Probit-normal models have attractive properties compared to logit-normal models. In particular, they allow for easy specification of marginal links of interest while permitting a conditional random effects structure. Moreover, programming fitting algorithms for probit-normal models can be trivial with the use of well-developed algorithms for approximating multivariate normal quantiles. In typical settings, the data cannot distinguish between probit and logit conditional link functions. Therefore, if marginal interpretations are desired, the default conditional link should be the most convenient one. We refer to models with a probit conditional link an arbitrary marginal link and a normal random effect distribution as link-probit-normal models. In this manuscript we outline these models and discuss appropriate situations for using multivariate normal approximations. Unlike other manuscripts in this area that focus on very general situations and implement Markov chain or MCEM algorithms, we focus on simpler, random intercept settings and give a collection of user-friendly examples and reproducible code. Marginally, the link-probit-normal model is obtained by a non-linear model on a discretized multivariate normal distribution, and thus can be thought of as a special case of discretizing a multivariate T distribution (as the degrees of freedom go to infinity). We also consider the larger class of multivariate T marginal models and illustrate how these models can be used to closely approximate a logit link.
Categorical Data Analysis
Caffo, Brian S. and Griswold, Michael , "A User-Friendly Introduction to Link-Probit-Normal Models" (August 2005). Johns Hopkins University, Dept. of Biostatistics Working Papers. Working Paper 85.