Density ratio model for multivariate outcomes

The Density Ratio Model is a semi-parametric regression model which allows analysis of data from any exponential family without making a parametric distribution assumption. For univariate outcomes several authors have shown desirable properties of this model including robustness to mis-specification and efficiency of the estimators within a suitable class. In this paper we consider analysis of multivariate outcomes with this model, where each marginal distribution is from an exponential family. We show that the model successfully analyzes data from mixed outcome types (continuous, integer, binary), providing valid tests of the joint effects of covariates. Furthermore, for continuous outcomes we provide a bootstrap technique which correctly estimates the underlying marginal regression parameters and provides appropriate coverage probabilities without specifying the covariance structure. The methods are demonstrated via simulation studies and analysis of healthcare data.