On estimation in the nested case-control design under nonproportional hazards

Analysis of time-to-event data using Cox's proportional hazards (PH) model is ubiquitous in scientific research. A sample is taken from the population of interest and covariate information is collected on everyone. If the event of interest is rare and covariate information is difficult to collect, the nested case-control (NCC) design reduces costs with minimal impact on inferential precision. Under PH, application of the Cox model to data from a NCC sample provides consistent estimation of the hazard ratio. However, under non-PH, the finite-sample estimates corresponding to the Cox estimator depend on the number of controls sampled and the censoring distribution. We propose two estimators based on a binary predictor of interest: one recovers the estimand corresponding to the Cox model under a simple random sample, while the other recovers an estimand that does not depend on the censoring distribution. We derive the asymptotic distribution and provide finite-sample variance estimators.