Beyond exchangeability: The other conditions for causal inference in medical research

Abstract

The assumption of exchangeability of the treated and the untreated – or, in general, of those subjects receiving different levels of the exposure – often gets most of the attention in discussions about causal inference. A key argument to prefer randomised experiments over observational studies is precisely that exchangeability is expected by design in the former. In contrast, investigators conducting observational studies need to use their expert knowledge to identify and measure many potential confounders. Their hope is to collect sufficient data to achieve exchangeability conditional on the measured covariates. Unfortunately, these investigators can never be certain that they have succeeded, even if they have actually succeeded. Exchangeability cannot be empirically tested in observational studies. Lack of exchangeability arises when the distributions of prognostic factors for the outcome differ between the treated and the untreated. There are two main reasons why these differences may occur. First, exchangeability will not generally hold if there are common causes of the treatment and the outcome. For example, having a high level of LDL-cholesterol increases the probability of initiating statin therapy and, independently, the probability of having a myocardial infarction.We then say that the association between statin therapy and myocardial infarction risk is confounded, or that there is confounding bias for the effect of statin therapy on myocardial infarction risk. This terminology explains that the assumption of conditional exchangeability given the measured variables is often referred to as the assumption of no unmeasured confounding. In randomised experiments confounding for the effect of assigned treatment is not expected because a random assignment of treatment results in balanced distributions of prognostic factors between the treated and the untreated. Second, exchangeability will not generally hold if the analysis is restricted to selected individuals and the selection process was affected by both the treatment and the outcome, or by causes of the treatment and the outcome. For example, suppose a third of the subjects were lost to follow-up during a study on statins and myocardial infarction, and the risk of being lost to follow-up was affected by both the presence of treatment’s side effects and of symptoms of coronary heart disease (i.e. angina). In this setting the analysis needs to be restricted to the selected two thirds who stayed in the study until their outcome was ascertained. We say that the association between statin therapy and myocardial infarction is biased in this selected group, or that there is selection bias for the effect of statin therapy on myocardial infarction risk. Confounding is a common source of lack of exchangeability in observational studies, but not in randomised experiments – as long as one sticks to an intention-to-treat analysis. On the other hand, selection bias due to loss to follow-up can happen in both observational studies and randomised experiments. Selection bias can be due to many forms of selection process besides loss to follow-up (e.g. self-selection, missing data, healthy worker). It may also arise when conventional methods for