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Abstract:We propose a Multivariate Logistic Distance (MLD) model for the analysis of multiple binary responses in the presence of predictors. The MLD model can be used to simultaneously assess the dimensional/factorial structure of the data and to study the effect of the predictor variables on each of the response variables. To enhance interpretation, the results of the proposed model can be graphically represented in a biplot, showing predictor variable axes, the categories of the response variables and the subjects' positions. The interpretation of the biplot uses a distance rule. The MLD model belongs to the family of marginal models for multivariate responses, as opposed to latent variable models and conditionally specified models. By setting the distance between the two categories of every response variable to be equal, the MLD model becomes equivalent to a marginal model for multivariate binary data estimated using a GEE method. In that case the MLD model can be fitted using existing statistical packages with a GEE procedure, e.g., the genmod procedure from SAS or the geepack package from R. Without the equality constraint, the MLD model is a general model which can be fitted by its own right. We applied the proposed model to empirical data to illustrate its advantages.

The ideal point classification (IPC) model was originally proposed for analysing multinomial data in the presence of predictors. In this paper, we studied properties of the IPC model for analysing bivariate binary data with a specific focus on three quantities: (1) the marginal probabilities; (2) the association structure between the two binary responses; and (3) the joint probabilities. We found that the IPC model with a specific class point configuration represents either the marginal probabilities or the association structure. However, the IPC model is not able to represent both quantities at the same time. We then derived a new parametrization of the model, the bivariate IPC (BIPC) model, which is able to represent both the marginal probabilities and the association structure. Like the standard IPC model, the results of the BIPC model can be displayed in a biplot, from which the effects of predictors on the binary responses and on their association can be read. We will illustrate our findings with a psychological example relating personality traits to depression and anxiety disorders.