Robust estimation in the nested case-control design under a misspecified covariate functional form

The Cox proportional hazards model is typically used to analyze time-to-event data. If the event of interest is rare and covariates are difficult or expensive to collect, the nested case-control (NCC) design provides consistent estimates at reduced costs with minimal impact on precision if the model is specified correctly. If our scientific goal is to conduct inference regarding an association of interest, it is essential that we specify the model a priori to avoid multiple testing bias. We cannot, however, be certain that all assumptions will be satisfied so it is important to consider robustness of the NCC design under model misspecification. In this manuscript, we show that in finite sample settings where the functional form of a covariate of interest is misspecified, the estimates resulting from the partial likelihood estimator under the NCC design depend on the number of controls sampled at each event time. To account for this dependency, we propose an estimator that recovers the results obtained using using the full cohort, where full covariate information is available for all study participants. We present the utility of our estimator using simulation studies and show the theoretical properties. We end by applying our estimator to motivating data from the Alzheimer's Disease Neuroimaging Initiative.