Likelihood Inferences on Semiparametric Odds Ratio Model

A flexible semiparametric odds ratio model has been proposed to unify and to extend both the log-linear model and the joint normal model for data with a mix of discrete and continuous variables. The semiparametric odds ratio model is particularly useful for analyzing biased sampling designs. However, statistical inference of the model has not been systematically studied when more than one nonparametric component is involved in the model. In this article, we study the maximum semiparametric likelihood approach to estimation and inference of the semiparametric odds ratio model. We show that the maximum semiparametric likelihood estimator of the odds ratio parameter is consistent and asymptotically normally distributed. We also establish statistical inference under a misspecified semiparametric odds ratio model, which is important when handling weak identifiability in conditionally specified models under biased sampling designs. We use simulation studies to demonstrate that the proposed approaches have satisfactory finite sample performance. Finally, we illustrate the proposed approach by analyzing multiple traits in a genome-wide association study of high-density lipid protein. Supplementary materials for this article are available online.