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. Non-parametric 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.
Applied Statistics | Biostatistics | Other Statistics and Probability
Ferreira, José A., "Confidence intervals for the treatment effect on the treated" (December 2014). COBRA Preprint Series. Working Paper 111.