In this article we describe a likelihood-based regression model appropriate for analyzing incomplete multivariate binary responses. We focus on “marginal models”; that is, models where the marginal mean or expectation of the binary response is related to a set of covariates. The association between the binary responses is modeled in terms ofconditionallog odds ratios. When the nonresponse mechanism isignorable, it is not necessary to specify a nonresponse model, and valid inferences can be obtained provided that the likelihood for the responses has been correctly specified. But when the nonresponse mechanism isnonignorable, valid inferences can only be obtained by incorporating a model for nonresponse. An unresolved issue with nonignorable models concerns the identifiability of the parameters. So far, no general and practically useful necessary and sufficient conditions for identifiability are available. Here we suggest some simple procedures for examining the identifiability status of nonignorable models when the response variable is discrete. Finally, we present results for an analysis of multiple informant data from the New Haven Child Survey and the Eastern Connecticut Child Survey.