Analytic Computation of the Expectation of the Linkage Disequilibrium Coefficient r

The squared correlation coefficient r2 (sometimes denoted ∆2) is a measure of linkage disequilibrium that is widely used, but computing its expectation E[r2] in the population has remained an intriguing open problem. The expectation E[r2] is often approximated by the standard linkage deviation σ2 d, which is a ratio of two expectations amenable to analytic computation. In this paper, a method of computing the population-wide E[r2] is introduced for a model with recurrent mutation, genetic drift and recombination. The approach is algebraic and is based on the diffusion process approximation. In the limit as the population-scaled recombination rate ρ approaches ∞, it is shown rigorously that the asymptotic behavior of E[r2] is given by 1/ρ + O(ρ−2), which, incidentally, is the same as that of σ2 d. A computer software that computes E[r2] numerically is available upon request.