AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 71:377-380 (1986) Correlations Between Relatives in Small Populations ALAN R. ROGERS Department of Anthropology, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 K E Y WORDS Population genetics, Altruism ABSTRACT Correlations between relatives in small, closed populations can be substantially smaller than predicted by the classical formulas of population genetics. This effect is especially pronounced for relatives whose most recent common ancestor is several generations removed. When the effective population size is small, correlations between even close relatives can be negative. This implies that in small populations conventional estimators of quantitative genetics parameters will be biased and that preferential treatment of close relatives will be less likely to evolve. Correlations between relatives measure the extent to which relatives tend to be more similar than random individuals drawn from some reference population. Classical population genetics theory predicts genetic correlations relative to a hypothetical infinite foundation stock. Unfortunately, in many contexts within agricultural genetics and evolutionary biology, the relevant reference population is small. Although it is often noted that these correlations are not equivalent (Wright, 1969; Canning and Thompson, 1981), the distinction between them is usually ignored. Correlations relative to a finite foundation stock are smaller than predicted by classical models, and correlations relative to the CUTrent generation are smaller still. In this paper, I investigate the magnitude of these differences and discuss their implications for quantitative genetics and evolutionary biology. CORRELATIONS RELATIVE TO A FINITE ANCESTRAL POPULATION Correlations between relatives are frequently predicted from pedigrees, using the formulas of Wright (1969) and Malecot (1969). As Jacquard (1975) has emphasized, these correlations measure similarity of individuals relative to the earliest generation in the pedigree (say generation 0). It is less widely appreciated, however, that these formulas are only approximate unless the population of generation 0 is finite. Consider two genes drawn a t random from individuals, X and Y , in generation t in a population of diploid individuals with dis- 0 1986 ALAN R. LISS, INC. crete generations and constant size N . Let x equal unity if the gene from X is a copy of allele A l , or zero otherwise, and define y similarly for the gene drawn from Y . The correlation between x and y relative to generation i is defined by where piisthe frequency ofAi in generation i. r; can be expressed in terms of &, the probability that genes drawn from individuals X and Y are “identical by descent” from generation i, i.e., that both genes are copies of a single gene inherited from a common ancestor in that generation. The possibility that X and Y may be identical by descent from some more remote generation does not contribute to this probability. For the moment, we are interested in ro, the correlation relative to the earliest generation in the pedigree. In calculating it, three possibilities need to be considered: The genes drawn from X and Y may be (a)copies of the same gene in generation 0, (b)copies of distinct genes from a common ancestor in generation 0, or (c) copies of genes from distinct individuals in generation 0. ro is the average of the correlations implied by (a), (b),and (c), weighted by the corresponding probabilities, which are 40, 40, and 1-240, respectively. Putting all this together, we have ro = 40 + +oro(b) + (1 - 24o)ro(~), Received October 14, 1985;revision accepted May 27, 1986 378 A.R. ROGERS where rkh)and r$c) are the correlations im- ents relative to the pool of gametes is thereplied by (b)and (c). To simplify matters, I will fore assume that mating is at random, implying that rJb) = r$) = r*, where r* is the average correlation between distinct genes is generation 0, relative to generation 0. Thus which is differs from the formula for correlations relative to generation 0 only in that N is replaced by N,. Thus changing N to N, in equation 2 produces a n expression for the which is analogous to Crow’s (1980)equation correlation of genes drawn from X and Y 4.In Malecot’s theory, genes are taken to be relative to the gamete pool produced by genindependent if they are not identical by de- eration 0. scent, yet distinct genes are not independent in finite populations. In a popualtion of size CORRELATIONS RELATIVE TO THE N , the average correlation between distinct CURRENT GENERATION genes, relative to their own generation, is r* Frequently, we are interested in rt, the cor= - l/(2N - 1).Substituting produces relation relative to the current generation, t. Equation 1 implies that 2N40 - 1 ro = ~ { x y =} PO ( 1 - p0)r-o + p20. 2N-1‘ When N is large, this reduces to ro = 40,the familiar large-population result. Note that if $0 < l / 2 N , ro is negative. Correlations between genes (relative to generation 0) are negative if the genes are less likely to be copies of the same gene in generation 0 than two genes drawn with replacement from that generation. Although I have made ro a correlation relative to the population of generation 0, it may be more useful to,use the correlation relative to the pool of gametes produced by that generation. Let l/2Ne be the probability that two distinct gametes produced by generation 0 are copies of the same parental gene. The probability that a gene drawn from this gamete pool is A1 is PO.The probability that a second gene is a copy of A l , but not a copy of the same parental gene, is PO- 1/2N,. If we condition on the fact that the genes are derived from different parental genes, the probability becomes Substituting this back into equation 1 produces rt = 1 - PO(1 - Po) - Pt - Po ( 1 - ro) Pt(1 - PA’ PtU - Pt) In the absence of selection, migration, and mutation, the expectation of pt is PO, and (Crow and Kimura, 1970:109).Taking a ratio of expectations as a n approximation to the expectation of a ratio, we have To apply this formula, values are needed for 40, t, and N,. If the complete pedigree of X and Y is known back to generation 0, $0 can be obtained using Wright’s (1969:177) well known formula. The value of t is the This is the probability that both genes are number of generations separating X and Y A1 given that the they are copies of different from their common ancestor. For example t = 1 for sibs, since they share ancestors in parental genes. The correlation of genes from distinct par- the previous generation, and t = 2 for cou- 379 CORRELATIONS IN SMALL POPULATIONS sins, since their common ancestors are grandparents. The correlations discussed here are between random genes drawn from two related individuals. In many applications, interest is focused instead on “genotypic values” of these individuals, equal to 2, 1, and 0 for genotypes AIA1, A1A2, and A2A2, respectively. When mating is at random, correlations between genotypic values are twice the genic correlations studied here (Crow and Kimura, 1970:138). DISCUSSION Equation 3 indicates that rt is a decreasing function oft, the number of generations separating individuals X and Y from their common ancestor. Thus the discrepancy between the infinite-population and small-population models depends on how closely the individuals are related. In the same population, the infinite-population model may provide a good approximation to the correlations between sibs but a poor approximation to correlations between more distant relatives. This effect is important when correlations are inferred from deep pedigrees. Table 1 shows how correlations between relatives are affected by effective population size, N,. For comparison, the last row contains the correlations under the conventional infinite-population model. Although the values of Ne in Table 1 are small, they are not so small as to be irrelevant. Ne is about onethird of N in many human population and would be much smaller in highly polygynous species or in many breeds of domestic animals. For example, Wright (1951) concluded that Ne was about 100 for British short-horn cattle during the nineteenth century. Table 1shows that popualtion size can have a large effect on correlations between relatives. Cor- relations in small populations are substantially smaller than those predicted by largepopulation theory, particularly for distant relatives. These results are accurate only for populations that are completely isolated. In other populations, correlations should lie somewhere between the prediction of equation 3 and that of the infinite-population model. Estimates of quantitative genetics parameters are generally based on correlations obtained under the infinite-population model. The results obtained here suggest that these estimates will be biased in small populations, especially if based on similarity between distant relatives. Estimates based on cousins would be biased by several percent even in populations of effective size 1,000. This bias should usually be negligible, since most estimates are based on similarity between first-degree relatives (sibs and parentoffspring pairs) in fairly large populations. With the advent of maximum-likelihood methods for estimating quantitative genetics parameters from deep pedigrees (Bulmer, 1980), however, the bias introduced by population size may become increasingly important. Hamilton (1964) showed that altruism can evolve if the benefit (B)to the recipient and the cost (C)to the altruist satisfy CIB < 24. In small populations, the 4 should be replaced by rt. Thus, Hamilton’s theory implies that selection within a population of effective size 10 should produce spite rather than altruism between cousins. On the other hand, selection between such groups might favor indiscriminate altruism toward members of the group (Wade, 1978) but would not favor selective altruism based on geneological reckoning. Subdivision of populations increases the opportunity for selection between TABLE 1. Genic correlations between relatives as a function of effective population size* First cousins ( t = 2) Full sibs ( t = 1) Ne 5 10 25 50 100 1,000 m Third cousins ( t = 4) )‘t 70 Tt % rt 5% 0.0741 0.1690 0.2191 0.2348 0.2424 0.2492 0.2500 30 68 88 94 97 100 100 -0.2860 -0.0935 0.0039 0.0338 0.0483 0.0611 0.0625 -458 - 150 6 54 77 98 100 -0.6869 -0.2873 -0.1020 -0.0474 -0.0214 0.0014 0.0039 -17,584 - 7,355 -2,610 - 1,214 -547 36 100 *Columns labeled “97” express E { r , ) as a percentage of its limiting large-population value. A.R. ROGERS 380 groups. At the same time, it reduces the opportunity for kin selection within groups by reducing correlations of relatives relative to local groups. This may be part of the reason why, among Hoogland's (1985, 1986) black-tailed prairie dogs, altruistic interactions are poorly correlated with pedigree kinship, and females often kill the offspring of close relatives. His study colony is polygynous and comprises only about 130 individuals, so its effective size is probably quite small. If so, selection favoring preferential treatment of relatives would be reduced. CONCLUSIONS The effects of finite population size on correlations between relatives are usually ignored. The results presented here make it possible to determine when this procedure is justified and when it is not. In small populations, correlations between relatives are smaller than the predictions of the conventional infinite-population model, and this effect is much greater for distant relatives than for close ones. For example, in a population of effective size 100, the correlations between sibs and third cousins are 98%and -547% of their large-population values, respectively. These results are approximate and refer to completely isolated populations. When there is immigration from the outside world, correlations will be larger than predicted by the model used here. These results imply that estimates of quantitative genetics parameters will be biased in small populations, especially if based on deep pedigrees Furthermore, they imply that seletion for altruism toward close kin will be reduced. This may help explain the absence of preferential treatment of close relatives among Hoogland's (1985, 1986) prairie dogs. ACKNOWLEDGMENTS I thank James F. Crow, Randall Fitzgerald, Steven Gaulin, Henry Harpending, Lynn Jorde, Russell Lande, and Michael Siege1 for their comments and suggestions. LITERATURE CITED Cannings, C, and Thompson EA (1981) Genealogical and genetic structure. New York: Cambridge Univeristy Press. Crow, JF (1980) The estimation of inbreeding from isonymy. Hum. Biol. 52:l-14. Crow, JF, and Kimura, M (1970) An introduction to population genetics theory. New York:Harper and Row. Hamilton, WD (1964) Genetical evolution of social behavior, I. J. Theor. Biol. 7:338-363. Hoogland, JL (1985) Infanticide in prairie dogs: Lactating females kill offsping of close kin. Science 230:10371040. Hoogland, JL (1986) Nepotism in prairie dogs (Cynomys ludouicianus) varies with competition not with kinship. Anim. Behav. 34:263-270. Jacquard, A (1975) Inbreeding: One word, several meanings. Theor. Population Biol. 7:338-363. Malecot, G (1969) The mathematics of heredity. San Francisco: H. Freeman. Wade, MJ (1978) A critical review of the models of group selection. Quart. Rev. Biol. 53:lOl-114. Wright, S (1951) The genetical structure of populations. Ann. Eugen. 15:323-354. Wright, S (1969) Evolution and the genetics of Populations 11. The Theory of Gene Frequencies. Chicago: University of Chicago Press.

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