A Multinomial Canonical Decomposition Model, with emphasis on the analysis of Multivariate Binary data

In this paper, we propose to decompose the canonical parameter of a multinomial model into a set of participant scores and category scores. Both sets of scores are linearly constraint to represent external information about the participants and categories. For the estimation of the parameters of the decomposition, we derive a majorization-minimization algorithm. We place special emphasis on the case where the categories represent profiles of binary response variables. In that case, the multinomial model becomes a regression model for multiple binary response variables and researchers might be interested in the relationship of an external variable for the participant (i.e., a predictor) and one of the binary response variable or in the relationship between this predictor and the association among binary response variables. We derive interpretational rules for these relationships in terms of changes in log odds or log odds ratios. Connections between our multinomial canonical decomposition and loglinear models, multinomial logistic regression, multinomial reduced rank logistic regression, and double constrained correspondence analysis are discussed. We illustrate our methodology with two empirical data sets.