Affiliation was introduced by Milgrom and Weber (1982) through the positive dependence intuition, that is, high value of one bidder’s estimate makes high values of the others’ estimates more likely. However, positive dependence has many alternative definitions; we show that affiliation is one of the most restrictive among them. This poses the question whether affiliation’s main implications (equilibrium existence and the revenue ranking of auctions) remain valid for other formalizations of positive dependence. We show that both implications can indeed be generalized in the context of private values, and give counterexamples for further generalizations.