A user-friendly introduction to link-probit-normal models

Brian Caffo, Michael Griswold

Research output: Contribution to journalReview articlepeer-review


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, 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 article we outline these models and discuss appropriate situations for using multivariate normal approximations for estimation. Unlike other articles in this area that focus on very general situations and implement Markov chain or MCEM algorithms, we focus on simpler settings and give a collection of user-friendly examples. Marginally, the link-probit-normal model is obtained by a nonlinear 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.

Original languageEnglish (US)
Pages (from-to)139-145
Number of pages7
JournalAmerican Statistician
Issue number2
StatePublished - May 2006


  • Binary outcomes
  • Generalized linear mixed models
  • Maximum likelihood
  • Nonlinear mixed effects models

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Statistics, Probability and Uncertainty


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