Multilevel Models for Ordinal and Nominal Variables

Abstract

IntroductionReflecting the usefulness of multilevel analysis and the importance of categorical outcomes in many areas of research, generalization of multilevel models for categorical outcomes has been an active area of statistical research. For dichotomous response data, several approaches adopting either a logistic or probit regression model and various methods for incorporating and estimating the influence of the random effects have been developed [9,21,34,37,103,115]. Several review articles [31,39,76,90] have discussed and compared some of these models and their estimation procedures. Also, Snijders and Bosker [99, chap. 14] provide a practical summary of the multilevel logistic regression model and the various procedures for estimating its parameters. As these sources indicate, the multilevel logistic regression model is a very popular choice for analysis of dichotomous data.Extending the methods for dichotomous responses to ordinal response data has also been actively pursued [4,29,30,44,48,58,106,113]. Again, developments have been mainly in terms of logistic and probit regression models, and many of these are reviewed in Agresti and Natarajan [5]. Because the proportional odds model described by McCullagh [71], which is based on the logistic regression formulation, is a common choice for analysis of ordinal data, many of the multilevel models for ordinal data are generalizations of this model. The proportional odds model characterizes the ordinal responses in C categories in terms of C−1 cumulative category comparisons, specifically, C−1 cumulative logits (i.e., log odds) of the ordinal responses. In the proportional odds model, the covariate effects are assumed to be the same across these cumulative logits, or proportional across the cumulative odds. As noted by Peterson and Harrell [77], however, examples of non-proportional odds are

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