Comparative Causal Effect Estimation and Robust Variance for Longitudinal Data Structures with Applications to Observational HIV Treatment

This dissertation discusses the application and comparative performance of double robust estimators for estimating the intervention specific mean outcome in longitudinal settings with time-dependent confounding as well as the corresponding estimator variances. Specifically, we focus on carefully defining target causal parameters to avoid known positivity issues, estimating these parameters using the asymptotically efficient and double robust targeted minimum loss-based estimation, comparing this to other double robust estimators of the same causal parameter, and estimating the corresponding variances in a way which demonstrates valid Type I errors while retaining statistical power. Chapter 1 begins by introducing the open problem in statistics. We present the International epidemiologic Databases to Evaluate AIDS, East Africa region (IeDEA-EA) cohort and the implementation of a low risk express care program implemented between 2007-2009. We continue in Chapter 2 by presenting the targeted learning road map for causal inference. This road map is applied, as a case study, to the IeDEA-EA cohort in evaluating the impact of the low risk express care program. Targeted minimum loss-based estimation is used to estimate the intervention specific mean outcome using data adaptive machine learning candidate estimators for the nuisance parameters. Practical issues are addressed, including carefully defining the causal parameters (and the corresponding causal contrasts) and remaining within the boundaries implied by the statistical model while using the machine learning algorithms. In Chapter 3, we compare additional estimators for the intervention specific mean outcome. The iterated conditional expectation estimator, inverse probability weighted estimator, augmented inverse probability weighted estimator, double robust iterated conditional expectation estimator, and targeted minimum loss-based estimator are presented. Additionally, variations on the double robust iterated conditional expectation estimator and targeted minimum loss-based estimator are reviewed and implemented. Simulations are conducted to analyze the finite sample performance of each estimator, in both correct and mis-specified models. The estimators are also applied to estimating the impact of enrollment into the low risk express care program in the IeDEA-EA cohort. Chapter 4 studies the estimation of estimator variance for estimators solving the efficient influence function. A robust approach of estimating the efficient influence function variance is presented, followed by approaches for estimating the derived expectation of the variance. This robust approach of estimating the EIF variance can be used to raise a red flag for unreliable statistical inference due to sparsity, thereby declaring that the target parameter is practically not identifiable from the data, and that the reported variance estimates (though large) will themselves be imprecise. We additionally present a bootstrap approach based on fitting the initial density of the data once, followed by a non-parametric bootstrap of the targeting step. This bootstrap approach can be used to estimate the variance of substitution based estimators solving the efficient influence function. Simulations are conducted, demonstrating the bias, variance, coverage, and statistical power resulting from each of the variance estimators. Standard errors and confidence intervals are calculated using each of the variance estimators in estimating the impact of enrollment into the low risk express care program in the IeDEA-EA cohort. The primary appendices present relevant proofs for the analyses conducted in this dissertation, while R code for implementing the various estimators are provided in secondary appendices.