Consistency of maximum likelihood estimators in general random effects models for binary data

We consider a general random effects model for repeated binary measures, assuming a latent linear model with any class of mixing distributions. The latent model is assumed to have the Laird-Ware structure, but the random effects may be from any specified class of multivariate distributions and the error vector may have any specified continuous distribution. Elementwise threshold crossing then gives the observed vector of binary outcomes. Special cases of this model include recently discussed mixed logistic regression and probit models, which have had either parametric (usually Gaussian) or nonparametric mixing distributions. We give sufficient conditions for identifiability of the mixing distribution and fixed effects and for convergence of maximum likelihood estimators for the mixing distribution and fixed effects. As expected, the conditions are much stronger for nonparametric mixing than for Gaussian mixing. We illustrate the conditions by applying them to a practical example.