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. External information about the participants or the categories can be used to restrict these scores. Therefore, we impose the constraint that the scores are linear combinations of the external variables. 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 effect of an external variable for the participant (i.e., a predictor) on a binary response variable or in the effect of this predictor on 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 use two empirical data sets, the first to show the relationships between a loglinear analysis approach and our modelling approach. The second data set is used as an illustration of our modelling approach and describes the model selection and interpretation in detail.