Multiply robust causal inference in the presence of an error-prone treatment

Numerous estimation procedures employed in causal inference often rely on accurately measured data. However, the prevalence of measurement errors in practical studies may yield biased effect estimates. It is common to employ validation samples to rectify such biases in the measurement error literature. This article focuses on the estimation of the average causal effect with a misclassified binary treatment in a primary population of interest. By leveraging a validation sample with covariates, an error-prone version of treatment and a true treatment recorded, we provide identifiability results under certain conditions. Building on identifiability, we explore three classes of estimators, each demonstrating consistency and asymptotic normality within distinct model sets. Furthermore, we propose a multiply robust estimation approach for the treatment effect based on the semiparametric theory framework. The multiply robust estimator retains consistent under any one of the listed model sets and achieves the semiparametric efficiency bound, provided all models are correct. We demonstrate the satisfactory performance of the proposed estimators through simulation studies and a real data analysis.