Abstract: There is a substantial literature in case-control logistic regression on whether or not non-confounding covariates should be adjusted for. However, only limited and ad hoc theoretical results are available on this important topic. A constrained maximum likelihood method was recently proposed, which appears to be generally more powerful than logistic regression methods with or without adjusting for non-confounding covariates. This note provides a theoretical clarification for the case-control logistic regression with and without covariate adjustment and the constrained maximum likelihood method on their relative performances in terms of asymptotic relative efficiencies. We show that the benefit of covariate adjustment in the case-control logistic regression depends on the disease prevalence. We also show that the constrained maximum likelihood estimator gives an asymptotically uniformly most powerful test.

Key words and phrases: Asymptotic relative efficiency, case-control design, constrained maximum likelihood, logistic regression, non-confounding covariate.