Generalized Linear Mixed Models for Categorical and Ordinal Responses

Abstract

AbstractAccording to Agresti (2013), a multinomial distribution is a generalization of a binomial distribution in cases with more than two possible ordered (ordinal) or unordered (nominal) outcomes. Given a response with more than two possible outcomes and independent trials with probabilities of similar category for each trial, the distribution of counts across categories follows a multinomial distribution. Quinn and Keough (2002) believe that several methods exist for multinomial data analysis. The most common form of categorical data analysis in biological sciences, which results in frequency counts, is creating cross-tabulations or contingency tables and chi-squared tests to examine associations between two or more categorical variables. However, such an approach is ill suited for a study aimed at estimating the response when there is a change in the explanatory variable(s), as contingency tables are used to analyze the association between variables without considering a predictor or response variable. In this analysis, the results are valid as long as less than 20% of the cells have an expected count less than five and none are less than one (Logan 2010). Fisher’s exact test extends the chi-squared test in studies involving small sample sizes.