Omitted-Variable Bias In Single-SNP Tests

Why a Single-SNP Estimate Can Depend on Covariate Adjustment

A single-SNP regression estimates the association between an allele dosage and a quantitative phenotype. But an estimated dosage coefficient can reflect more than the dosage–phenotype relationship represented by the model. If an omitted covariate is related to dosage and also predicts the phenotype, the unadjusted coefficient can absorb part of that covariate’s association with the phenotype. This is omitted-variable bias.

In genomic analyses, a covariate might represent ancestry, age, a technical batch, or another measured feature. The key issue is not whether a variable is labelled a “covariate”; it is whether the model has omitted a relevant predictor whose distribution varies with genotype. The coefficient change after adding a covariate is informative only when both models, their assumptions, and their target associations are understood.

The Bias Formula and Its Conditions

First consider one covariate. Suppose the conditional mean is correctly represented by

Here \(\beta_G\) is the dosage coefficient conditional on \(C\), and \(\gamma\) is the covariate’s conditional effect on the mean of \(Y\). The mean-zero condition implies that the error has zero covariance with \(G\). The unadjusted regression of \(Y\) on an intercept and \(G\) has population slope \(\operatorname{Cov}(G,Y)/\operatorname{Var}(G)\), provided \(\operatorname{Var}(G)>0\). Substituting the model gives

The intercept contributes zero covariance, and \(E(\varepsilon\mid G,C)=0\) gives \(\operatorname{Cov}(G,\varepsilon)=0\). Thus the unadjusted population slope equals the conditional dosage effect plus a bias term:

The formula has two requirements for nonzero bias: \(C\) must affect the conditional phenotype mean (\(\gamma\ne 0\)), and \(C\) must covary with dosage. If either factor is zero, this particular bias term is zero. With multiple omitted covariates collected in a vector \(\mathbf C\), and corresponding coefficient vector \(\boldsymbol{\gamma}\), the same argument gives \(\beta_G^{\mathrm{unadjusted}}=\beta_G+\boldsymbol{\gamma}^{T}\operatorname{Cov}(\mathbf C,G)/\operatorname{Var}(G)\).

This is a population relationship under the specified linear conditional-mean model, not a guarantee that estimates in a finite sample will equal their population values. The model assumes a correctly coded dosage, a linear additive dosage effect, an appropriate linear covariate effect, and no omitted features of the mean that invalidate the stated model. For interpreting an adjusted coefficient as a conditional association, the covariate model must be adequate and the included predictors must not be perfectly collinear. No causal interpretation follows from this algebra alone.

Direction, Magnitude, and Sign Reversal

The sign of the bias is the sign of the product \(\gamma\operatorname{Cov}(G,C)\), since \(\operatorname{Var}(G)\) is positive. A covariate can be positively associated with the phenotype but negatively associated with dosage; in that case, it shifts the unadjusted coefficient downward. If the shift is large enough, the unadjusted and conditional coefficients can even have opposite signs. This is not a mathematical contradiction: the regressions describe different associations.

In an actual analysis, the two estimates will usually not be exact population quantities. They vary because of sampling, measurement error, and model misspecification. A changed coefficient is therefore a clue to investigate, not by itself a diagnosis that a particular covariate caused bias.

Comparing Adjusted and Unadjusted Estimates

A useful comparison fits both models to the same individuals, using the same dosage coding and phenotype scale. The unadjusted model includes an intercept and \(G\); the adjusted model adds the covariate or covariates justified by the analysis. Compare the dosage coefficients, their standard errors, and the sample sizes. If missing covariate values cause different observations to be used, an apparent coefficient change can reflect the changed sample as well as adjustment.

For a comparison to diagnose the bias described by the formula, the covariate should be associated with dosage and have a conditional relationship with phenotype under the model. Check these relationships in the analysis sample, but do not treat separate marginal associations as proof of the full model assumptions. Also check whether the covariate is coded appropriately, whether important nonlinearities or interactions are plausible, and whether adding it creates severe collinearity. These checks help determine whether the adjusted coefficient is a credible conditional association.

Diagnostics, Alternatives, and Limitations

A practical comparison can be organized as follows.

Adjustment is the usual approach when covariates are measured and the desired model is conditional on them. Regression residualization, introduced in the earlier material on covariate adjustment by residualization, is an equivalent way to obtain the dosage coefficient when it uses the same linear model and covariate space. Stratifying by a covariate can also reveal within-group patterns, but may be inefficient or impossible when groups are small, and it does not automatically handle continuous covariates. Each approach depends on choosing an appropriate covariate model.

There are important limits. A covariate may be measured poorly, omitted, or represented with an inadequate functional form. Adding many correlated covariates can make estimates unstable. Furthermore, adjustment changes the coefficient’s interpretation to a conditional association; it is not always desirable to adjust for every available variable. In particular, whether a variable should be adjusted for depends on the scientific target and how it relates to the variables in the model. The omitted-variable formula describes a linear projection under explicit assumptions; it does not settle that scientific choice.

A common mistake is: “The adjusted coefficient differs from the unadjusted coefficient, so the unadjusted estimate was biased and the adjusted one is correct.” The correction is to check the model, sample, coding, and target first. A difference is consistent with omitted-variable bias, but can also reflect sampling variability, a changed analysis sample, or a misspecified adjusted model. Conversely, similar coefficients do not prove that confounding is absent; omitted effects can be small, poorly measured, or offset one another.