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A set-based association test in snpsettest

Jaehyun Joo

09 September, 2023

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 i1:p within a set s, the test statistic Qs is

Qs=pi=1z2i

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=Σ12Z (instead of Σ12, we could use any decomposition that satisfies Σ=AA with a p×p non-singular matrix A such that Y=A1Z). Then,

E(Y)=Σ12μVar(Y)=Σ12ΣΣ12=IpYN(Σ12μ, Ip)

Now, we posit U=Σ12(Zμ) so that

UN(0,Ip),  U=YΣ12μ

and express the test statistic Qs as a quadratic form:

Qs=pi=1z2i=ZIpZ=YΣ12IpΣ12Y=(U+Σ12μ)Σ(U+Σ12μ)

With the spectral theorem, Σ can be decomposed as follow:

Σ=PΛPΛ=diag(λ1,...,λp),  PP=PP=Ip

where P is an orthogonal matrix. If we set X=PU, X is a vector of independent standard normal variable XN(0,Ip) since

E(X)=PE(U)=0,  Var(X)=PVar(U)P=PIpP=Ip

Qs=(U+Σ12μ)Σ(U+Σ12μ)=(U+Σ12μ)PΛP(U+Σ12μ)=(X+PΣ12μ)Λ(X+PΣ12μ)

Under the null hypothesis, μ is assumed to be 0. Hence,

Qs=XΛX=pi=1λix2i

where X={x1,...,xp}. Thus, the null distribution of Qs is a linear combination of independent chi-square variables x2iχ2(1) (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

  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.