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Genet Epidemiol. Author manuscript; available in PMC 2010 April 1.

Published in final edited form as:

Genet Epidemiol. 2009 April; 33(3): 217–227.

doi: 10.1002/gepi.20372PMCID: PMC2745071

NIHMSID: NIHMS94213

Jianfeng Liu,^{1,}^{2} Yufang Pei,^{1,}^{2,}^{4} Chris J. Papasian,^{2} and Hong-Wen Deng^{1,}^{2,}^{3,}^{4}

Corresponding author: Hong-Wen Deng, Ph. D. Department of Basic Medical Science School of Medicine University of Missouri-Kansas City 2411 Holmes Street, Room: M3-CO3 Kansas City, MO 64108 Email: ude.ckmu@hgned Tel: 816-235-5354 Fax: 816-235-6517

See other articles in PMC that cite the published article.

Genome-wide association (GWA) study is becoming a powerful tool in deciphering genetic basis of complex human diseases/traits. Currently, the univariate analysis is the most commonly used method to identify genes associated with a certain disease/phenotype under study. A major limitation with the univariate analysis is that it may not make use of the information of multiple correlated phenotypes, which are usually measured and collected in practical studies. The multivariate analysis has proven to be a powerful approach in linkage studies of complex diseases/traits, but it has received little attention in GWA. In this study, we aim to develop a bivariate analytical method for GWAS, which can be used for a complex situation that a continuous trait and a binary trait measured are under study. Based on the modified extended generalized estimating equation (EGEE) method we proposed herein, we assessed the performance of our bivariate analyses through extensive simulations as well as real data analyses. In the study, to develop an EGEE approach for bivariate genetic analyses, we combined two different generalized linear models corresponding to phenotypic variables using a Seemingly Unrelated Regression (SUR) model. The simulation results demonstrated that our EGEE-based bivariate analytical method outperforms univariate analyses in increasing statistical power under a variety of simulation scenarios. Notably, EGEE-based bivariate analyses have consistent advantages over univariate analyses whether or not there exits a phenotypic correlation between the two traits. Our study has practical importance, as one can always use multivariate analyses as a screening tool when multiple phenotypes are available, without extra costs of statistical power and false positive rate. Analyses on empirical GWA data further affirm the advantages of our bivariate analytical method.

Genome-wide association (GWA) studies offer a powerful tool for identifying genes conferring modest disease risks [Risch and Merikangas 1996]. Currently, statistical methods developed for GWA data analyses were largely for single phenotypes. However, in practice, investigators often collect a set of correlated phenotypes that may share common environmental and/or genetic factors. The multivariate analysis is a natural way to take all phenotypes into consideration simultaneously. Multivariate analyses have been widely adopted in linkage studies [Allison, et al. 1998; Huang and Jiang 2003; Jiang and Zeng 1995; Lange and Whittaker 2001; Liu, et al. 2007a; Williams, et al. 1999], and there is a general consensus that multivariate analyses outperform univariate analyses in increasing statistical power and precision of parameter estimation. However, in GWA, multivariate analyses have not received sufficient attention, although there are some sporadic reports on methodology development of multi-trait analyses [Lange, et al. 2003; Lange, et al. 2002; Verzilli, et al. 2005]. In particular, as to the common situation when both binary (*e.g.*, disease status) and continuous data (*e.g.*, quantitative measures that are risk factors for the disease under study) are involved, little work has been done to evaluate the performance of the multivariate analytical method.

For a mixture of correlated phenotypes involving non-normal traits (such as binary or ordinal data), it is not straightforward to specify a full probability model, making it difficult to directly implement likelihood-based approaches [Lange and Whittaker 2001; Lange, et al. 2002]. To surmount this problem, several methods were developed [Catalano 1997; Gueorguieva and Agresti 2001; Gueorguieva and Sanacora 2006; Regan and Catalano 2000] based on the threshold model assumption, which transform a discrete phenotype to a “latent” normal variable. However, the potential concerns with these methods are the arbitrary and untestable distributional assumptions on the “latent” variable, and the difficulty with computational features of model fitting [Prentice and Zhao 1991].

Since the seminal work of Liang and Zeger [1986], a suite of methods [Hall 2001; Hall and Severini 1998; Liang, et al. 1992; Prentice and Zhao 1991; Zhao and Prentice 1990] based on the Generalized Estimating Equation (GEE) have been developed to address the situation of non-normally distributed variables in multivariate data analyses. GEE can be regarded as a multivariate version of generalized linear model (GLM), a method with reasonable statistical efficiency to analyze correlated data with non-normal distributions. A vast of literature on GEE theory and application has arisen over the past two decades [Zeger and Liang 1992; Ziegler, et al. 1998; Zorn 2001]. The most commonly used GEE-based approaches, referred to as GEE1 and GEE2, were introduced by Liang and Zeger [1986] and Zhao and Prentice [1990], respectively.

GEE1 and GEE2 have their respective limitations. In GEE1, regression and association parameters are presumed to be orthogonal which may not necessarily true, and the association parameter is treated as a nuisance [Liang and Zeger 1986; Liang, et al. 1992]. Because the relationship between the first- and second-order moment parameters is ignored, GEE1 has less efficient estimators than GEE2 if the working covariance model is correctly specified [Hall and Severini 1998; Liang, et al. 1992]. In contrast to GEE1, GEE2 removes the assumption of “orthogonality” initially imposed on GEE1. Thus, GEE2 can perform estimation for regression and association parameters simultaneously under a unified framework, leading to a significant increase in efficiency of parameter estimation. However, a limitation of GEE2 is that it may lose the consistency of regression parameter estimators if the covariance structure is misspecified [Hall 2001; Hall and Severini 1998; Liang, et al. 1992; Zhao and Prentice 1990].

To improve the performance of GEE1 and GEE2, an alternative GEE-based approach, named Extended Generalized Estimating Equation (EGEE), was proposed by Hall and Severini [1998]. EGEE was developed using the ideas of extended quasi-likelihood [McCullagh and Nelder 1989; Nelder and Pregibon 1987]. It retains many of the advantages of the original GEE1 and GEE2 methods while avoiding their potential limitations: (1) EGEE can offer consistent estimations of both regression and association parameters even under an incorrect covariance model; (2) EGEE is not required to assume third- and fourth-order moment “working” models which are necessary in GEE2; and (3) the estimation efficiency of EGEE is comparable to GEE2 and often outperforms GEE1 [Hall 2001; Hall and Severini 1998].

In the present study, using EGEE, being an improved GEE-based approach, we aim to develop bivariate gene/phenotype association analyses for population-based data when both continuous and binary traits are involved in the same studies. Since the original EGEE [Hall 2001; Hall and Severini 1998] cannot deal with mixed continuous and binary outcomes in a straightforward fashion, we modified the EGEE method for such situations. Our strategy is motivated by the usage of Seemingly Unrelated Regression (SUR) [Zellner 1962]. SUR procedure is an extension of ordinary least squares analyses for dealing with the system of linear equations with correlated error terms, which can be treated as a special case of generalized least squares analyses. Based on the idea of SUR models, two GLMs with different link functions can be incorporated into a unified equation system. Consequently, we developed a “specific” form of EGEE, which is well suited for the mixture of two phenotypes with different distributions.

We applied our modified EGEE method to assess the performance of bivariate analyses through extensive simulation studies. The results demonstrated that, compared with univariate analyses, bivariate analyses can substantially improve power while having comparable false-positive rates under almost all the scenarios simulated. The advantages of our EGEE-based bivariate analyses method was further validated in empirical data analyses using our genome-wide association scan data for osteoporosis and obesity. Given the emerging widespread GWA, our proposed method may have practical significance in the genetic epidemiology field in general.

We begin with the situation in which *N* unrelated random subjects are recruited in a population-based genetic association study, each having observations on two different types of phenotypes: one is in normal distribution (named *T*_{1}) and the other is a binary trait (named *T*_{2}). A 2×1 random observation vector for individual *i* (*i* = 1, …, *N*) is denoted as **y _{i}** = (

$${g}_{1}\left({\mu}_{i1}\right)={\mu}_{i1}={{\mathbf{x}}_{i}}^{\prime}{\beta}_{1},$$

[1]

$${g}_{2}\left({\mu}_{i2}\right)=\mathrm{ln}\left(\frac{{\mu}_{i2}}{1-{\mu}_{i2}}\right)={\mathbf{x}}_{i}^{\prime}{\beta}_{2}.$$

[2]

Here, for ease of incorporating the two different types of data into an overall EGEE analysis, we employ SUR model [Zellner 1962] to combine Equations [1] and [2] into a unified framework:

$$\mathbf{g}\left({\mu}_{i}\right)={{\mathbf{X}}^{\prime}}_{i}\beta ,$$

[3]

where ${\mathbf{X}}_{i}=\left(\begin{array}{cc}\hfill {\mathbf{x}}_{i}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathbf{x}}_{i}\hfill \end{array}\right)$, and **g**(*μ _{i}*) is a compound function vector in the form of (

Following the GLM theory, the respective variance functions of two distributions, *i.e.*, the normal and the binary, are given as:

$${v}_{1}\left({\mu}_{1}\right)=1,\phantom{\rule{thickmathspace}{0ex}}\text{and}\phantom{\rule{thickmathspace}{0ex}}{v}_{2}\left({\mu}_{i2}\right)={\mu}_{i2}(1-{\mu}_{i2}),$$

[4]

where *μ _{i}*

In the framework of GLM, probability distributions of the random variables are usually parameterized in terms of not only the means (e.g., *μ _{i}*

For specifying the marginal (co)variance model **V**_{i}, we further define the “working” correlation matrix $\mathbf{R}\left(\xi \right)=\left(\begin{array}{cc}\hfill 1\hfill & \hfill \xi \hfill \\ \hfill \xi \hfill & \hfill 1\hfill \end{array}\right)$, where *ζ* is the unknown association parameter falling within (-1.0, 1.0). It is noted that the regressors involved in the SUR model are allowed to have possibly different effects on either of the two regressands considered, with residuals from each regression assumed to be correlated [Verzilli, et al. 2005].

Given the variance functions, dispersion parameters and working correlation matrix defined above, the (co)variance matrix **V _{i}** between the two phenotypes for individual

$${\mathbf{V}}_{i}={\left(\begin{array}{cc}\hfill {\psi}_{1}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\psi}_{2}\hfill \end{array}\right)}^{\frac{1}{2}}{\left(\begin{array}{cc}\hfill {v}_{1}\left({\mu}_{i1}\right)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {v}_{2}\left({\mu}_{i2}\right)\hfill \end{array}\right)}^{\frac{1}{2}}\left(\begin{array}{cc}\hfill 1\hfill & \hfill \xi \hfill \\ \hfill \xi \hfill & \hfill 1\hfill \end{array}\right){\left(\begin{array}{cc}\hfill {v}_{1}\left({\mu}_{i1}\right)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {v}_{2}\left({\mu}_{i2}\right)\hfill \end{array}\right)}^{\frac{1}{2}}{\left(\begin{array}{cc}\hfill {\psi}_{1}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\psi}_{2}\hfill \end{array}\right)}^{\frac{1}{2}}.$$

[5]

On the basis of Equation [4] and the equality *ψ*_{2} 1, Equation [5] can be fully described as:

$${\mathbf{V}}_{i}=\left(\begin{array}{cc}\hfill {\psi}_{1}\hfill & \hfill \xi \sqrt{{\psi}_{1}{\mu}_{i2}(1-{\mu}_{i2})}\hfill \\ \hfill \xi \sqrt{{\psi}_{1}{\mu}_{i2}(1-{\mu}_{i2})}\hfill & \hfill {\mu}_{i2}(1-{\mu}_{i2})\hfill \end{array}\right).$$

[6]

From Equation [6], it is obvious that **V**_{i} is determined by **β**, * ζ* and

$${\mathbf{V}}_{i}=\left(\begin{array}{cc}\hfill {\phi}^{2}\hfill & \hfill \xi \phi \sqrt{{\mu}_{i2}(1-{\mu}_{i2})}\hfill \\ \hfill \xi \phi \sqrt{{\mu}_{i2}(1-{\mu}_{i2})}\hfill & \hfill {\mu}_{i2}(1-{\mu}_{i2})\hfill \end{array}\right).$$

[7]

For ease of presentation, we combine the second-order moment parameters in a 2×1 vector fashion, $\stackrel{~}{\alpha}={(\xi ,\phi )}^{\prime}$.

To examine whether the marker locus is associated with either of the two phenotypes within the EGEE framework, we take two steps (named in turn * Estimation Step* and

From the EGEE perspective, the unknown parameters, including the regression vector **β** and the association vector $\stackrel{~}{\alpha}$ aforementioned, can be estimated in a set of estimating equations. Here, we directly adopt the same form of the EGEEs as given in the original study [Hall 2001]:

$$\sum _{i=1}^{N}{U}_{i}(\beta ,\stackrel{~}{\alpha})=\sum _{i=1}^{N}\left(\begin{array}{cc}\hfill {\mathbf{D}}_{i}^{\prime}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathbf{F}}_{i}^{\prime}\hfill \end{array}\right)\left(\begin{array}{cc}\hfill {\mathbf{V}}_{i}^{-1}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \mathbf{I}\hfill \end{array}\right)\left(\begin{array}{cc}\hfill {\mathbf{y}}_{i}\hfill & \hfill -{\mu}_{i}\hfill \\ \hfill {\mathbf{s}}_{i}\hfill & \hfill -{\sigma}_{i}\hfill \end{array}\right)=0,$$

[8]

where, the 2×2*L* matrix **D**_{i} = **μ**_{i} / **β**’, and the 4×2 matrix ${\mathbf{F}}_{i}=\partial \left\{\mathrm{vec}{\mathbf{V}}_{i}^{-1}\right\}\u2215\partial {\stackrel{~}{\alpha}}^{\prime}$. The operator vec(•) here is a function for generating a column vector via stacking the columns (or rows) of the argument matrix. The 4×1 vectors **s**_{i} and **σ**_{i} are defined for estimating the second-order moment parameters, where **s**_{i} = vec{(**y**_{i} — **μ**_{i} )(**y**_{i} — **μ**_{i})’} and **σ**_{i} = *E*(**s**_{i} ) = vec**V**_{i}.

Obviously, Equation [8] regarding the unknowns **β** and $\stackrel{~}{\alpha}$ is hard to resolve straightforwardly due to its nonlinear property. Alternatively, a Fisher scoring algorithm may offer a natural way for obtaining the estimates of **β** and $\stackrel{~}{\alpha}$ . Starting with initial values **β**^{(0)} and ${\stackrel{~}{\alpha}}^{\left(0\right)}$, the updated parameter estimates **β**^{(l)} and ${\stackrel{~}{\alpha}}^{l}$ in the *l*th iteration are given by

$$\left(\begin{array}{c}\hfill {\beta}^{\left(l\right)}\hfill \\ \hfill {\stackrel{~}{\alpha}}^{\left(l\right)}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill {\beta}^{(l-1)}\hfill \\ \hfill {\stackrel{~}{\alpha}}^{(l-1)}\hfill \end{array}\right)+{\left(\sum _{i=1}^{N}\left(\begin{array}{cc}\hfill {\mathbf{D}}_{i}^{\prime (l-1)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathbf{F}}_{i}^{\prime (l-1)}\hfill \end{array}\right)\left(\begin{array}{cc}\hfill {{\mathbf{V}}_{i}}^{-{1}^{(l-1)}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \mathbf{I}\hfill \end{array}\right)\left(\begin{array}{cc}\hfill {{\mathbf{D}}_{i}}^{(l-1)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {{\mathbf{F}}_{i}}^{(l-1)}\hfill \end{array}\right)\right)}^{-1}\left(\sum _{i=1}^{N}{U}_{i}({\beta}^{(l-1)},{\stackrel{~}{\alpha}}^{(l-1)})\right).$$

[9]

The above iteration is repeated until convergence. In our study, we set a sufficiently rigid convergence criterion as

$$\left[\right({\beta}^{\prime \left(l\right)}-{\beta}^{\prime (l-1)},{\stackrel{~}{\alpha}}^{\prime \left(l\right)}-{\stackrel{~}{\alpha}}^{\prime (l-1)}\left){({\beta}^{\prime \left(l\right)}-{\beta}^{\prime (l-1)},{\stackrel{~}{\alpha}}^{\prime \left(l\right)}-{\stackrel{~}{\alpha}}^{\prime (l-1)})}^{\prime}\right]\u2215(2L+2)\le 10\mathrm{E}-12.$$

Accordingly, the updated values of **β** and $\stackrel{~}{\alpha}$ in the latest iteration will be treated as the solution of the EGEEs given by Equation [8], which are denoted as $\stackrel{~}{\beta}$ and $\widehat{\stackrel{~}{\alpha}}$, respectively. Accordingly, a sandwich-type estimate of the asymptotic (co)variance matrix of $\stackrel{~}{\beta}$ and $\widehat{\stackrel{~}{\alpha}}$, $\mathrm{var}(\stackrel{~}{\beta},\widehat{\stackrel{~}{\alpha}})$, is given by

$$\mathrm{var}(\widehat{\beta},\widehat{\stackrel{~}{\alpha}})={\left\{{U}_{.}^{\ast}(\widehat{\beta},\widehat{\stackrel{~}{\alpha}})\right\}}^{-1}\left\{\sum _{i=1}^{N}{U}_{i}(\widehat{\beta},\widehat{\stackrel{~}{\alpha}}){U}_{i}{(\widehat{\beta},\widehat{\stackrel{~}{\alpha}})}^{\prime}\right\}{\left\{{U}_{.}^{\ast}(\widehat{\beta},\widehat{\stackrel{~}{\alpha}})\right\}}^{-1},$$

[10]

$$\text{where}\phantom{\rule{thickmathspace}{0ex}}{U}_{.}^{\ast}(\beta ,\stackrel{~}{\alpha})=\sum _{i=1}^{N}\left(\begin{array}{cc}\hfill {\mathbf{D}}_{i}^{\prime}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathbf{F}}_{i}^{\prime}\hfill \end{array}\right)\left(\begin{array}{cc}\hfill {\mathbf{V}}_{i}^{-1}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \mathbf{I}\hfill \end{array}\right)\left(\begin{array}{cc}\hfill {\mathbf{D}}_{i}\hfill & \hfill 0\hfill \\ \hfill \partial {\sigma}_{i}\u2215\partial {\beta}^{\prime}\hfill & \hfill \partial {\sigma}_{i}\u2215\partial \stackrel{~}{\alpha}\hfill \end{array}\right).$$

We employ a Wald chi-square statistic to test whether the marker considered has an effect on either the continuous phenotype or the binary trait. With the assumption that the distribution of genotypes at the SNP locus is in Hardy-Weinberg equilibrium, the EGEEs for individual *i* can include the allele-based index variable *x _{i}*

$$\begin{array}{cc}\hfill {\chi}_{\mathit{SNP}-\mathit{EGEE}}^{2}=& ({\widehat{\beta}}_{m,1},{\widehat{\beta}}_{m,2})\mathrm{var}{({\widehat{\beta}}_{m,1},{\widehat{\beta}}_{m,2})}^{-1}{({\widehat{\beta}}_{m,1},{\widehat{\beta}}_{m,2})}^{\prime}\hfill \\ \hfill =& ({\widehat{\beta}}_{m,1},{\widehat{\beta}}_{m,2}){\left(\begin{array}{cc}\hfill \mathrm{Var}\left({\widehat{\beta}}_{m,1}\right)\hfill & \hfill \mathrm{Cov}({\widehat{\beta}}_{m,1},{\widehat{\beta}}_{m,2})\hfill \\ \hfill \mathrm{Cov}({\widehat{\beta}}_{m,1},{\widehat{\beta}}_{m,2})\hfill & \hfill \mathrm{Var}\left({\widehat{\beta}}_{m,2}\right)\hfill \end{array}\right)}^{-1}{({\widehat{\beta}}_{m,1},{\widehat{\beta}}_{m,2})}^{\prime}\sim {\xi}_{2}^{2},\hfill \end{array}$$

[11]

where ${\widehat{\beta}}_{m,1}$ and ${\widehat{\beta}}_{m,2}$ are the estimates of *β _{m}*

Given a significant threshold such as *α* = 0.05, if the calculated *p*-value for ${\chi}_{\mathit{SNP}-\mathit{EGEE}}^{2}$ is lower than *α*, we can declare that the SNP locus affects either or both of the two phenotypes. It is worth noting that we mainly focus on identifying the potential SNP effects on the traits through constructing the statistic ${\chi}_{\mathit{SNP}-\mathit{EGEE}}^{2}$ based on the estimates of regression parameters (*i.e.*, ${\widehat{\beta}}_{m,1}$ and ${\widehat{\beta}}_{m,2}$). A similar statistic can be easily developed in the same way to test the significance of the association parameter *ζ* if one is substantially interested in further exploring the correlation between two traits.

For comparison with univariate analyses, we separately detect SNP-phenotype associations based on traditional linear regression analysis (for *T*_{1}) and logistic regression analysis (for *T*_{2}). The respective *p*-values are denoted as *p*_{1} and *p*_{2}. Under this situation, we calculated the adjusted significance level (denoted as *α _{c}*) for multiple testing by two different methods, Bonferroni correction and M

To evaluate the performance of bivariate association analyses based on our developed EGEE method, we conduct extensive simulations for joint analyses of a range of mixtures of one continuous trait and one binary trait in different scenarios.

Under an additive model, we assume a biallelic locus under consideration has effects on both traits, *i.e.*, *T*_{1} with normal distribution and *T*_{2} in a binary pattern, within a random population. Firstly, for ease of describing the genetic effect on the binary phenotype, we adopt the penetrance function *f _{i}* as used in Liu

- Two alleles are sampled and paired to form a random individual according to the liability allele frequency
*q*. Following the theory of quantitative genetics [Falconer and Mackay 1996], under an additive model, the simulated genotype effect on*T*_{1}can be expressed aswhere$${a}_{1,j}=\{\begin{array}{cc}\hfill -\sqrt{\frac{{h}_{1}^{2}}{2q(1-q)(1-{h}_{1}^{2})}}\hfill & \hfill (j=0)\hfill \\ \hfill 0\hfill & \hfill (j=1),\hfill \\ \hfill \sqrt{\frac{{h}_{1}^{2}}{2q(1-q)(1-{h}_{1}^{2})}}\hfill & \hfill (j=2)\hfill \end{array}\phantom{\}}$$[12]*j*is the number of liability alleles assigned to the simulated subject. - We assume the locus simulated above also underlies a latent normal variable
*z*_{2}with mean*a*_{2, }*j*and variance ${\sigma}_{e,2}^{2}=1$, and the residual error correlation between*T*_{1}and*z*_{2}is set to*r*.The random variable*z*_{2}can be truncated to generate the binary trait*T*_{2}using a threshold of*t*. Based on our earlier study [Deng, et al. 2000],*a*_{2,j}and the threshold*t*can be determined via the aforementioned simulation parameters, including heterozygote*GRR*and the penetrance function*f*. To be concrete, we provide a detailed description in the APPENDIX section on how to bridge the gap between normal variable_{0}*z*_{2}and binary data*T*_{2}through heterozygote*GRR*and*f*_{0}as well as the threshold*t*. - For the random individual
*i*, we draw the related variables*T*_{1}and*z*_{2}conditionally on its genotype**G**_{j}from the bivariate normal distribution $\left(\begin{array}{c}\hfill {T}_{1}\hfill \\ \hfill {z}_{2}\hfill \end{array}\right)\sim \mathrm{N}\left(\left(\begin{array}{c}\hfill {a}_{1,j}\hfill \\ \hfill {a}_{2,j}\hfill \end{array}\right),\phantom{\rule{1em}{0ex}}\left(\begin{array}{cc}\hfill 1\hfill & \hfill r\hfill \\ \hfill r\hfill & \hfill 1\hfill \end{array}\right)\right)$. Accordingly, the respective binary phenotype*T*_{2}is truncated to a case (*z*_{2}≥*t*) or a control (*z*_{2}<*t*). - Through steps (1) ~ (3), we can obtain a random individual with a specific marker genotype as well as both the normal and binary phenotypic observations with correlation coefficient
*r*. To ensure the ratio of cases and controls involved in the binary data*T*_{2}, we repeat these processes till 0.5*N*cases and 0.5*N*controls are formed.

Under each of the specific simulation scenarios, we generate 20,000 independent samples for obtaining the robust estimates of statistical power and false-positive rates by the EGEE-based bivariate association analyses as well as by traditional univariate association analyses. For both bivariate and univariate analyses, we define the identical null hypothesis **H**_{0} as: the SNP marker has no association with either *T*_{1} or *T*_{2}, *i.e.*, ${h}_{1}^{2}=0$ and heterozygote *GRR* =1. The corresponding alternative hypothesis **H**_{a} is designed as: the SNP maker is associated with either or both of the two traits, *i.e.*, ${h}_{1}^{2}>0$ or/and heterozygote *GRR* >1. Accordingly, the power and the false-positive rate can be calculated as the proportions of significant SNP/phenotype associations reported in 20,000 independent replicates with the true **H**_{a} and **H**_{0}, respectively.

According to the combinations of various parameter settings for simulations, we perform extensive simulations involving 160 scenarios under the hypothesis **H**_{a} as well as 20 scenarios under the hypothesis **H**_{0}, defined above. To study the performance of detecting liability alleles based on the bivariate test, we conduct three association tests for each set of simulated phenotypes employing the EGEE-based bivariate analytical method (denoted as **B_EGEE**) and separate univariate analyses adjusted for multiple testing by the Bonferroni correction (denoted as **U_b**) and the M_{eff} method [Nyholt 2004] (denoted as **U_e**) . For all tests, we set the experiment-wise significance threshold *α* = 0.05.

Tables Tables11 and and22 present the power estimates of gene-phenotype associations with sample sizes of 150 and 300, respectively, for both bivariate and univariate association analytical methods. Under each setting, we highlight the maximal power in bold to emphasize the best performance among the three tests. It is obvious that the largest increases in power are consistently observed with **B_EGEE,** with the exception of only 3 data sets. This shows that the EGEE-based bivariate association test is highly preferable to the individual univariate tests adjusted either by the Bonferroni correction or by the M_{eff} method.

Power estimates of association tests for the mixture of the continuous and binary traits with the use of EGEE-based bivariate analyses and univariate analyses under the significance level *α* = 0.05 and the sample size *N* = 150.

Power estimates of association tests for the mixture of the continuous and binary traits with the use of EGEE-based bivariate analyses and univariate analyses under the significance level *α* = 0.05 and the sample size *N* = 300.

Comparing the power of **B_EGEE** across different levels of correlation coefficient *r* between the two traits, we can see that the performance of bivariate analyses varies along with *r* in the wide range of situations investigated. The effects of residual error correlation on the performance of **B_EGEE** are twofold. On one hand, for the situation in which only one phenotype, either *T*_{1} or *T*_{2}, is associated with the locus, the estimated power of **B_EGEE** has a trend of increasing with a rising absolute value of *r* from 0 to 0.45, regardless of its direction. Note that we set *r* > 0 or *r* < 0 herein, corresponding to the situations where it has either the identical or opposite direction, respectively, as that of correlation induced by the simulated pleiotropic QTL. On the other hand, under the scenario in which both phenotypes are associated with the gene considered, the performance of **B_GEE** is obviously influenced by both degree and direction of correlation *r*. Specifically, when the residual error correlation is opposite in sign to the correlation induced by QTL effects, **B_EGEE** has an increasing gain in power as the degree of *r* increases; otherwise, the power of **B_EGEE** decreases with the increase of *r*. The above trend of the power of **B_EGEE** is repeated over various design combinations of liability allele frequency and effect of QTL as well as sample size.

In Tables Tables11 and and2,2, comparing the performance of univariate tests with the M_{eff} method and the Bonferroni correction, we can clearly see that the former consistently outperforms the latter across all those simulation scenarios with *r* > 0. Furthermore, the higher degree *r* has, the more obvious the advantage of M_{eff}. Since M_{eff} may take non-independence between correlated phenotypes into consideration, it reasonably overcomes the conservativeness of the Bonferroni correction while having the ease of computation.

Results from Tables Tables11 and and22 further demonstrate that the highest power levels for both bivariate and univariate association analyses are achieved under the combinational design of the higher liability allele frequency at *q* = 0.15, a larger sample size at *N* = 300 and the highest QTL effects on both phenotypes. This trend logically complies with the routine power profiles in genetic association studies.

Table 3 offers the results of false-positive rate calculations for the three tests, *i.e.*, **B_EGEE**, **U_b** and **U_e**, under various design combinations at the significance level of 0.05. Overall, all estimated levels of false positive rates for both bivariate and univariate analyses are close to the threshold value of 0.05. This aspect further suggests that the increased power in **B_EGEE** is due primarily to the fact that it considers all phenotype data within a unified test framework rather than focusing on information of each single phenotype separately.

False-positive rate estimates of association tests for the mixture of the continuous and binary traits with the use of EGEE-based bivariate analyses and univariate analyses across various simulation scenarios under the significance level *α* = 0.05. **...**

Table 3 also shows that **B_EGEE** has a more stringent false-positive rate control for lower levels of both allele frequency *r* and sample size *N*. This result demonstrates that the convergence aspect of **B_EGEE** estimates toward their asymptotic distribution is affected, to some extent, by the sample characteristic mentioned above. For the two univariate analyses, it is obvious that **U_b** is slightly more conservative than **U_e** (when r≠0) since the former ignores the correlation between tests of the two phenotypes.

We perform real data analyses using our available GWA datasets for osteoporosis and obesity. Osteoporosis is a major public health problem characterized by low bone mineral density (BMD) and increased risk of fracture. Obesity is a condition of extra body weight defined as body mass index (BMI, weight/height^{2}) ≥30 kg/m^{2}. The relationship between osteoporosis and obesity has been widely studied, with conflicting results [Guney, et al. 2003; Radak 2004; Zhao, et al. 2008; Zhao, et al. 2007]. We recently found that osteoporosis and obesity share common genetic factors that may have inverse effects on BMD and BMI. To date, we are not aware of any studies investigating the relationship between obesity and low-trauma osteoporotic fractures (OF), the endpoint of osteoporosis.

Our GWA data are obtained using the Affymetrix 500K arrays on 700 Chinese Han subjects (350 with osteoporotic hip fractures and 350 controls) (Y. Guo, et. al., unpublished data). Among these, 598 also have BMI data.

To further validate the proposed method by assessing its empirical performance, we perform bivariate GWA using **B_EGEE** on the 598 subjects for which information is available for both OF and BMI. Before the association test, a suite of quality control procedures are performed. First, for the initial full-set of 500,568 SNPs, missing genotypes at each locus are imputed using HelixTree 5.3.1 (Golden Helix, Bozeman, MT), based on Expectation-Maximization (EM) algorithm. Second, we discard SNPs with a call rate < 95% (10,467 SNPs), those deviating from Hardy-Weinberg equilibrium (HWE) (*P*<0.001, 21,274 SNPs), and those having a minor allele frequency (MAF) < 0.05 in the total sample (141,945 SNPs). Eventually, 326,882 SNPs are available for subsequent analyses. At each locus, the allele-based association test is performed, and age, sex and age^{2} are included as covariates to adjust their respective effects on raw observations in the models for the **B_EGEE** bivariate analyses as well as respective univariate analyses.

We test whether either of the two phenotypes is associated with SNPs across the whole genome. For ease of presentation, we only select the top 100 SNPs with the strongest association signals, *i.e.*, the lowest *p*-values, from the **E_GEE**-based bivariate analyses. The profile of the *p*-values (in -*log* scale) is plotted in **Fig.** 1. For comparison, for these 100 SNPs, the respective raw *p*-values of univariate analyses and the *p*-values adjusted by Bonferroni correction for multiple traits are also presented in **Figs. 1a** and **1b** respectively. We also scrutinize the top 100 most significant SNPs based on the adjusted *p*-values of two univariate analyses. We find that there is a high overlap between these two sets of top 100 SNPs (89 out of 100), suggesting that the comparisons between the two methods are reasonable in terms of the top 100 SNPs chosen by **B_EGEE**.

Fig. 1a depicts the *p*-value profiles of three different analyses, *i.e.*, two separate traditional univariate analyses for the continuous trait BMI and the binary trait OF and **EGEE**-based bivariate analyses for the mixture of the both traits. It can be seen that the majority of the SNPs (66 out of 100) show the strongest association signals in **B_EGEE** analyses even compared with the raw *p*-values in univariate analyses. For the remaining SNPs (34 out of 100), the *p*-values of **B_EGEE** fall between the two raw *p*-values of the univariate analyses, but are closer to the smaller and more significant ones. The trend from Fig. 1a suggests that bivariate association analyses have practical significance. Specifically, when two or more traits are involved in GWA, we may experience the situations shown in Fig. 1a, *i.e.*, some causal variants with modest effects may show moderate signals (e.g., *p* = 1E-3) in univariate analyses while showing much stronger signals (such as *p* = 1E-7) in bivariate analyses. If only univariate associations are performed, these signals will probably not reach genome-wide significance. However, if a bivariate GWA screen is performed initially, some of these loci may achieve a genome-wide significance level. In the second stage, using separate univariate analyses to further confirm the association identified in bivariate GWA, we only need to adjust raw *p*-values of each univariate analysis for the number of phenotypes and specific handful markers screened out in the initial bivariate analyses. Therefore, those SNPs with moderate signals in univariate analyses may remain significant as the issue of multiple testing is minimized.

Fig. 1b further provides comparisons between the *p*-values of **E_GEE** statistic in bivariate analyses and the corresponding adjusted *p*-values for multiple testing of two phenotypes in univariate analyses. It shows that, with the exception of three loci, the *p*-value of **B_EGEE** statistic at each SNP locus is apparently lower than the corresponding adjusted *p*-value of univariate analyses. This further supports the conclusion that the performance of **B_EGEE** proposed herein greatly outperforms univariate analyses in studies in which multiple phenotypes are involved and multiple testing is adjusted for the number of phenotypes involved.

In addition, we selected 10 most significant SNPs based on *p*-values of **B_EGEE** statistic for our bivariate GWA data in Table 4. These identified SNPs could be treated as the potential genetic factors which are associated with the two traits investigated, i.e., BMI and OF. Due to the relatively small sample size of our GWA data, all of the association signals in our analysis did not reach the genome-wide significance threshold, such as 4.2×10^{-7} proposed by Lencz, *et al*. [2007]. However, our preliminary findings herein may offer evidence for further follow-up genetic studies on osteoporosis and obesity.

In this study, we have developed a bivariate association analytical method based on EGEE for GWA studies. To the best of our knowledge, this is the first study to develop specific methods designed for investigating the performance of bivariate genetic association analyses for population-based GWA data. The extensive simulation experiments and GWA real data analyses consistently demonstrated the advantages of our method over univariate analyses.

Here we consider a complex but common situation where both continuous and discrete phenotypic observations are available, for which, however, no existing analytical methods or public software are readily available. In this study, using a SUR model, we incorporated two GLMs, corresponding to different phenotypic distributions, into the same set of estimating equations; this was followed by parameter estimation and statistical testing based on the EGEE method. As pointed out by Hall and colleagues [Hall 2001; Hall and Severini 1998], EGEE generally performs better in estimation of both regression and association parameters compared to either GEE1 or GEE2 approaches. However, the original EGEE was developed for analyzing multiple related phenotypes with the same type of distributions that have the same dispersion parameter, such as longitudinal data obtained in clinical trials. The novelty of our modified EGEE is that it relaxes the constraint of the original EGEE and can handle response variables having different distributions or dispersion parameters. Hence, our modified EGEE method is well suited for bivariate association analyses on the mixture of a normal distributed continuous trait and a binary trait.

The advantages of multivariate analyses over univariate analyses may lie in two general aspects: (1) Multivariate analyses can incorporate information of correlation between traits into a single test statistic, which renders a gain in power. (2) Multivariate analyses can avoid or alleviate the issue of multiple testing. This becomes particularly important in circumstances in which the traits under investigation can be divided into several subgroups [Lange, et al. 2003]. Another similar situation is that statistical significance (especially when modest) of a potential causal locus may diminish if adjusted for the number of loci in a univariate GWA. However, it is likely that this locus will remain significant if genome-wide multivariate association analysis is performed and the genome-wide significance can be achieved, as described in the section of ANALYSES OF EMPIRICAL GWA DATA.

Two important aspects of this study worth further emphasis. First, in the circumstance in which both phenotypes are affected by the locus, the power of bivariate analyses is affected by both degree and direction of the residual correlation *r*; a larger power increase is always favored by the residual correlation *r* with opposite direction to the QTL-induced correlation. This is in agreement with studies of Jiang and Zeng [1995] and Allison *et al* [1998] for bivariate linkage studies. Further theoretical interpretations on our finding herein can trace to a recently report [Evans 2002], which pinpointed why the sign of corrections between variables affect the power of bivariate analyses. The rationale lies in that the opposite sign between QTL-induced and residual correlations can result in a corresponding positive term involved in the expression of non-centrality parameter (NCP), increasing the magnitude of NCP and the power; otherwise a negative term of NCP arises, decreasing the NCP and the power. As a result, the opposite direction of the correlations always contributes to the greater increase in power in contrast to the correlations with the same direction. Although the above interpretations are originally for bivariate linkage analyses, it should work in the similar way in association studies from statistical perspective. Second, even in the circumstance in which two phenotypes do not share genetic determinations at the locus investigated or residual effects, bivariate analyses still have obvious advantages over univariate analyses across various simulated scenarios due to alleviating the issue of multiple testing. This has also been observed in multivariate family-based association studies [Lange, et al. 2003]. These two features make the **B_EGEE** procedure an ideal screening tool for detecting gene-phenotype association no matter whether the phenotypic correlation between traits exists or not.

In **B_EGEE** method, we employed a Fisher scoring algorithm to estimate the unknown parameters due to nonlinear property of the equation system. The program implementing **B_EGEE** method is written in FORTRAN language, which is available on our website http://l.web.umkc.edu/liujian/. A potential concern is that our method may entail prohibitorily high computational demand for conducting the iterative calculation. To illustrate that the proposed method is computationally practical for GWA studies, we assessed the CPU time required by the program in simulation and empirical GWA data analyses. All the analyses were carried out on a computer with Intel® Pentium® 4 3.4GHz dual processors and 2.0GB RAM. It took 55hrs 16mins to complete simulation analyses for all 180 simulated scenarios and 14hrs 22mins for GWA data analyses. This indicates that the computation time required for simulation and empirical data analyses is acceptable, and thus our method is practical for bivariate analyses in the field of candidate gene or GWA studies.

For simplicity, we only conducted bivariate analyses for two correlated traits, with the purpose of demonstrating the advantages of **B_EGEE** over univariate analyses. It should be noted that our method can be easily extended to the situation of multiple correlated traits (>2) with more complex distributions, *e.g.*, Poisson distribution, Gamma distribution, etc. The basic idea of this extension is rather straightforward. One only needs to set up the respective GLM for each phenotypic variable via a specific link function according to its distribution feature [McCullagh and Nelder 1989]. Similar to **B_EGEE**, these GLMs then can be combined into a unified framework through the SUR model.

Another potential extension of **B_EGEE** is to incorporate haplotype information across multiple loci in the models. It has been shown that the haplotype-based strategy may be preferable in some situations [Akey, et al. 2001; Liu, et al. 2007b]. To test phenotype-haplotype association for multiple traits, the substantial modification needed for **B_EGEE** is to construct a new design matrix **X**_{i} which includes the index variables corresponding to the two haplotypes carried by individual *i*. For uncertainty of haplotypes from phase-unknown data, we may borrow the idea of haplotype trend regression [Zaykin, et al. 2002] in univariate analysis. We may incorporate the posterior probabilities of all uncertainties into the haplotype design matrix rather than most likely haplotype configurations. This will help to avoid a loss of information and potential bias in subsequent analyses [Liu, et al. 2007b].

As verified in simulation studies, Bonferroni correction is more conservative than the M_{eff} method when performing multiple testing corrections. Even so, in univariate analyses of real GWA data, we did not present the M_{eff}-adjusted *p*_values for multiple testing of the number of traits involved. The main reasons are as follows. On one hand, the M_{eff} method requires the correlation coefficients among phenotypic variables to estimate an effective number of independent variables [Nyholt 2004]. However, it is unfeasible to calculate the correlation between the continuous observation and the binary disease status using the traditional approach for our real data set, although the correlation coefficient is directly given as a parameter setting in the simulation studies; On the other hand, we borrowed the estimate of the association parameter *ζ* obtained in **B_EGEE** as an alternative of correlation coefficient between two phenotypic variables. At 100 most significant loci, the estimates of *ζ* vary from 0.016 through 0.023. Given these estimates of *ζ*, we implemented the M_{eff} method to estimate the effective number of independent tests, and the results range within the interval [1.9995, 1.9997]. This means the number of independent phenotypes is quite close to the true number of phenotype. This suggests that M_{eff}-adjusted *p*_values are almost the same as those by Bonferroni correction. Hence, to simplify our presentation, we just presented *p*-value comparisons between the proposed **B_EGEE** test and the adjusted univariate analyses using Bonferroni correction rather than using the M_{eff} method in Fig. 1b.

For population-based association studies, population stratification may cause false positive/negative results [Weiss, et al. 2001]. In both simulation and real data analyses conducted herein using **B_EGEE**, we did not consider or control population stratification. However, we point out that this can be done using the methods developed and commonly used for univariate analysis, including Structure [Pritchard, et al. 2000] and Principal Component methods [Price, et al. 2006]. For example, we can incorporate estimates of individual ancestry [Price, et al. 2006; Pritchard, et al. 2000] as covariates in each GLM. It has been shown that use of individual ancestry as a covariate is a promising way for adjusting the effects due to population stratification at most marker loci except those with very strong differentiation even among closely related populations and ethnic groups [Chen and Abecasis 2007]. In univariate association analysis, another way to control population stratification is to adjust test statistics using the genomic control method [Roeder, et al. 2006]; however, the direct use of this method for multiple dependent variables is not straightforward. We will explore the feasibility of extending the genomic control approach to multivariate analyses in further endeavors.

We are grateful to the anonymous reviewers for their insightful comments and constructive suggestions that greatly improved our manuscript. We thank Dr. Yongjun Liu for his help in revising our earlier manuscript and valuable discussions which led to improvements in the paper. Investigators of this work were partially supported by grants from NIH (R01 AR050496-01, R21 AG027110, R01 AG026564, and P50 AR055081). The study also benefited from grants from National Science Foundation of China, Huo Ying Dong Education Foundation, HuNan Province, Xi’an Jiaotong University, and the Ministry of Education of China.

Assume a SNP locus with two alleles: A_{1} and A_{2}, where A_{2} is the risk allele for the binary trait *T*_{2}. To keep consistent with previous description, we adopt the same symbols *a*_{2,j} (*j* = 0, 1, 2) as before to denote the genotype values corresponding to three genotypes A_{1}A_{1}, A_{1}A_{2} and A_{2}A_{2}. Under the additive model, we have

$${a}_{2,1}=0\phantom{\rule{thickmathspace}{0ex}}\text{and}\phantom{\rule{thickmathspace}{0ex}}{a}_{2,0}=-{a}_{2,2}.$$

[A1]

For the latent normal variable *z*_{2}, we assume it follows a mixture of normal distribution *z*_{2} ~ N (*a*_{2,j}, 1) (*j* = 0, 1, 2 ) conditionally on the genotype carried by each individual. Accordingly, we have the following expressions according to the theory of threshold theory:

$${f}_{0}=\mathrm{Pr}({T}_{2}=1\mid {\mathrm{A}}_{1}{\mathrm{A}}_{1})=\mathrm{Pr}({z}_{2}>t\mid {\mathrm{A}}_{1}{\mathrm{A}}_{1})=\Phi ({a}_{2,0}-t),$$

[A2]

$${f}_{1}={f}_{0}\cdot \mathit{GRR}=\mathrm{Pr}({T}_{2}=1\mid {\mathrm{A}}_{1}{\mathrm{A}}_{2})=\mathrm{Pr}({z}_{2}>t\mid {\mathrm{A}}_{1}{\mathrm{A}}_{2})=\Phi ({a}_{2,1}-t)=\Phi (-t),$$

[A3]

$${f}_{2}=\mathrm{Pr}({T}_{2}=1\mid {\mathrm{A}}_{2}{\mathrm{A}}_{2})=\mathrm{Pr}({z}_{2}>t\mid {\mathrm{A}}_{2}{\mathrm{A}}_{2})=\Phi ({a}_{2,2}-t),$$

[A4]

where, Φ(•) is the cumulative distribution function of a standard normal variable.

Given the value of penetrance function *f*_{0} for the genotype A_{1}A_{1} and heterozygote *GRR*, the status of the binary variable *T*_{2} can be determined conditionally by the genotype of each individual through the respective penetrance function *f _{i}* based on the above Equations A1~A4.

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