Why Burden Tests Favor Aligned Effects

Why Direction Matters for a Burden

A burden test compresses several variant dosages into one weighted genetic quantity and tests its association with a trait. This compression can be useful when individual variants are rare, but it combines their effects before testing. The key question is therefore not only whether variants affect the trait, but whether their effects point in a direction that the chosen burden can capture.

When the weights are nonnegative and variant effects all have the same sign, each variant can contribute to the burden–trait association in the same direction. Under suitable assumptions, those contributions add in the covariance between the burden and the trait. This alignment is one reason a burden test can have good power for a set of variants with a shared-direction effect pattern. It is not a guarantee of high power: the weights, genotype variability, residual variation, and sample size also matter.

Model and Burden Definition

Consider \(m\) variants measured in each of \(n\) independent individuals. Let \(G_j\) be the centered dosage for variant \(j\), with effect allele coding fixed consistently across individuals. Write \(\mathbf{G}=(G_1,\ldots,G_m)^\mathsf{T}\), and let \(\boldsymbol{\beta}=(\beta_1,\ldots,\beta_m)^\mathsf{T}\) denote the variant effects on a quantitative trait. For this derivation, use the additive model

Assume individuals are independent and the genotype covariance matrix \(\boldsymbol{\Sigma}=\operatorname{Var}(\mathbf{G})\) is well-defined. The burden weights \(w_j\) are fixed for the analysis, and the burden is \(B=\mathbf{w}^\mathsf{T}\mathbf{G}\). In the usual burden setup, weights are nonnegative: larger weights give a variant more influence on the combined measure. The weights might come from a specified frequency or functional weighting scheme; here the issue is what happens for a given scheme, not how to choose one.

This definition concerns effect direction, not effect size. Effects can be aligned even when their magnitudes differ. Also, the phrase “same direction” depends on coding: changing which allele is counted or reversing the trait scale can change coefficient signs. The interpretation must use one consistent coding convention.

How Alignment Enters the Burden Signal

The burden’s variance and its covariance with the trait follow directly from the model. Because the error has conditional mean zero, it has zero covariance with any function of the genotypes, including \(B\). Therefore,

These expressions also show why the simple “sum of variant contributions” requires an assumption about genotype dependence. If variants are independent, \(\boldsymbol{\Sigma}\) is diagonal. Writing \(v_j=\operatorname{Var}(G_j)\), the covariance becomes

If each included variant has \(v_j>0\), each nonzero term \(w_j\beta_jv_j\) has the common sign of the effects when the weights are nonnegative and the effects are aligned. The terms then add rather than opposing one another in the covariance. This is the precise sense in which aligned effects reinforce the burden signal under independence.

With correlated variants, the covariance is instead \(\mathbf{w}^\mathsf{T}\boldsymbol{\Sigma}\boldsymbol{\beta}\). Cross-variant covariance terms matter, so the independent-variant sum cannot simply be used. Alignment of the \(\beta_j\) values and nonnegative weights by themselves do not establish the sign or size of this expression for every possible covariance matrix.

Burden Regression and the Relevant Residual Variance

A burden-only regression relates \(Y\) to \(B\), rather than fitting every variant separately. For a quantitative trait, its population slope is the least-squares projection coefficient

The residual from this projection is \(R=Y-E(Y)-\theta_B\{B-E(B)\}\). Its variance is

This is the residual variance after projecting the trait on the burden. It is not automatically the error variance \(\operatorname{Var}(\varepsilon)\) in the multi-variant trait model. If variant effects are not fully represented by the burden, genetic variation left outside the burden can remain in the regression residual.

Under independent individuals, a nonzero burden variance, and the usual large-sample regression conditions, the burden slope has approximate standard error \(\tau_B/\sqrt{n\operatorname{Var}(B)}\) when the regression residual is homoscedastic. Thus the approximate standardized signal under the alternative is

The magnitude \(|\delta_B|\) describes the approximate separation of the estimated burden slope from zero in standard-error units; its sign gives the direction of association. This is an approximation to regression-test power, not a p-value. It depends on the residual variance for the burden regression and on the assumptions behind its standard error. If residual variance changes with burden, a robust standard error may be needed, and this simple expression need not describe the test’s standardization.

In the special case \(Y=\mu+\theta B+\varepsilon\), with error variance \(\sigma^2\) and errors satisfying the regression assumptions, the burden-regression residual variance is \(\tau_B^2=\sigma^2\). Only in such a case, or under a stated approximation that justifies it, can \(\sigma^2\) be substituted for \(\tau_B^2\). A score test has its own null-model variance and calibration; the trait-model error variance should not be substituted without justification.

What Alignment Does—and Does Not—Establish

For independent variants and nonnegative weights, aligned effects make the terms \(w_j\beta_jv_j\) add with a common sign. A larger covariance can increase the burden test’s signal, but power depends on the full standardized quantity: covariance relative to burden variability and residual variability. A weighting scheme can emphasize variants with larger effects, but it can also give substantial weight to variants with little trait effect. Such variants can increase burden variability without contributing much covariance.

Alignment is a modeling assumption about the effects in the set, not a conclusion supplied by the burden construction. Functional or frequency-based weights do not, by themselves, prove that included effects share a direction. In addition, correlated variants affect both the burden variance and its covariance with the trait. A burden can therefore be weak even when some effects have the same sign.

The calculation above concerns a quantitative-trait burden regression. For a binary trait, the link function and null-model variance used in a logistic score test change the test’s calculation and calibration. The same broad idea—that a burden score must capture a coherent direction of association—remains relevant, but the quantitative-trait formula should not be carried over as though it were the binary-trait test statistic.

Common Mistake

• Wrong reasoning: “The trait model has error variance \(\sigma^2\), so the burden test’s standard error always uses \(\sigma^2\).”
• Correction: A burden-only regression leaves any trait variation not explained by the burden in its residual. Use the residual variance after projecting the trait on that burden, \(\tau_B^2\), under the regression conditions. A score test instead uses the variance specified by its null-model calibration.
• Wrong reasoning: “All effects are positive, so any nonnegative burden must have high power.”
• Correction: Alignment makes contributions reinforce in the independent-variant covariance formula, but it does not determine their size relative to burden and residual variation. Power also depends on the weights, genotype variability, sample size, and test calibration.