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

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

Penalized likelihood estimation of a trivariate additive probit model

This article proposes a penalized likelihood method to estimate a trivariate probit model, which accounts for several types of covariate effects (such as linear, nonlinear, random, and spatial effects), as well as error correlations. The proposed approach also addresses the difficulty in estimating accurately the correlation coefficients, which characterize the dependence of binary responses conditional on covariates. The parameters of the model are estimated within a penalized likelihood framework based on a carefully structured trust region algorithm with integrated automatic multiple smoothing parameter selection. The relevant numerical computation can be easily carried out using the SemiParTRIV() function in a freely available R package. The proposed method is illustrated through a case study whose aim is to model jointly adverse birth binary outcomes in North Carolina

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

Copula Regression Spline Sample Selection Models: The R Package SemiParSampleSel

Sample selection models deal with the situation in which an outcome of interest is observed for a restricted non-randomly selected sample of the population. The estimation of these models is based on a binary equation, which describes the selection process, and an outcome equation, which is used to examine the substantive question of interest. Classic sample selection models assume a priori that continuous covariates have a linear or pre-specified non-linear relationship to the outcome, and that the distribution linking the two equations is bivariate normal. We introduce the R package SemiParSampleSel which implements copula regression spline sample selection models. The proposed implementation can deal with non-random sample selection, non-linear covariate-response relationships, and non-normal bivariate distributions between the model equations. We provide details of the model and algorithm and describe the implementation in SemiParSampleSel. The package is illustrated using simulated and real data examples

Preliminary results from an application of PTV to bed-load grains

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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

Copula selection models for non-Gaussian responses that are missing not at random

Missing not at random (MNAR) data poses key challenges for statistical inference because the model of interest is typically not identifiable without imposing further (e.g., distributional) assumptions. Sample selection models have been routinely used for handling MNAR by jointly modelling the outcome and selection variables assuming that these follow a bivariate normal distribution. Recent studies have advocated parametric selection model approaches, for example estimated by multiple imputation and maximum likelihood, that are more robust to departures from the normality assumption. However, the proposed methods have been mostly restricted to a specific joint distribution (e.g., bivariate t-distribution). This paper discusses a flexible copula-based selection approach (which accommodates a wide range of non-Gaussian outcome distributions and offers great flexibility in the choice of functional form specifications for both the outcome and selection equations) and proposes a flexible imputation procedure that generates plausible imputed values from the copula selection model. A simulation study characterises the relative performance of the copula model compared with the most commonly used selection models for estimating average treatment effects with MNAR data. We illustrate the methods in the REFLUX study, which evaluates the causal effect of laparoscopic surgery compared to usual medical management on long-term quality of life in patients with reflux disease. We provide software code for implementing the proposed copula framework using the R package GJRM