Average Collapsibility of Some Association Measures

TL;DR

This paper explores the concept of average collapsibility for various association measures, providing conditions for when conditional and marginal measures align, with applications to multiple regression models.

Contribution

It introduces the concept of average collapsibility, extending traditional collapsibility, and derives sufficient conditions for its occurrence in different statistical models.

Findings

Derived conditions for average collapsibility of association measures.

Constructed counter-examples illustrating the concept.

Discussed extensions to multivariate covariates.

Abstract

Collapsibility deals with the conditions under which a conditional (on a covariate W) measure of association between two random variables X and Y equals the marginal measure of association, under the assumption of homogeneity over the covariate. In this paper, we discuss the average collapsibility of certain well-known measures of association, and also with respect to a new measure of association. The concept of average collapsibility is more general than collapsibility, and requires that the conditional average of an association measure equals the corresponding marginal measure. Sufficient conditions for the average collapsibility of the measures under consideration are obtained. Some difficult, but interesting, counter-examples are constructed. Applications to linear, Poisson, logistic and negative binomial regression models are addressed. An extension to the case of multivariate covariate W is also discussed.