Discovery-Sample Ancestry Composition and Weight Precision

Why Discovery Composition Matters

A GWAS weight is an estimated association, not a known genetic effect. Its precision depends on how much information the discovery sample contributes at that variant. Two features matter directly: how many informative individuals are included and how much the genotype varies among them. Because allele frequencies can differ across ancestry groups, equal group sizes need not provide equal information for every variant.

This creates an important distinction. A score may be less accurate in a target population because the genetic effects differ, because the tagging of causal variation differs, or simply because the discovery data estimated some weights imprecisely. The focus here is the last pathway: sampling uncertainty in weights arising from discovery-sample size and composition. The central comparison assumes effects transfer; unequal precision alone can still produce unequal score accuracy.

Sampling Precision of a GWAS Weight

Consider one variant in one discovery group. Let \(G\) be its allele dosage, coded as 0, 1, or 2 copies, and let \(Y\) be a quantitative trait. A simple association model is

Here \(\beta\) is the change in the mean trait per additional effect-allele copy under this model. The ordinary least-squares estimate is denoted \(\hat{\beta}\). For the familiar variance formula below, assume independent sampled individuals, a correctly specified linear conditional mean, finite relevant moments, and homoskedastic errors: \(E(\varepsilon_i\mid G_i)=0\) and \(\operatorname{Var}(\varepsilon_i\mid G_i)=\sigma^2\). Also assume \(0<\operatorname{Var}(G)<\infty\). Under the usual large-sample conditions, the approximate variance and standard error are

The units of \(\hat{\beta}\) are trait units per allele copy. Larger \(N\) and larger genotype variance reduce its sampling uncertainty. Under Hardy–Weinberg equilibrium, if the effect-allele frequency is \(p\), then \(\operatorname{Var}(G)=2p(1-p)\). This expression is conditional on that population-genetic assumption; in practice, genotype variance can instead be estimated from the observed sample.

A Justification—and the Role of the Assumptions

With an intercept in the regression, let \(G_c=G-E(G)\). The OLS slope is based on the sample covariance of genotype and outcome divided by the sample genotype variance. When the conditional mean is linear, the leading random part of the slope comes from the term \(G_c\varepsilon\). Under homoskedasticity, its variance is

The sample average of these terms has variance approximately \(\sigma^2\operatorname{Var}(G)/N\). Dividing by the squared limiting denominator \(\{\operatorname{Var}(G)\}^2\) gives \(\sigma^2/[N\operatorname{Var}(G)]\), as stated. This argument shows why conditional homoskedasticity matters: an unconditional residual variance alone does not make the genotype-weighted residual variance equal to \(\sigma^2\operatorname{Var}(G)\).

If \(\operatorname{Var}(\varepsilon\mid G)\) varies with genotype, the usual large-sample variance for the slope instead has the form

A heteroskedasticity-robust standard error estimates this variance without imposing a constant conditional residual variance. With covariates, the corresponding calculation uses genotype variation remaining after accounting for those covariates, together with the appropriate residual variance. Thus, the simple formula is useful, but only when its assumptions—or a suitable adjusted version—are justified.

How Composition Contributes to Information

Suppose a discovery sample contains groups indexed by \(k\), with \(N_k\) individuals in group \(k\). To estimate a common effect while allowing group-specific means, consider a model with an intercept \(\alpha_k\) for each group and a shared slope \(\beta\). Assume the same per-allele effect in every group, independent observations, and residual variance \(\sigma_k^2\) within group \(k\), with conditional mean zero and conditional homoskedasticity there. With genotype centered within each group, the approximate variance of the shared-slope estimate is

Each group contributes information in proportion to its sample size and within-group genotype variance, and inversely to its residual variance. If residual variances are equal, this reduces to the reciprocal of the sum of \(N_k\operatorname{Var}(G\mid k)/\sigma^2\). A group can therefore add many participants but relatively little information at a particular variant if genotype variation there is low. Conversely, a smaller group can contribute meaningfully at a variant where genotype variation is larger.

Centering within groups is important to this interpretation: the association is estimated from genotype differences among individuals within groups, rather than from differences in group-average genotype and group-average outcome. If the true effects are not shared, the common-slope estimate need not represent the effect in every group. The calculation describes precision under a shared-effect model; it does not establish that the model is biologically appropriate.

From Weight Uncertainty to Score Accuracy

At a set of variants, write the centered genotype vector in a target population as \(G\), the vector of true effects as \(\beta\), and the estimated weights as \(\hat{\beta}=\beta+u\), where \(u\) is estimation error. The difference between the estimated and true scores is \(G^\mathsf{T}u\). Let \(\Sigma_T=\operatorname{Cov}(G)\) in the target population, and let \(C=\operatorname{Cov}(u)\) describe uncertainty in the estimated weights. If weight errors have mean zero and are independent of target genotypes, then the expected variance of the score error is

This expression connects discovery precision to target score error. In the special case of independent loci and independent weight errors, it becomes

Thus, weight uncertainty at a variant contributes more score noise when that variant also varies substantially in the target population. If loci or estimation errors are correlated, the full covariance expression is needed; the sum across loci is not generally valid. These equations concern uncertainty from estimated weights, not LD mismatch or effect heterogeneity.

Common Mistakes and Practical Interpretation

• Assuming sample size alone determines precision. Two groups with the same \(N\) can have different genotype variances and hence different standard errors at a variant. Use \(N\operatorname{Var}(G)\), with residual variance also accounted for, under the stated model.
• Using the homoskedastic formula with only an unconditional residual variance. Knowing \(\operatorname{Var}(\varepsilon)=\sigma^2\) does not establish \(\operatorname{Var}(\varepsilon\mid G)=\sigma^2\). Check conditional homoskedasticity or use an appropriate robust variance estimate.
• Interpreting an imprecise weight as a small effect. A large standard error says the data estimate the effect less precisely; it does not by itself imply the true effect is small or absent.
• Calling unequal score performance proof of non-transferable effects. Lower discovery representation can increase weight noise and reduce accuracy even when effects are shared. Other mechanisms may also matter, so performance differences alone do not identify their cause.
• Treating the total number of participants as the whole story. Relatedness, missing genotypes, covariate adjustment, phenotype residual variance, and the effective genotype variation can all alter the information available beyond the nominal sample count.