Measures for Assessing Causal Effect Heterogeneity Unexplained by Covariates

There has been considerable interest in estimating heterogeneous causal effects across individuals or subpopulations. Researchers often assess causal effect heterogeneity based on the subjects' covariates using the conditional average causal effect (CACE). However, substantial heterogeneity may persist even after accounting for the covariates. Existing work on causal effect heterogeneity unexplained by covariates mainly focused on binary treatment and outcome. In this paper, we introduce novel heterogeneity measures, P-CACE and N-CACE, for binary treatment and continuous outcome that represent CACE over the positively and negatively affected subjects, respectively. We also introduce new heterogeneity measures, P-CPICE and N-CPICE, for continuous treatment and continuous outcome by leveraging stochastic interventions, expanding causal questions that researchers can answer. We establish identification and bounding theorems for these new measures. Finally, we show their application to a real-world dataset.