Semiparametric Estimation Exploiting Covariate Independence in Two‐Phase Randomized Trials

Summary Recent results for case–control sampling suggest when the covariate distribution is constrained by gene‐environment independence, semiparametric estimation exploiting such independence yields a great deal of efficiency gain. We consider the efficient estimation of the treatment–biomarker interaction in two‐phase sampling nested within randomized clinical trials, incorporating the independence between a randomized treatment and the baseline markers. We develop a Newton–Raphson algorithm based on the profile likelihood to compute the semiparametric maximum likelihood estimate (SPMLE). Our algorithm accommodates both continuous phase‐one outcomes and continuous phase‐two biomarkers. The profile information matrix is computed explicitly via numerical differentiation. In certain situations where computing the SPMLE is slow, we propose a maximum estimated likelihood estimator (MELE), which is also capable of incorporating the covariate independence. This estimated likelihood approach uses a one‐step empirical covariate distribution, thus is straightforward to maximize. It offers a closed‐form variance estimate with limited increase in variance relative to the fully efficient SPMLE. Our results suggest exploiting the covariate independence in two‐phase sampling increases the efficiency substantially, particularly for estimating treatment–biomarker interactions.