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.

翻译：本文提出将多项模型中的典型参数分解为一组参与者得分与类别得分。两组得分均受线性约束，以体现参与者与类别的外部信息。针对分解参数的估计，我们推导出一种优化-最小化算法。我们特别关注类别表示二值响应变量剖面的情形。在此情况下，多项模型转化为多元二值响应变量的回归模型，研究者可能关注参与者外部变量（即预测变量）与单个二值响应变量的关系，或该预测变量与二值响应变量间关联性的关系。我们针对这些关系推导出基于对数发生比或对数发生比变化的解释规则。本文探讨了所提出的多项典型分解与对数线性模型、多项逻辑回归、多项降秩逻辑回归及双重约束对应分析之间的理论联系。最后通过两个实证数据集展示了该方法的应用。