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 r(2) 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 r(2) is approximately 0.43051, and that r(2) can exceed a threshold of 4/5 in only approximately 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 r(2) is greatest when the known locus has a minor allele frequency of approximately 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 r(2) allowing for the possibility of recombination between the loci exceeds the constrained maximum assuming that no recombination has occurred. Finally, we use r(max)(2) to examine the connection between r(2) and the D(') measure of linkage disequilibrium, finding that r(2)/r(max)(2)=D('2) for approximately 72.683% of the space of allowed values of (p(a),p(b),p(ab)). Our results concerning the properties of r(2) have the potential to inform the interpretation of unusual LD behavior and to assist in the design of LD-based association-mapping studies.

Genotype, haplotype and copy-number variation in worldwide human populations

Genome-wide patterns of variation across individuals provide a powerful source of data for uncovering the history of migration, range expansion, and adaptation of the human species. However, high-resolution surveys of variation in genotype, haplotype and copy number have generally focused on a small number of population groups. Here we report the analysis of high-quality genotypes at 525,910 single-nucleotide polymorphisms (SNPs) and 396 copy-number-variable loci in a worldwide sample of 29 populations. Analysis of SNP genotypes yields strongly supported fine-scale inferences about population structure. Increasing linkage disequilibrium is observed with increasing geographic distance from Africa, as expected under a serial founder effect for the out-of-Africa spread of human populations. New approaches for haplotype analysis produce inferences about population structure that complement results based on unphased SNPs. Despite a difference from SNPs in the frequency spectrum of the copy-number variants (CNVs) detected--including a comparatively large number of CNVs in previously unexamined populations from Oceania and the Americas--the global distribution of CNVs largely accords with population structure analyses for SNP data sets of similar size. Our results produce new inferences about inter-population variation, support the utility of CNVs in human population-genetic research, and serve as a genomic resource for human-genetic studies in diverse worldwide populations.