For set-based association tests, the snpsettest package employed the statistical model described in VEGAS (versatile gene-based association study) [1], which takes as input variant-level p values and reference linkage disequilibrium (LD) data. Briefly, the test statistics is defined as the sum of squared variant-level Z-statistics. Letting a set of $Z$ scores of individual SNPs $z_i$ for $i \in 1:p$ within a set $s$, the test statistic $Q_s$ is
$$Q_s = \sum_{i=1}^p z_i^2$$
Here, $Z = {z_1,...,z_p}'$ is a vector of multivariate normal distribution with a mean vector $\mu$ and a covariance matrix $\Sigma$ in which $\Sigma$ represents LD among SNPs. To test a set-level association, we need to evaluate the distribution of $Q_s$. VEGAS uses Monte Carlo simulations to approximate the distribution of $Q_s$ (directly simulate $Z$ from multivariate normal distribution), and thus, compute a set-level p value. However, its use is hampered in practice when set-based p values are very small because the number of simulations required to obtain such p values is be very large. The snpsettest package utilizes a different approach to evaluate the distribution of $Q_s$ more efficiently.
Let $Y = \Sigma^{-\frac12}Z$ (instead of $\Sigma^{-\frac12}$, we could use any decomposition that satisfies $\Sigma = AA'$ with a $p \times p$ non-singular matrix $A$ such that $Y = A^{-1}Z$). Then,
$$ \begin{gathered} E(Y) = \Sigma^{-\frac12} \mu \ Var(Y) = \Sigma^{-\frac12}\Sigma\Sigma^{-\frac12} = I_p \ Y \sim N(\Sigma^{-\frac12} \mu,~I_p) \end{gathered} $$
Now, we posit $U = \Sigma^{-\frac12}(Z - \mu)$ so that
$$U \sim N(\mathbf{0}, I_p),~~U = Y - \Sigma^{-\frac12}\mu$$
and express the test statistic $Q_s$ as a quadratic form:
$$ \begin{aligned} Q_s &= \sum_{i=1}^p z_i^2 = Z'I_pZ = Y'\Sigma^{\frac12}I_p\Sigma^{\frac12}Y \ &= (U + \Sigma^{-\frac12}\mu)'\Sigma(U + \Sigma^{-\frac12}\mu) \end{aligned} $$
With the spectral theorem, $\Sigma$ can be decomposed as follow:
$$ \begin{gathered} \Sigma = P\Lambda P' \ \Lambda = \mathbf{diag}(\lambda_1,...,\lambda_p),~~P'P = PP' = I_p \end{gathered} $$
where $P$ is an orthogonal matrix. If we set $X = P'U$, $X$ is a vector of independent standard normal variable $X \sim N(\mathbf{0}, I_p)$ since
$$E(X) = P'E(U) = \mathbf{0},~~Var(X) = P'Var(U)P = P'I_pP = I_p$$
$$ \begin{aligned} Q_s &= (U + \Sigma^{-\frac12}\mu)'\Sigma(U + \Sigma^{-\frac12}\mu) \ &= (U + \Sigma^{-\frac12}\mu)'P\Lambda P'(U + \Sigma^{-\frac12}\mu) \ &= (X + P'\Sigma^{-\frac12}\mu)'\Lambda (X + P'\Sigma^{-\frac12}\mu) \end{aligned} $$
Under the null hypothesis, $\mu$ is assumed to be $\mathbf{0}$. Hence,
$$Q_s = X'\Lambda X = \sum_{i=1}^p \lambda_i x_i^2$$
where $X = {x_1,...,x_p}'$. Thus, the null distribution of $Q_s$ is a linear combination of independent chi-square variables $x_i^2 \sim \chi_{(1)}^2$ (i.e., central quadratic form in independent normal variables). For computing a probability with a scalar $q$,
$$Pr(Q_s > q)$$
several methods have been proposed, such as numerical inversion of the characteristic function [2]. The snpsettest package uses the algorithm of Davies [3] or saddlepoint approximation [4] to obtain set-based p values.
References
Liu JZ, Mcrae AF, Nyholt DR, Medland SE, Wray NR, Brown KM, et al. A Versatile Gene-Based Test for Genome-wide Association Studies. Am J Hum Genet. 2010 Jul 9;87(1):139–45.
Duchesne P, De Micheaux P. Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods. Comput Stat Data Anal. 2010;54:858–62.
Davies RB. Algorithm AS 155: The Distribution of a Linear Combination of Chi-square Random Variables. J R Stat Soc Ser C Appl Stat. 1980;29(3):323–33.
Kuonen D. Saddlepoint Approximations for Distributions of Quadratic Forms in Normal Variables. Biometrika. 1999;86(4):929–35.
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