SKAT, weights, and projections

Thomas Lumley

The original SKAT statistic for linear and generalised linear models is \[Q=(y-\hat\mu)'GW^2G'(y-\hat\mu)=(y-\hat\mu)'K(y-\hat\mu)\] where \(G\) is \(N\times M\) genotype matrix, and \(W\) is a weight matrix that in practice is diagonal. I’ve changed the original notation from \(W\) to \(W^2\), because everyone basically does. The Harvard group has a factor of 1/2 somewhere in here, the BU/CHARGE group doesn’t.

When the adjustment model isn’t ordinary linear regression, there is a second weight matrix, which I’ll write \(\Sigma\), giving the metric that makes \(y\mapsto y-\hat\mu\) the projection orthogonal to the range of \(X\). That is \[\hat\mu = \left(\Sigma^{-1/2}X(X^T\Sigma^{-1} X)^{-1}X^T\Sigma^{-1/2}\right)Y\] Note that both \[\left(\Sigma^{-1/2}X(X^T\Sigma^{-1} X)^{-1}X^T\Sigma^{-1/2}\right)\] and \[I-\left(\Sigma^{-1/2}X(X^T\Sigma^{-1} X)^{-1}X^T\Sigma^{-1/2}\right)\] are projections.

The matrix whose eigenvalues are needed for SKAT is \(H=P_0^{1/2}KP_0^{1/2}\) (or \(K^{1/2}P_0K^{1/2}\)) where \[P_0=V^{1/2}\left[I-\left(\Sigma^{-1/2}X(X^T\Sigma^{-1} X)^{-1}X^T\Sigma^{-1/2}\right)\right]V^{1/2}\] is the covariance matrix of the the residuals, with \(V=\mathop{var}[Y]\). Usually \(V=\Sigma\), but that’s not necessary.

famSKAT has test statistic \[Q=(y-\hat\mu)'V^{-1}GW^2G'V^{-1}(y-\hat\mu)=(y-\hat\mu)'V^{-1}KV^{-1}(y-\hat\mu)\] so the matrix \(H\) is \(H=P_0^{1/2}V^{-1}KV^{-1}P_0^{1/2}\).

When we want to take a square root of \(P_0\) it helps a lot that the central piece is a projection, and so is idempotent: we can define \[\Pi_0=\left[I-\left(\Sigma^{-1/2}X(X^T\Sigma^{-1} X)^{-1}X^T\Sigma^{-1/2}\right)\right]\] and write \(P_0=V^{1/2}\Pi_0V^{1/2}=V^{1/2}\Pi_0\Pi_0V^{1/2}\).

Now consider \(\tilde G\), with \(H=\tilde G^T\tilde G\). We can take \(\tilde G = WG'V^{-1}V^{1/2}\Pi_0=WG'V^{-1/2}\Pi_0\) where \(G\) is sparse, \(W\) is diagonal. The projection \(\Pi_0\) was needed to fit the adjustment model, so it will be fast. In family data where \(V=\Sigma\) is based on expected relatedness from a pedigree, the Cholesky square root \(R=V^{1/2}=\Sigma^{1/2}\) will be sparse.

Let \(f\) be the size of the largest pedigree. We should still be able to multiply a vector by \(\tilde G\) in \(O(MN\alpha+Nf^2)\) time where \(\alpha\ll 1\) depends on the sparseness of \(G\). If so, we can compute the leading-eigenvalue approximation in \(O(MNk\alpha+Nkf^2)\) time. (In fact, we can replace \(f^2\) by the average of the squares of number of relatives for everyone in the sample)

The relevant bits of the code, the functions that multiply by \(\tilde G\) and \(\tilde G^T\), look like

rval <- list(
    mult = function(X) {
      base::qr.resid(qr,as.matrix(solve(CholSigma,(spG %*% X))))
    tmult = function(X) {
      crossprod(spG, solve(t(CholSigma), base::qr.resid(qr,X)))