F ST and the triangle inequality for biallelic markers

The population differentiation statistic F , introduced by Sewall Wright, is often treated as a pairwise distance measure between populations. As was known to Wright, however, F is not a true metric because allele frequencies exist for which it does not satisfy the triangle inequality. We prove that a stronger result holds: for biallelic markers whose allele frequencies differ across three populations, F never satisfies the triangle inequality. We study the deviation from the triangle inequality as a function of the allele frequencies of three populations, identifying the frequency vector at which the deviation is maximal. We also examine the implications of the failure of the triangle inequality for four-point conditions for placement of groups of four populations on evolutionary trees. Next, we study the extent to which F fails to satisfy the triangle inequality in human genomic data, finding that some loci produce deviations near the maximum. We provide results describing the consequences of the theory for various types of data analysis, including multidimensional scaling and inference of neighbor-joining trees from pairwise F matrices.