SKAT Power Under Heterogeneous Variant Effects

Why Effect Heterogeneity Matters

A rare-variant set may contain effects that differ in magnitude, and some may increase a quantitative phenotype while others decrease it. A burden score adds variant contributions, so opposing effects can cancel. SKAT instead uses a weighted sum of squared score contributions. Squaring prevents cancellation of opposite signs, but it does not guarantee high power: power also depends on the total signal, variant weights, genotype correlation, and the null distribution of the statistic.

The focus here is how those pieces combine under heterogeneous alternatives. The derivation uses a quantitative-trait score model. It is a clean setting for seeing the mechanism; binary-trait score tests have analogous large-sample behavior but require their own null-model variance calculations.

Model, Statistic, and Assumptions

Let \(y\) be an \(n\)-vector of quantitative phenotypes, \(C\) an \(n\times q\) covariate matrix, and \(G\) an \(n\times m\) matrix of genotypes for \(m\) variants. Consider the model

Here \(\gamma\) contains nuisance covariate effects and \(\beta\) contains the variant effects. For the exact finite-sample results below, treat \(C\) and \(G\) as fixed, assume \(C\) has full column rank, and assume independent Gaussian errors with common variance \(\sigma^2\). The null hypothesis is \(H_0:\beta=0\). Define the residual projection matrix

The score vector for the variant effects under the null is \(U=G'P_Cy\). With nonnegative variant weights \(w_j\), let \(W=\operatorname{diag}(w_1,\ldots,w_m)\). The SKAT quadratic statistic is \(Q=U'WU\). The weights can reflect prespecified variant properties; they must be chosen without using the observed phenotype in a way that invalidates the test’s calibration.

Under \(H_0\), \(U\) has mean zero and covariance matrix \(\sigma^2H\). Under the specified alternative, its mean is \(H\beta\). Thus the statistic’s behavior changes not only through the effect vector but also through the genotype information matrix \(H\). A zero or nearly zero eigenvalue of the weighted information matrix means that a corresponding genotype combination is unobserved or weakly informed by the data.

Why Squared Scores Avoid Sign Cancellation

A useful first case is orthogonal, equally informative variants: take \(H=I_m\), \(W=I_m\), and suppose \(\sigma^2\) is known. Then the score coordinates are independent, with \(U_j/\sigma\sim N(\beta_j/\sigma,1)\). Therefore,

The noncentrality parameter \(\lambda\) is the total squared standardized signal in this special case. The signs of the effects do not appear in it. A burden score, by contrast, adds the score coordinates before testing, so positive and negative contributions can offset each other. This contrast explains a key way SKAT can gain power: it can detect a set with real effects even when their directions are mixed.

General Alternative Distribution

With correlation among variants or unequal weights, the orthogonal case no longer describes the full distribution. Set \(A=W^{1/2}HW^{1/2}\) and \(z=W^{1/2}U/\sigma\). Then \(Q/\sigma^2=z'z\), and under the model \(z\) has covariance \(A\) and mean \(W^{1/2}H\beta/\sigma\). Let \(a_j\) be the positive eigenvalues of \(A\), with corresponding unit eigenvectors \(v_j\). Resolving \(z\) in these eigen-directions gives the alternative distribution

Here each \(\chi^2_1(\delta_j)\) is a noncentral chi-square variable with one degree of freedom and noncentrality \(\delta_j\), and the variables in the sum are independent. Under the null, all \(\delta_j=0\); the null distribution is therefore a weighted sum of central chi-square variables. This is the mixture chi-square calibration for SKAT.

The formula follows by diagonalizing \(A\). In the eigenvector direction \(v_j\), the coordinate \(v_j'z\) has variance \(a_j\) and mean \(v_j'W^{1/2}H\beta/\sigma\). Dividing that coordinate by \(\sqrt{a_j}\) gives a unit-variance normal variable with squared mean \(\delta_j\). Its square is \(\chi^2_1(\delta_j)\), and its contribution to \(z'z\) is \(a_j\) times that square. Eigen-directions with \(a_j=0\) have no random score variation; under this model they also have zero alternative mean and contribute nothing.

Interpreting Heterogeneity, Weights, and Power

A random-effects description often used for SKAT represents variant effects as mean-zero random effects with a variance related to the variant weights. In that formulation, the variance component describes the scale of effect variation, while the score test assesses whether that component is zero. This connects SKAT’s quadratic form to alternatives with dispersed effects, including mixed signs. It does not mean that every variant must have a nonzero effect, or that the fitted model identifies each variant’s direction.

Weights matter because they change both the contributions to \(Q\) and the eigenvalues that determine its null reference distribution. A weight that emphasizes truly associated variants can improve sensitivity; a weight that emphasizes null or weakly associated variants can dilute it. Likewise, LD can combine variant signals into eigen-directions. Effects aligned with well-informed directions may contribute differently from effects aligned with poorly informed directions. The eigenvalue calculation, rather than a simple count of positive effects, captures this interaction.

Power is the probability that the test rejects \(H_0\) under a specified alternative. It is not determined by a \(p\)-value observed in one sample. For SKAT it depends on the effect vector, the genotype covariance, the weights, the residual variance, the sample size, and the significance threshold. The introductory material on random-effects models for variant-set association gives the variance-component perspective; here the key power implication is that the quadratic score accumulates squared signal, subject to the covariance structure.

Diagnostics, Alternatives, and Common Pitfalls

Before interpreting a power calculation, verify that its model reflects the analysis. Check that the tested variant set and weights were specified as intended, that the covariate adjustment is adequate, and that genotype coding and quality control are appropriate. Inspect whether the effective eigenvalue spectrum is concentrated in a few directions or spread across many; this helps explain what information the statistic can use. If the Gaussian linear-model assumptions are implausible, use a score variance and null calibration appropriate to the phenotype model rather than borrowing the quantitative-trait formula uncritically.

The main alternatives answer different questions. A burden test compresses variant scores into a signed aggregate and can be efficient when effects point in a common direction, but can lose signal through cancellation. SKAT retains squared contributions and is suited to alternatives with varying directions, though its power can be lower when the signal is highly aligned and a burden score captures it efficiently. Single-variant tests localize evidence variant by variant but do not pool weak signals across a set in the same way. These contrasts describe their mechanisms; they do not provide a universal selection rule.

• Mistake: “Mixed signs guarantee SKAT power.” Mixed signs prevent cancellation in the squared statistic, but weak effects, poor weights, limited sample size, or low-information genotype directions can still produce low power.
• Mistake: “More effect heterogeneity always raises SKAT power.” With orthogonal equal-information variants, equal sums of squared standardized effects give the same noncentrality, regardless of sign pattern or distribution of magnitudes.
• Mistake: “The noncentrality parameter is the power.” It indexes signal strength, but power also depends on the null critical value and the alternative distribution. In correlated settings, a weighted sum of noncentral chi-square variables may be needed.
• Mistake: “A large expected \(Q\) proves high power.” Power depends on the full distribution relative to the rejection threshold, not its mean alone. Two alternatives with different variance or eigenvalue structure can have different rejection probabilities even when their means are similar.