A gene-based test of association through an orthogonal decomposition of genotype scores

Abstract

The burden test and the sequence kernel association test (SKAT) are two popular methods for detecting association with rare variants. Treated as two different sources of association information, they are adaptively combined to form an optimal SKAT (SKAT-O) method for optimal power. We show that the burden test is part of rather than independent of the SKAT. We introduce a new test statistic that is the sum of the burden statistic and a statistic asymptotically independent of the burden statistic. The performance of this new test statistic is demonstrated through extensive simulation studies and applications to a Genetic Analysis Workshop 17 data set and the Ocular Hypertension Treatment Study data.

Appendix: Proofs

Proof of Proposition 2.1

Under the model assumption, asymptotically, $\tilde{y}\sim N(0,1)$. The covariance between $Q_{\text{B}}$ and $Q_{\text{s}}$ is

$$ \begin{aligned} {\text{Cov}}\left( {Q_{\text{B}} ,Q_{\text{s}} } \right) = 2{\text{trace}}\left[ {G11^{\varvec{T}} G^{T} GG^{T} } \right] = 2{\text{trace}}\left[ {G11^{\varvec{T}} G^{T} \mathop \sum \limits_{j = 1}^{m} \lambda_{j} v_{j} v_{j}^{T} } \right] \hfill \\ = 2\mathop \sum \limits_{j = 1}^{m} \lambda_{j} {\text{trace}}\left[ {G11^{\varvec{T}} G^{T} v_{j} v_{j}^{T} } \right] = 2\mathop \sum \limits_{j = 1}^{m} \lambda_{j} {\text{trace}}\left[ {1^{\varvec{T}} G^{T} v_{j} v_{j}^{T} G1} \right] = 2\mathop \sum \limits_{j = 1}^{m} \lambda_{j} (v_{j}^{T} G1)^{2} > 0. \hfill \\ \end{aligned} $$

Therefore, $Q_{\text{B}}$ and $Q_{\text{s}}$ are correlated in general.

Proof of Theorem 2.1

Since $\tilde{G} = G - \tilde{v}_{0} \tilde{v}_{0}^{T} G$, $\tilde{v}_{0} = \frac{G1}{{\sqrt {1^{T} G^{T} G1} }},$ and $\tilde{G}\tilde{G}^{T} \tilde{v}_{0} = 0$. Therefore, $\tilde{v}_{0}$ is orthogonal to the space spanned by the column vectors of $\tilde{G}\tilde{G}^{T}$. Hence, for each $\tilde{v}_{j} > 0$, we have $\tilde{v}_{j}^{T} \tilde{v}_{0} = 0$. The covariance between $\tilde{v}_{j}^{T} \tilde{y}$ and $\tilde{v}_{k}^{T} \tilde{y}$ is ${\text{Cov}} ( {\tilde{v}_{j}^{T} \tilde{y},\tilde{v}_{k}^{T} \tilde{y}} ) = \tilde{v}_{j}^{T} \tilde{v}_{k} = 0\; {\text{if}}\; j \ne k, \;{\text{and}}\; j,k = 0, 2, \ldots ,m$, since under the null hypothesis $\tilde{v}_{j}^{T} \tilde{y}$ has asymptotic standard normal distribution: $E[\tilde{v}_{j}^{T} \tilde{y}] = 0, \;{\text{and}}\; {\text{Var}}[\tilde{v}_{j}^{T} \tilde{y}] = 1.$

Proof of Theorem 2.2

Under the null hypothesis, asymptotically, both p values P ₁ and P ₂ from Q ₁, and Q ₂, respectively, have uniform distribution between 0 and 1. Using quantile transformation as described in the text, both variables $\left( {\chi_{1}^{2} } \right)^{ - 1} \left( {P_{i} } \right) (i = 1,2)$ asymptotically and independently follow a Chi-square distribution with a degree of freedom 1. Therefore, the new test statistic $Q_{\text{new}} = \mathop \sum \nolimits_{i = 1}^{2} \left( {\chi_{1}^{2} } \right)^{ - 1} (P_{i} )$ has an asymptotic Chi-square distribution with degrees of freedom 2.