Burden Tests and SKAT Under Different Effect Patterns

Why Effect Patterns Matter

A gene-level rare-variant test combines evidence across multiple variants. The choice of how to combine that evidence matters: a test that gains signal when effects point in the same direction can lose signal when they oppose one another. Another test can retain evidence from variants acting in different directions, but may pay a price when effects are uniform and many variants contribute little or no signal.

The burden test and the sequence kernel association test (SKAT) use different combinations of variant-level scores. A burden test first combines scores into one weighted sum and then tests that sum. SKAT instead forms a weighted sum of squared scores. This difference explains why their relative power depends on the pattern of variant effects. It does not mean that either test is always more powerful.

The subscript 0 denotes a variance or distribution under the null hypothesis. In an analysis with covariates or correlated variant scores, the burden variance and the null distribution of SKAT must reflect that structure. For example, if the null score vector \(U=(U_1,\ldots,U_m)'\) has covariance matrix \(V\), then \(\operatorname{Var}_0(B)=a'Va\). SKAT’s null distribution is determined by the eigenvalues of the appropriately weighted score covariance matrix; it is not generally an ordinary chi-square distribution with \(m\) degrees of freedom.

For a first comparison, use a deliberately simplified score model. Suppose the scores are independent and have equal information \(I>0\), so that under a local alternative \(U_j\sim N(I\beta_j,I)\). Here \(\beta_j\) is the effect for variant \(j\) on the model’s effect scale, and \(I\) is the common information scale. This approximation isolates the effect-pattern comparison; real analyses must account for covariates, linkage disequilibrium, and the appropriate trait model.

What Each Statistic Retains

Under the simplified model, the weighted burden score has mean \(I\sum_j a_j\beta_j\) and variance \(I\sum_j a_j^2\). Its standardized score therefore has a one-degree-of-freedom noncentral chi-square distribution after squaring:

The quantity \(\lambda_B\) is the noncentrality parameter in this simplified model. It depends on the squared weighted sum of the effects. Consequently, effects of opposite signs can cancel before the score is squared.

For SKAT, the same simplified model gives a different distribution. The component scores are independent, and \(U_j/\sqrt I\sim N(\sqrt I\beta_j,1)\). Thus each squared standardized score has a noncentral chi-square distribution:

When weights differ, this is a weighted sum of noncentral chi-square variables, not in general one noncentral chi-square distribution with a single noncentrality parameter. The quantity \(I\sum_j a_j^2\beta_j^2\) is a useful weighted signal measure, but it is not, in general, a noncentrality parameter for the whole SKAT distribution. If all weights equal a common value \(c\), then \(Q_S/(c^2I)\sim\chi^2_m(I\sum_j\beta_j^2)\). In particular, with all weights equal to 1, \(Q_S/I\sim\chi^2_m(I\sum_j\beta_j^2)\).

A Projection View of the Burden Comparison

With equal weights, the burden test compresses the vector of effects into its sum, while the SKAT weighted signal measure depends on the sum of squared effects. This can be made precise using the Cauchy–Schwarz inequality:

For equal weights, the burden noncentrality is \(I(\sum_j\beta_j)^2/m\), while the corresponding SKAT weighted signal measure is \(I\sum_j\beta_j^2\). The inequality shows that the burden signal is the part of the effect pattern captured by the all-ones direction; it is largest relative to the squared-effect signal when the effects are aligned and equal. Equality holds when all \(\beta_j\) are equal. This comparison helps explain signal retention, but it does not by itself establish which test has greater power: the tests have different null distributions and critical values.

Mixed Directions and Cancellation

When some variants increase the modeled trait and others decrease it, a burden sum may conceal their separate contributions. The sign convention for each effect must be consistent—for example, the same allele coding and effect scale across variants—before describing effects as concordant or mixed. SKAT squares each variant score, so a negative score does not cancel a positive one in its quadratic statistic. That feature can be useful when directions genuinely vary, although it does not guarantee high power in every mixed-effect setting.

Sparse Effects and Dilution

A different mismatch occurs when only a small subset of the included variants affects the trait. With equal weights, a burden score averages its evidence across all \(m\) variants through its standardization. If many included variants have no effect, the combined score can be less concentrated than it would be if the active variants were known in advance. SKAT can retain squared evidence from the active component scores without requiring their signs to align.

Conditions Behind the Comparison

These examples use a simplified independent, equal-information score model to make effect-pattern consequences transparent. In practice, several features can change the comparison:

• Effect alignment and weights. A burden test is most naturally suited to effects that align with the chosen allele coding and weights. If the weights emphasize variants with larger effects and the signs are aligned, the burden projection can be efficient. Weights that poorly match the effects can weaken that advantage.
• Correlation among variants. Linkage disequilibrium can make score components correlated. Then the burden variance uses the full covariance \(a'Va\), and SKAT’s null distribution depends on the weighted covariance eigenvalues. Treating correlated scores as independent can give incorrect calibration.
• Trait model and covariates. The score vector and its covariance must come from the specified trait model, with covariates handled appropriately. The binary- and quantitative-trait formulations differ in their score construction and variance; the effect-pattern comparison does not remove the need for correct model specification.
• Variant set and weighting. Both tests depend on which variants are included and how they are weighted. If many variants are uninformative or the chosen weights are poorly matched to the effects, performance can change. A test’s theoretical advantage under one effect pattern may not carry over to another analysis.

The tests also answer the same broad gene-level association question through different alternatives. A significant result from either test is evidence against its gene-level null under the analysis assumptions; it does not, by itself, identify which variant is causal or establish the direction of every variant’s effect.

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