Confidence intervals for the treatment effect on the treated

The average effect of the treatment on the treated is a quantity of interest in observational studies in which no definite parameter can be used to quantify the treatment effect, such as those where only a random subset of the data obtained by stratification can be used for analysis. Nonparametric confidence intervals for this quantity appear to be known only in the case where the responses to the treatment are binary and the data fall into a single stratum. We propose nonparametric confidence intervals for the average effect of the treatment on the treated in studies involving one or more strata and general numerical responses.