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

TL;DR

This paper develops methods to estimate bounds on the probability of tiered benefits in observational studies with ordered outcomes, addressing challenges of partial identification and nonregular inference.

Contribution

It introduces a stabilized one-step correction procedure with stratum-specific matrices for robust inference on bounds of tiered benefits, extending existing techniques.

Findings

Bounds suggest moderate chances of benefit and harm for certain subgroups.

Strong monotonicity does not guarantee point identification for three or more tiers.

The proposed method improves inference accuracy over existing approaches.

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.