Semiparametric Regression in Likelihood-Based Models

Abstract A weighted likelihood is used to estimate the parameters in a semiparametric model involving two covariates and allowing an association between the covariates. The development is for arbitrary but specified densities of the observations. The estimators are consistent and asymptotically normal. Hypothesis testing of the parametric component can be performed using a Wald test. Simulations and analysis of data with Bernoulli observations demonstrate the estimators' application. Speckman developed kernel estimators where the conditional density of the observations is normal with p parametric covariates. Speckman's estimators and the new estimators are asymptotically equivalent, with the bias of Speckman's estimators being smaller. As an example, we study the relationship between a binary response indicating the occurrence of an intraoperative cardiac complication (ICC) in vascular surgery patients and two risk factors: duration of the operation (OR) and ASA score, which is an evaluation of the patient's overall health prior to surgery. ASA score is modeled in the parametric portion, because it appears valid to assume that ASA is linearly related to the logit of the probability of an ICC. The functional relationship between OR duration and the logit of the probability of an ICC is unknown, so it is modeled nonparametrically.