  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.
