In many randomized controlled trials the outcome of interest is a time to event, and one measures on each subject baseline covariates and time-dependent covariates until the subject either drops-out, the time to event is observed, or the end of study is reached. The goal of such a study is to assess the causal effect of the treatment on the survival curve. Standard methods (e.g., Kaplan-Meier estimator, Cox-proportional hazards) ignore the available baseline and time-dependent covariates, and are therefore biased if the drop-out is affected by these covariates, and are always inefficient. We present a targeted maximum likelihood estimator of the causal effect of treatment on survival fully utilizing all the available covariate information, resulting in a double robust locally efficient substitution estimator that will be consistent and asymptotically linear if either the censoring mechanism is consistently estimated, or if the maximum likelihood based estimator is already consistent. In particular, under the independent censoring assumption assumed by current methods, this TMLE is always consistent and asymptotically linear so that it provides valid confidence intervals and tests. Furthermore, we show that when both the censoring mechanism and the initial maximum likelihood based estimator are mis-specified, and thus inconsistent, the TMLE exhibits stability when inverse probability weighted estimators and double robust estimating equation based methods break down The TMLE is used to analyze the Tshepo study, a study designed to evaluate the efficacy, tolerability, and development of drug resistance of six different first-line antiretroviral therapies. Most importantly this paper presents a general algorithm that may be used to create targeted maximum likelihood estimators of a large class of parameters of interest for general longitudinal data structures.
Stitelman, Ori M.; De Gruttola, Victor; and van der Laan, Mark J., "A General Implementation of TMLE for Longitudinal Data Applied to Causal Inference in Survival Analysis" (April 2011). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 281.