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CALIBRATING THE CLOCK: USING STOCHASTIC PROCESSES TO MEASURE THE RATE OF EVOLUTION 122 The Ewens Sampling Formula Motivated by the realization that mutations in DNA sequences could lead to an essentially infinite number of alleles at the given locus, Kimura and Crow (1964) advocated modeling the effects of mutation as an infinitely- many-alleles model. In this process, a gene inherits the type of its ancestor if no mutation occurs and inherits a type not currently (or previously) existing in the population if a mutation does occur. In such a process the genes in the sample are thought of as unlabeled, so that the experimenter knows whether two genes are different, but records nothing further about the identity of alleles. In this case the natural statistic to record about the sample is its configuration Cn â¡ (C1, C2,. . ., Cn), where Cj = number of alleles represented j times. Of course, C1 + 2C2+ . . . + nCn = n, and the number of alleles in the sample is Kn â¡ C1 + C2 + . . . +Cn. (5.3) The sampling distribution of Cn was found by Ewens (1972): (5.4) for a = (a1,a2,. . .,an) satisfying aj ⥠0 for j = 1,2,. . .,n and and where θ (n)⡠θ (θ + 1)â¦(θ+ vâ 1). From (5.4) it follows that
CALIBRATING THE CLOCK: USING STOCHASTIC PROCESSES TO MEASURE THE RATE OF EVOLUTION 123 (5.5) and (5.6) being the Stirling number of the first kind. From (5.5) and (5.4) it follows that Kn is sufficient for θ, so that the information in the sample relevant for estimating θ is contained just in Kn. This allows us (Ewens, 1972, 1979) to calculate the maximum likelihood (and moment) estimator of θ as the solution of the equation (5.7) where k is the number of alleles observed in the sample. In large samples, the estimator has variance given approximately by (5.8) For the pyrimidine sequence data described above in the ''Overview" section, there are k = 24 alleles. Solving equation (5.7) for gives = 10.62, with a variance of 9.89. An approximate 95 percent confidence interval for θ is therefore 10.62 ± 6.29. This example serves to underline the variability inherent in estimating θ from this model. The pyrimidine region comprises 201 sites, so that the per site substitution rate is estimated to be 0.053 ± 0.031. The goodness of fit of the model to the data may be assessed by using the sufficiency of Kn for θ: given Kn, the conditional distribution of the allele frequencies is independent of θ. Ewens (1972, 1979) gives further details on this point. To describe alternative goodness-of-fit methods, we return briefly to the probabilistic structure of mutation in the coalescent.
