Trait polygenicity is usually treated like a single number, but this paper shows that four familiar measures are really just points along a continuous scale—one that changes how strongly major and minor variants count. Understanding the genomic basis of phenotypic variation is key to many questions in evolutionary biology and human genetics.

Genome-wide association studies have revealed that the vast majority of traits are polygenic, with variances shaped by a diverse combination of variants across the genome. A variant’s contribution depends on both its effect size and its frequency in the population, generating a wide range of relative contributions across variants.

The extent to which a trait is polygenic is often discussed but rarely explicitly defined, making it difficult to compare polygenicity across traits. A rigorous measure must account not only for the number of contributing variants, but also for the relative contribution of each variant to the trait’s genetic variation.

Quasi-arithmetic means have been proposed for measuring polygenicity because they satisfy mathematical properties considered desirable for a polygenicity measure. Four particular quasi-arithmetic means summarize how many variants contribute to the genetic variation.

Hill polygenicities depend on the vector p, where each p sub i is the relative contribution of variant i to the genetic variance of a trait in a sample. They also depend on a continuous parameter, or order, a greater than or equal to zero, which tunes the emphasis on major versus minor variants.

The Hill polygenicity of order a is defined by the expression shown here. The four polygenicity measures of O’Connor and Sella are instances of the Hill polygenicities. Specifically, Hill polygenicities of order zero, one, two, and infinity are equivalent to the four discrete polygenicity measures of O’Connor and Sella.

First-degree homogeneity means that rescaling a function’s inputs by a constant is equivalent to rescaling its output by the same constant. Here, the function’s inputs are effective polygenicities, one over p sub i, whose units are a number of variants. If a polygenicity measure is also to be a number of variants, it must use a homogeneous function.

Because power means are the only homogeneous quasi-arithmetic means, Hill numbers are the only polygenicity measures among all possible quasi-arithmetic means that yield an effective number of variants. For this scenario, the four discrete measures are ten thousand, approximately nine thousand fifty-one, approximately seven thousand three hundred ninety-six, and two thousand for orders zero, one, two, and infinity.

Order zero captures that ten thousand variants contribute in some way to the trait. The infinity order indicates that there are effectively two thousand contributing variants when only the most dominant contribution is considered.

The gradual taper between order zero and infinity captures many variants with substantial contributions, but contributions that are not completely even. Hill polygenicities are defined for any order a greater than or equal to zero, not only for the four measures that were the focus of O’Connor and Sella.

This makes it possible to plot a polygenicity profile, with order a on the x-axis and the corresponding Hill polygenicity on the y-axis. A trait whose variants have equal contributions has a horizontal profile, while all other genetic architectures have monotonically declining profiles.

A mixture of large and small contributions starts at the number of variants and decays toward the reciprocal of the largest contribution, with more dominated traits showing faster decay. In the plotted points, scenario B falls from about nine thousand at order one to about seven thousand four hundred at order two, while scenario C is already near one at both orders.

At the infinite-order endpoint, the curves sit near ten thousand for A, two thousand for B, and one for C, making the contrast between the scenarios especially clear. The four focal measures correspond to the zero, one, two, and infinity orders of the Hill numbers, a family of ecological diversity statistics.

Viewing them as Hill numbers reveals four points along a continuum that emphasizes major versus minor variants, and measures emphasizing minor variants yield higher polygenicity estimates. Hill numbers extend the four focal measures into a continuous family, enabling polygenicity profiles that facilitate comparisons across traits.

The Hill numbers can be directly integrated into the estimation framework of O’Connor and Sella by using f of x equal to x to the power of one minus a in their equation fifteen. The Hill numbers provide a unified, continuous framework for trait polygenicity: they explain the existing measures, preserve an effective-number interpretation, and let researchers compare traits through full polygenicity profiles.