  Assessing the causal effects of interventions on ordinal outcomes is an
important objective of many educational and behavioral studies. Under the
potential outcomes framework, we can define causal effects as comparisons
between the potential outcomes under treatment and control. However,
unfortunately, the average causal effect, often the parameter of interest, is
difficult to interpret for ordinal outcomes. To address this challenge, we
propose to use two causal parameters, which are defined as the probabilities
that the treatment is beneficial and strictly beneficial for the experimental
units. However, although well-defined for any outcomes and of particular
interest for ordinal outcomes, the two aforementioned parameters depend on the
association between the potential outcomes, and are therefore not identifiable
from the observed data without additional assumptions. Echoing recent advances
in the econometrics and biostatistics literature, we present the sharp bounds
of the aforementioned causal parameters for ordinal outcomes, under fixed
marginal distributions of the potential outcomes. Because the causal estimands
and their corresponding sharp bounds are based on the potential outcomes
themselves, the proposed framework can be flexibly incorporated into any chosen
models of the potential outcomes, and are directly applicable to randomized
experiments, unconfounded observational studies, and randomized experiments
with noncompliance. We illustrate our methodology via numerical examples and
three real-life applications related to educational and behavioral research.
