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Statistical test in snpsettest

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Jaehyun Joo edited this page Mar 14, 2021 · 2 revisions

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 likage 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

  1. 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.

  2. 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.

  3. 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.

  4. Kuonen D. Saddlepoint Approximations for Distributions of Quadratic Forms in Normal Variables. Biometrika. 1999;86(4):929–35.

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