Multivalued Treatment Effects

Treatment Effect Model and Population Parameters

A general statistical treatment effect model with multivalued treatment assignments is easily described in the context of the classical potential outcomes model. This model assumes that each unit i in a population has an underlying collection of potential outcome random variables

, where denotes the collection of possible treatment assignments. The random variables Yi(t) are usually called potential outcomes because they represent the random outcome that unit i would have under treatment regime. For each unit i and for any two treatment levels, t1 and t2, it is always possible to define the individual treatment effect given by, which may or may not be a degenerate random variable. However, because units are not observed under different treatment regimes simultaneously, such comparisons are not feasible. This idea, known as the fundamental problem of causal inference, is formalized in the model by assuming that for each unit i only (Yi, Ti) is observed, where Yi = Yi(Ti) and Ti. In words, for each unit i, only the potential outcome for treatment level Ti = t is observed while all other (counterfactual) outcomes are missing. Of course, in most applications, which treatment each unit has taken up is not random and hence further assumptions would be needed to identify the treatment effect of interest.

A binary treatment effect model has

, a finite multivalued treatment effect model has for some positive integer J, and a continuous treatment effect model has. (Note that the values in are ordinal, that is, they may be seen just as normalizations of the underlying real treatment levels in a given application.) Many applications focus on a binary treatment effects model and base the analysis on the comparison of two groups, usually called treatment group (Ti = 1) and control group (Ti = 0). A multivalued treatment may be collapsed into a binary treatment, but this procedure usually would imply some important loss of information in the analysis. Important phenomena such as nonlinearities, differential effects across treatment levels or tipping points, cannot be captured by a binary treatment effect model.

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