Causal inference under multiple versions of treatment

Abstract: In this article, we discuss causal inference when there are multiple versions of treatment. The potential outcomes framework, as articulated by Rubin, makes an assumption of no multiple versions of treatment, and here we discuss an extension of this potential outcomes framework to accommodate causal inference under violations of this assumption. A variety of examples are discussed in which the assumption may be violated. Identification results are provided for the overall treatment effect and the effect of treatment on the treated when multiple versions of treatment are present and also for the causal effect comparing a version of one treatment to some other version of the same or a different treatment. Further identification and interpretative results are given for cases in which the version precedes the treatment as when an underlying treatment variable is coarsened or dichotomized to create a new treatment variable for which there are effectively “multiple versions”. Results are also given for effects defined by setting the version of treatment to a prespecified distribution. Some of the identification results bear resemblance to identification results in the literature on direct and indirect effects. We describe some settings in which ignoring multiple versions of treatment, even when present, will not lead to incorrect inferences.