Bivariate traits association analysis using generalized estimating equations in family data

Mariza de Andrade; Mauricio A. Mazo Lopera; Nubia E. Duarte

doi:10.1515/sagmb-2019-0030

Article

Bivariate traits association analysis using generalized estimating equations in family data

Mariza de Andrade , Mauricio A. Mazo Lopera and Nubia E. Duarte

Published/Copyright: May 5, 2020

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From the journal Statistical Applications in Genetics and Molecular Biology Volume 19 Issue 2

Abstract

Genome wide association study (GWAS) is becoming fundamental in the arduous task of deciphering the etiology of complex diseases. The majority of the statistical models used to address the genes-disease association consider a single response variable. However, it is common for certain diseases to have correlated phenotypes such as in cardiovascular diseases. Usually, GWAS typically sample unrelated individuals from a population and the shared familial risk factors are not investigated. In this paper, we propose to apply a bivariate model using family data that associates two phenotypes with a genetic region. Using generalized estimation equations (GEE), we model two phenotypes, either discrete, continuous or a mixture of them, as a function of genetic variables and other important covariates. We incorporate the kinship relationships into the working matrix extended to a bivariate analysis. The estimation method and the joint gene-set effect in both phenotypes are developed in this work. We also evaluate the proposed methodology with a simulation study and an application to real data.

Keywords: bivariate analysis; gene-set test; generalized estimating equations; family data

Corresponding author: Mariza de Andrade, Department of Health Sciences Research, Mayo Clinic, Rochester, MN, 55905, USA, E-mail: mandrade@mayo.edu

Funding source: Foundation of the São Paulo State (FAPESP)

Award Identifier / Grant number: 2007/58,150-7

Appendix A Defining the eigenvalues for the score statistic Tz distribution

Following the results of Wang et al. (2013), we denote P=∑i=1NDiTVi−1Di and partition P as Px˜x˜, Px˜z˜, Pz˜x˜ and Pz˜z˜ according to the dimensions of β and α from model (1). Let us denote Q=(Pz˜x˜Px˜x˜−1, I) and

W=∑i=1NDiTVi−1Cov(Yi)Vi−1Di

where I is the identity matrix of dimensions (2qℒ2q) and Cov(Yi) is estimated by [Yi−μ0,i][Yi−μ0,i]T, where μ0,i is obtained under the null hypothesis. When N→∞, Tz→d∑i=1qλiχi,12, where χi,12 are independent chi-squred random variables with one degree of freedom and (λ1, …, λq) are the eigenvalues of W1/2QTQW1/2, where W1/2 denotes the Cholesky decomposition of W. For more details with respect to this result see Wang et al. (2013).

Appendix B Simulating multivariate correlated phenotypes

Appendix B.1 Multivariate binary data

Let Yb,ij and Yb,ik denote two binary response variables for the j-th and k-th members of the i-th family, such that P(Yb,ij=1)=pij, P(Yb,ik=1)=pik and P(Yb,ij=1, Yb,ik=1)=pijk.

The correlation coefficient, ri,jk, for Ybij and Ybik can be written as:

rijk=pijk−pijpikpij(1−pij)pik(1−pik),

and from this equation:

(B.1)pijk=rijkpij(1−pij)pik(1−pik)+pijpik.

Given the probabilities pij and pik, and the correlation rijk, we use two latent random variables, normally distributed with variances equal to 1, Lij and Lik, to simulate Yb,ij and Yb,ik, respectively. The latent variables are obtained by equating:

pijk=P(Lij>0, Lik>0)=P(L¯ij>−μij, L¯ik>−μik),

where μij=E(Lij), μik=E(Lik), L¯ij=Lij−μij and L¯ik=Lik−μik.

Let us assume that L¯ij and L¯ik have a standard bivariate normal distribution with correlation ρijk, and joint distribution:

P(L¯ij≥a, L¯ik≥b)=∫a∞∫b∞12π(1−ρijk) exp (−lij2−2ρijklijlik+lik22(1−ρijk2))dlijdlik.

Patel and Read (1982) tabulated the values of P(L¯ij≥a, L¯ik≥b) and Leisch et al. (1998) use Monte Carlo simulations to estimate these values and provide the following algorithm to obtain the correlated binary random variables:

Choose the marginal probabilities pij and pik, and the correlation rijk.
Set μij=Φ−1(pij) and μik=Φ−1(pik), where Φ is the standard normal distribution function.
Find pijk replacing pij, pik and rijk in Eq. (B. 1).
Obtain ρijk, by equating pijk=P(L¯ij≥μij, L¯ik≥μik).
Generate the bivariate normal random variables (Lij, Lik) using the parameters μij, μik and ρijk.
Set Ybij=1⇔Lij>0 and Ybik=1⇔Lik>0.

The general process to simulate a multivariate binary random vector is available in Leisch et al. (1998) and it is implemented in the package “bindata” of R (version 3.2.2) software (Leisch et al. 2012).

Appendix B.2 Multivariate continuous data

To simulate a multivariate continuous vector, Yc, with mean μc and covariance matrix Σc, we use a vector of independent standard normal variables, u. The vector Yc is obtained using the equation:

Yc=μc+u′C,

where Σc=CC′.

Appendix B.3 Correlated multivariate continuous and binary vectors

Let Yb,ij and Yc,ik be two random variables, binary and a continuous, respectively. We have that

Cov(Yc,ik, Yb,ij)=pijE(Yc,ik|Yb,ij=1)−pijE(Yc,ik)

and from this equation

(B.2)Cov(Yc,ik, Yb,ij)pij=E(Yc,ik|Yb,ij=1)−E(Yc,ik).

On the other hand, following the ideas of Cox and Wemuth (1992),

(B.3)E(Yc,ik|Yb,ij=1)=μc,ik+Corr(Yc,ik, Lij)ψc ϕ(αc,ij)Φ(−αij)

where Lij∼N(αij, 1), Yc,ik∼N(μc,ik, ψc), αij=Φ−1(pij), μc,ik=E(Yc,ik) and

Yb,ij={1,Lij≥αij0,Lij<αij.

Replacing Eq. (B. 3) in Eq. (B. 2):

Cov(Yc,ik, Yb,ij)pij=Corr(Yc,ik, Lij)ψcϕ(αc,ij)Φ(−αij).

And thus:

Cov(Yc,ik, Yb,ij)pijℒΦ(−αij)ϕ(αc,ij)=Corr(Yc,ik, Lij)ψc=Cov(Yc,ik, Lij).

These result can be generalized for the vectors Yc,i=(Yc,i1, …, Yc,ini), Yb,i=(Yb,i1, …, Yb,ini) and Li=(Li1, …, Lini), i. e.,

(B.4)Cov(Yc,i, Li)=Cov(Yc,i, Yb,i) Di,

where

Di=diag{Φ(−αi1)pi1ℒϕ(αi1), …, Φ(−αini)piniℒϕ(αini)}.

Appendix C Pedigree for the reference families

Figure C.1 shows the pedigree graphics for the eight reference families used in the simulation study and that were obtained of the Baependi Heart Study (Oliveira et al. 2008).

Figure C.1:

Pedigree graphics for the eight reference families used in the simulation study.

Figure C.2:

Linkage disequilibrium (LD) plot obtained from Haploview for the selected regions for PPARG (top) and CDKAL1 (bottom) SNPs of the Baependi dataset. Each diagonal represents a definite SNP and each square indicates the pairwise magnitude of LD or correlation structure between 2 SNPs corresponding to the crossing diagonals.

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Published Online: 2020-05-05

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https://doi.org/10.1515/sagmb-2019-0030

Keywords for this article

bivariate analysis; gene-set test; generalized estimating equations; family data