The Relation of Collapsibility and Confounding to Faithfulness and Stability

A probability distribution may have some properties that are stable under a structure (e.g., a causal graph) and other properties that are unstable. Stable properties are implied by the structure and thus will be shared by populations following the structure. In contrast, unstable properties correspond to special circumstances that are unlikely to be replicated across those populations. A probability distribution is faithful to the structure if all independencies in the distribution are logical consequences of the structure. We explore the distinction between confounding and noncollapsibility in relation to the concepts of faithfulness and stability. Simple collapsibility of an odds ratio over a risk factor is unstable and thus unlikely if the exposure affects the outcome, whether or not the risk factor is associated with exposure. For a binary exposure with no effect, collapsibility over a confounder also requires unfaithfulness. Nonetheless, if present, simple collapsibility of the odds ratio limits the degree of confounding by the covariate. Collapsibility of effect measures is stable if the covariate is independent of the outcome given exposure, but it is unstable if the covariate is an instrumental variable. Understanding stable and unstable properties of distributions under causal structures, and the distinction between stability and faithfulness, yields important insights into the correspondence between noncollapsibility and confounding.