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Existing theory for inferences about linkage disequilibrium is restricted to a measure defined on gametic frequencies. Unless gametic frequencies are directly observable, they are inferred from genotypic frequencies under the assumption of random union of gametes. Primary emphasis in this paper is given to genotypic data, and disequilibrium coefficients are defined for all subsets of two or more of the four genes, two at each of two loci, carried by an individual. Linkage disequilibrium coefficients are defined for genes within and between gametes, and methods of estimating and testing these coefficients are given for gametic data. For genotypic data, when coupling and repulsion double heterozygotes cannot be distinguished. Burrows' composite measure of linkage disequilibrium is discussed. In particular, the estimate for this measure and hypothesis tests based on it are compared to the usual maximum likelihood estimate of gametic linkage disequilibrium, and corresponding likelihood ratio or contingency chi-square tests. General use of the composite measure, whether or not random union of gametes is an appropriate assumption, is recommended. Attention is given to small samples, where the non-normality of gene frequencies will have greatest effect on methods of inference based on normal theory. Even tools such as Fisher's z-transformation for the correlation of gene frequencies are found to perform quite satisfactorily.

SUMMARYAn infinite population practising a constant amount of selfing and random mating is studied. The effects of the mating system on two linked loci with an arbitrary number of neutral alleles are determined. Expressions are obtained for the two-locus descent measure, and hence genotypic frequencies and disequilibria functions, in any generation. The nature of the equilibrium population is deduced. The special cases of pure selfing or pure random mating and completely linked or completely unlinked loci are considered separately.

The analysis of gene frequencies for a nested structure of genes within individuals, individuals within subpopulations, and subpopulations within populations is considered. Alternative parameterizations are provided by measures of correlation and of identity by descent, but the latter parameters provide more flexibility. The effects of population size, mating system, mutation, and migration can be incorporated into transition equations for identity measures and the structure of equilibrium populations can be determined; the procedures are illustrated for a finite island model. With parameters dermed before estimation procedures are developed, problems of estimates depending on the numbers of sampled subpopulations are avoided, while the descent measures also avoid the approximations found in other treatments.In the analysis of gene frequencies in natural populations, it is important to have a parametric model elucidating the kinds of variation to be encountered. In this way, various assumptions about the model are clarified and estimation can be guided by the model. Without any information, just correlations and variances often must suffice. Even so, these should also be accurately parameterized as a guide to the appropriate analysis (1, 2).In some cases, the correlations bear the interpretation of identity by descent parameters. These parameters are very useful in studies of the consequences of mating system, finite population size, migration, and even mutation in certain circumstances.The purpose of this note is to relate correlation and identity by descent parameters and to provide an illustration of the versatility of identity by descent parameters for a finite island model at equilibrium with respect to migration and mutation. Genic StructureSince individual genes are identified, the genic hierarchical structure to be considered is genes within individuals, individuals within subpopulations, and subpopulations within populations. This means that distinct pairs of genes fall into the following categories: genes within individuals, genes in different individuals in the same subpopulation, genes in different subpopulations in the same population, and genes in different independent replicate populations. Variance and Correlation ParametersThis development is the same as that of Cockerham (1, 2). Essential details will be reviewed for completeness with some extensions. We utilize a measure xi, = 1 if the gene is Al, the lth allele, where i identifies the location of the gene in the hierarchy, and xi, = 0 if the gene is another allele, Ak, k # l. Then, for a random gene Exi = pi, where 56 denotes expectation and pi is the parametric gene frequency. The variance among random independent genes is %xij -(Gx,1)2 = plPi The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.

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