Alternative measures of the explanatory power of multivariate probit models with continuous or ordinal responses

In this paper R2-type measures of the explanatory power of multivariate linear and categorical probit models proposed in the literature are reviewed and their deficiencies are discussed. It is argued that a measure of the explanatory power should take into account the components which are explicitely modeled when a regression model is estimated while it should be indifferent to components not explicitely modeled. Based on this view three different measures for multivariate probit models are proposed. Results of a simulation study are presented designed to compare two measures in various situations and evaluate the BCa bootstrap technique for testing the hypothesis that the corresponding measure is zero and to calculate approximate confidence intervals. The BCa bootstrap technique turned out to work quite well for a wide range of situations, but may lead to misleading results if the true values of the corresponding measure is close to zero.