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Abstract In studying allele-frequency variation across populations, it is often convenient to classify an allelic type as “rare,” with nonzero frequency less than or equal to a specified threshold, “common,” with a frequency above the threshold, or entirely unobserved in a population. When sample sizes differ across populations, however, especially if the threshold separating “rare” and “common” corresponds to a small number of observed copies of an allelic type, discreteness effects can lead a sample from one population to possess substantially more rare allelic types than a sample from another population, even if the two populations have extremely similar underlying allele-frequency distributions across loci. We introduce a rarefaction-based sample-size correction for use in comparing rare and common variation across multiple populations whose sample sizes potentially differ. We use our approach to examine rare and common variation in worldwide human populations, finding that the sample-size correction introduces subtle differences relative to analyses that use the full available sample sizes. We introduce several ways in which the rarefaction approach can be applied: we explore the dependence of allele classifications on subsample sizes, we permit more than two classes of allelic types of nonzero frequency, and we analyze rare and common variation in sliding windows along the genome. The results can assist in clarifying similarities and differences in allele-frequency patterns across populations. 

Spatial population genetic data often exhibits ‘isolation-by-distance,’ where genetic similarity tends to decrease as individuals become more geographically distant. The rate at which genetic similarity decays with distance is often spatially heterogeneous due to variable population processes like genetic drift, gene flow, and natural selection. Petkova et al., 2016 developed a statistical method called Estimating Effective Migration Surfaces (EEMS) for visualizing spatially heterogeneous isolation-by-distance on a geographic map. While EEMS is a powerful tool for depicting spatial population structure, it can suffer from slow runtimes. Here, we develop a related method called Fast Estimation of Effective Migration Surfaces (FEEMS). FEEMS uses a Gaussian Markov Random Field model in a penalized likelihood framework that allows for efficient optimization and output of effective migration surfaces. Further, the efficient optimization facilitates the inference of migration parameters per edge in the graph, rather than per node (as in EEMS). With simulations, we show conditions under which FEEMS can accurately recover effective migration surfaces with complex gene-flow histories, including those with anisotropy. We apply FEEMS to population genetic data from North American gray wolves and show it performs favorably in comparison to EEMS, with solutions obtained orders of magnitude faster. Overall, FEEMS expands the ability of users to quickly visualize and interpret spatial structure in their data.