Partial Compliance, Effect of Treatment on the Treated and Instrumental Variables

Under the assumption that treatment assignment has no direct effect on the response, a non parametric probabilistic model of the distribution involving the latent confounder under partial compliance leads to a generalized definition of the effect of treatment on the treated and reveals that the instrumental variable estimand equals a suitable average of such causal effects only when certain restrictions hold. An application to a popular data set concerning reduction of cholesterol level is used as an illustration.