Effects of Adjusting for Instrumental Variables on the Bias and Precision of Propensity Score Weighted Estimators: Analysis Under Complete, Near, and No Positivity Violations

Purpose: To demonstrate that using an instrumental variable (IV) with monotonicity reduces the accuracy of propensity score (PS) weighted estimators for the average treatment effect (ATE). Methods: Monotonicity in the relationship between a binary IV and a binary treatment variable is an important assumption to identify the ATE for compliers who would only take treatment when encouraged by the IV. We perform theoretical and numerical investigations to study the impact of using the IV that satisfies monotonicity on the PS of treatment in terms of the positivity assumption, which requires that the PS be strictly between 0 and 1, and the accuracy of PS weighted estimators. Two versions of monotonicity that result in one-sided or two-sided noncompliance are considered. Results: The PS adjusting for the IV always violates the positivity assumption when noncompliance occurs in one direction (one-sided noncompliance) and is more extreme than without the IV under two-sided noncompliance. These results are valid if the probability of being encouraged to get treatment and the compliance score, the probability of being a complier, are strictly between 0 and 1. Conclusion: Using a binary IV with monotonicity as a covariate for the PS model makes the estimated PSs unnecessarily extreme, reducing the accuracy of the PS weighted estimators.