  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.
