Copula‐based regression models for a bivariate mixed discrete and continuous outcome

Abstract

This paper is concerned with regression models for correlated mixed discrete and continuous outcomes constructed using copulas. Our approach entails specifying marginal regression models for the outcomes, and combining them via a copula to form a joint model. Specifically, we propose marginal regression models (e.g. generalized linear models) to link the outcomes' marginal means to covariates. To account for associations between outcomes, we adopt the Gaussian copula to indirectly specify their joint distributions. Our approach has two advantages over current methods: one, regression parameters in models for both outcomes are marginally meaningful, and two, the association is 'margin-free', in the sense that it is characterized by the copula alone. By assuming a latent variable framework to describe discrete outcomes, the copula used still uniquely determines the joint distribution. In addition, association measures between outcomes can be interpreted in the usual way. We report results of simulations concerning the bias and efficiency of two likelihood-based estimation methods for the model. Finally, we illustrate the model using data on burn injuries.

Mixed-Outcome Data

“…Another involves the interpretation of the dependence parameter of copula functions, a serious problem for discrete distributions. De Leon and Wu [18] have developed copula-based regression models for mixed outcomes by adopting a latent-variable formulation of the discrete outcomes and using Gaussian copulas to "glue" mixed-outcome marginal regression models. Work on generalizations of their approach to high-dimensional settings with possibility of incorporating random effects would be worthwhile.…”

Surrogacy assessment using principal stratification and a Gaussian copula model

“…The correlation coefficients ρ 00 and ρ 11 are the Pearson correlation coefficients between S̃ (0) and the normally transformed T (0) and between S̃ (1) and the normally transformed T (1), respectively, and can be seen as a proxy for the polyserial correlations between T (0) and S (0) and between T (1) and S (1). 18 These polyserial correlations are estimable from the data. 23 Because only one of the counterfactual pairs of outcomes is observed for each subject, ρ s , ρ t , ρ 01 , and ρ 10 are not identifiable.…”

“…$f_{{\mathbf{y}}_{\mathbf{i}}}(k_0,k_1,t_0,t_1)=\frac{{\partial }^2}{\partial t_0\partial t_1}P(S_i(0)=k_0,S_i(1)=k_1,T_i(0)\le t_0,T_i(1)\le t_1)$, where the derivative of F ỹ i ( α k 0 , α k 1 , t 0 , t 1 ) with respect to t 1 and t 0 is given by: 18 …”

“…The application of a Gaussian copula model to jointly model bivariate discrete and continuous outcomes was examined by de Leon and Wu. 18 Here, we extend the use of this Gaussian copula model to a four dimensional model, two for the potential surrogate marker values under each treatment arm and two for the potential outcomes under each treatment arm. The Gaussian copula model was chosen as opposed to another type of copula because it allows for a multivariate correlation structure with separate coefficients for the six pairwise correlations.…”

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.…”