The Hill numbers synthesize and generalize measures of trait polygenicity

Characterizing the genetic architecture of phenotypic variation is central to human genetics, yet trait polygenicity---the diversity of genetic variants responsible for a heritable trait's variation in a sample---is rarely quantified formally. O'Connor and Sella (2026) addressed this need by proposing four polygenicity measures that are quasi-arithmetic means and therefore satisfy a set of principles they argue are desirable for polygenicity measures. Here, I show that these four measures are points on a continuum corresponding to the Hill numbers, a family of ecological diversity statistics offering a tunable emphasis on variants with small versus large contributions. I further show that the Hill numbers are the only quasi-arithmetic means suitable for measuring polygenicity because they are the only ones that are "effective numbers" of variants. The connection to the Hill numbers clarifies the relationships among O'Connor and Sella's four polygenicity measures: all four are effective numbers of variants, and those that give greater weight to variants with small contributions yield higher estimates of polygenicity. It also connects polygenicity to results in the ecological diversity literature, including the replication principle, a criterion for how such measures should behave when independent sets of variants are combined. Together, these results offer a unified framework for interpreting, comparing, and selecting measures of trait polygenicity.