Messer et al. Respond to "Positivity in Practice"
American Journal of Epidemiology
Westreich and Cole's commentary (1) on positivity and on the 2 accompanying articles by us (2) and Cheng et al. (3) is clear, thoughtful, and helpful. We appreciate their ability to define positivity, explain its role in causal inference, and highlight procedures for diagnosis and methodological advancement. It is their distinction between deterministic and random violations of positivity that we wish to say more about. From our perspective, violations of positivity pose a very serious threat
... credible causal inference in, and advancement of, social epidemiology. Use of regression models to overcome confounding can mean that key comparisons are based on very sparse or even fully interpolated (i.e., modeldependent) data. This is especially true in multilevel "neighborhood effects" research, where social stratification so clearly clusters individuals and exposures. As Westreich and Cole explain (1), the problem boils down to a tradeoff between confounder control and empirical support for inferences: Efforts to overcome confounding undermine empirical support for comparisons. It is worth noting that while results from such sparse-cell analyses may not be wrong, heroic modeling assumptions are required to support them. It is obvious that not having a uterus or using a 2-headed coin in randomization can yield deterministic violations of positivity. It is also obvious that any given sample from a target population can suffer a larger or smaller violation, especially given model specifications. The key difference between deterministic and random violations is the sense that random violations could be overcome by collecting more data. With a sufficient sample size, one can imagine overcoming the problem of random violations entirely. It follows that random violations should be more associated with sampling and frequentist inference than with fundamental identification problems. On the other hand, deterministic violations of positivity lead to structural confounding, a condition for which the collection of more data is not helpful because observable counterfactual substitutes do not exist in the target population.