Empirical and conditional likelihoods for two‐phase studies

Two‐phase, response‐dependent sampling is often used in regression settings that involve expensive covariate measurements. Conditional maximum likelihood (CML) is an attractive approach in many cases as it avoids modelling of the covariate distribution, unlike full maximum likelihood. Scott & Wild (2011) introduced an augmented CML approach which is semi‐parametric efficient in certain settings with a discrete response variable. We consider general regression models and show the Scott–Wild estimator of covariate effects has the same asymptotic efficiency as two empirical likelihood estimators, and that these estimators dominate the CML estimator. We compare the efficiencies of various estimators in simulation studies and illustrate the methodology in a two‐phase genetics study.