Separating Stratification From Ancestry-Specific Effects

Why Separate Stratification From Effect Differences?

A genotype can be associated with an outcome in a pooled sample even when it has no association within any ancestry stratum. This can happen when the strata differ in both genotype frequencies and average outcome. The pooled association then partly reflects the mixture of groups, not a genotype effect within groups.

But a different pattern is also possible: genotype effects may genuinely differ across strata. In that case, a single common-effect estimate can conceal important variation. The key is to estimate effects within strata and then ask whether their differences are larger than sampling uncertainty would reasonably explain. An ancestry-stratified analysis helps distinguish these situations, but it does not by itself establish why effects differ or make ancestry a biological cause.

How Pooling Can Create an Association

Let \(G\) be genotype dosage coded as 0, 1, or 2 copies of the allele of interest, let \(Y\) be a quantitative outcome, and let \(S\) denote ancestry stratum. A simple within-stratum linear model is \(Y=\alpha_s+\beta_sG+\varepsilon\) for individuals in stratum \(s\). Here, \(\alpha_s\) is the stratum-specific intercept, \(\beta_s\) is the change in mean outcome per additional allele copy, and the errors have conditional mean zero. A common-effect model assumes \(\beta_s=\beta\) for every stratum; an interaction model allows the slopes to differ.

For a pooled linear regression with an intercept, the estimated slope is the pooled covariance of \(G\) and \(Y\), divided by the pooled variance of \(G\). The covariance can be decomposed into within-stratum and between-stratum components:

where \(w_s\) is the proportion of observations in stratum \(s\), \(\mu_{G,s}\) and \(\mu_{Y,s}\) are its genotype and outcome means, and \(\mu_G\) and \(\mu_Y\) are the pooled means. The first term summarizes within-stratum association. The second summarizes how the stratum means vary together. Thus, even if every within-stratum covariance is zero, the pooled covariance need not be zero.

Estimating Effects and Testing Heterogeneity

For two strata, a regression with an interaction can be written using an indicator \(I_B\), which equals 1 in stratum B and 0 in A:

The genotype slope in A is \(\beta\); the slope in B is \(\beta+\delta\). The parameter \(\gamma\) allows the groups to have different outcome means when \(G=0\). Testing \(H_0:\delta=0\) tests whether the two genotype slopes are equal in this model. With more than two strata, include a genotype-by-stratum interaction and test the joint null that all interaction coefficients are zero.

It is also possible to estimate a slope separately in each stratum, then test a contrast between estimates. For two independent strata, if \(\hat{\beta}_A\) and \(\hat{\beta}_B\) have standard errors \(SE_A\) and \(SE_B\), their difference has estimated standard error \(\sqrt{SE_A^2+SE_B^2}\), provided the estimates are independent. A Wald statistic divides the estimated difference by this standard error. Under the null and the usual large-sample approximation, it is compared with a standard normal distribution; its square is compared with a chi-square distribution with 1 degree of freedom. In a joint model, use the estimated covariance of the contrast rather than assuming independence when that assumption does not hold.

Assumptions and Interpretation

These calculations are meaningful only when the analysis is sufficiently well specified. For the linear model, the conditional mean of \(Y\) should be adequately represented by the chosen intercepts, genotype terms, and any included covariates. Genotype should be coded consistently across strata, and the dosage effect assumes a linear change in the mean outcome per additional copy. If that coding or functional form is inappropriate, a slope comparison may not capture the relevant pattern.

For the usual standard errors and Wald tests, observations should be independent at the level assumed by the model, and the standard-error calculation must reflect the outcome variability and study design. If relatives or other clusters are present, a procedure that accounts for their dependence is needed. The normal approximation also requires enough information for the coefficient estimates to be reasonably approximated by a normal distribution. In small or sparse strata, estimates and tests may be unstable.

An interaction test is scale-specific. For a quantitative outcome, it tests equality of mean differences per dosage copy under the chosen model. For a binary outcome, an interaction in a logistic regression tests equality of log-odds effects, not equality of risk differences. Effects can be equal on one scale and differ on another. The scale and interpretation should therefore be stated with the result.

Stratification can help identify a pooled association driven by differences between groups, while an interaction test evaluates whether the modeled within-stratum effects differ. It cannot by itself establish the biological mechanism of a difference. Measurement, sampling, environmental exposures, model misspecification, and limited precision may all contribute. Nor does a nonsignificant interaction show that effects are equal: a wide confidence interval can reflect low power to detect heterogeneity.

Common Mistake

A common mistake is to say, “The genotype effect is significant in stratum A but not significant in stratum B, so the effects differ.” This is not a valid heterogeneity test. Statistical significance depends on both the estimated effect and its standard error; two similar estimates can fall on opposite sides of a significance threshold if their precisions differ. Conversely, two individually nonsignificant estimates can still differ significantly from each other.

The correction is to test the contrast directly, such as \(H_0:\beta_A-\beta_B=0\), and report its estimate, standard error, p-value, and confidence interval. A second mistake is to treat a pooled slope as the within-stratum effect. The covariance decomposition shows why that can fail: pooled association includes both within-stratum and between-stratum contributions.