Mathematical properties of the r2 measure of linkage disequilibrium

Statistics for linkage disequilibrium (LD), the non-random association of alleles at two loci, depend on the frequencies of the alleles at the loci under consideration. Here, we examine the r2 measure of LD and its mathematical relationship to allele frequencies, quantifying the constraints on its maximum value. Assuming independent uniform distributions for the allele frequencies of two biallelic loci, we find that the mean maximum value of r2 is 0.43051, and that r2 can exceed a threshold of 4/5 in only 14.232% of the allele frequency space. If one locus is assumed to have known allele frequencies–the situation in an association study in which LD between a known marker locus and an unknown trait locus is of interest–we find that the mean maximum value of r2 is greatest when the known locus has a minor allele frequency of 0.30131. We find that in 1/4 of the space of allowed values of minor allele frequencies and haplotype frequencies at a pair of loci, the unconstrained maximum r2 allowing for the possibility of recombination between the loci exceeds the constrained maximum assuming that no recombination has occurred. Finally, we use to examine the connection between r2 and the D′ measure of linkage disequilibrium, finding that for 72.683% of the space of allowed values of (pa,pb,pab). Our results concerning the properties of r2 have the potential to inform the interpretation of unusual LD behavior and to assist in the design of LD-based association-mapping studies.