Combining dependent F-tests for robust association of quantitative traits under genetic model uncertainty

Long Qu

doi:10.1515/sagmb-2013-0001

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Combining dependent F-tests for robust association of quantitative traits under genetic model uncertainty

Long Qu

Veröffentlicht/Copyright: 4. März 2014

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Aus der Zeitschrift Statistical Applications in Genetics and Molecular Biology Band 13 Heft 2

Abstract

In association mapping of quantitative traits, the F-test based on an assumed genetic model is a basic statistical tool for testing association of each candidate locus with the trait of interest. However, the true underlying genetic model is often unknown, and using an incorrect model may cause serious loss of power. For case-control studies, it is known that the combination of several tests that are optimal for different models is robust to model misspecification. In this paper, we extend the test combination approach to quantitative trait association. We first derive the exact correlations among transformed test statistics and discuss interesting special cases. We then propose and evaluate a multivariate normality based approximation to the joint distribution of test statistics, such that the marginal distributions and pairwise correlations among test statistics are accounted for. Through simulations, we show that the sizes of the resulting approximate combined tests are accurate for practical purposes under a variety of situations. We find that the combination of the tests from the additive model and the genotypic model performs well, because it demonstrates both robustness to incorrect models and satisfactory power. A mouse lipoprotein data set is used to demonstrate the method.

Keywords: combination of p-values; dominance; genetic association; model misspecification; ratio of quadratic forms

Corresponding author: Long Qu, Department of Mathematics & Statistics, Wright State University, Dayton, OH 45435, USA, e-mail: long.qu@wright.edu

Appendices

An example of QR-decomposition

As a simple example, we consider the QR-decomposition of

X0=[1x1⋮⋮1xn]=[1n, x]

with rank 2 and n>2. Let r denote the residual from regressing x over 1_n, i.e., r=x−x¯1n where x¯=x⋅/n with x⋅=∑i=1nxi. Then the Q₀₀ matrix can be given by [1n1n, 1||r||r].

The corresponding R₀ matrix is then the upper Cholesky root of

X0′X0=[nx⋅x⋅||x||2],

i.e.,

R0=[n1nx⋅0||r||].

It is trivial to verify that Q₀₀R₀=X₀ and that Q₀₀′Q₀₀ is identity.

The columns of Q₀₁ can be any set of orthogonal basis vectors perpendicular to the space spanned by columns of Q₀₀. The choice is not unique, since any valid Q₀₁ can be right multiplied by an arbitrary (n–2)×(n–2) orthonormal matrix P and the resulting Q₀₁P can also be used to compute residual contrasts.

Proof of Theorem 1

First, we derive the formula for the second moment of the ratio of quadratic forms in normal random variables, which is a consequence of Theorem 6 of Magnus (1986).

Lemma 1Letx be ann˜×1 vector following multivariate normal distribution with zero mean and identity covariance matrix. LetA₁ andA₂ ben˜×n˜ symmetric real matrices. Then,

1.1) Fori=1 or 2,

E{(x′Aixx′x)2}={tr(Ai)}2+2tr(Ai2)n˜2+2n˜.

1.2)

E{(x′A1xx′x)(x′A2xx′x)}=tr(A1)tr(A2)+2tr(A1A2)n˜2+2n˜.

Proof. Define Δ(z)=(1+2z)^–1/2I, and R(z)=Δ(z)AΔ(z)=(1+2z)^–1A, where A can be either A₁ or A₂. Then it directly follows from Theorem 6 of Magnus (1986) that

E{(x′Axx′x)2}=∑n∈Uγ2(n)∫0∞z|Δ(z)|∏j=12[tr{R(z)j}]njdz,

where U={(n1,n2)′:∑j=12jnj=2, and nj is a nonnegative integer. }={(0,1)′, (2,0)′} and

γ2(n)=2!⋅22∏j=12{nj!(2j)nj}−1.

As γ₂{(0, 1)′}=2 and γ₂{(2, 0)′}=1, we have

E{(x′Axx′x)2}=2∫0∞z|Δ(z)|[tr{R(z)}]0[tr{R(z)2}]dz+∫0∞z|Δ(z)|[tr{R(z)}]2[tr{R(z)2}]0dz=∫0∞z|Δ(z)|(2tr{R(z)2}+[tr{R(z)}]2)dz.

Because |Δ(z)|=(1+2z)−n˜/2, [tr{R(z)}]²=(1+2z)^–2{tr(A)}², and tr{R(z)²}=(1+2z)^–2tr(A²), we have

E{(x′Axx′x)2}=[{tr(A)}2+2tr(A2)]∫0∞z(1+2z)n˜/2+2dz=[{tr(A)}2+2tr(A2)]21−n˜/2−2)z−122(n˜/2+2−1)(n˜/2+2−2)(2z+1)n˜/2+2−1|z=0∞={tr(A)}2+2tr(A2)n˜2+2n˜.

This completes the proof of part 1.1) of the lemma.

Next, notice that, by applying 1.1),

E{(x′A1xx′x±x′A2xx′x)2}=E[{x′(A1±A2)xx′x}2]={tr(A1±A2)}2+2tr{(A1±A2)2}n˜2+2n˜.

Also, for any a, b∈ℜ, 4ab=(a+b)²–(a–b)². Thus,

E{(x′A1xx′x)(x′A2xx′x)}=14(E[{x′(A1+A2)xx'x}2]−E[{x′(A1−A2)xx'x}2])=14(n˜2+2n˜)[{tr(A1+A2)}2+2tr{(A1+A2)2}−{tr(A1−A2)}2−2tr{(A1−A2)2}]=14(n˜2+2n˜)[{tr(A1+A2)}2−{tr(A1−A2)}2+2tr{(A1+A2)2−(A1−A2)2}]=14(n˜2+2n˜){tr(2A1)tr(2A2)+2tr(2A1A2+2A2A1)}=1(n˜2+2n˜){tr(A1)tr(A2)+2tr(A1A2)}.

Proof of Theorem 1. For i, j=1, 2, …, M, □

Bi+Bj=y˜′(PX˜i+PX˜j)y˜y˜′y˜=(y˜/σe) ′(PX˜i+PX˜j)(y˜/σe)(y˜/σe) ′(y˜/σe),

which is a ratio of quadratic forms in normal random variables. In addition, when the null hypothesis holds, B_i+B_j has the same distribution as

z′(PX˜i+PX˜j)zz′z,

where z is a vector of independent standard normal random variables. Similarly, B_i–B_j is identically distributed as the random variable

z′(PX˜i−PX˜j)zz′z.

By applying Lemma 1, we have

E(BiBj)=tr(PX˜i)tr(PX˜j)+2tr(PX˜iPX˜j)n˜2+2n˜=νiνj+2tr(PX˜iPX˜j)n˜2+2n˜.

Therefore,

ρij={E(BiBj)−μiμj}τiτj=n˜νiνjrirj{tr(PX˜iPX˜j)−νiνjn˜}.

Further, if we perform a QR-decomposition for X˜m=Q˜mR˜m, where Q˜m is n˜×νm such that Q˜mm′Q˜mm=I and R˜m is ν_m×ν_m upper triangular matrix, then tr(PX˜iPX˜j)=‖Q˜i′Q˜j‖F2 where ||‧||_F denotes the Frobenius norm. Thus, the lower bound of ρ_ij occurs when the norm is zero. It follows that ρij≥−νiνjrirj. □

Computational details of μ^ij(ϱij)

To facilitate the numerical integration involved in μ^ij(ϱij), we perform the following change of variables. First let z˜i=zi and z˜j=zj−ϱijzi1−ϱij2. It can be shown that z˜i and z˜j are independently and standard normally distributed if z_i and z_j follow the bivariate standard normal distribution with correlation ϱ_ij. Therefore,

μ^ij(ϱij)=∫−∞∞∫−∞∞ℏ(z˜i, z˜j; ϱij)dz˜idz˜j,

where

ℏ(z˜i, z˜j; ϱij)=Ξi−1{Φ(z˜i)}Ξj−1{Φ(ϱijz˜i+1−ϱijz˜j)}ϕ(z˜i)ϕ(z˜j).

Further, to avoid the indefinite interval over which the integration is performed, we let z¯i=arctan(z˜i) and z¯j=arctan(z˜j), such that μ^ij(ϱij) is equal to

∫−π2π2∫−π2π2ℏ(tan(z¯i),tan(z¯j); ϱi,j)cos2(z¯i)cos2(z¯j)dz¯idz¯j.

Alternatively, the integration can be performed by letting b¯i=Φ(z˜i) and b¯j=Φ(z˜j). Then b¯i and b¯j are independently and uniformly distributed and μ^ij(ρij) is given by

∫01∫01Ξi−1(b¯i)Ξj−1[Φ{ϱijΦ−1(b¯i)+1−ϱijΦ−1(b¯j)}]db¯idb¯j.

Either of the above two formulas is in the form to be readily fed to common numerical integration routines, e.g., the adapt Integrate function in the cubature package of R.

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Published Online: 2014-3-4

Published in Print: 2014-4-1

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https://doi.org/10.1515/sagmb-2013-0001

Schlagwörter für diesen Artikel

combination of p-values; dominance; genetic association; model misspecification; ratio of quadratic forms