Average Collapsibility of Some Association Measures

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.