Understanding admixture fractions: theory and estimation of gene-flow

Estimation of admixture proportions has become one of the most commonly used computational tools in population genomics. However, there is remarkably little population genetic theory on statistical properties of these variables. We develop theoretical results that can accurately predict means and variances of admixture proportions within a population using models with recombination and genetic drift. Based on established theory on measures of multilocus disequilibrium, we show that there is a set of recurrence relations that can be used to derive expectations for higher moments of the admixture proportions distribution. We obtain closed form solutions for some special cases. Using these results, we develop a method for estimating admixture parameters from estimated admixture proportions obtained from programs such as Structure or Admixture. We apply this method to HapMap 3 data and find that the population history of African Americans, as expected, is not best explained by a single admixture event between people of European and African ancestry. The model of constant gene flow starting at 8 generations and ending at 2 generations before present gives the best fit.