Why the Effect Architecture Matters
A rare-variant set can affect a phenotype in different ways. Several variants may have effects in the same direction, or their effects may differ in size and direction. A burden test and a SKAT-style variance-component test are sensitive to different patterns: a burden test combines variant scores in a specified direction, whereas SKAT-style tests accumulate squared scores and do not require all effects to point the same way.
The choice is therefore about the alternative patterns a test can detect well—not simply which test is more powerful in general. Biological knowledge about the variants, the phenotype, and the set definition can inform that choice. A pattern in the current data may also be suggestive, but noisy scores do not reliably reveal the true effect architecture on their own.
The Two Score-Test Targets
Suppose a null model for the phenotype includes the relevant covariates, and let \(U\) be the vector of efficient score statistics for the variants in a set under that model. Let \(V\) be the covariance matrix of \(U\) under the null. Efficient scores account for estimation of nuisance parameters, such as covariate effects. For a vector of prespecified, nonnegative variant weights, let \(W\) be the diagonal matrix containing those weights.
The following statistics show the central difference. The burden test projects the scores onto the fixed weighted burden direction \(W\mathbf{1}\); the SKAT-style statistic measures their weighted squared magnitude. Their numerical calibration must use the appropriate null distribution for the model and covariance structure.
Under the null, if the score vector is approximately multivariate normal with mean zero and covariance \(V\), \(T_B\) is approximately chi-square with one degree of freedom. The null distribution of \(Q\) is generally a weighted sum of independent chi-square variables with one degree of freedom; its weights are the eigenvalues of \(WVW\). Thus, linkage disequilibrium affects calibration through the covariance among variant scores. Treating correlated scores as independent can produce incorrect p-values.
In a simple setting where the weighted score coordinates are independent standard normal variables under the null, write them as \(Z_1,\ldots,Z_m\). With equal weights, the burden statistic is \(Z_B=(\sum_j Z_j)/\sqrt m\), and \(Q=\sum_j Z_j^2\). Under the alternative, suppose \(Z_j\) is approximately \(N(\delta_j,1)\), where \(\delta_j\) is its noncentral mean. This simplified model makes the contrast precise.
Why Directional Agreement Favors Burden Tests
A burden score is a signed sum. If the component effects point in the same direction, their score contributions tend to reinforce one another. If they point in opposing directions, positive and negative contributions can cancel before the statistic is squared. In the independent-score setting, the burden statistic has noncentrality parameter
For the SKAT-style statistic, the expected value under the same alternative is
The first expression depends on the signed sum of the effects; the second depends on their squared magnitudes. This explains why a shared-direction alternative often favors the burden test, while mixed directions can make SKAT-style testing more effective. It does not guarantee which test will have the smaller p-value in a particular sample: power also depends on effect sizes, weights, covariance, and sample size.
Worked Examples
Worked Example: A Plausible Shared-Direction Pattern
Consider four hypothetical, independent, equally weighted variant-score coordinates. For illustration, suppose their observed values are \(Z=(1.8,1.6,1.3,1.1)\), and use the standard normal and chi-square null calibrations for this simplified setting. All four scores are positive. This alignment is compatible with a shared-direction signal, although these scores alone do not prove that the underlying effects have the same direction.
The burden statistic is
For a two-sided burden test, the p-value is approximately \(2P(N(0,1)\geq 2.90)=0.0037\). The SKAT-style statistic is
Under the simplified null, \(Q\sim\chi^2_4\), so its upper-tail p-value is approximately \(0.069\). The burden test is more significant here because the observed scores reinforce one another in the specified direction. The comparison is illustrative rather than a universal rule: with real linkage disequilibrium, the SKAT-style null distribution would use the relevant covariance matrix rather than \(\chi^2_4\).
Worked Example: Opposing Scores and Burden Cancellation
Now suppose four hypothetical independent, equally weighted score coordinates are \(Z=(2.4,2.2,-2.1,-2.0)\). The scores include strong evidence in both directions, so a single shared-direction burden may be poorly aligned with the pattern.
The burden statistic is
Its two-sided standard normal p-value is approximately \(0.802\). The near-zero sum does not mean that every variant has a negligible score: it results from cancellation. In contrast,
Under the independent-score null, the upper-tail probability for \(\chi^2_4\) is \(e^{-19.01/2}(1+19.01/2)\), approximately \(0.00078\). Here the SKAT-style test detects a pattern of large scores without requiring a common sign. It still does not identify which variants are causal or establish the direction of any individual effect.
Worked Example: When the Score Pattern Is Not Decisive
Suppose another hypothetical set has independent, equally weighted scores \(Z=(1.2,-0.4,1.1,0.2)\). Some coordinates are positive and others are negative, but most are modest. Calculate both statistics before drawing a conclusion about architecture.
For the burden test,
giving a two-sided p-value of approximately \(0.294\). For the SKAT-style statistic,
The \(\chi^2_4\) upper-tail p-value is \(e^{-2.85/2}(1+2.85/2)\), approximately \(0.583\). Neither test gives strong evidence against the null. The mixed signs are not, by themselves, evidence of a genuinely heterogeneous architecture: sampling noise can create that pattern. It would be unwarranted to declare either architecture from these data.
Making the Choice in Practice
A defensible choice starts with the alternative that is scientifically plausible, then checks whether the analysis assumptions and score calibration support the comparison. A practical sequence is:
Choose variants and any functional weights using information defined independently of the tested phenotype, or account for the extra selection. The burden direction must reflect the allele orientation and the direction the analysis is designed to detect.
If prior biological or functional information supports mostly concordant effects, a burden test is a natural targeted test. If effect magnitudes or directions may vary, a SKAT-style test is a reasonable choice.
The null phenotype model, covariate adjustment, genotype coding, and score covariance must be appropriate. Use a calibration suitable for the design and data; sparse binary outcomes, for example, may require more than a simple normal approximation.
Report the chosen test and its assumptions. If both tests are used, account for testing both when making a single inferential claim. A smaller unadjusted p-value does not establish which architecture is true.
The assumptions behind the displayed null distributions matter. The phenotype null model must be adequately specified, including relevant covariates and any design features that affect the score variance. Score approximations must be sufficiently accurate for the sample size and outcome distribution, and \(V\) must represent dependence among variants. In binary-phenotype analyses with sparse data, ordinary asymptotic calibration can be inaccurate; the sparse-data methods covered earlier are relevant in that setting. Weights and the tested variant set should not be chosen after inspecting the same outcome scores unless the selection is incorporated into inference.
Diagnostics, Alternatives, and Limitations
Before interpreting a test, inspect whether the assumptions used for its calibration are credible. For a SKAT-style statistic, check that the null distribution incorporates linkage disequilibrium; a chi-square reference with one degree of freedom per variant is justified only in the simple independent, standardized setting. For a burden test, verify that the allele coding and weights make the proposed direction meaningful. In either case, check that covariate adjustment and the phenotype null model match the study design. A score pattern dominated by one coordinate or a handful of carriers deserves cautious interpretation: the aggregate test alone does not show that the architecture is broadly shared across the set.
The main alternative to choosing one test in advance is to use a procedure designed to combine or adapt across burden-like and variance-component signals. Earlier material on combined multivariate and collapsing tests illustrates a different way of combining signals; adaptive burden-variance procedures are another option. Such approaches have their own calibration and interpretation requirements. Trying several tests and reporting only the smallest p-value is not an equivalent substitute: it can inflate the false-positive rate unless the selection is accounted for.
Neither choice removes the limitations of a set-level association test. A significant result does not prove causation, establish the direction of every effect, or identify a causal variant. A nonsignificant result does not show that all effects are zero: the test may have low power for the actual pattern, the set or weights may be poorly aligned, or the score approximation may be inadequate. Interpret the result as evidence from the specified set, model, and test—not as a definitive diagnosis of the biological architecture.
Common Pitfall
Correction: Mixed observed signs can arise from sampling variation, and comparing p-values selected from multiple tests without adjustment changes the error rate. Choose the primary test using prespecified scientific reasoning, or use an approach that accounts for adaptation. Treat the observed score pattern as descriptive evidence, not proof of the true effect architecture.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In the independent-score setting, why can a burden statistic be small when several variant scores have large absolute values?
- For \(Z=(1,1,1,1)\), calculate \(Z_B\) and \(Q\).
- Why does linkage disequilibrium affect the null calibration of a SKAT-style statistic?
- What is wrong with selecting whichever of two tests gives the smaller unadjusted p-value and reporting that p-value as if the test had been chosen in advance?
- If a set has modest, mixed-sign scores and neither test is significant, what can and cannot be concluded about its effect architecture?
Solutions. (1) The burden test sums signed scores, so positive and negative contributions can cancel; \(Q\) instead sums squared scores. (2) \(Z_B=(1+1+1+1)/\sqrt{4}=2\), and \(Q=1^2+1^2+1^2+1^2=4\). (3) Linkage disequilibrium makes score coordinates correlated, changing the eigenvalues that determine the null mixture distribution. (4) The choice uses the observed data and can increase the probability of a false positive; the analysis must prespecify the test or account for selection. (5) The data do not provide strong evidence against the null under either test, but they do not establish that all effects are zero or reveal the true architecture; noise, limited power, and test-alternative mismatch remain possible.