Targeted Minimum Loss Based Estimation of Causal Effects of Multiple Time Point Interventions

We consider estimation of the effect of a multiple time point intervention on an outcome of interest, where the intervention nodes are subject to time-dependent confounding by intermediate covariates. In previous work van der Laan (2010) and Stitelman and van der Laan (2011a) developed and implemented a closed form targeted maximum likelihood estimator (TMLE) relying on the log-likelihood loss function, and demonstrated important gains relative to inverse probability of treatment weighted estimators and estimating equation based estimators. This TMLE relies on an initial estimator of the entire probability distribution of the longitudinal data structure. To enhance the finite sample performance of the TMLE of the target parameter it is of interest to select the smallest possible relevant part of the data generating distribution, which is estimated and updated by TMLE. Inspired by this goal, we develop a new closed form TMLE of an intervention specific mean outcome based on general longitudinal data structures. The target parameter is represented as an iterative sequence of conditional expectations of the outcome of interest. This collection of conditional means represents the relevant part, which is estimated and updated using the general TMLE algorithm. We also develop this new TMLE for other causal parameters, such as parameters defined by working marginal structural models. The theoretical properties of the TMLE are also practically demonstrated with a small scale simulation study. The proposed TMLE is building upon a previously proposed estimator Bang and Robins (2005) by integrating some of its key and innovative ideas into the TMLE framework.