# Sharp bounds on the relative treatment effect for ordinal outcomes

## TL;DR

This paper derives sharp bounds for the relative treatment effect in ordinal outcomes, allowing for arbitrary dependence between potential outcomes, which enhances interpretability when the average treatment effect is ill-defined.

## Contribution

It provides the first derivation of sharp bounds on the relative treatment effect for ordinal outcomes without assuming independence of potential outcomes.

## Key findings

- Derived sharp bounds on the relative treatment effect for ordinal outcomes.
- Bounds are identifiable from observed data and accommodate arbitrary dependence.
- Enhances interpretability of treatment effects in ordinal outcome studies.

## Abstract

For ordinal outcomes, the average treatment effect is often ill-defined and hard to interpret. Echoing Agresti and Kateri (2017), we argue that the relative treatment effect can be a useful measure especially for ordinal outcomes, which is defined as $\gamma = \mathrm{pr}\{ Y_i(1) > Y_i(0) \} - \mathrm{pr}\{ Y_i(1) < Y_i(0) \}$, with $Y_i(1)$ and $Y_i(0)$ being the potential outcomes of unit $i$ under treatment and control, respectively. Given the marginal distributions of the potential outcomes, we derive the sharp bounds on $\gamma,$ which are identifiable parameters based on the observed data. Agresti and Kateri (2017) focused on modeling strategies under the assumption of independent potential outcomes, but we allow for arbitrary dependence.