Maximum likelihood estimation of logistic regression parameters under two-phase, outcome-dependent sampling

Outcome-dependent sampling increases the efficiency of studies of rare outcomes, examples being case-control studies in epidemiology and choice-based sampling in econometrics. Two-phase or double sampling is a standard technique for drawing efficient stratified samples. We develop maximum likelihood estimation of logistic regression coefficients for a hybrid two-phase, outcome-dependent sampling design. An algorithm is given for determining the estimates by repeated fitting of ordinary logistic regression models. Simulation results demonstrate the efficiency loss associated with alternative pseudolikelihood and weighted likelihood methods for certain data configurations. These results provide an efficient solution to the measurement error problem with validation sampling based on a discrete surrogate.