Choosing Variant Weights in SKAT

Why Variant Weights Matter

A gene-level SKAT analysis combines evidence across variants, but it need not give every variant equal influence. A variant’s weight determines how strongly its score contribution enters the test statistic. Giving greater weight to variants believed to be more informative can improve sensitivity when that belief matches the underlying effect pattern. If it does not, the weighting can reduce sensitivity. The weights are therefore part of the test specification, not a cosmetic rescaling.

This tutorial focuses on two common sources of weights: allele frequency and functional annotation. Frequency-based weights typically give greater influence to rarer variants. Annotation-based weights can increase or decrease influence according to prespecified information about a variant, such as whether it is predicted to alter protein function. These choices affect both the observed statistic and the null distribution used to calibrate it.

Definition: Let $G$ be the $n$-by-$m$ genotype matrix for $m$ variants, with one column per variant, and let $U_j$ be the score contribution for variant $j$ after accounting for the null model and covariates. A weighted SKAT statistic can be written as $$ Q=\sum_{j=1}^{m} w_j U_j^2=U^\mathsf{T}WU, $$ where $U=(U_1,\ldots,U_m)^\mathsf{T}$, $W=\operatorname{diag}(w_1,\ldots,w_m)$, and $w_j\geq 0$ is the kernel multiplier for variant $j$. Equivalently, the genotype kernel is $K=GWG^\mathsf{T}$. A larger $w_j$ gives that variant’s squared score a larger contribution to $Q$.

The weights are commonly fixed before examining the outcome association evidence. For example, they may be determined from allele frequency or external annotation. The score vector is calculated under the null model, including the relevant covariates, and the weighted quadratic form is then evaluated. The usual score-test calibration assumes that the null model is adequately specified and that the score vector has its stated null distribution (often a normal approximation). The genotype data and chosen weights are treated as fixed for that calibration.

A useful distinction is between a kernel multiplier $w_j$ and a multiplier used to scale a genotype column. If a genotype column is multiplied by $a_j$, its outer product is multiplied by $a_j^2$. Thus the corresponding kernel multiplier is $w_j=a_j^2$, not $a_j$. Software and papers may describe either convention as a “variant weight,” so the actual kernel definition should always be checked.

Allele-Frequency Weights

A widely used frequency weighting function is the beta density evaluated at the minor-allele frequency, often with parameters $1$ and $25$. For minor-allele frequency $f_j$, its unnormalized value is

b_j=\operatorname{BetaDensity}(f_j;1,25)=25(1-f_j)^{24}.

This function is larger at small frequencies: for instance, a variant with frequency $0.01$ receives a larger value than one with frequency $0.20$. To make the relative multipliers easier to read, one may divide all $b_j$ values by their maximum. Multiplying every kernel weight by the same positive constant rescales $Q$ and all null-distribution eigenvalues by that constant, so it does not change the calibrated p-value. The relative weights, rather than their overall scale, determine the allocation of influence among variants.

Worked Example: Applying Beta Allele-Frequency Weights

Consider three hypothetical variants with minor-allele frequencies $0.01$, $0.05$, and $0.20$. Suppose their null-model score contributions are $U=(1.5,1.0,2.0)^\mathsf{T}$. Use beta density weights with parameters $1$ and $25$, normalized so that the largest weight is one.

The unnormalized weights are approximately $25(0.99)^{24}=19.64$, $25(0.95)^{24}=7.300$, and $25(0.80)^{24}=0.1181$. Dividing each by the maximum gives relative weights approximately $1$, $0.37164$, and $0.0060106$. The weighted statistic is

$$ Q=1(1.5^2)+0.37164(1.0^2)+0.0060106(2.0^2) =2.25+0.37164+0.0240424 =2.6456824. $$

With equal weights, the statistic for these same scores would be $1.5^2+1.0^2+2.0^2=7.25$. The difference reflects the chosen relative weighting: the score of the variant at frequency $0.20$ has much less influence under the beta weighting. It does not, by itself, establish which scheme gives the smaller p-value; each scheme has its own null distribution.

Combining Frequency and Functional-Annotation Weights

A functional annotation can be represented by an additional nonnegative multiplier $a_j$. For example, an analysis might assign greater multipliers to variants with a prespecified damaging annotation and smaller multipliers to variants with weaker evidence of function. If the frequency component is $b_j$, one possible combined kernel weight is

$$ w_j=b_j a_j. $$

This formula defines $a_j$ as a multiplier of the kernel weight. If instead an annotation value is used to scale the genotype column, the resulting kernel multiplier is its square; that alternative convention must be stated explicitly. Neither an annotation score nor a frequency weight is universally correct. The choice encodes a scientific expectation about which variants should contribute most strongly.

A combined scheme should also be interpreted in terms of its final $w_j$ values. A variant can be very rare yet receive little overall influence if its annotation multiplier is small. Conversely, an annotation multiplier can partly offset a modest frequency weight. Reporting only the name of a weighting method may conceal these consequences; describing the formula or the resulting weights makes the analysis reproducible.

Worked Example: Functional Multipliers Change the Statistic

Suppose four hypothetical variants have score contributions $1.8$, $1.8$, $0.9$, and $1.4$. For illustration, take the first two kernel weights as $1$, and assign the third and fourth variants prespecified functional-annotation multipliers of $0.101526$ and $0.00006098$. These values are hypothetical multipliers, not a universal annotation scale.

The statistic is calculated by multiplying each squared score by its kernel weight:

\begin{aligned} Q_{\mathrm{annotated}} &=1(1.8^2)+1(1.8^2)+0.101526(0.9^2)+0.00006098(1.4^2)\\ &=3.24+3.24+0.101526(0.81)+0.00006098(1.96)\\ &=3.24+3.24+0.08223606+0.0001195208\\ &=6.5623555808\approx 6.562. \end{aligned}

The first two terms sum to $6.48$, because $2(1.8^2)=2(3.24)=6.48$. With equal weights, the statistic would instead be $6.48+0.81+1.96=9.25$. Here, the annotation multipliers reduce the contributions from the third and fourth variants. The observed statistic is therefore smaller under this scheme, but it is not valid to compare $6.562$ and $9.25$ as if they shared a null distribution. The weights alter that distribution too.

How Weighting Changes Null Calibration

Under the null hypothesis, suppose the score vector is approximately multivariate normal with mean zero and covariance matrix $\Sigma_U$. Then the weighted quadratic statistic has a mixture chi-square distribution:

Q\ \overset{d}{=}\ \sum_{k=1}^{m}\lambda_k Z_k^2,\qquad Z_k\sim N(0,1)\ \text{independently},

where the $\lambda_k$ are the eigenvalues of $W^{1/2}\Sigma_U W^{1/2}$, including any zero eigenvalues. This gives a direct justification for recalibrating when the weights change: changing $W$ generally changes the eigenvalues and hence the reference distribution. A p-value compares the observed $Q$ with the null distribution for that same weighting scheme, not with the distribution from another scheme.

Start with the weighted statistic.
Write $Q=U^\mathsf{T}WU$, using the score vector and kernel weights for the specified scheme.

Apply the null covariance.
Under the null approximation $U\sim N(0,\Sigma_U)$, the covariance of the score vector determines the variability of the quadratic form.

Find the mixture weights.
Diagonalizing $W^{1/2}\Sigma_U W^{1/2}$ gives eigenvalues $\lambda_k$, so the null statistic has the distribution of $\sum_k\lambda_k Z_k^2$.

Calibrate the observed value.
Calculate the upper-tail probability for the observed $Q$ under this scheme-specific mixture distribution.

The derivation in Step 3 follows from writing $U=\Sigma_U^{1/2}Z$, where $Z$ has independent standard normal components. Then

Q=Z^\mathsf{T}\Sigma_U^{1/2}W\Sigma_U^{1/2}Z.

A real symmetric nonnegative quadratic-form matrix can be diagonalized by an orthogonal transformation. Because an orthogonal transformation of a standard multivariate normal vector is still standard multivariate normal, the resulting quadratic form is a sum of eigenvalues times independent squared standard normal variables. The nonzero eigenvalues are the same as those of $W^{1/2}\Sigma_U W^{1/2}$, yielding the stated mixture representation.

Worked Example: Distinguishing Weight Multipliers From Genotype Scaling

For two hypothetical variants, suppose the unscaled score contributions are $U_1=1.2$ and $U_2=2.0$. An analyst wants to scale the genotype columns by $a_1=2$ and $a_2=0.5$. Scaling the genotype columns scales their score contributions by the same factors, giving $2(1.2)=2.4$ and $0.5(2.0)=1.0$. The resulting statistic is

$$ Q=(2.4)^2+(1.0)^2=5.76+1.00=6.76. $$

Equivalently, use kernel multipliers $w_1=a_1^2=4$ and $w_2=a_2^2=0.25$:

$$ Q=4(1.2^2)+0.25(2.0^2)=4(1.44)+0.25(4)=5.76+1.00=6.76. $$

Using $a_1$ and $a_2$ directly as kernel multipliers would give $2(1.44)+0.5(4)=4.88$, which is a different kernel and a different statistic. The calculation depends on whether the numbers are defined as genotype-column scalers or as kernel multipliers.

Choosing and Reporting a Weighting Scheme

Weights should correspond to a defensible expectation about variant effects. Frequency weighting is useful when rarer variants are expected, on average, to be more informative or more likely to have effects. Functional multipliers are useful when prespecified annotation supports different degrees of functional relevance. Neither choice guarantees higher power: the benefit depends on whether the weighting aligns with the actual pattern of effects and the study’s data.

For a transparent analysis, report the frequency function and its parameters, how weights were normalized, the definition and scale of any annotation multiplier, and whether it multiplies a genotype column or the kernel directly. Also report the variant set and the method used to obtain the p-value. These details make clear which test was performed and how its null distribution was calibrated.

Common Mistake: Comparing Statistics Without Their Null Distributions

A common error is to see that one weighting scheme produces a larger $Q$ and conclude that it provides stronger evidence of association. That conclusion does not follow from the statistic alone. A weighting change affects both the observed quadratic form and its null eigenvalues, so the p-value must be recomputed under the corresponding null distribution. Similarly, a smaller $Q$ under one scheme does not necessarily mean a less significant result.

Another mistake is to treat a frequency or annotation weight as an estimated effect size. In SKAT, the weight controls the contribution to the kernel; it is not the estimated direction or magnitude of a variant’s association. SKAT sums weighted squared score contributions, so the statistic does not indicate whether an individual variant increases or decreases the trait.

Key takeaway: In SKAT, the kernel weight $w_j$ multiplies the squared score contribution $U_j^2$. Frequency and annotation choices change the relative influence of variants and generally change the null mixture distribution as well. Interpret an association result using the p-value calibrated for the exact weights used.

Check Your Understanding

Answer each question by referring to the kernel definition and the calculations above.

If a genotype column is scaled by $a_j=3$, what kernel multiplier corresponds to that scaling?
Why does a beta frequency weighting function with parameters $1$ and $25$ give greater relative weight to a variant at frequency $0.01$ than to one at frequency $0.20$?
In the functional-multiplier example, which arithmetic gives the first two variants’ combined contribution of $6.48$?
Why can the p-value not be inferred by comparing the observed $Q$ values from two different weighting schemes?
What information about an annotation weight should be reported to make its meaning unambiguous?