Concerning the consistency assumption in causal inference.

Cole and Frangakis (Epidemiology. 2009;20:3-5) introduced notation for the consistency assumption in causal inference. I extend this notation and propose a refinement of the consistency assumption that makes clear that the consistency statement, as ordinarily given, is in fact an assumption and not an axiom or definition. The refinement is also useful in showing that additional assumptions (referred to here as treatment-variation irrelevance assumptions), stronger than those given by Cole and Frangakis, are in fact necessary in articulating the ordinary assumptions of ignorability or exchangeability. The refinement furthermore sheds light on the distinction between intervention and choice in reasoning about causality. A distinction between the range of treatment variations for which potential outcomes can be defined and the range for which treatment comparisons are made is discussed in relation to issues of nonadherence. The use of stochastic counterfactuals can help relax what is effectively being presupposed by the treatment-variation irrelevance assumption and the consistency assumption.