Causal Inference Under Multiple Versions of Treatment

The potential outcomes framework for causal inference employs a number of assumptions (Neyman, 1923; Rubin, 1974, 1990; Robins, 1986). One of the assumptions that is generally made, either implicitly or explicitly, is an assumption that is sometimes described as the “no-multiple-versions-of-treatment assumption”; the assumption is part of what Rubin defined as the “Stable Unit Treatment Value Assumption” or SUTVA (Rubin, 1980, 1986). The assumption is made so that the potential outcomes for each individual under each possible treatment are well defined and take on a single value. If there are multiple versions of treatment present, as might arise for surgery treatment say if there are different surgeons who perform the surgery, and if these different versions of treatment give rise to different potential outcomes, then this assumption will be violated. To circumvent the problems created by multiple versions of treatment in such contexts, one might restrict inference to a single version of treatment or, more generally, redefine each version of treatment as a different treatment. For example, one could consider the effect of surgery conducted by each particular surgeon rather that the effect of surgery generally. Such redefinition of the treatment of interest would make the no-multiple-versions-of-treatment assumption more reasonable and we do consider this approach below (see Proposition 1). However, redefining each version of treatment as a different treatment may not always be possible or desirable. One may not have data on which version each patient received. Moreover, if a patient needs to decide whether or not to undergo surgery but has no control over the choice of the surgeon who will actually perform the surgery, the average effect of the surgery treatment generally, rather than the average effect of surgery for each particular surgeon, may be what is most relevant. Such average effects of a particular surgical procedure (averaged over the surgeon who administers it) may also be of interest from a policy perspective. If, for example, in the treatment of cancer patients we are comparing the effects of radiation versus surgery, although different surgeons (i.e. different versions of treatment) may have different effects on survival, from a policy perspective, we could not simply select the most competent surgeon to perform all of the surgeries as the number of surgeries needed would be far too numerous for one surgeon to undertake. The policy question of interest here would be evaluating the overall survival rates of radiation versus surgery, taking into account the fact that not all surgeons are equally skilled. Motivated by the above considerations, the purpose of this article is to consider causal inference under violations of the no-multiple-versions-of-treatment assumption. We will consider two broad settings: one in which the treatment precedes the version and one in which the version precedes the treatment. In the first setting, treatment is well-defined even before the version has been assigned but an individual’s potential outcomes are not fixed by the treatment alone; the potential outcome may be different under different versions of treatment. For the purposes of identification, we draw a distinction between confounders of the treatment-outcome relationship and the treatment-version relationship. We show that if data is available on version and one is simply interested in causal effects comparing one version to another, then adjustment does not need to be made for treatment-version confounders in the estimation of such causal effects (Proposition 1). We also show that if there is only one version of the control condition, then to estimate the effect of treatment on the treated, adjustment likewise does not need to be made for treatment-version confounders (Proposition 2). However, under multiple versions of treatment, to estimate overall causal effects for a population, adjustment needs to be made both for treatment-version confounders and for treatment-outcome confounders (Proposition 3) and failure to control for treatment-version confounders can result in bias. Propositions 1 and 3 follow also from theory for direct effects and from causal diagrams (e.g. Pearl, 2009; cf. Petersen, 2011) when applied to the multiple versions of treatment context. Further discussion of accommodating multiple versions of treatment using causal diagrams, and of the relative advantages of using the approach like that presented here versus expanded causal diagrams, is given elsewhere (Pearl, 2010; Hernan and VanderWeele, 2011; Petersen, 2011). In this setting in which treatment precedes version, additional identification results are given for several other types of causal effects which compare outcomes under various settings under which the version of treatment is randomly assigned from some specified distribution (Propositions 4–7). Some of these results (Propositions 5 and 6) have analogues in the literature on stochastic interventions in the context of direct effects (Didelez et al., 2006; Geneletti, 2007). In the Appendix, we also consider analogous results when there may be a time-dependent confounder that is affected by treatment assignment but confounds the version-outcome relationship (Propositions 9–11). Finally, we give a result for the perhaps even more common setting in which the version precedes the treatment variable or in which the actual treatment variable used in the analysis is in fact a coarsening of the underlying version of treatment variable. We show that use of the coarsened variable in the analysis yields effect estimates that can be interpreted as the effect of interventions in which the underlying version of treatment is randomly selected from certain distributions naturally arising in the population (Proposition 8). We discuss the relation of these results to those obtained from the treatment then version setting. Although some of the results follow in a straightforward way from existing theory we believe it is of benefit for applied research to carefully consider, using the results in this paper as guidance, what different versions of treatment might be present and how this may, or may not, affect inferences. In the setting of treatment-then-version, ignoring multiple versions of treatment and potential treatment-version confounders can result in biased estimates of the overall treatment effect. In the version-then-treatment setting, if multiple versions of treatment are ignored, similar problems can arise with regard to bias due to inadequate control for confounding. Moreover, in this case, even if adequate control for confounding is made, the interpretation of the effect estimates are considerably more subtle than they at first may seem. The remainder of the paper is organized as follows. In section 2 we review the potential outcomes framework and discuss how it can accommodate settings of multiple versions of treatment; we discuss the definition of causal effects under multiple versions of treatment. In section 3, we discuss identification of these effects when first the treatment is assigned and then a particular version of treatment is assigned. We discuss the interpretation of causal effect estimates under multiple versions and we consider when multiple versions of treatment can be ignored. In section 4, we will consider what new questions might be of substantive interest when multiple versions of treatment are present; some of the identification results bear certain resemblances to the analysis of direct and indirect effects (Robins and Greenland, 1992; Pearl, 2001; Didelez et al., 2006), though, as discussed further below, the precise technical details are distinct. In section 5, we discuss cases in which the ordering is version then treatment rather than treatment then version. An illustration is given in section 6. In section 7, we offer some concluding remarks and discuss how the extensions given in this paper to allow for violations of the no-multiple-versions-of-treatment assumption parallels in certain ways extensions described elsewhere for violations of the other major component of SUTVA, the no-interference assumption (Sobel, 2006; Hong and Raudenbush, 2006; Rosenbaum, 2007; Hudgens and Halloran, 2008; VanderWeele and Tchetgen Tchetgen, 2011a b; Halloran and Hudgens, 2012; Tchetgen Tchetgen and VanderWeele, 2012).