Converting Binary-Trait Estimates to the Liability Scale

Why a Binary-Trait Estimate Needs a Scale

For a binary trait, the recorded outcome is usually coded 0 for controls and 1 for cases. Its observed variance depends on the fraction of cases in the data. If cases are deliberately sampled more often than they occur in the population, that variance reflects the sample composition, not population prevalence. A heritability estimate on this observed 0–1 scale therefore cannot be interpreted directly as the fraction of liability variance explained by the SNP set.

The liability-threshold model represents disease status as the result of an unobserved, continuous liability crossing a threshold. The conversion uses two distinct quantities: the population prevalence, which locates the threshold, and the sample case fraction, which determines the observed binary phenotype variance in the analyzed data. Confusing these quantities can substantially change the converted estimate, especially for uncommon traits.

The Threshold-Model Conversion

Let \(G\) be the centered additive genetic value attributable to the SNP set, and let \(E\) collect the remaining liability components. For the standard derivation, assume \(G\) and \(E\) are independent, centered normal variables and \(L=G+E\) is standardized to have mean 0 and variance 1. Thus, \(\operatorname{Var}(G)=h^2_{SNP,L}\), the SNP heritability on the liability scale. The threshold is chosen so that \(P(L>t)=K\).

The binary phenotype has observed-scale variance \(\hat{p}(1-\hat{p})\) in a sample with case fraction \(\hat{p}\). In the usual threshold-model conversion, the genetic variance on this scale is approximated by \(z^2\operatorname{Var}(G)\). The resulting relationship and its rearrangement are:

Here \(h^2_{SNP,obs}\) is the observed-scale estimate, defined relative to the binary phenotype variance in the analyzed sample; \(h^2_{SNP,L}\) is the liability-scale SNP heritability, a proportion of the unit liability variance. All quantities in the conversion factor are dimensionless. The formula is often called the observed-to-liability scale conversion.

The displayed expression uses \(K^2(1-K)^2\) rather than simply \(K(1-K)\) because \(z\), the density at the population threshold, is related to prevalence through the local change in case probability near that threshold. The formula follows from the threshold model and the standard linear or small-effect approximation for translating liability genetic variance to the binary scale. It is not an exact equality for every possible distribution of genetic effects or arbitrarily large effects.

Deriving the Density-at-the-Threshold Term

A short derivation explains why the standard normal density \(z\) enters the formula. Since \(G\) and \(L\) are centered and jointly normal, with \(\operatorname{Var}(L)=1\) and \(\operatorname{Cov}(G,L)=\operatorname{Var}(G)\), their conditional expectation is:

Because \(G\) is centered, \(\operatorname{Cov}(Y,G)=E[(Y-K)G]=E[G\,1(L>t)]\). Conditioning on \(L\) and integrating over the liability tail gives:

For the last equality, \(\phi'(l)=-l\phi(l)\), so \(\int_t^\infty l\phi(l)\,dl=\phi(t)\). The slope of the best linear prediction of \(Y\) from \(G\) is therefore \(\operatorname{Cov}(Y,G)/\operatorname{Var}(G)=z\). Its explained variance is \(z^2\operatorname{Var}(G)\), which motivates the observed-scale relationship above. The conversion additionally uses the binary variance appropriate to the sample and the prevalence-defined threshold.

Worked Conversions

How to Apply and Interpret the Result

A reliable calculation keeps the inputs separate and in context. Use the population prevalence to determine where the liability threshold lies; use the analyzed sample’s case fraction for the observed binary variance. Then multiply the observed-scale estimate by the conversion factor. The resulting liability-scale estimate refers to the SNP set used for the original estimate, not to all possible genetic influences.

The result is an estimate, not a probability that a particular person will develop the trait. It does not mean that a stated percentage of an individual’s liability is genetic. It describes the estimated fraction of liability variance attributable to the specified SNP set in the population and under the model. The estimate is also specific to the population and prevalence used: changing the prevalence changes the threshold and therefore the conversion.

Common Mistakes and Model Limits

• Using the sample case fraction as the population prevalence. The threshold is set by population prevalence \(K\); the observed binary variance uses sample fraction \(\hat{p}\). They coincide only when sampling reflects population prevalence.
• Using the wrong binary variance. For an analyzed sample with case fraction \(\hat{p}\), the observed variance is \(\hat{p}(1-\hat{p})\), not automatically \(K(1-K)\).
• Reversing the conversion. The liability estimate is the observed estimate multiplied by \(K^2(1-K)^2/\{\hat{p}(1-\hat{p})z^2\}\). Inverting this factor converts in the opposite direction.
• Treating the relationship as exact in every model. The standard conversion uses a linear or small-effect approximation. Strong departures from the threshold-model assumptions, or effects too large for the approximation, can make it less reliable.
• Leaving the prevalence undefined. Prevalence must refer to the population relevant to the trait and analysis. A mismatched prevalence produces a mismatched threshold and conversion.