Maximum likelihood estimation for probit-linear mixed models with correlated random effects

The probit-normal model for binary data (McCulloch, 1994, Journal of the American Statistical Association 89; 330-335) is extended to allow correlated random effects. To obtain maximum likelihood estimates, we use the EM algorithm with its M-step greatly simplified under the assumption of a probit link and its E-step made feasible by Gibbs sampling. Standard errors are calculated by inverting a Monte Carlo approximation of the information matrix rather than via the SEM algorithm. A method is also suggested that accounts for the Monte Carlo variation explicitly. As an illustration, we present a new analysis of the famous salamander mating data. Unlike previous analyses, we find it necessary to introduce different variance components for different species of animals. Finally, we consider models with correlated errors as well as correlated random effects.