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

Anders Huitfeldt,Mats J Stensrud,Etsuji Suzuki

doi:10.1186/s12982-018-0083-9

Abstract

The relationship between collapsibility and confounding has been subject to an extensive and ongoing discussion in the methodological literature. We discuss two subtly different definitions of collapsibility, and show that by considering causal effect measures based on counterfactual variables (rather than measures of association based on observed variables) it is possible to separate out the component of non-collapsibility which is due to the mathematical properties of the effect measure, from the components that are due to structural bias such as confounding. We provide new weights such that the causal risk ratio is collapsible over arbitrary baseline covariates. In the absence of confounding, these weights may be used for standardization of the risk ratio.

Highlights

IntroductionA measure of association (such as the risk difference or the risk ratio) is said to be collapsible if the marginal measure of association is equal to a weighted average of the stratum-specific measures of association [1]
A measure of association is said to be collapsible if the marginal measure of association is equal to a weighted average of the stratum-specific measures of association [1]
We show that the causal risk ratio RR(−) is collapsible over arbitrary covariates V if we use the weights wv = Pr(V = v|Y a=0 = 1), i.e. weights determined by the distribution of the baseline covariates among those individuals who would have been cases if they, possibly contrary to fact, were not treated with drug A: Our goal is to show that a=1 =1|V a=0 =1|V

Summary

Introduction

A measure of association (such as the risk difference or the risk ratio) is said to be collapsible if the marginal measure of association is equal to a weighted average of the stratum-specific measures of association [1]. We argue that the concept of collapsibility can be made clearer if we frame the discussion in terms of causal effect measures based on counterfactual variables. We are interested in the effect of a binary exposure A (e.g. a drug), on a binary outcome Y (e.g. a side effect). The associational risk difference is Pr(Y = 1|A = 1) − Pr(Y = 1|A = 0) whereas the causal risk difference (RD) is Pr(Y a=1 = 1) − Pr(Y a=0 = 1). These effect measures may be defined within levels of covariates V

Objectives

Conclusion