Why Genetic Architecture Matters for Test Choice
A gene-based test combines evidence across multiple variants, but different tests combine that evidence in different ways. A burden test is most sensitive when included variants tend to affect the outcome in a common direction. SKAT, a variance-component test, can retain evidence when effects point in different directions or only a subset of variants is causal. SKAT-O combines burden-like and SKAT-like alternatives, with a calibration step that accounts for evaluating more than one alternative.
These are expectations about power under different effect patterns, not guarantees about the result of any particular analysis. A small p-value does not identify the causal variants or establish the assumed architecture. The examples below use simplified score data to make the contrast transparent; real analyses must use the score covariance, variant weights, and null distribution appropriate to the study.
What the Tests Accumulate
A score \(U_j\) summarizes evidence for association at variant \(j\), conditional on the covariates in the null model. Its sign indicates the direction of the score relative to the chosen allele coding and outcome model. In actual analyses, scores are generally not independent: linkage disequilibrium, covariate adjustment, and the study design affect \(V\). A valid test accounts for this covariance rather than treating the raw scores as independent.
For weights \(w_j\), a standardized burden statistic is the squared weighted sum divided by its null variance. Under the null and suitable regularity conditions, it has an approximate \(\chi^2_1\) distribution. A weighted SKAT statistic is a sum of squared scores; its null distribution is generally a weighted sum of chi-square variables, with weights determined by the covariance and variant weights. In the simplified setting \(V=I\) with unit weights, that distribution is \(\chi^2_m\).
The burden statistic adds signed scores before squaring. Opposite signs can therefore cancel. SKAT squares each weighted score before adding, so positive and negative contributions do not cancel, although a large score at a variant with little weight contributes less. Neither construction is universally preferable: the useful question is which pattern is plausible for the variant set and scientific context.
Worked Comparisons Across Three Architectures
The score vectors in the next examples are hypothetical, not results from a real study. They are chosen to isolate three patterns: effects aligned in direction, effects opposing one another, and a single strong contribution among otherwise null variants. Each coordinate is a standardized score. The arithmetic illustrates test behavior under the stated simplified null; it is not a substitute for fitting and checking an appropriate model.
Worked Example: Aligned Effects Favor a Burden Test
Suppose five variants have the hypothetical score vector \(U=(2,2,2,2,2)^\mathsf{T}\). All scores point in the same direction. Assume the null scores are independent standard normals and use equal weights, so \(V=I\).
The burden score sum is \(2+2+2+2+2=10\), and its null variance is \(5\). Thus
Compared with \(\chi^2_1\), this gives an upper-tail p-value of approximately \(7.74\times10^{-6}\). The SKAT statistic is
Compared with \(\chi^2_5\), its upper-tail p-value is approximately \(0.00125\). In this constructed example, the burden test gives stronger evidence because all five scores reinforce one another in the linear sum. If the true effects tend to share a direction, this is the pattern that can make burden testing efficient. It does not prove that all five variants are causal or that their effects have the same size.
Worked Example: Opposing Effects Favor SKAT
Now suppose the hypothetical scores are \(U=(2\sqrt{2},-2\sqrt{2},0,0,0)^\mathsf{T}\), again with independent standard-normal null scores and unit weights. The two nonzero scores oppose each other.
The burden sum is \(2\sqrt{2}-2\sqrt{2}=0\), so \(T_B=0^2/5=0\). Its upper-tail p-value is 1: this burden statistic shows no evidence against its null. In contrast, SKAT accumulates the squared scores:
The simplified SKAT reference distribution is \(\chi^2_5\), giving an upper-tail p-value of approximately \(0.006844\). The contrast arises because the signed burden sum cancels, while SKAT retains both contributions. This supports choosing a variance-component approach when effects may have opposing directions. It does not establish that the two nonzero scores represent genuine effects; chance variation can also produce a large quadratic statistic.
Worked Example: Sparse Effects Can Favor SKAT
Suppose only one of five variants has a strong score: \(U=(4,0,0,0,0)^\mathsf{T}\). Keep the same simplified assumptions. The burden score sum is \(4\), with null variance \(5\), so
Against \(\chi^2_1\), the burden p-value is approximately \(0.0737\). SKAT gives \(Q_S=4^2=16\). Against \(\chi^2_5\), its upper-tail p-value is approximately \(0.006844\). Here SKAT retains the large contribution from the one variant, while the burden statistic’s variance includes all five variants.
This example illustrates a possible advantage when only a small fraction of variants are causal. It does not imply that SKAT is always more powerful for sparse effects: the result depends on effect sizes, weights, covariance, and sample size. A different weighting scheme or a different pattern of scores could change the comparison.
Why SKAT-O Requires Calibration
SKAT-O addresses uncertainty about whether the effect pattern is more burden-like or more variance-component-like. In a common construction, a parameter \(\rho\) indexes a family of quadratic score statistics. At one endpoint the statistic is SKAT-like, and at the other it is burden-like; intermediate values represent combinations of the two. The test evaluates evidence across this family and calibrates its omnibus p-value to account for that search.
The important interpretation is that SKAT-O is not simply whichever of two ordinary p-values happens to be smaller. Those p-values are dependent because they use the same scores. Reporting their minimum as if only one test had been conducted would generally fail to account for that dependence and would overstate evidence. SKAT-O uses a null calibration for its adaptive procedure. The resulting p-value tests the omnibus null; it does not, by itself, identify which \(\rho\) or genetic architecture is true.
Justification: Why Direction Changes the Burden Signal
The contrast can be seen directly in the burden numerator. Suppose, for illustration, two score contributions have equal magnitude \(a\). If they align, their sum is \(a+a=2a\), and the squared sum is \(4a^2\). If they oppose, their sum is \(a-a=0\), and the squared sum is zero. The SKAT contribution in either case is the sum of squared scores, \(a^2+a^2=2a^2\). Thus changing a sign can eliminate the burden signal without changing this SKAT statistic.
This calculation justifies the direction-sensitivity comparison, not a universal ranking. In real data, the score variance and covariance also enter the test. A covariance-aware statistic can differ from the simple sums above, and a poorly calibrated null distribution can invalidate either p-value. The architecture argument is useful only when paired with a correctly specified and calibrated analysis.
Diagnostics, Alternatives, and Limitations
Before interpreting a contrast among tests, check whether its assumptions and implementation are credible. Review the null-model fit and covariate adjustment; verify allele coding and score signs; inspect whether the selected variants and weights match the prespecified definition; and confirm that the covariance and null calibration reflect relatedness, population structure, and case-control sampling as relevant. Variant-level score contributions and null-calibration diagnostics can help reveal influential variants or systematic inflation. A test comparison is difficult to interpret if one method is calibrated and another is not.
A practical comparison is:
| Test | Signal it accumulates | Potential advantage | Key caution |
|---|---|---|---|
| Burden | Signed, weighted sum, squared and standardized | Can be powerful when included effects mostly align | Opposing effects can cancel; results depend on weights and set membership |
| SKAT | Weighted sum of squared variant scores | Retains evidence across directions and can suit sparse effects | Null distribution is not generally a simple chi-square; covariance and weights matter |
| SKAT-O | Calibrated adaptive combination across burden-like and SKAT-like statistics | Can be useful when the architecture is uncertain | Adaptation costs calibration and does not reveal the true architecture |
Alternatives should be selected for a defined scientific question, not just because they produce the smallest observed p-value. If multiple tests are run and the most favorable result is reported, the selection must be accounted for in the inference or the result must be clearly treated as exploratory. A prespecified SKAT-O test is one way to address uncertainty within its defined family, not a correction for every possible set definition, weighting scheme, phenotype, or analysis tried.
All three approaches test a set-level null and can be limited by weak power, inaccurate variant annotation, poor imputation, population differences, or a misspecified outcome model. A significant set-level result does not show which variants are causal, establish a biological mechanism, or imply that each included variant has an effect. A nonsignificant result is also not proof of no association: it may reflect limited information, a mismatched test, or an imperfect set definition.
Common Mistake
Wrong reasoning: “I ran burden and SKAT, saw that SKAT had the smaller p-value, and can report that p-value because SKAT fit the data better.” This treats the selected result as though the choice had been made before seeing the data. It also mistakes a smaller p-value for proof that the SKAT effect model is true.
Correction: Choose and justify a primary test in advance when possible, or use an adaptive procedure with its appropriate calibration. Interpret the p-value as evidence against the tested null under the test’s assumptions. A smaller SKAT p-value may be consistent with opposing or sparse effects, but does not establish that architecture or identify causal variants.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- With five independent standard-normal null scores and unit weights, what is the burden statistic for \(U=(1,1,1,1,1)^\mathsf{T}\)?
- Why can scores \(a\) and \(-a\) produce no burden signal but a positive SKAT contribution?
- For \(U=(4,0,0,0,0)^\mathsf{T}\), what are the simplified burden statistic and SKAT statistic?
- Why is the minimum of an ordinary burden p-value and an ordinary SKAT p-value not automatically a valid SKAT-O p-value?
- Name one diagnostic or design feature that could invalidate the examples’ simple chi-square reference distributions in a real study.
Solutions. (1) The score sum is 5 and its null variance is 5, so \(T_B=5^2/5=5\). (2) The signed sum is \(a+(-a)=0\), so the burden numerator is zero; the SKAT contribution is \(a^2+(-a)^2=2a^2\). (3) \(T_B=4^2/5=3.2\), while \(Q_S=4^2=16\). (4) Both p-values use the same data and are dependent; selecting the smaller after inspection changes its null distribution. SKAT-O calibrates the adaptive procedure rather than treating that minimum as a single prespecified test. (5) Examples include correlated variant scores from linkage disequilibrium, related participants, population structure, or a misspecified null model. These can change the score covariance or invalidate the assumed reference distribution unless properly handled.