Locally Efficient Estimation of a Multivariate Survival Function in Longitudinal Studies

We consider estimation of the joint distribution of multivariate survival times ~ T = (T1, . . . , Tk), which are subject to right censoring by a common censoring variable C. Two estimators are proposed: an initial inverse-probability-ofcensoring weighted (IPCW) estimator, and a 1-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 1-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 improves 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 processe, our locally effcient one-step estimator, unlike the MLE of Van der Laan (1996a,b) but like the Dabrowska , Prentice-Cai (1992a, b), and Bickel (Dabrowska,1988) estimators, does not require smoothing and so will perform well in moderate size samples even if k is large, say 7 or 8; further unlike all