Covariate-Adjusted Score Tests and Nuisance Projection

Why Covariate Adjustment Changes a Score Test

A rare-variant burden combines genotypes across a defined variant set into one subject-level score. In earlier material on burden tests, the score is tested for association with a phenotype. But a burden can also track age, ancestry, batch, or another covariate that predicts the phenotype. If that structure is not accounted for, a burden score may appear associated even when it has no association conditional on those covariates.

Adjustment is not just a matter of subtracting a covariate effect from the phenotype. The score test must account for how much of the burden itself is predictable from the covariates. This is the nuisance projection: it removes the component of the genetic score that lies in the covariate space, and it reduces the test’s information accordingly.

Model and Assumptions

Let \(G\) be the \(n \times m\) matrix of genotype dosages for \(n\) subjects and \(m\) variants. Let \(w\) be a prespecified vector of variant weights, and define the burden vector \(b=Gw\). Thus \(b_i\) is subject \(i\)’s weighted burden. Let \(C\) be an \(n \times q\) covariate design matrix, including an intercept, and let \(\gamma\) denote its nuisance coefficients.

For a clear general derivation, suppose the phenotype follows a correctly specified canonical generalized linear model. Under the null hypothesis of no burden effect, its mean satisfies \(g(\mu_i)=C_i^\mathsf{T}\gamma\), where \(g\) is the link function and \(C_i^\mathsf{T}\) is row \(i\) of \(C\). Under the alternative, add a burden term: \(g(\mu_i)=C_i^\mathsf{T}\gamma+\beta b_i\). The score test evaluates \(H_0:\beta=0\), allowing the nuisance coefficients \(\gamma\) to be fitted under the null.

The derivation assumes independent subjects, a correctly specified null mean model, a full-rank covariate design, and finite, nonsingular information for the parameters being tested. The burden and its weights should be defined independently of the phenotype analysis, or any data-dependent selection must be accounted for. For a canonical model, write \(r=y-\hat{\mu}\) for the residual vector at the null fit and \(W\) for the diagonal matrix of null information weights. The null fit satisfies \(C^\mathsf{T}r=0\), and the information matrix for the linear predictor is \(C^\mathsf{T}WC\) for covariates and corresponding cross-products involving \(b\).

Why the Projection Appears

The joint score for the burden coefficient and nuisance coefficients, evaluated under the null, is \(U_\beta=b^\mathsf{T}r\) and \(U_\gamma=C^\mathsf{T}r\). Their information blocks are \(I_{\beta\beta}=b^\mathsf{T}Wb\), \(I_{\beta\gamma}=b^\mathsf{T}WC\), and \(I_{\gamma\gamma}=C^\mathsf{T}WC\). The nuisance score is not a separate signal to test: it represents directions in the data that can be explained by changing the fitted covariates.

To see the adjustment, consider removing from the burden the part that can be predicted by \(C\), using the information weights \(W\). The weighted least-squares projection of \(b\) onto the columns of \(C\) is \(C(C^\mathsf{T}WC)^{-1}C^\mathsf{T}Wb\). Therefore the residualized burden is

This is a weighted projection: \(\tilde b\) is orthogonal to every covariate column in the \(W\)-weighted inner product, since \(C^\mathsf{T}W\tilde b=0\). Its squared weighted length is the adjusted information:

The efficient score can be written as \(\tilde b^\mathsf{T}r\). Substituting the projection gives \(\tilde b^\mathsf{T}r=b^\mathsf{T}r-b^\mathsf{T}WC(C^\mathsf{T}WC)^{-1}C^\mathsf{T}r\). At the fitted null, \(C^\mathsf{T}r=0\), so this equals \(b^\mathsf{T}r\). Thus the fitted nuisance model makes the score numerator orthogonal to nuisance-score directions; the projection is essential for the variance even when it leaves the numerator unchanged.

Equivalently, the nuisance-adjusted information is the Schur complement of \(I_{\gamma\gamma}\) in the joint information matrix. It is the information about \(\beta\) left after allowing the nuisance coefficients to vary. If \(I_{\mathrm{eff}}=0\), the burden is entirely explained by the covariates in the information geometry, so the data cannot distinguish a burden effect from a covariate effect.

Confounding and Loss of Identifiability

An unadjusted score can be confounded when both the burden and the phenotype vary systematically with an omitted covariate. Under a null model with mean determined by \(C\), the expected phenotype residual is zero conditional on \(C\). But if the fitted null model omits an important covariate, its residuals can retain a systematic pattern. A burden correlated with that pattern can then have a nonzero score even without a conditional genetic effect.

A simple expected-value illustration makes the risk clear. Suppose there are 100 subjects in each of two groups, \(z=0\) and \(z=1\). Under no genetic effect, the outcome probability is \(0.1\) for \(z=0\) and \(0.5\) for \(z=1\). Suppose burden is 0 in the first group and 1 in the second. An intercept-only model would fit an overall probability of \(0.3\). The expected unadjusted burden score is then \(100(1)(0.5-0.3)=20\), not zero. The apparent signal reflects the omitted covariate’s association with both burden and outcome, rather than a burden effect conditional on group.

Including \(z\) in the null model removes this mean structure, but in this illustration the burden is exactly \(z\). Its residualized value is zero, so \(I_{\mathrm{eff}}=0\): no conditional burden test is identifiable. More generally, strong correlation between burden and covariates can leave very little adjusted information. Adjustment can prevent confounding, but it cannot create information that the observed design does not contain.

Practical Checks and Alternatives

The projection formula is appropriate only when its null model and information matrix are appropriate. Check that the covariate design has full rank, the null model fits without separation or other major convergence problems, and the adjusted information is positive and not nearly zero. A very small \(I_{\mathrm{eff}}\) means the burden is nearly predictable from covariates; the standardized statistic can then be unstable and the test may have little power. A residualized burden with no variation means the conditional effect is not estimable.

Also check whether the null model captures relevant structure. An omitted ancestry, batch, or relatedness effect can leave residual dependence or a residual mean pattern. The independent-subject information \(W\) used above is not a remedy for such misspecification. If observations are dependent or the mean model is inadequate, an appropriate covariance or robust analysis may be needed; the simple projection and chi-square reference should not be assumed to remain valid without justification.

The principal alternatives answer different questions. An unadjusted score test is simpler but does not test association conditional on covariates and can be confounded. A likelihood-ratio or Wald test fits the alternative model, whereas a score test fits only the null; they may agree approximately in large samples but can differ in finite samples. For a single scalar burden, the efficient score is a one-dimensional test. For a vector of variant-specific scores, nuisance projection likewise adjusts the score vector and its covariance matrix, but the choice of a multivariate or variance-component statistic is a separate testing decision.

A common mistake is to residualize the phenotype and burden with ordinary, unweighted regression and then use an unadjusted variance. The correction is to use the null-model information weights \(W\), form the weighted residualized burden, and calculate its variance as \(\tilde b^\mathsf{T}W\tilde b\). Another mistake is to say that adjustment always changes the score numerator. At a correctly fitted null, the nuisance score is zero, so the efficient-score numerator equals \(b^\mathsf{T}r\); the adjustment changes the information and therefore the calibration of the statistic.