Probit models are commonly used to describe the relation between one or more independent variables and a quantal response (e.g., alive or dead; mild, moderate, or severe degree of illness). They are often motivated from the idea of a latent continuous variable underlying the quantal response and the existence of thresholds, such that if the unobserved continuous variable is below a certain threshold then the observed quantal variable is in a certain response category. For example, if toxicity is below the critical threshold the subject is alive, otherwise the subject is dead. However, the threshold concept and the existence of an underlying continuous variable are not necessary for model interpretability. The usual probit model is used when the sample consists of n independent cases with a single quantal outcome variable, while multivariate probit models are extending the usual probit models to cases of several quantal outcome variables observed on the same individual. In contrast to probit models for independent data, correlated probit models are useful in analyzing one or more quantal response variables when these variables are repeatedly measured within clusters or over time. For example, it may be of interest to model the probability of the presence of a particular illness or symptom in the members of the same family or on the same individual over the course of follow-up in a longitudinal study. As measurements on individuals within the same family or repeated measurements on the same individual are likely to be more highly correlated than measurements on unrelated individuals, special statistical methods are necessary to account for this correlation.

The goals of this entry are to familiarize the reader with correlated probit models and to provide enough information and relevant references so that these models can be successfully applied using available statistical software. To achieve this goal we first introduce the probit, multivariate probit, and correlated probit models, followed by a discussion on model interpretation, identifiability, and model fitting. An illustrative example of a correlated probit model with both multiple response variables and clustering is provided, followed by a brief overview of available statistical software packages that can be used for model fitting.