Stable balancing weighted effect curves: uniform inference and robust selection of an imbalance parameter

Abstract Propensity score (PS) methods are central to estimating treatment effects in observational studies, but conventional approaches such as inverse-probability-of-treatment weighting (IPTW) are vulnerable to extreme PSs. These limitations can lead to unstable estimates and poor generalizability. Stable balancing weighting (SBW) offers an alternative by directly minimizing weight variability while controlling covariate imbalance. However, current practice typically selects a single tuning parameter $$\:\delta\:$$ , potentially overlooking valuable information across the range of bias-variance trade-offs. We propose a novel framework that evaluates treatment effects across a continuum of $$\:\delta\:$$ values, generating a Stable Balancing Weighted Effect (SBWE) curve. This curve characterizes the trajectory of causal estimates as covariate balance requirements are tightened. To facilitate inference, we developed uniform confidence bands for the SBWE curve, allowing for a comprehensive selection of the optimal $$\:\delta\:$$ and a robust test of the null hypothesis of no treatment effect. We validated our method using two real-world datasets. In the first case, the proposed SBWE framework supported the optimal $$\:\delta\:$$ chosen by the existing selection method. In the second case, however, the existing method selected a $$\:\delta\:$$ near the lower bound of the range, while our method identified an optimal value at the upper bound. Notably, the confidence interval produced by our method was nested within that produced by the existing method. The SBWE curve approach enhances the robustness of SBW by shifting the focus from a single point estimate to a comprehensive curve. This allows researchers to visualize the stability of treatment effects and select tuning parameters that yield more reliable and precise causal estimates.