Mixed binary‐continuous copula regression models with application to adverse birth outcomes

Abstract

Bivariate copula regression allows for the flexible combination of two arbitrary, continuous marginal distributions with regression effects being placed on potentially all parameters of the resulting bivariate joint response distribution.Motivated by the risk factors for adverse birth outcomes, many of which are dichotomous, we consider mixed binary-continuous responses that extend the bivariate continuous framework to the situation where one response variable is discrete (more precisely, binary) whereas the other response remains continuous. Utilizing the latent continuous representation of binary regression models, we implement a penalized likelihood-based approach for the resulting class of copula regression models and employ it in the context of modeling gestational age and the presence/absence of low birth weight. The analysis demonstrates the advantage of the flexible specification of regression impacts including nonlinear effects of continuous covariates and spatial effects. Our results imply that racial and spatial inequalities in the risk factors for infant mortality are even greater than previously suggested. KEYWORDSadverse birth outcomes, copula, latent variable, mixed discrete-continuous distributions, penalized maximum likelihood, penalized splines Statistics in Medicine. 2019;38:413-436. wileyonlinelibrary.com/journal/sim

Copula geoadditive modelling of anaemia and malaria in young children in Kenya, Malawi, Tanzania and Uganda

“…We propose the use of a bivariate copula regression model to jointly model anaemia and malaria. The model is based on a pair of responses and a copula specification for the dependence structure between the two responses [ 24 ]. Copulas are functions that enable the separation of the marginal distributions from the dependence structure of a given multivariate distribution [ 25 ].…”

“…If Y i 1 and Y i 2 were both continuous, the copula C would be unique. However, in the case of both outcomes being binary, the copula is no longer uniquely defined [ 24 ]. As such, we make use of the latent (unobserved) variable representation of binary models where we define a continuous latent variable , where η ij is the linear predictor consisting of fixed and random effects as well as non-linear and spatial effects, and ε ij is an error term.…”

“…Two districts are considered neighbours if they share a border. More information on the specification and estimation of each marginal model can be found in [ 24 ].…”

Copula modeling of receiver operating characteristic and predictiveness curves

“…Judging by both rugs, it is notable that the largest and the minimum values of the marker in the ER-negative and ER-positive groups, respectively, are considerably far from their respective clusters, suggesting the presence of outliers. Figure 2 shows the bagplot 52 and boxplots of the conditional residuals r D | M,i and r M | D,i obtained from the Gaussian copula model with the mixture of two skewed normal distributions set as the margin for M, as specified by Equation (8). The plots show that the two extreme outliers are noticeable in the set of residuals corresponding to the conditional distribution of the marker given the outcome.…”

“…The correct specification of the joint distribution of the marker and the outcome should lead to the adequate characterization of both curves. In Song 6 and de Leon and Wu, 7 classes of multivariate models for mixed continuous and discrete random variables were constructed by joining prespecified marginal models with a copula, and in Klein et al 8 a specialization was employed in the mixed binary‐continuous responses framework for modeling the presence/absence of low birth weight and gestational age. In the current marker assessment context, the adoption of such classes of copula‐based models is appealing since the construction of the joint model is conveniently formulated by linking a binomial distribution for the outcome and a continuous distribution for the marker, with the copula model encoding the dependence structure between the two random variables, similar to the usual copula setting for constructing families of continuous bivariate distributions, where the copula joins a bivariate distribution function to its continuous univariate margins.…”