The Probability of Tiered Benefit: Partial Identification with Robust and Stable Inference

Abstract

We define the Probability of Tiered Benefit in scenarios with a binary exposure and an outcome that is either categorical with $K \geq 2$ ordered tiers or continuous partitioned by $K-1$ fixed thresholds into disjoint intervals. Similarly to other pure counterfactual queries, this parameter is not $g$-identifiable without additional assumptions. We demonstrate that strong monotonicity does not suffice for point identification when $K \geq 3$ and provide sharp bounds both with and without such constraint. Inference and uncertainty quantification for these bounds are challenging tasks due to potential nonregularity induced by ambiguities in the underlying individualized optimization problems. Such ambiguities can arise from immunities or null treatment effects in subpopulations with positive probability, affecting the lower bound estimate and hindering conservative inference. To address these issues, we extend the available Stabilized One-Step Correction (S1S) procedure by incorporating stratum-specific stabilizing matrices. Through simulations, we illustrate the benefits of this approach over existing alternatives. We apply our method to estimate bounds on the probabilities of tiered benefit and harm from pharmacological treatment for ADHD upon academic achievement, employing observational data from diagnosed Norwegian schoolchildren. Our findings indicate that while girls and children with low prior test performance could have moderate chances of both benefit and harm from treatment, a clear-cut recommendation remains uncertain across all strata.