Omitted-Variant Bias in Association

Why Leaving Out a Correlated Variant Matters

At a locus, two variants can carry overlapping information because their genotype dosages are correlated. A regression of phenotype on just one variant therefore describes its marginal association, not necessarily its effect after accounting for the other variant. If the omitted variant is associated with the phenotype and correlated with the included variant, its association can contribute to the marginal coefficient.

The key result is simple: in a two-variant linear model, the shift in the coefficient for the included variant is the omitted variant’s conditional coefficient multiplied by the relationship between the two genotypes. The relationship can be written as a covariance for unstandardized genotypes or a correlation for standardized genotypes. This is a statement about regression coefficients; by itself, it does not establish a causal effect.

Two-Variant Model and Omitted-Variant Bias

For the formula below, assume the conditional mean is correctly specified by the two-variant model, both genotype variances are nonzero, and the residual has mean zero conditional on both genotypes: \(E(\varepsilon\mid G_1,G_2)=0\). There is no intercept in the displayed equation because the genotypes are centered; equivalently, an intercept can be included when genotypes are not centered. The coefficient is a linear association coefficient under this model. Interpreting it as a causal effect requires additional assumptions not supplied by this regression setup.

Write \(\operatorname{Var}(G_1)\) for the variance of \(G_1\), and \(\operatorname{Cov}(G_1,G_2)\) for the covariance of the two genotype dosages. The population marginal slope is the covariance of \(G_1\) with \(Y\), divided by the variance of \(G_1\). Substituting the two-variant model gives the central identity.

The last equality uses \(E(\varepsilon\mid G_1,G_2)=0\), which implies \(\operatorname{Cov}(G_1,\varepsilon)=0\). Thus the omitted-variant bias, defined as \(\beta_{1,\mathrm{marg}}-\beta_1\), is \(\beta_2\operatorname{Cov}(G_1,G_2)/\operatorname{Var}(G_1)\). It is zero if the omitted variant has no conditional association with the phenotype (\(\beta_2=0\)) or if the genotypes are uncorrelated. Otherwise, its direction and size depend on both the conditional coefficient and genotype covariance.

Correlation Form and the Role of Linkage Disequilibrium

Let \(\sigma_1\) and \(\sigma_2\) be the standard deviations of the two genotype dosages, and let \(r_{12}=\operatorname{Corr}(G_1,G_2)\). Since \(\operatorname{Cov}(G_1,G_2)=r_{12}\sigma_1\sigma_2\), the same identity can be written \(\beta_{1,\mathrm{marg}}=\beta_1+\beta_2r_{12}\sigma_2/\sigma_1\). If both genotypes are standardized to have variance one, the ratio of standard deviations is one, so the bias term simplifies to \(r_{12}\beta_2\).

In association analysis, linkage disequilibrium (LD) is the non-independence of alleles at different variants. Genotype-dosage correlation is a convenient way to quantify how the observed dosages co-vary in this regression. Positive correlation means individuals with above-average dosage at one variant tend also to have above-average dosage at the other; negative correlation means that above-average dosage at one tends to accompany below-average dosage at the other. Neither sign alone determines the bias: the sign of \(\beta_2\) matters as well.

Worked Calculations

Why the Formula Holds in a Sample

The population identity follows from covariance algebra, but the same idea can be seen directly in a finite sample. Let \(x_1\) and \(x_2\) be vectors of centered genotype dosages, and let \(y\) and \(e\) be the centered phenotype and residual vectors. The simple-regression slope of \(y\) on \(x_1\) is \(x_1^\mathsf{T}y/(x_1^\mathsf{T}x_1)\). If the observed phenotype satisfies \(y=\beta_1x_1+\beta_2x_2+e\), substitution gives the exact decomposition below.

The second term is the sample analogue of omitted-variant bias: the sample genotype cross-product divided by the included genotype sum of squares, multiplied by \(\beta_2\). The final term is the sample association between \(G_1\) and the residual. Under the conditional-mean assumption it has expectation zero across repeated samples, but it need not be exactly zero in any one sample. Consequently, the population formula describes the systematic shift; a realized sample coefficient can also differ because of residual variation.

Interpretation, Scope, and a Common Pitfall

Omitted-variant bias is important because a marginal association can reflect more than the conditional association assigned to the included variant in a joint model. The formula identifies the ingredients of that difference: the omitted variant’s conditional coefficient and the degree of genotype co-variation. A marginal coefficient is therefore not automatically an estimate of a variant-specific conditional coefficient. In particular, a strong marginal association does not by itself establish which correlated variant accounts for the association.

Common mistake: “The variants are correlated, so omitting one must increase the coefficient of the other.” This reasoning ignores both the sign of the genotype correlation and the sign of the omitted variant’s conditional coefficient. The correction is to calculate the product: \(\beta_2\operatorname{Cov}(G_1,G_2)/\operatorname{Var}(G_1)\). For standardized genotypes, calculate \(r_{12}\beta_2\). The product can be positive, negative, or zero.

The identity also depends on the model and assumptions stated above. If the conditional mean is not adequately represented by the two-variant linear model, or if residuals are associated with genotype, the displayed decomposition does not isolate the entire difference in the same way. Additional covariates and more variants change the regression setting; the two-variant formula should not be applied without checking that the intended model matches its setup.