Efficiency of the maximum partial likelihood estimator for nested case control sampling

TL;DR

This paper investigates the efficiency of the maximum partial likelihood estimator (MPLE) in nested case-control sampling within Cox models, showing it approaches optimal efficiency under certain stratified and covariate independence conditions as sample size grows.

Contribution

It demonstrates that MPLE and Breslow estimator attain the information lower bound in large samples under specific sampling and covariate independence scenarios.

Findings

MPLE approaches efficiency in highly stratified situations.

Both estimators achieve the information lower bound asymptotically.

Efficiency gains occur when covariate dependence on past is minimal.

Abstract

In making inference on the relation between failure and exposure histories in the Cox semiparametric model, the maximum partial likelihood estimator (MPLE) of the finite dimensional odds parameter, and the Breslow estimator of the baseline survival function, are known to achieve full efficiency when data is available for all time on all cohort members, even when the covariates are time dependent. When cohort sizes become too large for the collection of complete data, sampling schemes such as nested case control sampling must be used and, under various models, there exist estimators based on the same information as the MPLE having smaller asymptotic variance. Though the MPLE is therefore not efficient under sampling in general, it approaches efficiency in highly stratified situations, or instances where the covariate values are increasingly less dependent upon the past, when the covariate distribution, not depending on the real parameter of interest, is unknown and there is no censoring. In particular, in such situations, when using the nested case control sampling design, both the MPLE and the Breslow estimator of the baseline survival function achieve the information lower bound both in the distributional and the minimax senses in the limit as the number of cohort members tends to infinity.