On the collapsibility of measures of effect in the counterfactual causal framework

TL;DR

This paper explores the concept of collapsibility in causal effect measures within the counterfactual framework, identifying which measures are inherently collapsible and providing conditions and weights for collapsibility.

Contribution

It introduces a distinction between two definitions of collapsibility and shows how causal measures can be manipulated to be collapsible, except for the odds ratio.

Findings

Causal risk difference and risk ratio are collapsible with appropriate weights.

The odds ratio cannot be made collapsible with any weights.

Mathematical properties of effect measures influence their collapsibility.

Abstract

A measure of association is said to be collapsible over a set of baseline covariates if the marginal value of the measure of association is equal to a weighted average of the stratum-specific measures of association. In this paper, we consider two subtly different definitions of collapsibility, and show that by considering causal measures of effect based on counterfactual variables it is possible to separate out the component of non-collapsibility which is due to the mathematical properties of the effect measure. We provide weights such that the causal risk difference and the causal risk ratio are collapsible over arbitrary baseline covariates, and demonstrate that such general weights do not exist for the odds ratio.