Copula regression spline models for binary outcomes

We introduce a framework for estimating the effect that a binary treatment has on a binary outcome in the presence of unobserved confounding. The methodology is applied to a case study which uses data from the Medical Expenditure Panel Survey and whose aim is to estimate the effect of private health insurance on health care utilization. Unobserved confounding arises when variables which are associated with both treatment and outcome are not available (in economics this issue is known as endogeneity). Also, treatment and outcome may exhibit a dependence which cannot be modeled using a linear measure of association, and observed confounders may have a non-linear impact on the treatment and outcome variables. The problem of unobserved confounding is addressed using a two-equation structural latent variable framework, where one equation essentially describes a binary outcome as a function of a binary treatment whereas the other equation determines whether the treatment is received. Non-linear dependence between treatment and outcome is dealt using copula functions, whereas covariate-response relationships are flexibly modeled using a spline approach. Related model fitting and inferential procedures are developed, and asymptotic arguments presented

A joint regression modeling framework for analyzing bivariate binary data in R

We discuss some of the features of the R add-on package GJRM which implements a flexible joint modeling framework for fitting a number of multivariate response regression models under various sampling schemes. In particular,we focus on the case inwhich the user wishes to fit bivariate binary regression models in the presence of several forms of selection bias. The framework allows for Gaussian and non-Gaussian dependencies through the use of copulae, and for the association and mean parameters to depend on flexible functions of covariates. We describe some of the methodological details underpinning the bivariate binary models implemented in the package and illustrate them by fitting interpretable models of different complexity on three data-sets

Mixed Binary-Continuous Copula Regression Models with Application to Adverse Birth Outcomes

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 a study examining the risk factors of adverse birth outcomes, we consider mixed binary-continuous responses that extend this framework to the situation where one response variable is discrete (more precisely binary) while 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 modelling jointly gestational age and the presence/absence of low birth weight. The analysis strongly benefits from the flexible specification of regression effects 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

Implicit Copulas from Bayesian Regularized Regression Smoothers

We show how to extract the implicit copula of a response vector from a Bayesian regularized regression smoother with Gaussian disturbances. The copula can be used to compare smoothers that employ different shrinkage priors and function bases. We illustrate with three popular choices of shrinkage priors --- a pairwise prior, the horseshoe prior and a g prior augmented with a point mass as employed for Bayesian variable selection --- and both univariate and multivariate function bases. The implicit copulas are high-dimensional, have flexible dependence structures that are far from that of a Gaussian copula, and are unavailable in closed form. However, we show how they can be evaluated by first constructing a Gaussian copula conditional on the regularization parameters, and then integrating over these. Combined with non-parametric margins the regularized smoothers can be used to model the distribution of non-Gaussian univariate responses conditional on the covariates. Efficient Markov chain Monte Carlo schemes for evaluating the copula are given for this case. Using both simulated and real data, we show how such copula smoothing models can improve the quality of resulting function estimates and predictive distributions

Estimating the Binary Endogenous Effect of Insurance on Doctor Visits by Copula-Based Regression Additive Models

This paper seeks to estimate the causal effect of having health insurance on health care utilization, while accounting for potential endogeneity bias. The topic has impor- tant policy implications, because health insurance reforms implemented in U.S. in recent decades have focused on extending coverage to the previously uninsured. Consequently, understanding the effects of those reforms requires an accurate estimate of the causal effect of insurance on utilization. However, obtaining such an estimate is complicated by the discreteness inherent in common measures of health care usage. This paper presents a flexible estimation approach, based on copula functions, that consistently estimates the coefficient of a binary endogenous regressor in count data settings. The relevant numeri- cal computations can be easily carried out using the freely available GJRM R package. The empirical results find significant evidence of favorable selection into insurance. Ignoring such selection, insurance appears to increase doctor visit usage by 62%, but adjusting for it, the effect increases to 134%

Did the ACA's "guaranteed issue" provision cause adverse selection into nongroup insurance? Analysis using a copula-based hurdle model

Prior to the Affordable Care Act (ACA), insurance companies could charge higher premiums, or outright deny coverage, to people with preexisting health problems. But the ACA's “guaranteed issue” provision forbids such price discrimination and denials of coverage. This paper seeks to determine whether, after implementation of the ACA, nongroup private insurance plans have experienced adverse selection. Our empirical approach employs a copula-based hurdle regression model, with dependence modeled as a function of dimensions along which adverse selection might occur. Our main finding is that, after implementation of the ACA, nongroup insurance enrollees with preexisting health problems do not appear to exhibit adverse selection. This finding suggests that the ACA's mandate that everyone acquire coverage might have attracted enough healthy enrollees to offset any adverse selection

Bivariate copula additive models for location, scale and shape

In generalized additive models for location, scale and shape (GAMLSS), the response distribution is not restricted to belong to the exponential family and all the model’s parameters can be made dependent on additive predictors that allow for several types of covariate effects (such as linear, non-linear, random and spatial effects). In many empirical situations, however, modeling simultaneously two or more responses conditional on some covariates can be of considerable relevance. The scope of GAMLSS is extended by introducing bivariate copula models with continuous margins for the GAMLSS class. The proposed computational tool permits the copula dependence and marginal distribution parameters to be estimated simultaneously, and each parameter to be modeled using an additive predictor. Simultaneous parameter estimation is achieved within a penalized likelihood framework using a trust region algorithm with integrated automatic multiple smoothing parameter selection. The introduced approach allows for straightforward inclusion of potentially any parametric marginal distribution and copula function. The models can be easily used via the copulaReg() function in the R package SemiParBIVProbit. The proposal is illustrated through two case studies and simulated data