0095
,0105
.0066
.0075
.0085
.0096
,0107
.0067
.0076
.0086
.0097
.0065
.0077
.0087
.0098
.0109
.0069
.0078
,0038
.0099
.0110
.oooo .m .oooo .m
.oooo .m .oooo .oooo .o001
.0040
.0047
.0055
.0063
.0072
.0082
.0092
.0103
.009
.006
.OoO5
.0002
.0003
.o005
.0015
,0108
.008
.oOOo
.om1
.0002
.0004
82
LAURENCE H. SNYDER AND CHARLES W. COTTERMAN
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
.0111 .0113
.0124 .0125
.0136 .0138
.0150 .0151
.0164 .0166
.0114
.0126
.0139
.0153
.0167
.0115
.0127
.0140
.0154
.0169
.0116 .0117
.0129 .0130
.0142 .0143
.0156 .0157
.0170 .0172
.0119
.0131
.0145
.0159
.0173
.0120
.0133
.0146
.0160
.0175
.0121
.0134
.0147
.0161
.0176
.0122
.0135
.0149
.0163
.0178
.2SO
.260
.270
.280
.290
.0179
.0195
.0212
.0229
.0248
.0181
.0197
.0214
.0231
.0250
.OH3
.0199
.0215
.0233
.0251
.ON4
.0200
.0217
.0235
.0253
.ON6
.0202
.0219
.0237
.0255
.OH7
.0204
.0221
.0238
.0257
.OH9 .0191
.0205 .0207
.0222 .0224
.0240 .0242
.0259 ,0261
.0192
.0209
.0226
.0244
.0263
.0194
.0210
.0228
.0246
.0265
.300 .0267
.310 .0287
.320 .0308
.330 .0329
.340 .0352
.0269
.OB9
.0310
.0332
.0354
.0271
.0291
.0312
.0334
.0357
.0273
.0293
.0314
.0336
.0359
.0275
.0295
.0316
.0338
.0361
,0277
.0297
.0318
.0340
.0364
.0279 .0281
.0299 .0301
.0320 .0323
.0343 .0345
.0366 .0365
.0283
.0303
.0325
.0347
.0371
.0285
.0305
.0327
.0350
.0373
.3SO
.360
.370
.380
.390
.0375 .0378
.0400 .0403
.0425 .0428
.0452 .0455
.0479 .0482
.400
.0508
.410
.420
.430
.OS38
.OS68
.Om0
.440
.0633
.0380 .0383
.0385
.0410
.0431 .0433 .0436
.0457 .0460 .0463
.0485 .0488 .0491
.0405
.OS11 .OS14
.OS41 .OS44
.OS72 .OS75
.0604 .Om7
.0637 . O M
.OM8
.OS17
.OS20 .OS23 .OS26
.OS47 .OS50 .OS53 .OS56
.OS78 .OS81 .OS84 .0587
.0610 .0613 .0617 .0620
.0644 .0647 .0650 .0654
.4SO .0668 .0671 .0675 .0678
.460 .0703 .0707 .0710 .0714
.470 .0740 .0744 .0747 .0751
.a0
.0778 .0782 .0786 .0789
.490 .OS17 .0821 .0825 .0829
.so0
.0858
.510 .o900
.S20 ,0944
.S30 .0989
.540 .lo35
.5SO
.560
.570
.580
.590
.lo84
.1134
.1185
.1239
.1294
.0388 .0390
.0413 .0415
.0439 .0441
.0466 .0468
.0494 .o497
.0682
.0718
.0755
.0793
,0832
.OS29 .0532
.OS59 .OS62
.OS91 .OS94
.0623 .0627
.0657 .0661
.OS35
.OS65
.OS97
.0630
.0664
.0685 .0689
.0721 .0725
.0758 .0762
.0797 .0801
.0837 .0841
.0692 .0696 .0699
.0729 .0732 .0736
.0766 .0770 .0774
.0805 .0809 .OS13
.OS46 .0550 .0854
.0879
.0922
.0966
.lo12
.lo59
.0883
.0926
.0970
.lo16
.lo64
.0887
.0930
.0975
.lo21
.lo69
.0891 .0896
.0935 .0939
.o980 .0984
.lo26 .lo31
.lo74 .lo79
.1103 .1108
.1154 .1159
.1206 .1212
.1260 .1266
.1316 .1322
.1113
.1164
.1217
.1271
.1328
.1118
.1169
.1222
.1277
.1334
.1123 .1128
.1175 .1180
.1228 .1233
.1283 .1288
.1339 .1345
.0862
.0904
.o948
.0993
.lo40
.0866 .0870 .0875
.0909 .0913 .0917
.0952 .0957 .0961
.0998 .lo03 .lo07
.lo45 .lo50 .lo54
.lo89
.1139
.1190
.1244
.1299
.lo93
.1144
.1196
.1249
.1305
.lo98
.1149
.1201
.1255
.1311
.0393 .0395 .0398
.0418 .0420 .0423
.0444 .0447 .0449
.0471 .0474 .0476
.0499 .OS02 .OSOS
83
STUDIES IN HUMAN INHERITANCE XI11
TABLE
1. (Continued)
Table of Values of
~=((I-dl-~)'
.000
.001
.002
.003
.004
.005
.006
.007
.008
.009
.1351
.1410
.1471
.1534
.1600
.1357
.1416
.1477
.1541
.1607
.1363
.1422
.1484
.1547
.1613
.1368
.1428
.1490
.1554
.1620
.1374
.1434
.1496
.1560
.1627
.1380
.1440
.1502
.1567
.1634
.1386
.1446
.1509
.1574
.1640
.1392
.1453
.1515
.1580
.1647
.1398
.1459
.152!
.1587
.1654
.1404
.1465
.1528
.1593
.1661
.1668
.1738
.1811
.a0 .1886
.690 .1964
.1675
.1745
.1818
.1894
.1972
.1682
.1752
.1826
.1902
.1980
.1689
.1760
.1833
.1909
.1988
.1696
.1767
.1841
.1917
.1997
.1703
.1774
.1848
.1925
.2005
.1710
.1781
.1856
.1933
.2013
.1717
.1789
.1863
.1941
.2021
.1724
.1796
.1871
.1949
.2029
.1731
.1803
.1879
.1957
.2037
.700
.710
.720
.730
.740
.2046
.2130
.2217
.2308
.2402
.2054
.2138
.2226
.2317
.2412
.2062
.2147
.2235
.2326
.2421
.2070
.2156
.2244
.2336
.2431
.2079
.2164
.2253
.2345
.2441
.2087
.2173
.2262
.2354
.2450
.2096
.2182
.2271
.2364
.2460
.2104
.2190
,2280
,2373
.2470
.2113
.2199
.2289
.2383
.2480
.2121
.2208
.2298
.2392
.2490
.750
.760
.770
.780
.790
.2500
.2602
.2708
.2819
.2935
.2510
.2612
.2719
.2831
,2947
.2520
.2623
.2730
.2842
.2959
.2530
.2633
.2741
.2853
.2971
.2540
.2644
.2752
.2865
.2983
.2551
,2655
.2763
.2876
.2995
.2561
.2665
.2774
.2888
.3007
.2571
,2676
.2785
.2900
.3019
.2581
.2637
.2797
.2911
.3031
.2592
.2698
.2808
.2923
.3043
.800
.810
.820
.830
.840
.3056
.3182
.3315
.3454
.3600
.3068
.3195
.3328
.3468
.3615
.3081
.3208
.3342
.3482
.3630
.3093
.3221
.3356
.3497
.3645
.3106
.3234
.3370
.3511
.3661
.3118
.3248
.3383
.3526
.3676
.3131
.3261
.3397
.3540
.3691
.3144
.3274
.3411
.3555
.3707
.3156
.3288
.3425
.3570
.3723
.3169
.3301
.3440
,3585
,3738
.850
,860
.870
.880
.890
.3754
.3917
.4089
.4272
.4467
,3770
.3933
.4107
.4291
.4487
.3786
.3950
.4125
.4310
.4507
.3802
.3967
.4143
.4329
.4528
.3818
.3984
.4161
.4348
.4548
.3834
.4002
.4179
.4368
.4569
.3851
.4019
.4197
.4387
.4590
.3867
.4036
.4216
.4407
.4611
.3883
.4054
.4234
.4427
.4633
.3900
.4071
.4253
.4447
.4654
.900
.910
.920
.930
.940
.4675
.4900
.5143
.5408
.5701
.4697
.4923
.5169
.5436
,5732
.4719
.4947
.5194
.5465
.5763
.4741
.4971
.5220
.5493
.5795
.4763
.4995
S246
.5522
.5827
.4786
.SO19
.5273
.5551
.5860
.4808
.SO43
.5299
.5580
.5892
.a31
.SO68
.5326
.5610
S926
.4854
.SO93
.5353
5640
.5959
.4877
.5118
.5381
.5670
.5993
.6028
.6063
.6440
.6884
.7433
.8193
.6098
.6481
.6933
.7497
.8291
.6134
.6523
.6984
.7562
.8397
.6170
.6565
.7035
.7630
.SS11
.6207
.6608
.7088
.7701
.8636
.6245
.6652
.7142
.7774
.8775
.6283
.6697
.7197
.7850
.8935
.6321
.6742
.7254
.7929
.9126
.6360
.6789
.7312
.8012
.9378
.600
.610
.620
.630
.640
.650
.660
.670
.950
.960
.970
.980
.990
.6400
.6836
.7372
.8100
LITERATURE CITED
FISHER,
R. A., 1930 Statistical methods for research workers. Edinburgh, Oliver and Boyd.
SNYDER,L. H. and YINGLING,
H., 1935: Human Biology. 7: 608-615.