Targeted Minimum Loss-Based Estimation of Causal Effects in Right-Censored Survival Data with Time-Dependent Covariates: Warfarin, Stroke, and Death in Atrial Fibrillation

Abstract

Causal effects in right-censored survival data can be formally defined as the difference in the marginal cumulative event probabilities under particular interventions. Conventional estimators, such as the Kaplan-Meier (KM), fail to consistently estimate these marginal parameters under dependent treatment assignment or dependent censoring. Several modern estimators have been developed that reduce bias under both dependent treatment assignment and dependent censoring by incorporating information from baseline and time-dependent covariates. In the present article we describe a recently developed targeted minimum loss-based estimation (TMLE) algorithm for general longitudinal data structures and present in detail its application in right-censored survival data with time-dependent covariates. The treatment-specific marginal cumulative event probability is defined via a series of iterated conditional expectations in a time-dependent counting process framework. The TMLE involves an initial estimator of each conditional expectation and sequentially updates these such that the resulting estimator solves the efficient influence curve estimating equation in the nonparametric statistical model. We describe the assumptions required for consistent estimation of statistical parameters and additional assumptions required for consistent estimation of the causal effect parameter. Using simulated right-censored survival data, the mean squared error, bias, and 95% confidence interval coverage probability of the TMLE is compared with those of the conventional KM and the inverse probability of censoring weight estimating equation, conventional maximum likelihood substitution estimator, and the double robustaugmented inverse probability of censoring weighted estimating equation. We conclude the article with estimation of the causal effect of warfarin medical therapy on the probability of “stroke or death” within a 1-year time frame using data from the ATRIA-1 observational cohort of persons with atrial fibrillation. Our results suggest that a fixed policy of warfarin treatment for all patients would result in 2% fewer deaths or strokes within 1-year as compared with a policy of withholding warfarin from all patients.

Keywords

• targeted minimum loss-based estimation
• survival analysis
• dependent right-censoring
• efficient influence curve
• targeted maximum likelihood estimation