Ancestry-Specific Versus Shared Effect Estimates

Why Compare Shared and Ancestry-Specific Estimates?

A GWAS can estimate an effect separately within each ancestry group, or constrain the effect to be the same across groups. The shared-effect analysis can use information from all groups to estimate one coefficient, often with greater precision than any one group could achieve alone. But that precision answers a particular question: what common effect is compatible with the model? If effects genuinely differ, a precise shared estimate may not describe the effect in the ancestry where a score will be used.

The distinction matters for polygenic scores because the weights are intended to represent genetic effects relevant to a target population. The earlier material on effect-size heterogeneity distinguishes differences in true effects from differences in linkage disequilibrium or allele frequencies. Here the focus is narrower: how the choice to share or separate an effect changes the estimate, its precision, and its interpretation.

Model and Precision

For clarity, consider a quantitative phenotype \(Y\) and a single, harmonized diploid allele dosage \(G\), coded 0, 1, or 2 copies of the same effect allele. Let \(a\) index ancestry groups. Within group \(a\), consider the linear model

Here \(\beta_a\) is the per-allele change in the mean phenotype in group \(a\), in phenotype units per effect-allele copy. The group-specific intercept \(\mu_a\) allows mean phenotype levels to differ across groups. For the simple standard-error approximation below, assume independent unrelated individuals within and between groups, a correctly specified linear mean model, constant residual variance \(\sigma_a^2\) within each group, and no unaccounted confounding of genotype and phenotype. The genotype is measured comparably and coded in the same direction in each group. Covariates can be included; then the relevant genotype information is the variation remaining after adjustment for those covariates.

In the simple model, the genotype information is approximately \(n_a\operatorname{Var}(G_a)\). Consequently, the standard error of the within-group estimate is approximately

This expression shows three distinct influences on precision: sample size, dosage variance, and residual noise. Allele frequency alone does not determine dosage variance. Under Hardy–Weinberg equilibrium for a well-measured diploid genotype, with effect-allele frequency \(f_a\), the dosage variance is \(2f_a(1-f_a)\). Outside that assumption, use the actual dosage variance (or its covariate-adjusted counterpart), rather than substituting this Hardy–Weinberg expression.

Estimating a Shared Effect

Suppose the effect really is common, so \(\beta_a=\beta\) in every group. If the ancestry-specific estimates are independent and approximately normal, with \(v_a=SE(\hat{\beta}_a)^2\), the shared estimate that uses their sampling precision is

Its standard error under the common-effect model is \(1/\sqrt{\sum_a w_a}\). Thus, more precise estimates receive more weight. In a simple linear regression with group-specific intercepts, this weighting corresponds to the genotype information in each group divided by its residual variance. It does not mean that groups should be weighted equally, nor that the largest sample must always contribute the most information.

The precision result follows directly from the sampling model. If \(\hat{\beta}_a\) is normally distributed with mean \(\beta\) and variance \(v_a\), the log likelihood, up to a constant, is

Differentiating with respect to \(\beta\) and setting the derivative to zero gives \(\sum_a(\hat{\beta}_a-\beta)/v_a=0\). Solving yields the weighted estimate above. Independence gives its variance:

What a Pooled Estimate Means Under Heterogeneity

If the true effects differ, write them as \(\beta_a\) rather than a common \(\beta\). The same weighted calculation then has expectation

It estimates an information-weighted average of the group effects, not each group’s effect. The weights depend on precision, which in turn depends on sample size, genotype variation, residual variance, and covariate adjustment. Therefore the pooled value need not equal an unweighted average, and it need not be close to the effect in a particular target ancestry. A large or highly informative group can dominate the estimate.

Under heterogeneity, a small standard error can still accurately describe the sampling variability of the weighted average. It does not make that average a good substitute for every \(\beta_a\). In particular, the uncertainty about a target group’s effect is not captured by reporting only the standard error of a shared estimate. This distinction is important when the estimate will be used as a polygenic-score weight for a specific ancestry.

Dosage Variance Is Not Always \(2f(1-f)\)

The Hardy–Weinberg expression is useful only when its assumptions are appropriate. Let \(G\) be a diploid dosage and \(f=E[G]/2\) the effect-allele frequency. The general variance is \(E[G^2]-E[G]^2\); frequency specifies \(E[G]\), but does not by itself specify \(E[G^2]\). Genotype distributions can differ from Hardy–Weinberg proportions, for example because of population structure, inbreeding, or sampling.

Choosing Between Shared and Ancestry-Specific Estimates

A shared estimate is attractive when the scientific target is a common effect, the groups have compatible phenotype definitions and allele coding, and the model adequately accounts for structure and relevant covariates. It can reduce sampling noise, especially when one ancestry-specific GWAS is small or has little genotype variation at a variant. The gain is most useful when differences between observed group estimates are plausibly due to sampling error rather than meaningful differences in effects.

An ancestry-specific estimate directly targets the effect within that group, but may be imprecise when its sample has limited size or dosage information. Greater precision is not the only criterion: if true effects vary, separate estimates preserve that variation, while a shared estimate compresses it into one weighted value. For score construction, the appropriate choice depends on the target estimand established for the intended population, not simply on which estimate has the smallest standard error.

In either approach, population structure and differences in phenotype or covariate measurement can distort comparisons if left unaddressed. A pooled model should account for group-specific baseline differences and the design’s relevant covariates; merely combining records does not ensure a valid shared-effect estimate. If samples overlap or estimates are correlated, the independent-estimate variance formula also needs modification. These checks concern whether the model and its uncertainty calculation fit the data, not just whether a pooled number can be computed.

Common Mistake

Wrong reasoning: “The shared estimate has the smallest standard error, so it is the best effect estimate for every ancestry.” A small standard error describes precision for the parameter being estimated under the model; it does not establish that all groups share a true effect.

Correction: First identify the target. If the goal is a common effect and the shared-effect assumption is defensible, pooling can improve precision. If effects are heterogeneous and the goal is prediction in a particular ancestry, the pooled weighted average may be a poor substitute for that ancestry’s effect. Also calculate dosage information from the dosage distribution or justify Hardy–Weinberg equilibrium before using \(2f(1-f)\).