The Consistency Statement in Causal Inference: A Definition or an Assumption?

Abstract

Three assumptions sufficient to identify the average causal effect are consistency, positivity, and exchangeability (ie, “no unmeasured confounders and no informative censoring,” or “ignorability of the treatment assignment and measurement of the outcome”). The exchangeability assumptions are well known territory for epidemiologists and biostatisticians. Briefly, to be satisfied, these 2 exchangeability assumptions that require exposed and unexposed subjects, and censored and uncensored subjects have equal distributions of potential outcomes, respectively. Indeed, the so-called fundamental problem of causal inference1 is directly linked to the first exchangeability assumption. In contrast, the consistency and positivity assumptions are less well known. The positivity assumption states that there is a nonzero (ie, positive) probability of receiving every level of exposure for every combination of values of exposure and confounders that occur among individuals in the population.2,3 It remains unclear why the consistency and positivity assumptions are less well known. Optimistically, perhaps these assumptions are less important with respect to an impact on estimation of the average causal effect. Pessimistically, these assumptions are less well known because there is little alarming evidence of a departure from either of these assumptions in observational studies without explicitly looking for the departure. Here we will focus on the preliminary issue of clarifying the consistency assumption. The consistency assumption is often stated such that an individual's potential outcome under her observed exposure history is precisely her observed outcome.4 Methods for causal inference require that the exposure is defined unambiguously. Specifically, one needs to be able to explain how a certain level of exposure could be hypothetically assigned to a person exposed to a different level. This requirement is known as consistency. Consistency is guaranteed by design in experiments, because application of the exposure to any individual is under the control of the investigator. Consistency is plausible in observational studies of medical treatments, because one can imagine how to manipulate hypothetically an individual's treatment status. However, consistency is problematic in observational studies with exposures for which manipulation is difficult to conceive. Consistency is especially difficult when the exposure is a biologic feature, such as body weight, insulin resistance, or CD4 cell count.5,6 For example, there are many competing ways to assign (hypothetically) a body mass index of 25 kg/m2 to an individual, and each of them may have a different causal effect on the outcome. To state consistency formally, let us first define individual j's potential outcome Yj(x) under exposure x as the outcome that would have been observed if individual j had received exposure x. The variable Yj(x) is known as a potential outcome because it describes subject j's outcome value that would have been observed under a potential exposure value that the subject may or may not have actually experienced. For a more detailed definition of potential outcomes, please see references.7,8 Many authors,7,9–12 but not all,13 find the use of potential outcomes central to the definition of causation and causal effects. Indeed, potential outcomes in the form of “causal response types” have a long history of assisting epidemiologists in the clarification of concepts such as confounding,14 effect measure modification,15 and mediation.16 Consistency is often stated as where x denotes the argument of the potential outcome function, and (1) is read “the observed outcome for individual j is the potential outcome, as a function of intervention, when the intervention is set to the observed exposure.” Those who are willing to accept potential outcomes at least as a useful framework may see statement (1) as an axiom, an identity, or even a tautology. Indeed, discussions of inferences from applications of methods directly using potential outcomes rarely account for possible departures from the consistency assumption.11,12,17 Some of these same applications routinely discuss departures from the exchangeability assumptions stated previously, or even use analyses quantifying the sensitivity of inferences to the exchangeability assumptions. The point of the present commentary is that the statement of an assumption should explicitly show what is being assumed. More precisely, if statement (1) is to be an “assumption” (as it is often referred to) and not a definition, it must be possible to be incorrect given the concepts that have been already accepted before stating (1). The problem is that precisely in much of the work that takes (1) to be an assumption, there is no stated concept based on which this assumption could be incorrect, thus making it a definition or axiom. So, what is (1) supposed to “assume” if it is an assumption and not a definition? To clarify, we may