Probit and Logit Models: Differences in the Multivariate Realm

Current opinion regarding the selection of link function in binary response models is that the probit and logit links give essentially similar results. This seems to be true for univariate binary response models; however, for multivariate binary response models such advice is misleading. We address a gap in the literature by empirically examining the relationship between link function selection and model fit in two classes of multivariate binary response models. We find clear evidence that model fit can be improved by the selection of the appropriate link even in small data sets. In multivariate link function models, the logit link provides better fit in the presence of extreme independent variable levels. Conversely, model fit in random effects models with moderate size data sets is improved generally by selecting the probit link.