The estimator of the optimal measure of allelic association: mean, variance and probability distribution when the sample size tends to infinity.

ABSTRACT

The allelic association or linkage disequilibrium between two loci is a parameter of fundamental interest in modern population genetics for evolutionary inference and association mapping studies. Among the many measures available, the optimal measure of allelic association rho presents a strong evolutionary theory basis and is modeled on the physical distance along the chromosome with the Malécot equation for isolation by distance. Moreover, rho is equal to the absolute value of D', the standardized measure of gametic disequilibrium. We studied here the statistical properties of the rho sample estimator. We derived its asymptotic probability distribution and showed that it is neither asymptotically normal nor unbiased when rho=0 or when allelic frequencies are equal at both loci, in contrast to previous claims. This asymptotic study leads to propose a new test for absence of linkage disequilibrium. We compared it to Pearson's Chi2 test for independence in a contingency table and showed by simulations that the range in power of these two tests depends on the sign of D'. The new test outperformed slightly the Chi2 test, when D', polarized with respect to major alleles, is negative. Finally, we derived the asymptotic bias and information of the rho estimator that are due to the experimental sampling and showed by simulation that its bias is large in small samples. The consequences of these findings on applications using the rho measure are then discussed in particular for constructing LD unit maps, and call for a revised statistical treatment.