Locally Efficient Estimation of a Multivariate Survival Function in Longitudinal Studies

Abstract

We consider estimation of the joint distribution of multivariate survival times T = (T1,…,Tk), which are subject to right-censoring by a common censoring variableC. Two estimators are proposed: an initial inverse-probability-of-censoring weighted (IPCW) estimator and a one-step estimator. Both estimators incorporate information on available time-independent and time-dependent prognostic factor (covariate) data. The IPCW estimator is consistent and asymptotically normal (CAN) under coarsening at random (CAR) and a correct specification of a model for the hazard of censoring given the past covariate and failure data. The one-step estimator is a locally efficient doubly robust estimator. That is, (i) it is CAN under the assumption of CAR and either (but not necessarily both) correct specification of a model for the hazard of censoring given the past or correct specification of a model for the conditional distribution of T given past failure and covariate information, and (ii) it is efficient when both these models are correctly specified. The proposed methodology does not require that the time variables T1,…,Tk be ordered, although our methods cover this important special case. In particular, our estimators can be used to estimate the gap time distributions associated with an ordered series of events. The proposed methodology is an improvement over currently available approaches in a number of ways. Specifically, when censoring and failure are dependent because the hazard of censoring depends on both past failure and covariate history, our one-step estimator is the only estimator with the double-robustness property. When censoring can be assumed to be independent of the failure and covariate processes, our locally efficient one-step estimator, unlike the maximum likelihood estimator (MLE) of van der Laan but like the estimators of Dabrowska, Prentice and Cai, and Bickel, does not require smoothing and so will perform well in moderate size samples even if k is large, say 7 or 8; furthermore, unlike all previous estimators, our estimator exploits the information available in past covariate as well as failure history and so will be efficient (nearly efficient) even when the components of T are highly dependent, whenever the specified model for the conditional distribution of T given past failure and covariate information is correct (nearly correct). We examine the finite sample performance of our estimators in a simulation study. Finally, we apply our estimators to data on time to wound excision and time to wound infection in a population of burn victims.