Search

We present two novel approaches to variance estimation of semi-parametric efficient point estimators of the treatment-specific mean: (i) a robust approach that directly targets the variance of the influence function (IF) as a counterfactual mean outcome and (ii) a modified non-parametric bootstrap-based approach. The performance of these approaches to variance estimation is compared to variance estimation based on the sample variance of the empirical IF in simulations across different levels of positivity violations and treatment effect sizes. In this article, we focus on estimation of the nuisance parameters using correctly specified parametric models for the treatment mechanism in order to highlight the challenges posed by violation of positivity assumptions (distinct from the challenges posed by non-parametric estimation of the nuisance parameters). Results demonstrate that (1) variance estimation based on the empirical IF may provide highly anti-conservative confidence interval coverage (as reported previously), (2) the proposed robust approach to variance estimation in this setting provides conservative coverage, and (3) the proposed modified bootstrap maintains close to nominal coverage and improves power. In the appendix, we (a) generalize the robust approach of estimating variance to marginal structural working models and (b) provide a proof of the consistency of the targeted minimum loss-based estimation bootstrap.

This paper describes a targeted maximum likelihood estimator (TMLE) for the parameters of longitudinal static and dynamic marginal structural models. We consider a longitudinal data structure consisting of baseline covariates, time-dependent intervention nodes, intermediate time-dependent covariates, and a possibly time-dependent outcome. The intervention nodes at each time point can include a binary treatment as well as a right-censoring indicator. Given a class of dynamic or static interventions, a marginal structural model is used to model the mean of the intervention-specific counterfactual outcome as a function of the intervention, time point, and possibly a subset of baseline covariates. Because the true shape of this function is rarely known, the marginal structural model is used as a working model. The causal quantity of interest is defined as the projection of the true function onto this working model. Iterated conditional expectation double robust estimators for marginal structural model parameters were previously proposed by Robins (2000, 2002) and Bang and Robins (2005). Here we build on this work and present a pooled TMLE for the parameters of marginal structural working models. We compare this pooled estimator to a stratified TMLE (Schnitzer et al. 2014) that is based on estimating the intervention-specific mean separately for each intervention of interest. The performance of the pooled TMLE is compared to the performance of the stratified TMLE and the performance of inverse probability weighted (IPW) estimators using simulations. Concepts are illustrated using an example in which the aim is to estimate the causal effect of delayed switch following immunological failure of first line antiretroviral therapy among HIV-infected patients. Data from the International Epidemiological Databases to Evaluate AIDS, Southern Africa are analyzed to investigate this question using both TML and IPW estimators. Our results demonstrate practical advantages of the pooled TMLE over an IPW estimator for working marginal structural models for survival, as well as cases in which the pooled TMLE is superior to its stratified counterpart.