A global view on parametric and nonparametric approaches to the analysis of ordered categorical data

Rank approaches are very common in the analysis of ordered categorical data but can only be interpreted on an experiment-wise level. Therefore, parametric tests from linear models, although based on metric structures, are used frequently to analyze this type of data. So the questions arise 1. what parametric tests measure in this context and 2. whether the rank approach could be modified to achieve a global level of interpretation. A possible solution to question 2. offers the so called ridit approach, which is based on known reference distributions. In this paper we discuss a global view that shows how rank analysis and ridit analyses are related and how parametric procedures fit into the same framework. The use of the uniform distribution as a reference in the ridit approach gives an explanation to question 1. The asymptotic multivariate normality of the effect estimators is shown and robust test statistics are discussed. Type I and type II error rates are examined in simulation studies and the approach is applied to a toxicological example.