Consider estimation of causal parameters in a marginal structural model for the discrete intensity of the treatment specific counting process (e.g. hazard of a treatment specific survival time) based on longitudinal observational data on treatment, covariates and survival. We assume the sequential randomization assumption (SRA) on the treatment assignment mechanism and the so called experimental treatment assignment assumption which is needed to identify the causal parameters from the observed data distribution. Under SRA, the likelihood of the observed data structure factorizes in the auxiliary treatment mechanism and the partial likelihood consisting of the product over time of conditional distributions of covariate and survival at a specific time, given the past. Due to the curse of dimensionality, without assuming lower dimensional models for either the partial likelihood or the treatment mechanism, there exist no regular asymptotically linear estimators with reasonable practical performance (van der Laan and Robins ). In this article, we define three estimators the Inverse Probability of Treatment weighted (IPTW) estimator based on a maximum likelihood estimator of the treatment mechanism according to a model, the maximum likelihood estimator (MLE) based on a maximum likelihood estimator of the partial likleihood according to a model, and a double robust (DL) estimator based on the maximum likelihood estimator of the treatment mechanism and the maximum likelihood estimator of the partial likelihood. The double robust estimator is obtained by following a general methodology for constructing double robust estimating functions in censored data models as described in van der Laan and Robins . We propose specific implementation of this estimator based on Monte-Carlo simulation methods, which makes the estimator computationally tractable and maximally robust. The double-robust estimator is consistent and asymptotically linear when either the treatment mechanism or the partial likelihood of the observed data is consistently estimated. We extend the estimator to handle informative censoring. We illustrate the practical performance of the DR estimator relative to the IPTW and ML estimators in a simulation study. The proposed methodology is also applied to estimate the causal effect of exercise on physical functioning in a longitudinal study of seniors in Sonoma County.
Longitudinal Data Analysis and Time Series | Numerical Analysis and Computation | Statistical Methodology | Statistical Theory | Survival Analysis
Yu, Zhuo and van der Laan, Mark J., "Double Robust Estimation in Longitudinal Marginal Structural Models" (June 2003). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 132.