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The multinomial probit model differs only slA polychoric correlation matrix is the correlation matrix between the latent variables in a multivariate probit model with no covariates. This type of matrix is commonly used as input for structural equation models when ordinal data are present. How could the Gibbs sampler for the multivariate probit model be modified to obtain the polychoric correlation matrix rather than regression coefficients and an error correlation matrix?ightly from the multivariate probit model. In the multinomial probit model, the outcomes in each dimension are all dichotomous and are mutually exclusive; that is, an individual can only take a 1 on (at most) a single outcome variable. The model is generally used in social sciences for predicting a nominal level outcome—the outcome variable is broken into a series of dummy variables with one omitted as the reference. The mutual exclusivity constraint leads to only two differences between the multinomial probit and the multivariate probit we have already discussed. First, a slightly different approach to sampling the latent data thought to underlie the observed response must be undertaken. Second, only one of the diagonal elements of the error covariance matrix must be constrained to 1 to identify the model. Regarding the simulation of the latent data, individuals are assumed to have latent traits, the maximum of which is the one for which the respondent’s observed outcome is “1.” That latent trait must be above 0. Latent traits for the other outcomes may also be above 0 but cannot be larger than that one. If an individual does not take a “1” value on any of the outcomes (i.e., his/her response is the reference outcome), all latent traits must be sampled from below 0. Develop a multinomial probit model algorithm and compare the results with those obtained using a multinomial logit procedure in another software package (see Imai and van Dyk 2005 for an in-depth consideration of various extant MCMC approaches to this model).

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