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Abstract:
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Under the potential outcomes framework, causal effects are defined as comparisons between the treatment and control potential outcomes.However, the average causal effect, generally the parameter of interest, is not well defined for ordinal outcomes. To address this problem, we propose two new causal parameters, i.e., the probabilities that the treatment is beneficial and strictly beneficial for the experimental units, which are well defined for any outcomes and of particular interest for ordinal outcomes. We define causal effects based on potential outcomes themselves without invoking any parametric models. These two new causal parameters, though of scientific importance and interest, depend on the association between the potential outcomes and therefore without any further assumptions they are not identifiable from the observed data. In this paper for ordinal outcomes, we derive sharp bounds of the two new causal parameters using only the marginal distributions, without imposing any assumptions on the joint distribution of the potential outcomes.
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