Survival analysis under the Cox proportional hazards model with pooled covariates

For a continuous time-to-event outcome and an expensive-to-measure exposure, we develop a pooling design and propose a likelihood-based approach to estimate the hazard ratios (HRs) of a Cox proportional hazards (PH) model. Our proposed approach fits a PH model based on pooled exposures with individually observed time-to-event outcomes. The design and estimation exploits the equivalence of the conditional logistic likelihood functions arising from a matched case-control study and the partial likelihood function of a riskset-matched, nested case-control (NCC) subset of a cohort study. To create the pools, we first focus on an NCC subcohort. Pools are formed at random while keeping the matching intact. Pool-level exposure and confounders are then evaluated and used in the likelihood to estimate the HR and the standard error of the estimates. The estimators are MLEs, provide consistent estimates of the individual-level HRs, and are asymptotically normal. Our simulation results indicate that the pooled estimates are comparable to the estimates obtained from the NCC subcohort. The units of analysis for the pooled design are the pools and not the individual participants. Hence the effective sample size is reduced. Therefore, the variance of the HR estimate increases with increasing poolsize. However, this variance inflation in small samples can be offset by including more matched controls per case within the NCC subcohort. An application is demonstrated with the Second Manifestations of ARTerial disease (SMART) study.