Joint Regression Analysis of Correlated Data Using Gaussian Copulas

Abstract

This article concerns a new joint modeling approach for correlated data analysis. Utilizing Gaussian copulas, we present a unified and flexible machinery to integrate separate one-dimensional generalized linear models (GLMs) into a joint regression analysis of continuous, discrete, and mixed correlated outcomes. This essentially leads to a multivariate analogue of the univariate GLM theory and hence an efficiency gain in the estimation of regression coefficients. The availability of joint probability models enables us to develop a full maximum likelihood inference. Numerical illustrations are focused on regression models for discrete correlated data, including multidimensional logistic regression models and a joint model for mixed normal and binary outcomes. In the simulation studies, the proposed copula-based joint model is compared to the popular generalized estimating equations, which is a moment-based estimating equation method to join univariate GLMs. Two real-world data examples are used in the illustration.

Goodness-of-fit test for specification of semiparametric copula dependence models

“…All of them have been investigated extensively in a vast literatures; see for example, Song (2000), Chen and Fan (2005), Cossin and Schellhorn (2007), Song et al (2009) and Genest et al (2009), just to name a few. The former two copulas are prominent examples of the elliptical families and the latter two are mostly used Archimedean copulas.…”

Discrete Longitudinal Data Modeling with a Mean-Correlation Regression Approach

“…An appealing approach for modeling correlated discrete longitudinal variables is the copula construction (Song, et al, 2009). Sklar's theorem ensures that a multivariate distribution can be determined jointly by the univariate marginal distributions and a copula, a multivariate function of these marginals responsible for dependence.…”

“…With these notations, we intend to develop models that can handle general unbalanced longitudinal data. Existing methods, for example, the one in Song, et al (2009) and that in Gaskins, et al (2014), both work on balanced and equally spaced longitudinal data.…”

“…Actually, directly calculating the distribution function of each subject i specified by (2) requires 2 m i summations of lower dimensional distribution functions as in the approach of Song, et al (2009), thus the computational cost grows exponentially with m i .…”

Gaussian Copula Regression in R

“…This paper focuses on Gaussian copula regression method where dependence is conveniently expressed in the familiar form of the correlation matrix of a multivariate Gaussian distribution (Song 2000;Pitt, Chan, and Kohn 2006;Masarotto and Varin 2012). Gaussian copula regression models have been successfully employed in several complex applications arising, for example, in longitudinal data analysis (Frees and Valdez 2008;Sun, Frees, and Rosenberg 2008;Shi and Frees 2011;Song, Li, and Zhang 2013), genetics (Li, Boehnke, Abecasis, and Song 2006;He, Li, Edmondson, Raderand, and Li 2012), mixed data (Song, Li, and Yuan 2009;de Leon and Wu 2011;Wu and de Leon 2014;Jiryaie, Withanage, Wu, and de Leon Well-known limits of the Gaussian copula approach are the impossibility to deal with asymmetric dependence and the lack of tail dependence. These limits may impact the use of Gaussian copulas to model forms of dependence arising, for example, in extreme environmental events or in financial data.…”