Regression models for multivariate ordered responses via the Plackett distribution. (English) Zbl 1152.62031
Summary: We investigate the properties of a class of discrete multivariate distributions whose univariate marginals have ordered categories, all the bivariate marginals, like in the Plackett distribution, have log-odds ratios which do not depend on cut points and all higher-order interactions are constrained to 0. We show that this class of distributions may be interpreted as a discretized version of a multivariate continuous distribution having univariate logistic marginals. Convenient features of this class relative to the class of ordered probit models (the discretized version of the multivariate normal) are highlighted. Relevant properties of this distribution like quadratic log-linear expansion, invariance to collapsing of adjacent categories, properties related to positive dependence, marginalization and conditioning are discussed briefly. When continuous explanatory variables are available, regression models may be fitted to relate the univariate logits (as in a proportional odds model) and the log-odds ratios to covariates.
MSC:
| 62H10 | Multivariate distribution of statistics |
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
| 62J12 | Generalized linear models (logistic models) |
| 60E15 | Inequalities; stochastic orderings |
Keywords:
Plackett distribution; marginal models; global logits; proportional odds; multivariate ordered regressionReferences:
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