Ann. Hum Qenet., Lond. (1976), 39, 471
471
Printed in Ured Brilain
Test of the neutrality hypothesis Written on behalf of GEORGE R. PRICE* *Note. Dr Price died unexpectedly in January 1976 while he was working on the Kimura neutral gene hypothesis. He had been investigating methods of programming and using a significance teat, due to W. J. Ewens (1972), and had been applying them to the date on enzyme polymorphisms collected in the Galton Laboratory. Although many of his papers have gone astray, some computer print-outs have been saved. I n view of the work he had put into these, the natural importance of the subject, the generous support given to him by the Medical Research Council, and the affection of his many friends, it would seem worth while publishing these results. No doubt he himself would have presented them differently; but the following paper attempts t o set them out fairly. I am grateful for help, suggestions, and criticism from Professor H. Harris, Dr C. J. Hilditch, and especially from Professor W. J. Ewens, and also to Dr W. J. Hamilton who has succeeded in saving some of Dr Price’s incomplete manuscripts. CEDRIC A. B. SMITH
Suppose that we take at random a sample of n individuals from a population (large in size compared with the sample). Suppose that at a particular locus we find k different alleles, with respective frequencies nl,n2,...,nk,so that Xns = 2n. Suppose further that the following conditions held: (1) These alleles have virtually no selective effect, so that the population has reached its present state purely by the introduction of new alleles by rare mutations and disappearance by random extinction; (2) No two mutations are identical, and different mutants are recognizable and distinguishable phenotypically; (3) The population is stationary and of constant size. Then Ewens (1972) showed that under random mating the n, have the following distribution
Pr(n,ln,Ic)= (2n)!/(k!Zknl, ...,nk) within wide limits independently of the population size, or mutation rate. I n principle it should be possible to see whether the observed allele frequencies are compatible with such a distribution. I n fact, two methods were proposed by Ewens for this. The first one used the statistic k
B=
z [(nrPn)In (ntl2nn)l. i- 1
Ewens gives formulas for the expectation b(B) and variance var(B) of B on the neutrality hypothesis. Hence, on that hypothesis, B - b(B) ,Ivar (Bj has zero expectation and unit standard deviation. This can be used as a crude test of the neutrality hypothesis so that values of ILI exceeding 2.0 will show deviations from neutrality roughly significant at the 5 % level. But because the distribution can be very non-Gaussian, considerable caution must be exercised in interpreting these values. More accurately, Ewens gives a variable P which has distribution nearly equal to that of the variance ratio with determinable degrees of freedom. For details, see Ewens (1972).
GEORGER. PRICE
472
Table 1 Sample size
No. of alleles
Locus
n
k
PGM, PGM, PGM, PHI TPI GPD, LDH, ICD, MOR, PGD ADH, ADH, DIA SOD, GOT, GOT, GPT AK ADA NP ACP, ACP, PL PEPA PEPB PEPC PEPD ESD
10333 10333
7 4
I937
2 2
1550
1750 555
3
1015
2
718 516 4939 598 847 I975 11237 I I95 I054 498 6760 4798 I542 7887 91I 3244 8798 7041 4536 4262 454
3
2
2
3 2
2
6 2
3 2
4 3
4 3 3 2
14 7 4 4 4 2
B
0.55889 0'00349 0'57095 0*00290 0.00298 0.01317 0.00784 0.02077 0'00770 0.09514 0' I3474 0.67575 0.03812 0.00281 0.00737 0.08612 0.72365 0.16868 0.20929 0.00579 0.83427 0.00468 0.96026 0,00486 0*00930 0.06799 0'04700 0.3I844
L
P
-0.4853
0.6661 0.0058 12.7464
- 1.3141 1.6996 -0.8257 - 1.2129 - 0.8792 - 0.8409 - 1.2779 -0.9112 - 0.7776 -0.3440 2.0497
- 2.0202
- 0.6988 - 1.2477 - 0'4955 0.4804 -0.4886 -0.7568 - 1.2191 1.8763 - 0.8648 - 1.1605 - 2.0546 - 1.3378 - I -2070 - I .2808 0'4172
Upper tail Prob ( I /P) 0.352 0'02 I
0.876
0'01 I I
0.311
0.0058
0.047
0.0427
0.302
0.0281
0.320
0.0350
0.069 0.269
0.0244
0'2341 0.5399 92.5221
0.301
0.534 0'942
0.03I I
0'002
0.0141 0'0134 0.3509 1'5534 0.4670 0.3691
0.429 0.054
0'01I 0
0.055
8.2895 0.0I 64 0.5406 0.0041 0.0148 0.1062 0.07I 6 I 4304.4
0.932 0.290 0.130
0.511
0.676 0.430 0.283
0'0002
0.029 0' I 0 0
0.07I 0.682
Jn real populations the assumptions (2) and (3) above will not strictly hold good. But one may hope that they will not seriously upset the appropriate validity of the results. With this in mind, an improved computer program was devised and applied to the data on enzyme systems so far obtained in the Galton Laboratory (Harris, Hopkinson & Robson, 1974). The results were as shown in Table 1. A first inspection shows that among the systems with more than one allele studied, 3 have ILI > 2, whereas at first sight we would expect about 1.4 on the basis of a Gaussian distribution However, on using the more accurate P test PGM,, DIA and PEPA all give an upper tail probability of less than 0.025 and TPI,PEPB, ADH, GOT,, N P and PEPI) all achieve or approach an upper-tail of 0.05. Also, on the sample sizes chosen, Ewens hart pointed out (in a personal communication) that when k = 2 it is not possible to obtain an uppcr tail as small as 0.05; so that it is only in the 17 cases with k > 3 that significance could be achieved, and half of these have roughly a 6 yo one-tail significance.
Test of the neutrality hypothesis
473
SUMMARY
An examination of the frequency of certain enzyme system variants in biochemical data provisionally suggests that at least half of those with more than two alleles observed are subject to selection. REFERENCES
EWENS,W. J. (1972). The sampling theory of selectively neutral alleles. Theoretical Population Biology 3 , 87-112. HARRIS, H., HOPKINSON, D. A. & ROBSON, E. B. (1974). The incidence of rare alleles determining electrophoretic variants; data on 43 enzyme loci in man. Annul8 of Human Qenetics 37, 237-253.
471
Printed in Ured Brilain
Test of the neutrality hypothesis Written on behalf of GEORGE R. PRICE* *Note. Dr Price died unexpectedly in January 1976 while he was working on the Kimura neutral gene hypothesis. He had been investigating methods of programming and using a significance teat, due to W. J. Ewens (1972), and had been applying them to the date on enzyme polymorphisms collected in the Galton Laboratory. Although many of his papers have gone astray, some computer print-outs have been saved. I n view of the work he had put into these, the natural importance of the subject, the generous support given to him by the Medical Research Council, and the affection of his many friends, it would seem worth while publishing these results. No doubt he himself would have presented them differently; but the following paper attempts t o set them out fairly. I am grateful for help, suggestions, and criticism from Professor H. Harris, Dr C. J. Hilditch, and especially from Professor W. J. Ewens, and also to Dr W. J. Hamilton who has succeeded in saving some of Dr Price’s incomplete manuscripts. CEDRIC A. B. SMITH
Suppose that we take at random a sample of n individuals from a population (large in size compared with the sample). Suppose that at a particular locus we find k different alleles, with respective frequencies nl,n2,...,nk,so that Xns = 2n. Suppose further that the following conditions held: (1) These alleles have virtually no selective effect, so that the population has reached its present state purely by the introduction of new alleles by rare mutations and disappearance by random extinction; (2) No two mutations are identical, and different mutants are recognizable and distinguishable phenotypically; (3) The population is stationary and of constant size. Then Ewens (1972) showed that under random mating the n, have the following distribution
Pr(n,ln,Ic)= (2n)!/(k!Zknl, ...,nk) within wide limits independently of the population size, or mutation rate. I n principle it should be possible to see whether the observed allele frequencies are compatible with such a distribution. I n fact, two methods were proposed by Ewens for this. The first one used the statistic k
B=
z [(nrPn)In (ntl2nn)l. i- 1
Ewens gives formulas for the expectation b(B) and variance var(B) of B on the neutrality hypothesis. Hence, on that hypothesis, B - b(B) ,Ivar (Bj has zero expectation and unit standard deviation. This can be used as a crude test of the neutrality hypothesis so that values of ILI exceeding 2.0 will show deviations from neutrality roughly significant at the 5 % level. But because the distribution can be very non-Gaussian, considerable caution must be exercised in interpreting these values. More accurately, Ewens gives a variable P which has distribution nearly equal to that of the variance ratio with determinable degrees of freedom. For details, see Ewens (1972).
GEORGER. PRICE
472
Table 1 Sample size
No. of alleles
Locus
n
k
PGM, PGM, PGM, PHI TPI GPD, LDH, ICD, MOR, PGD ADH, ADH, DIA SOD, GOT, GOT, GPT AK ADA NP ACP, ACP, PL PEPA PEPB PEPC PEPD ESD
10333 10333
7 4
I937
2 2
1550
1750 555
3
1015
2
718 516 4939 598 847 I975 11237 I I95 I054 498 6760 4798 I542 7887 91I 3244 8798 7041 4536 4262 454
3
2
2
3 2
2
6 2
3 2
4 3
4 3 3 2
14 7 4 4 4 2
B
0.55889 0'00349 0'57095 0*00290 0.00298 0.01317 0.00784 0.02077 0'00770 0.09514 0' I3474 0.67575 0.03812 0.00281 0.00737 0.08612 0.72365 0.16868 0.20929 0.00579 0.83427 0.00468 0.96026 0,00486 0*00930 0.06799 0'04700 0.3I844
L
P
-0.4853
0.6661 0.0058 12.7464
- 1.3141 1.6996 -0.8257 - 1.2129 - 0.8792 - 0.8409 - 1.2779 -0.9112 - 0.7776 -0.3440 2.0497
- 2.0202
- 0.6988 - 1.2477 - 0'4955 0.4804 -0.4886 -0.7568 - 1.2191 1.8763 - 0.8648 - 1.1605 - 2.0546 - 1.3378 - I -2070 - I .2808 0'4172
Upper tail Prob ( I /P) 0.352 0'02 I
0.876
0'01 I I
0.311
0.0058
0.047
0.0427
0.302
0.0281
0.320
0.0350
0.069 0.269
0.0244
0'2341 0.5399 92.5221
0.301
0.534 0'942
0.03I I
0'002
0.0141 0'0134 0.3509 1'5534 0.4670 0.3691
0.429 0.054
0'01I 0
0.055
8.2895 0.0I 64 0.5406 0.0041 0.0148 0.1062 0.07I 6 I 4304.4
0.932 0.290 0.130
0.511
0.676 0.430 0.283
0'0002
0.029 0' I 0 0
0.07I 0.682
Jn real populations the assumptions (2) and (3) above will not strictly hold good. But one may hope that they will not seriously upset the appropriate validity of the results. With this in mind, an improved computer program was devised and applied to the data on enzyme systems so far obtained in the Galton Laboratory (Harris, Hopkinson & Robson, 1974). The results were as shown in Table 1. A first inspection shows that among the systems with more than one allele studied, 3 have ILI > 2, whereas at first sight we would expect about 1.4 on the basis of a Gaussian distribution However, on using the more accurate P test PGM,, DIA and PEPA all give an upper tail probability of less than 0.025 and TPI,PEPB, ADH, GOT,, N P and PEPI) all achieve or approach an upper-tail of 0.05. Also, on the sample sizes chosen, Ewens hart pointed out (in a personal communication) that when k = 2 it is not possible to obtain an uppcr tail as small as 0.05; so that it is only in the 17 cases with k > 3 that significance could be achieved, and half of these have roughly a 6 yo one-tail significance.
Test of the neutrality hypothesis
473
SUMMARY
An examination of the frequency of certain enzyme system variants in biochemical data provisionally suggests that at least half of those with more than two alleles observed are subject to selection. REFERENCES
EWENS,W. J. (1972). The sampling theory of selectively neutral alleles. Theoretical Population Biology 3 , 87-112. HARRIS, H., HOPKINSON, D. A. & ROBSON, E. B. (1974). The incidence of rare alleles determining electrophoretic variants; data on 43 enzyme loci in man. Annul8 of Human Qenetics 37, 237-253.
