A note on the robustness of quantile treatment effect estimands

This note examines the robustness of two quantile treatment effect estimands to a perturbation away from the common effect assumption. The first estimand QY1−Y0(τ) is the τ-quantile of the difference between the potential outcomes and the second estimand QY1(τ)−QY0(τ) is the difference between the τ-quantiles of the potential outcomes. To this end, this note provides a simple “trembling hand” example whereby the treatment effect deviates from the common effect β for only one individual in a large population. As a result, each estimand deviates from β, and may have the opposite sign than β, in a distinct range of τ. In general, this perturbation leads QY1−Y0(⋅) to differ from β more severely but only in a small and extreme range of τ whereas it leads QY1(⋅)−QY0(⋅) to differ from β less severely but over a substantial and central range of τ. This example suggests that researchers should carefully evaluate estimates or bounds for QY1−Y0(τ) and especially QY1(τ)−QY0(τ) over a large range of τ.