Empirical and weighted conditional likelihoods for matched case-control studies with missing covariates

In clinical and epidemiological studies, matched case-control designs have been used extensively to investigate the relationships between disease/response and exposure/covariate. Due to the retrospective nature of the study, some covariates may not be observed for all study subjects and missing covariate information may create bias and reduce the efficiency of the parameter estimates. We explore the use of profile empirical likelihood (EL) to cope with this situation by combining unbiased estimating equations when the number of estimating equations is greater than the number of unknown parameters. For high dimensional covariates, we propose a weighted conditional likelihood (WCL) method to solve the computational problem of the profile EL method. The proposed EL and WCL methods can achieve semiparametric efficiency if the probability of missingness is correctly specified. Based on the EL and WCL functions, we also develop Wilks’ type tests and corresponding confidence regions for the model parameters. A simulation study is conducted to assess the performance of the proposed methods in terms of robustness and efficiency.