The nonlinear structure of linkage disequilibrium

The allele frequency dependence of the ranges of all measures of linkage disequilibrium is well-known. The maximum values of commonly used parameters such as r2 and D vary depending on the allele frequencies at each locus. However, though this phenomenon is recognized and accounted for in many studies, the comprehensive mathematical framework underlying the limits of linkage disequilibrium measures at various frequency combinations is often heuristic or empirical. Here, it is demonstrated that underlying this behavior is the fundamental shift between linear and nonlinear dependence in the linkage disequilibrium structure between loci. The proportion of linear and nonlinear dependence can be estimated and it demonstrates how even the same values of r2 can have different implications for the nature of the overall dependence. One result of this is the value of D′, when defined as only a positive number, has a minimum value of |r|. Understanding this dependence is crucial to making correct inferences about the relationships between two loci in linkage disequilibrium.