A Flexible General Class of Marginal and Conditional Random Intercept Models for Binary Outcomes Using Mixtures of Normals

Random intercept models for binary data are useful tools for addressing betweensubject heterogeneity. Unlike linear models, the non-linearity of link functions used for binary data force a distinction between marginal and conditional interpretations. This distinction is blurred in probit models with a normally distributed random intercept because the resulting model implies a probit marginal link as well. That is, this model is closed in the sense that the distribution associated with the marginal and conditional link functions and the random effect distribution are all of the same family. In this manuscript we explore another family of random intercept models with this property. In particular, we consider instances when the distributions associated with the conditional and marginal link functions and the random effect distribution are mixtures of normals. We show that this flexible family of models is related to several others presented in the literature. Moreover, we also show that this family of models offers considerable computational benefits. A diverse series of examples illustrates the wide applicability of the approach.