The SKAT Statistic as a Weighted Genotype Kernel

From Variant Scores to Individual Similarities

SKAT is often introduced through a score for each variant: each score measures how strongly that variant’s genotype pattern aligns with the outcome residuals under the null model. The SKAT statistic combines these scores by squaring and weighting them. That construction has an equivalent interpretation: it measures how the residuals align with a matrix of weighted genetic similarities among individuals.

The equivalence is useful because it changes the viewpoint without changing the statistic. In the variant view, the calculation aggregates evidence across genotype columns. In the individual view, it compares pairs of genotype profiles. The kernel matrix makes that second view explicit.

For an additive genotype coding, \(G_{ij}\) is commonly a dosage from 0 to 2, representing the number of copies of a specified allele. Other codings are possible, but the coding must be fixed and interpreted consistently. In a binary-trait analysis, \(r\) is based on the difference between each observed outcome and its null-model expectation. More generally, \(r\) denotes the residual vector entering the score. Examples below use hypothetical residuals and focus on the algebra, not on fitting a null model.

The weights in \(W\) are multipliers in the statistic: variant \(j\)'s score contributes \(w_j U_j^2\). They must be nonnegative for the usual kernel interpretation below. Different conventions may define a weight before it is squared or transformed; the \(w_j\) here always mean the final nonnegative multiplier in \(Q\).

The Kernel Matrix as Weighted Genotype Similarity

Write the genotype profile of individual \(i\) as the row vector \(G_i=(G_{i1},\ldots,G_{im})\). The \(i,k\) entry of \(K\) is

Thus, \(K_{ik}\) is a weighted inner product of two individuals’ genotype profiles. Equivalently, form a weighted profile \(z_i=(\sqrt{w_1}G_{i1},\ldots,\sqrt{w_m}G_{im})\); then \(K_{ik}=z_i^{T}z_k\). A variant with a larger weight contributes more to the similarity, while a variant with weight zero contributes nothing.

With uncentered genotype dosages, this is an uncentered inner product: it is not automatically a correlation or a distance. Individuals can have a positive kernel entry because they share nonzero dosages at weighted variants. Centering genotype columns before constructing the kernel instead measures inner products of deviations from the column centers. That is a different kernel construction, so the choice must be stated rather than silently assumed.

Why the Two Forms of the Statistic Are Equal

The variant-score form and the individual-kernel form are two matrix expressions for the same quantity. The identity follows directly from substituting \(U=G^{T}r\) and regrouping the products.

The last expression shows how the statistic combines residuals across individuals, weighted by their genotype similarities. The contributions are not all necessarily positive: if \(r_i\) and \(r_k\) have opposite signs, their product is negative. It is the complete quadratic form, not each pairwise term separately, that equals the nonnegative sum of weighted squared variant scores.

Reading the Kernel Through Individuals

The diagonal entry \(K_{ii}\) is the weighted squared length of individual \(i\)'s genotype profile. An off-diagonal entry \(K_{ik}\) is the weighted overlap between individuals \(i\) and \(k\). These entries describe genotype relationships under the chosen coding and weights; they do not describe similarity of outcomes.

The kernel also makes clear why SKAT uses squared variant scores. Each variant contributes its own squared alignment with the residual vector, and the weights scale those contributions before they are added. In the individual representation, those same contributions are assembled into \(K\), and the residual vector is evaluated against that matrix. This is a change in representation, not a different test.

What the Kernel View Does—and Does Not—Establish

The kernel view matters because it connects a variant-based score statistic to a structured comparison among individuals. It can make the role of weights and genotype coding easier to inspect, and it provides the matrix used in the quadratic-form representation. The statistic can be computed either from the variant scores or from \(r^{T}Kr\); the preferable calculation depends on the data dimensions and implementation.

The algebra alone does not determine a p-value. Inferential calibration depends on the null model and its assumptions, including how the residuals and their dependence are handled. Nor does a large statistic identify a particular causal variant: \(Q\) aggregates contributions across the weighted genotype columns. The null distribution and p-value calculations require separate treatment.

Common Mistake

• Wrong reasoning: “Because \(K_{ik}\) is a similarity, every term \(r_i r_k K_{ik}\) must be positive, so the statistic is just a sum of positive pairwise similarities.”
• Correction: The residual product \(r_i r_k\) can be negative, so individual pairwise terms can be negative. The complete quadratic form is nonnegative because it equals \(\sum_j w_j U_j^2\) when the weights are nonnegative.
• Another pitfall: Calling \(K_{ik}\) a phenotype similarity or interpreting it as a correlation. It is a weighted genotype inner product under the chosen coding; it is not generally normalized, centered, or bounded between \(-1\) and \(1\).