An exact formula for the contribution of sampling error to r2, a common measure of linkage disequilibrium

Interest in quantifying linkage disequilibrium (LD, non-random associations of alleles at different loci) has skyrocketed in recent years as researchers have focused on use of LD in genome-wide association studies (GWAS), for studying historical demography, and for estimating effective population size (Ne). The most widely used LD metric is r2 = the squared correlation of alleles at a pair of loci. Despite a half century of efforts, developing an unbiased expectation of r2 as a function of the many factors that can affect it (physical linkage, genetic drift, selection, migration, mutation, mating systems) remains elusive. Furthermore, even when all of these other factors are absent, empirical estimates of r2 are upwardly biased by sampling a finite number (S) of individuals, and that must be accounted for if one wants to focus on the desired signal of LD. Previous approaches to estimate ${r}_{sample}^2$ have been shown to be biased to greater or lesser degrees. The purpose of this short paper is to demonstrate that a simple and exact expression for $E\left({r}_{sample}^2\right)$ does exist for the special case where sampling error is the only factor contributing to r2, in which case $E\left({r}_{sample}^2\right)=1/\left(S-1\right)$. When other factors contribute heavily to LD, $E\left({r}_{sample}^2\right)$ shrinks toward 0 as empirical r2 ➔ 1. However, for estimating contemporary Ne with unlinked markers, empirical r2 will generally be small and $1/\left(S-1\right)$ will provide a robust estimate of $E\left({r}_{sample}^2\right)$.