Sampling variance and distribution of the D' measure of overall gametic disequilibrium between multiallelic loci

The development of the theory of estimation of gametic disequilibrium for multiallelic systems is particularly necessary, since a large number of the genetic markers available at present are highly polymorphic multiallelic systems. The D' coefficient is one of the most commonly used measures of the extent of overall disequilibrium between all possible pairs of alleles at two multiallelic loci. Nevertheless, the sampling properties of this measure of overall disequilibrium, are to date, unknown. In this work, we have derived explicit expressions by large-sample theory to compute the approximate sampling variance of Dhat' between pairs of multiallelic loci, when samples of haplotypes are taken from populations. Formulae for calculating the asymptotic sampling variance were checked by Monte Carlo simulation. In addition, the magnitude of the sampling variance of Dhat' was investigated under different scenarios of disequilibrium between multiallelic loci. Extensive simulations were also carried out for describing the sampling distribution of Dhat', conditioned on the sample size, number of alleles and their frequencies, and disequilibrium components. It was found that the sampling distribution of Dhat' generally approaches well the theoretical normal distribution for experimental sample sizes, particularly when loci have many alleles. Disequilibrium data between microsatellite loci of human chromosome 11p are used for illustration. These investigations increase substantially our knowledge about this widely used measure of overall disequilibrium, which is relevant to evaluate disequilibrium between multiallelic loci in populations.