Generalized partial linear varying multi-index coefficient model for gene-environment interactions

Xu Liu; Bin Gao; Yuehua Cui

doi:10.1515/sagmb-2016-0045

Published by De Gruyter December 19, 2016

Generalized partial linear varying multi-index coefficient model for gene-environment interactions

Xu Liu , Bin Gao and Yuehua Cui

From the journal Statistical Applications in Genetics and Molecular Biology

https://doi.org/10.1515/sagmb-2016-0045

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Abstract

Epidemiological studies have suggested the joint effect of simultaneous exposures to multiple environments on disease risk. However, how environmental mixtures as a whole jointly modify genetic effect on disease risk is still largely unknown. Given the importance of gene-environment (G×E) interactions on many complex diseases, rigorously assessing the interaction effect between genes and environmental mixtures as a whole could shed novel insights into the etiology of complex diseases. For this purpose, we propose a generalized partial linear varying multi-index coefficient model (GPLVMICM) to capture the genetic effect on disease risk modulated by multiple environments as a whole. GPLVMICM is semiparametric in nature which allows different index loading parameters in different index functions. We estimate the parametric parameters by a profile procedure, and the nonparametric index functions by a B-spline backfitted kernel method. Under some regularity conditions, the proposed parametric and nonparametric estimators are shown to be consistent and asymptotically normal. We propose a generalized likelihood ratio (GLR) test to rigorously assess the linearity of the interaction effect between multiple environments and a gene, while apply a parametric likelihood test to detect linear G×E interaction effect. The finite sample performance of the proposed method is examined through simulation studies and is further illustrated through a real data analysis.

Keywords: B-spline; backfitting; generalized likelihood ratio; kernel smoothing; single index model; varying coefficient model

Acknowledgement

The authors wish to thank three anonymous referees for their constructive comments that greatly improved the manuscript. This work was supported in part by grants from National Science Foundation (DMS-1209112 and IOS-1237969) and from National Natural Science Foundation of China (31371336), and by Program for Innovative Research Team of Shanghai University of Finance and Economics.

Appendix: Proofs

Notations: For any vector 𝝃=(ξ1,⋯,ξs)T∈ℛ∫, denote ||𝝃||∞=max1≤l≤s|ξl|. For any nonzero matrix As×s, denote its L_r norm as ||A||r=max𝝃∈ℝs,𝝃≠0||A||r||𝝃||r−1. For any matrix A=(Aij)i,j=1s,t, denote ||A||∞=maxi≤i≤s∑j=1t|Aij. Let C(p)[a,b]={ψ:ψ(p)∈C[a,b]} be the space of the pth-order smooth functions. Denote the space of Lipschitz continuous functions for any fixed constant c₀ as Lib([a,b],c0)={ψ:|ψ(x1)−ψ(x2)|≤c0|x1−x2|,∀x1,x2∈[a,b]}. The following assumptions are required to show the consistency and asymptotic normality of our estimators.

For each l = 0, 1, the density function fU(βl)(⋅) of random variable U(𝜷l)=𝜷lTX is bounded away from 0 on Ω_l and there exists a constant 0<c0<∞ such that fU(βl)(⋅)∈Lib([a,b],c0) for 𝜷_l in the neighborhood of 𝜷l0, where Ωl={𝜷lTX,X∈𝒳} and 𝒳 is a compact support of X.
The nonparametric function ml∈C(r)[a,b], l = 0, 1.
The variance var(Y|V=v)<c1 for some 0<c1<∞.
∂3g(u)/∂u3 and ∂2V(u)/∂u2 are continuous.
There exist constants 0<c21≤c22<∞ such that cz≤Q(x)=E(Z~Z~T|X=x)≤Cz for all x∈𝒳.
The kernel function K(⋅) is a symmetric density function with compact support [−1,1] and K∈Lib([a,b],cK) for some constant c_K.
The function u3K(u) and u3K′(u) are bounded and ∫u4K(u)du<∞.

Let Yz,i=Yi−ZiT𝜶00−ZiT𝜶10Gi, Yz=(Yz,1,⋯,Yz,n)T, e=(ε1,⋯,εn)T, 𝕏=(X1,⋯,Xn)T, ℤ=(Z1,⋯,Zn)T, G=(G1,⋯,Gn)T, 𝔾=(f1n,G) and Λ^(𝜽)=(𝜶^T,λ^(𝜷)T)T. Let Z~=(Z~1,⋯,Z~n)T with Z~i=(ZiT,ZiTGi)T, defined in Section 2.1. Let 𝚯 be the parametric space of 𝜽. Let qj(x)=(∂j/∂xj)Q{g−1(x),y}, j = 1, 2, 3. Then q1(x)={y−g−1(x)}ρ1(x) and q2(x)={y−g−1(x)}ρ1′(x)−ρ2(x), and ρj(x)={dg−1(x)/dx}j/V{g−1(x)}. Denote qk(η~i) by qk{η~(Vi;𝜽0,λ(𝜷0))}, k = 1,2, i = 1, ⋯, n. Let qk=(qk(η~1),⋯,qk(η~n))T and Wq2 be a diagonal matrix with diagonal elements q2{η~}. Define

(A.1)U(𝜷)=E[q2(η~i)Di(𝜷)Di(𝜷)T],U^(𝜷)=1nD(𝜷)TWq2D(𝜷),U(Z~,𝜷)=E[q2(η~i)Di(Z~,𝜷)Di(Z~,𝜷)T],U^(Z~,𝜷)=1nD(Z~,𝜷)TWq2D(Z~,𝜷)

where Di(𝜷)=(Di,sl(𝜷l)𝔾il,1≤s≤Jn,l=0,1)T and D(𝜷)=(D1(𝜷),⋯,Dn(𝜷))T, which is a n×2Jn matrix.

Proof of Theorem 1: Because the observations V1,⋯,Vn are independent, by Lindeberg-Feller central limit theorem, it is easy to prove that

n−1/2∑i=1n(Z~i−P~(Zi)Xm,i−P~(Xi))q1{η~i}⟶ℒN(,Σ~),

where Σ=E[ρ2{η(V,𝜽0)}ϕ(V,𝜷0)⊗2]. Then combining Lemma S.6 in the Supplementary material, the proof of Theorem 1 can be completed by Slusky’s theorem.

Proof of Theorem 2: Because for any ul∈[al,bl], Bs,l(ul), s=1,⋯,Jn,l=0,1, have the banded first derivatives, by (S.9) and (S.7) in the Supplementary material, and Theorem 1, we have, for any ul∈[al,bl],

|m~l(ul,𝜷^)−m~l(ul,𝜷0)|=|D(𝜷^)Tλ^(𝜷^)−D(𝜷0)Tλ(𝜷0)|≤|D(𝜷0)T{λ^(𝜷^)−λ(𝜷0)}|+|{D(𝜷^)−D(𝜷0)}Tλ^(𝜷^)|≤|n−1D(𝜷0)TU^(𝜷0)−1D(𝜷0)Tq1{η~}|+Op(n−1/2)=Op(N/n).

Then, combining Lemma S.4 in the Supplementary material, we have

This completes the proof of Theorem 2.

Proof of Theorem 4: Due to nhl5=O(1), we have nhln−2/5=oo(1). By Theorem 3, we have

nhl{m^l(ul,𝜷^)−ml(ul)−bl(ul)hl2}=nhl{m^lO(ul,𝜷^)−ml(ul)−bl(ul)hl2}+op(1).

Thus Theorem 4 can be shown straightforwardly by Lemma S.7 in the Supplementary material.

Proof of Theorem 5: It is easy to see that

ℓn1(H0)−ℓn1O(H0)=∑i=1nq1(η^H0O(Vi;𝜽^)){η^H0(Vi;𝜽^)−η^H0O(Vi;𝜽^)}−1/2∑i=1nq2(η^H0O(Vi;𝜽^)){η^H0(Vi;𝜽^)−η^H0O(Vi;𝜽^)}2=Op(1),

and similarly

ℓn1(H1)−ℓn1O(H1)=∑i=1nq1(η^H1O(Vi;𝜽^)){η^H1(Vi;𝜽^)−η^H1O(Vi;𝜽^)}−1/2∑i=1nq2(η^H1O(Vi;𝜽^)){η^H1(Vi;𝜽^)−η^H1O(Vi;𝜽^)}2=Op(1),

As the same proof of Lemma S.12 in the Supplementary material, T1=T1O+o(1) holds by showing following claims that

ℓn1(H1)−ℓn1(H0)=ℓn1O(H1)−ℓn1O(H0)+Op(1).

Thus, T₁ and T1O have the same distribution, which completes the proof of Theorem 5 by Lemma S.12 in the Supplementary material.

Proof of Theorem 7: This proof is similar to Liu et al. (2016). So accordingly, we only provide a sketch of a proof here. Let Bn=AΓn−1AT, where where A is defined in Section 3.2 and

Γn=∑i=1nq2(η(Vi;𝜽0))𝚿^i(X,Z)𝚿^i(X,Z)T,with𝚿^i(X,Z)=(Xm^,i−P~n(Xi)Z~i−P~n(Zi)).

By Lemma S.5 in the Supplementary material, T₃ can be decomposed as

2(ℓn3(H1)−ℓn3(H0))= 2∑i=1nq1(η^3,H0(Vi;𝜽^)){η^3,H1(Vi;𝜽^)−η^3,H0(Vi;𝜽^)}−∑i=1nq2(η^3,H0(Vi;𝜽^)){η^3,H1(Vi;𝜽^)−η^3,H0(Vi;𝜽^)}2+op(1)= 2∑i=1nq1(η^3,H0O(Vi;𝜽^)){η^3,H1O(Vi;𝜽^)−η^3,H0O(Vi;𝜽^)}−∑i=1nq2(η^3,H0O(Vi;𝜽^)){η^3,H1O(Vi;𝜽^)−η^3,H0O(Vi;𝜽^)}2+op(1)=(𝜽^H0−𝜽^H1)TΓn(𝜽^H0−𝜽^H1)+op(1),

where

η^3,H0(Vi;𝜽^)=m^0(𝜷^0,H0TXi,𝜽^H0)+𝜶^0,H0TZi+(m^1(𝜷^1,H0TXi,𝜽^H0)+𝜶^1,H0TZi)Gi,η^3,H1(Vi;𝜽^)=m^0(𝜷^0,H1TXi,𝜽^H1)+𝜶^0,H1TZi+(m^1(𝜷^1,H1TXi,𝜽^H1)+𝜶^1,H1TZi)Gi,η^3,H0O(Vi;𝜽^)=m^0O(𝜷^0,H0TXi,𝜽^H0)+𝜶^0,H0TZi+(m^1O(𝜷^1,H0TXi,𝜽^H0)+𝜶^1,H0TZi)Gi,η^3,H1O(Vi;𝜽^)=m^0O(𝜷^0,H1TXi,𝜽^H1)+𝜶^0,H1TZi+(m^1O(𝜷^1,H1TXi,𝜽^H1)+𝜶^1,H1TZi)Gi.

As the estimators for 𝜽 under the null and alternative hypotheses have the relation

𝜽^H0=𝜽^H1+Γn−1ATBn−1(𝜸−A𝜽^H1),

we have

T3=(𝜸−A𝜽^H1)TBn−1AΓn−1∑i=1n𝚿^i(X,Z)⊗2Γn−1ATBn−1(𝜸−A𝜽^H1)+op(1)=(𝜸−A𝜽^H1)TBn−1(𝜸−A𝜽^H1)+op(1)

Therefore, under null hypothesis, T3⟶ℒχk2 and under alternative hypothesis T₃ asymptotically follows a noncentral χ² distribution with k degrees of freedom and noncentrality parameter ϕ. This completes the proof of Theorem 7.

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Supplemental Material

The online version of this article (DOI: 10.1515/sagmb-2016-0045) offers supplementary material, available to authorized users.

Published Online: 2016-12-19

Published in Print: 2017-3-1

Generalized partial linear varying multi-index coefficient model for gene-environment interactions

Abstract

Acknowledgement

Appendix: Proofs

References

Supplemental Material

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