Effect Interpretation After a Significant Burden Test

What a Significant Burden Result Says

A significant burden test is evidence of association between a phenotype and a particular constructed summary of genotypes. The summary is determined by which variants were included, how their alleles were coded, and what weights were assigned. Thus the result is not an unqualified statement that “the gene affects the phenotype.” Its meaning is conditional on those choices and on the model used to test the burden.

The earlier material on variant-set definition, dosage coding, and burden scores provides the ingredients for this interpretation. Here the central distinction is between an association with a weighted aggregate and an effect estimate for any one variant. A significant aggregate test does not, by itself, identify which included variants are associated or establish that they all have effects in the same direction.

The weights and allele coding give the score its units and meaning. If \(G_{ij}\) counts copies of the minor allele and the weights are all positive, a higher score means a larger weighted dosage of those coded alleles. If weights are frequency- or annotation-based, a one-unit increase may not correspond to one additional allele copy; it means one unit of that weighted score. The variant set matters just as much: adding or removing variants changes the quantity being tested.

The Model Behind the Interpretation

A burden test is commonly understood through a model with one coefficient for the constructed score. For a quantitative phenotype, a simple version is \(Y_i=\alpha+\gamma B_i+\boldsymbol{x}_i^\mathsf{T}\boldsymbol{\eta}+\varepsilon_i\), where \(\boldsymbol{x}_i\) contains covariates and \(\varepsilon_i\) has mean zero and constant variance conditional on the predictors for the usual linear-model standard errors. Here \(\gamma\) is the change in the conditional mean of \(Y\) per one-unit increase in burden, holding covariates fixed.

For a binary phenotype, a common model is \(\operatorname{logit}\{P(Y_i=1\mid B_i,\boldsymbol{x}_i)\}=\alpha+\gamma B_i+\boldsymbol{x}_i^\mathsf{T}\boldsymbol{\eta}\). Under this model, \(\exp(\gamma)\) is the conditional odds ratio per one-unit increase in burden. This is a model-based association, not a causal effect. Its interpretation also depends on the outcome coding: the event \(Y_i=1\) must be specified.

Both models impose a strong structure if viewed as models for the individual variant dosages. Substituting \(B_i=\sum_j w_jG_{ij}\) gives a genetic linear predictor \(\sum_j(\gamma w_j)G_{ij}\). Thus the one-coefficient burden model constrains the variant coefficients to be proportional to the specified weights. A burden score test can have power when this directional structure is a useful approximation; it does not prove that the constraint is true.

Why the Score Test Has an Aggregate Direction

The score-test calculation makes the aggregate nature explicit. Consider a logistic model under the null hypothesis \(H_0:\gamma=0\), with covariates included in the null fit. Let \(\widehat{\mu}_{0i}\) be the fitted event probability and \(W\) the diagonal matrix with entries \(\widehat{\mu}_{0i}(1-\widehat{\mu}_{0i})\). Let \(X\) be the null-model covariate matrix, including the intercept, and \(B\) the vector of individual burden scores. The covariate-adjusted burden is \(\widetilde B=B-X(X^\mathsf{T}WX)^{-1}X^\mathsf{T}WB\). The score and its null variance are

This form assumes the null model is correctly specified for the covariate-adjusted mean, the burden and covariates are available as modeled, and the variance calculation and calibration are appropriate. Under a large-sample normal approximation, \(Z\) is compared with a standard normal distribution under \(H_0\). With sparse binary outcomes, that approximation can be inaccurate; an appropriate sparse-data calibration should be used. Regardless of calibration, a positive \(U_\gamma\) indicates that higher adjusted burden tends to occur with positive null residuals, meaning more observed events than expected under the null. It does not reveal which variant contributes that pattern.

The basic algebra behind this test also explains its interpretation. Under the null, the expected residual vector is zero, so \(E(U_\gamma\mid X,B)=0\). Its variance is \(V_\gamma\) under the working logistic model. Standardizing the signed score therefore tests for a one-dimensional departure associated with the burden contrast. The test is about that contrast, not about \(m\) separately estimated variant coefficients.

A score test does not directly provide a fully fitted effect estimate. In a one-parameter model, \(U_\gamma/V_\gamma\) is a first-order, score-based approximation to the coefficient estimate near the null. For a reported coefficient and confidence interval, fit the burden model and report its estimate, standard error, and model-based interval, with the burden units clearly stated. A score-test p-value alone should not be converted into a variant-specific estimate.

Worked Interpretations

Why a Burden Result Cannot Identify Variant Effects

A single aggregate statistic compresses information across the selected variants. Even if the burden test is highly significant, many patterns of variant-specific effects can be compatible with an association in the burden direction. In particular, a significant result is not proof that every variant has an effect, that all effects have the same sign, or that the assigned weights match the true effect sizes.

A simple projection illustrates this limitation. Suppose two independent genotype variables have equal variance \(\sigma_G^2\), and a quantitative outcome follows \(Y=\beta_1G_1+\beta_2G_2+\varepsilon\), with \(\varepsilon\) uncorrelated with both genotypes. A regression using only the unweighted burden \(B=G_1+G_2\) has population slope

If \(\beta_1=0.80\) and \(\beta_2=-0.30\), the burden slope is \((0.80-0.30)/2=0.25\), despite the opposing variant effects. If estimated precisely enough, the aggregate association could be significant, while the burden coefficient still would not equal either variant-specific effect. This calculation depends on the stated independence and equal-variance conditions; with linkage disequilibrium or unequal variances, the projection changes. The general lesson remains: the aggregate coefficient summarizes association with the chosen score rather than separately estimating every \(\beta_j\).

Diagnostics, Alternatives, and Limitations

Before interpreting a result, verify that the variant-set definition, allele orientation, weights, and phenotype coding are the intended ones. Check that genotype quality and missingness are acceptable, that covariates were included appropriately, and that the test calibration suits the sample size and phenotype sparsity. A sign that changes after allele harmonization, or a result driven by an unintended weight or variant inclusion, is an interpretation problem even if the p-value is small.

A useful diagnostic is to report or inspect the burden distribution and the contributions \(w_jG_{ij}\), so it is clear which variants and individuals can substantially change the score. Sensitivity analyses using defensible alternative set definitions or weights can reveal dependence on analytic choices. Such analyses should be described as additional analyses, not as grounds to choose the most favorable result without accounting for that selection. If a set was selected after screening many sets, the reported significance also needs to be interpreted in light of that search.

The main alternative for estimating variant-specific effects is a model that includes the individual variant dosages rather than collapsing them into one score. That approach estimates more parameters and can be unstable when variants are rare or highly correlated; it answers a different question. The burden approach uses fewer parameters and can be efficient when its directional structure is reasonable, but it can obscure heterogeneity and cancellation. Neither a burden coefficient nor its p-value alone resolves causal interpretation, functional mechanism, or which variant should be followed up.

Common Pitfall