Scoring functions for ordered classifications in statistical analysis

The problem of scoring ordered classifications prior to the further statistical analysis is discussed. A review of some methods of scoring is provided. This includes linear transformations of integer scores, where previous applications to two way classifications are introduced. Also reviewed are scores based on canonical correlations, maximum likelihood scores under assumed logistic distributions for variables, ridits, and conditional mean scoring functions. The latter are shown to satisfy a reasonable set of postulates, and demonstrates that some earlier attempts to do this were incomplete. Examples of the conditional mean scoring function under different distributional assumptions are given. Methods based on compounded functions of proportions for categorical data are applied to many of the scores reviewed and introduced. Appropriate algorithms for these methods are introduced and exemplified. Through the medium of a range of existing data sets the sensitivity of their results to differing scoring systems applied to two way classifications is examined. It is seen that apart from data arising from highly skewed distributions little is to be lost by using simple integer scores.