An affiliation network is a particular type of two-mode social network that consists of a set of `actors' and a set of `events' where ties indicate an actor's participation in an event. Although networks describe a variety of consequential social structures, statistical methods for studying affiliation networks are less well developed than methods for studying one-mode, or actor-actor, networks. One way to analyze affiliation networks is to consider one-mode network matrices that are derived from an affiliation network, but this approach may lead to the loss of important structural features of the data. The most comprehensive approach is to study both actors and events simultaneously. In this paper, we extend the bilinear mixed-effects model, a type of latent space model developed for one-mode networks, to the affiliation network setting by considering the dependence patterns in the interactions between actors and events and describe a Markov chain Monte Carlo algorithm for Bayesian inference. We use our model to explore patterns in extracurricular activity membership of students in a racially-diverse high school in a Midwestern metropolitan area. Using techniques from spatial point pattern analysis, we show how our model can provide insight into patterns of racial segregation in the voluntary extracurricular activity participation profiles of adolescents.