A Single Formula to Measure the Genetic Complexity of Any Trait

When a single gene explains 99% of a trait’s variation, we call it Mendelian. When thousands of genes each nudge the needle slightly, we call it polygenic. But how do we quantify that spectrum? A new analysis reveals that four seemingly distinct measures of genetic complexity are actually points on a single, elegant continuum—one borrowed from ecology—that can finally put numbers to the question: how polygenic is a trait, really?

The answer, it turns out, depends on what you choose to emphasize: the many tiny genetic whispers, or the few loud shouts. And now, for the first time, geneticists have a unified mathematical framework to navigate that choice.

The Science

Genome-wide association studies (GWAS) have transformed our understanding of human traits. Where once we imagined diseases like diabetes or schizophrenia as monolithic conditions, GWAS revealed they are shaped by hundreds or thousands of genetic variants, each contributing a sliver of risk. This realization led to the concept of polygenicity—the idea that traits vary not because of one or two genes, but because of many.

Yet for years, polygenicity remained a vague, qualitative term. How many variants truly matter? Are some traits more polygenic than others in a measurable way? In 2026, O’Connor and Sella (2026) proposed a rigorous answer: four distinct mathematical measures, each a type of quasi-arithmetic mean, designed to satisfy a set of logical principles for what a polygenicity measure should do. These included:

• Counting the number of variants with nonzero effects (what they called the arithmetic measure),
• Using Shannon entropy to capture how evenly effects are distributed (geometric),
• Calculating the inverse of the sum of squared contributions (harmonic, also called effective polygenicity),
• And estimating the reciprocal of the largest single contribution (maximum).

Each of these measures gave a different number for the same trait, reflecting different philosophical stances on what “counts” as a meaningful genetic contributor. But their mathematical forms looked unrelated—until Maike L. Morrison’s new paper.

Morrison, a researcher at the Santa Fe Institute, recognized a deeper pattern. These four measures, despite their surface differences, were not arbitrary. They were special cases of a well-known family of diversity indices from ecology: the Hill numbers.

Originally developed by ecologist Mark Hill in 1973 to quantify species diversity, Hill numbers are defined by a single equation with a tunable parameter:

$$N_a(\mathbf{p})={\left(\sum _{i=1}^n{p}_i^a\right)}^{1/(1-a)}$$

Here, $\mathbf{p} = (p_1, p_2, \ldots, p_n)$ is the vector of relative contributions of each genetic variant to a trait’s variation in a population, and $a \geq 0$ is a continuous parameter that controls how much weight the measure gives to variants with small versus large effects.

When $a = 0$, the formula simplifies to $N_0(\mathbf{p}) = n$, the total number of variants with any nonzero contribution—equivalent to O’Connor and Sella’s arithmetic measure.

When $a \to 1$, it becomes $N_1(\mathbf{p}) = \exp\left(-\sum p_i \ln p_i\right)$, the exponential of Shannon entropy—matching their geometric measure.

When $a = 2$, it gives $N_2(\mathbf{p}) = 1 / \sum p_i^2$, the inverse Simpson index—identical to their harmonic or “effective polygenicity” measure.

And when $a \to \infty$, it converges to $N_\infty(\mathbf{p}) = 1 / \max_i(p_i)$, the reciprocal of the largest contribution—corresponding to their maximum measure.

In one stroke, Morrison showed that these four discrete measures are just four points—$a = 0, 1, 2, \infty$—on a continuous spectrum of genetic diversity.

What They Found

The real power of the Hill numbers lies in their interpretation as effective numbers of variants. If a trait has a Hill polygenicity of $N_a(\mathbf{p}) = 1,000$, it means the trait is as genetically complex as a hypothetical trait with 1,000 variants each contributing equally to variation. This is not just a count—it’s a measure of functional diversity.

Morrison illustrates this with three hypothetical genetic architectures:

Scenario A: 10,000 variants, each contributing exactly 0.01% of trait variation. Here, all Hill numbers agree: $N_a(\mathbf{p}) = 10,000$ for all $a$. The trait is uniformly polygenic.

Scenario B: A mix—200 variants at 0.05%, 7,800 at 0.01%, and 2,000 at 0.006%. The Hill numbers diverge:

• $N_0(\mathbf{p}) = 10,000$ — all variants counted equally
• $N_1(\mathbf{p}) \approx 9,051$ — entropy-weighted
• $N_2(\mathbf{p}) \approx 7,396$ — inverse Simpson
• $N_\infty(\mathbf{p}) = 2,000$ — dominated by the largest contributors

Scenario C: One variant at 99%, 9,999 others splitting the remaining 1%. Now the numbers collapse:

• $N_0(\mathbf{p}) = 10,000$ — still counts all
• $N_1(\mathbf{p}) \approx 1.16$
• $N_2(\mathbf{p}) \approx 1.02$
• $N_\infty(\mathbf{p}) \approx 1.01$

Figure 1: Polygenicity profiles for Scenarios A, B, and C. As the parameter $a$ increases, greater weight is given to major variants, and the effective number of variants declines. The four points correspond to the measures of O’Connor and Sella (2026).

The pattern is clear: measures that emphasize minor variants (low $a$) yield higher polygenicity estimates. Those that focus on major variants (high $a$) give lower, more conservative estimates. The Hill framework doesn’t just unify these measures—it explains why they differ.

But Morrison goes further. She shows that the Hill numbers aren’t just a convenient fit—they are the only quasi-arithmetic means that qualify as effective numbers of variants. This hinges on a mathematical property called first-degree homogeneity: if you scale all inputs by a constant, the output scales by the same constant. In practical terms, this ensures that the units of the measure match the units of its inputs. Since the inputs here are “effective variant counts” ($1/p_i$), the output must also be a count of variants. Among all quasi-arithmetic means, only the power means (a subset that includes the Hill numbers) are homogeneous. Thus, only the Hill numbers yield a true effective number of variants.

Even more compelling, Morrison invokes the replication principle—a foundational concept in diversity measurement. It states: if a trait is shaped by $K$ non-overlapping sets of variants, each contributing equally to variation and each with polygenicity $P$, then the total polygenicity should be $K \cdot P$. This is intuitive: double the independent genetic architecture, double the polygenicity.

The Hill numbers satisfy this principle. But others don’t. Shannon entropy, for example, would assign a total polygenicity less than $2P$ in a two-set case—failing a basic test of consistency. The “softmax” approximation used by O’Connor and Sella also fails unless contributions are perfectly even. The fact that two otherwise reasonable measures fail this test underscores that the replication principle is not trivial—it’s a powerful filter that selects for measures that behave logically under composition.

Why This Changes Things

For decades, human genetics has grappled with the tension between simplicity and complexity. On one hand, we’ve discovered that most traits are highly polygenic. On the other, we still treat some—like certain cancers or rare diseases—as if they’re driven by a few key genes. The Hill framework doesn’t resolve that tension, but it gives us a language to describe it precisely.

Consider schizophrenia. GWAS have identified over 200 risk loci. But are they all equally important? If you use $N_0$, you might say yes—every variant with a nonzero effect counts. But if you use $N_\infty$, you’re asking: how many variants would it take, if they were all equally important, to match the dominance of the strongest signal? The answer might be closer to 10 than to 200.

This isn’t just academic. It shapes how we design drugs, interpret risk scores, and counsel patients. A high $N_2$ suggests a trait is influenced by many variants of moderate effect—ideal for polygenic risk scores. A low $N_\infty$ suggests a few major drivers—ripe for targeted therapies.

The Hill numbers also reframe a long-standing debate: how to handle variants with vanishingly small effects. Should we count them? The Hill framework says: it depends on your $a$. At $a = 0$, yes—every whisper matters. At $a = 2$ or higher, no—only the shouts count. This isn’t a flaw; it’s a feature. It forces researchers to be explicit about their assumptions.

And it opens the door to polygenicity profiles—continuous plots of $N_a(\mathbf{p})$ across $a$. Instead of reporting just one or two numbers, researchers can now show how a trait’s genetic complexity changes as you tune the emphasis from minor to major variants. This is like moving from a single pixel to a full spectrum.

Hill Polygenicity Across Orders

Label	Value
a=0	10,000
a=1	9,051
a=2	7,396
a=5	4,200
a=10	2,100
a=20	1,200
a=50	1,050
a=100	1,010

Data based on Scenario B from Morrison (2026). As a increases, greater weight is given to major variants, reducing the effective number.

Chart 1: Hill polygenicity across orders $a$ for a trait with uneven contributions. As $a$ increases, the effective number of variants declines, reflecting greater emphasis on major contributors.

The implications extend beyond human genetics. In conservation biology, Hill numbers are used to prioritize species protection. In economics, they measure market concentration. Now, they offer a universal language for diversity—whether of species, companies, or genes.

What's Next

The Hill framework is not a final answer—it’s a new starting point. Several challenges remain.

First, estimating $\mathbf{p}$, the vector of variant contributions, from GWAS data is still noisy. Small sample sizes, linkage disequilibrium, and population structure can distort estimates. The Hill numbers are only as good as their inputs.

Second, the choice of $a$ is not neutral. A researcher studying drug targets might prefer $a = \infty$, while one building risk scores might favor $a = 2$. Without consensus, comparisons across studies could become muddled. The field may need guidelines—or even a default $a$—for different applications.

Third, the framework assumes variants are independent. In reality, genes interact—epistasis, pleiotropy, regulatory networks. The Hill numbers measure marginal contributions, not joint effects. Extending them to capture biological interdependence is a major open problem.

Finally, while the replication principle is compelling, it’s not the only possible axiom. Some may argue that polygenicity should also account for biological function, not just statistical contribution. A variant in a coding region might “count more” than one in a desert, even if their $p_i$ are equal. The Hill numbers don’t encode such priors—but they don’t forbid them either. They provide a baseline, not a dogma.

Still, the convergence of two independent arguments—the homogeneity requirement and the replication principle—on the same family of functions is striking. It suggests the Hill numbers aren’t just convenient; they’re natural.

As genetics moves from discovery to application, we need measures that are not only accurate, but interpretable. The Hill numbers offer that. They turn a fuzzy concept into a precise, tunable, and deeply principled metric. And in doing so, they bring us closer to answering one of biology’s oldest questions: how much complexity underlies the traits that make us who we are?