The population genetics of the self-incompatibility polymorphism in Papaver rhoeas. IV. The estimation of the number of alleles in a population

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Three methods of estimating the number of S-alleles in a population have been proposed in the literature (Bateman, 1947; Whitehouse, 1949; Paxman, 1963). These methods, all of which assume that genotype frequencies in the population are equal, are described briefly and are used to estimate the number of alleles in populations of Trifolium pratense (Williams and Williams, 1947), Oenothera organensis (Emerson, 1939) and Papaver rhoeas (Campbell and Lawrence, 1981b; Lawrence and O'Donnell, 1981). The estimates yielded by Bateman's and White-house's methods are similar to those given by Paxman's maximum likelihood method with the Trifolium and Oenothera data where there is little reason to suppose that the allele frequencies are other than equal. Bateman's method, however, breaks down when used on the Papaver data in which the S-allele frequencies are known to be unequal; and Whitehouse's and the maximum likelihood methods yield estimates which are biased downwards when used on these data. An attempt has been made, therefore, to devise two new estimators of the number of S-alleles in a population which do not assume that their frequencies are equal. The properties of these estimators has been investigated with data from eight populations generated on the computer in which the numbers and frequencies of alleles are known. One of these new estimators (E2) yields estimates which are less biased downwards than those given by Paxman's method when allele frequencies are unequal, but gives estimates which are biased upwards when these frequencies are equal. The other estimator (E1) is generally less satisfactory than the first, particularly when the number of alleles in the population is large. Though neither of these new estimators are wholly satisfactory, there is some justification for using E2 when allele frequencies are known to be unequal. Estimates given by E2 when used on the Papaver data range from 34 to 42 alleles which, bearing in mind that these estimates are still likely to be biased downwards, suggests that the number of alleles in natural populations of this species is likely to be between 40 and 45. A new procedure for calculating confidence intervals for maximum likelihood estimates, assuming equal allele frequencies, is also described and applied to the Oenothera and Papaver data.