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 zi for i ∈ 1 : p within a set s, the test statistic Qs is
$$Q_s = \sum_{i=1}^p z_i^2$$
Here, Z = {z1, ..., zp}′ is a vector of multivariate normal distribution with a mean vector μ and a covariance matrix Σ in which Σ represents LD among SNPs. To test a set-level association, we need to evaluate the distribution of Qs. VEGAS uses Monte Carlo simulations to approximate the distribution of Qs (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 Qs more efficiently.
Let $Y = \Sigma^{-\frac12}Z$ (instead of $\Sigma^{-\frac12}$, we could use any decomposition that satisfies Σ = AA′ with a p × p non-singular matrix A such that Y = A−1Z). 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 Qs 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, Σ 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 ∼ N(0, Ip) since
E(X) = P′E(U) = 0, Var(X) = P′Var(U)P = P′IpP = Ip
$$ \begin{aligned} Q_s &= (U + \Sigma^{-\frac12}\mu)'\Sigma(U + \Sigma^{-\frac12}\mu) \\ &= (U + \Sigma^{-\frac12}\mu)'P\Lambda P'(U + \Sigma^{-\frac12}\mu) \\ &= (X + P'\Sigma^{-\frac12}\mu)'\Lambda (X + P'\Sigma^{-\frac12}\mu) \end{aligned} $$
Under the null hypothesis, μ is assumed to be 0. Hence,
$$Q_s = X'\Lambda X = \sum_{i=1}^p \lambda_i x_i^2$$
where X = {x1, ..., xp}′. Thus, the null distribution of Qs is a linear combination of independent chi-square variables xi2 ∼ χ(1)2 (i.e., central quadratic form in independent normal variables). For computing a probability with a scalar q,
Pr(Qs > 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.