Causal Effects Versus Tag Associations Across Populations

Why a Tag Association Is Not the Causal Effect

A genetic association is often observed at a marker that is correlated with, rather than identical to, the variant affecting the outcome. That marker is a tag SNP. Its estimated association can be useful for detecting a genomic signal, but it is not automatically an estimate of the biological effect of the causal variant. The distinction matters when comparing ancestry groups: a causal effect can be stable while the tag association changes because the correlation between the two variants differs.

The relationship below concerns a single causal variant and a marginal tag-SNP regression within each population. “Marginal” means that the tag is the genetic predictor in the regression; the causal variant is not additionally included as a predictor. Covariates may be included, but the relevant genotype variation and correlation are then those remaining after adjustment for those covariates.

The Single-Causal-Variant Relationship

Consider a quantitative outcome \(Y\) in population \(k\), with the additive model

Here \(X_i\) is a vector of covariates, and \(\varepsilon_i\) is the residual. For the tag association to represent the relationship derived below, assume that this model is correctly specified, the causal effect is additive on the outcome scale, and \(E(\varepsilon_i\mid C_i,T_i,X_i)=0\). Also assume that ancestry-related confounding and other sources of bias have been adequately addressed by the design and covariates. A causal interpretation of \(\beta_{C,k}\) requires the relevant causal identification assumptions; the regression algebra alone does not establish causality.

Let \(\widetilde C,\widetilde T,\widetilde Y\) denote residuals after separately regressing \(C,T,Y\) on the intercept and the same covariates \(X\), within population \(k\). By the Frisch–Waugh–Lovell result, the coefficient of \(T\) in the regression of \(Y\) on \(T\) and \(X\) equals the slope from regressing \(\widetilde Y\) on \(\widetilde T\). Under the model above, residualizing removes the intercept and covariate contribution, giving \(\widetilde Y=\beta_{C,k}\widetilde C+\widetilde\varepsilon\). Therefore,

The conditional mean-zero assumption makes the final covariance zero. Thus the tag coefficient is

where \(r_{TC,k}\) is the correlation between the residualized genotypes, and \(s_{C,k}\) and \(s_{T,k}\) are their standard deviations. The tag coefficient is measured in outcome units per additional copy of the tag effect allele. It depends not only on the causal effect but also on the LD and the residualized genotype scales.

To express effects per standard deviation, define the standardized causal effect as \(\beta_{C,k}s_{C,k}\), using the raw-dosage causal coefficient \(\beta_{C,k}\). The coefficient for a one-standard-deviation increase in residualized tag dosage is then

Equivalently, if \(\beta_{C,k}\) is already defined per standard deviation of the residualized causal genotype, the standardized-tag coefficient is \(\beta_{C,k}r_{TC,k}\). That shortened expression does not apply to a raw per-allele causal coefficient without the factor \(s_{C,k}\). With covariate adjustment, the standard deviations and correlation in these standardized expressions are those of the residualized genotypes.

Three Ways Portability Can Be Misread

A comparison across populations must keep the causal and tag quantities separate, and must use consistent allele coding and effect scales. The following examples show distinct patterns that can otherwise be conflated.

These examples concern estimated effect coefficients, not statistical significance. A smaller coefficient can be estimated precisely in a large sample, while a larger coefficient can be imprecise in a small sample. Differences in allele frequency also change genotype variance and may affect standard errors and power. Those changes should not be mistaken for changes in the causal effect.

Assumptions, Diagnostics, and Alternatives

The derivation is exact for the stated linear model and a single causal predictor, with the same covariate adjustment used to residualize the genotypes and outcome. In practice, check whether the assumed setting is plausible before interpreting the formula:

• Confirm allele alignment. The effect allele at each locus must be identified consistently. A flipped tag allele reverses the sign of its coefficient and of the reported LD correlation. The earlier material on allele harmonization is directly relevant.
• Use population-specific LD and genotype scales. Do not substitute a pooled correlation for \(r_{TC,k}\) when comparing population-specific tag effects. With covariate adjustment, use the residualized rather than unadjusted genotype correlation and standard deviations.
• Check whether a single causal variant is credible. If several causal variants contribute, the tag coefficient generally combines their effects and their covariances with the tag. The one-variant formula is then incomplete.
• Assess confounding and model fit. Residual association of the tag with ancestry-related factors, omitted covariates, or violations of the additive model can produce a tag coefficient not explained by the stated LD relationship.
• Check variation and precision. If the tag has little residualized variation, its coefficient can be unstable. A low-precision estimate does not demonstrate that the true association is zero or that the causal effect differs.

The direct alternative is to estimate the association at the causal variant itself, if that variant is known, accurately measured, and adequately represented in each population. In the single-variant model, its conditional coefficient targets \(\beta_{C,k}\), whereas a marginal tag coefficient is LD-dependent. Joint or conditional analyses can help distinguish correlated signals, but their interpretation depends on correctly specifying and measuring the variants included. A tag association remains useful when the causal variant is unknown or unavailable; it should simply be described as a marker association rather than as the causal effect.

The linear derivation should not be transferred mechanically to every outcome model. For binary outcomes analyzed with logistic regression, marginal and conditional odds ratios can differ for reasons beyond LD, including noncollapsibility. Effect scales also need careful interpretation when populations differ in outcome distributions or covariate distributions. Finally, the causal effect itself can vary because of genuine effect modification, environmental context, or differences in the relevant biological mechanism. LD differences are one explanation for changing tag associations, not a universal explanation.

Common Mistake