Methods for the Analysis of Sampled Cohort Data in the Cox Proportional Hazards Model
Annals of Statistics
B. Langholz University of Southern California Methods are provided for regression parameter and cumulative baseline hazard estimation in the Cox model when the cohort is sampled according to a predictable sampling probability law. It is shown how a marked point process representation of cohort sampling naturally leads to the derivation of a partial likelihood which may be used for the estimation of regression parameters. Standard counting process techniques are used to show that this partial
... hat this partial likelihood may be treated as a likelihood in that, at the true parameter, the expectation of the score is sero and the variance ofthe score is the expected information. Generalisations of the Breslow estimator of the cumulative baseline hazard and estimators of the associated variance processes are provided. The results are used to derive partial likelihoods for three new sampling designs, stratified and quota sampling from the risk sets and nested case-control sampling with number of controls dependent on the failure's exposure status, as well as for simple nested casecontrol and case-cohort sampling. Baseline hazard estimators are given for simple and stratified nested case-control sampling. General asymptotic theory is developed for the maximum partial likelihood estimator and cumulative baseline hazard estimator and is used to derive the asymptotic: distributions for estimators from simple and stratified nested case-control sampling. Generalisations to stratified populations or multistate problems and the Aalen linear regression model are given. ?.