Linkage disequilibrium - equivalence of the correlation and probability approaches

Amongst measures of linkage disequilibrium, the correlation measure r2, due originally to Sewall Wright, has several advantages. Although basically a haploid statistic, it can readily be estimated using covariance and correlation from diploid data. Its population expectation can also be found from probability methods. We show that the probability of linked identity by descent, LIBD, directly estimates the expectation for r2 for Wright-Fisher models. The simplest such model for two loci is the haploid model, where each haplotype is chosen independently from haplotypes of the previous generation. This gives an equilibrium between recombination (recombination frequency c) and drift (population size N) where the expected value of r2 is approximately 1/(1+4Nc). For diploid models, the expectation is similar for low values of c, but contains a term in c2 and rises to double the haploid value for the limiting value of c = 0.5. An increase in r2 is also expected for the model of monogamous mating, as opposed to random mating for each offspring. The expectation for r2 is unaffected by the use of diploid rather than haploid data, an exception being the case of monogamous mating. Computer simulation shows close agreement of r2 with probability expectations, with minor discrepancies observed for unlinked loci. Using the probabilistic definition, r2 can be intuitively understood as a parameter shaping the effective sample size in genome-wide association studies.