ABSTRACT
This chapter generalizes the simplest α for nominal data to phenomena that exhibit various orderings. It does not require the reader to be familiar with the previous chapters. It repeats the definition of the canonical form of reliability data and the construction of observed and expected coincidences to focus on what matters here.
Given the matrices of observed and expected coincidences that are distinguished only by whether coincidences are in the off-diagonal cells (summed to the observed and expected disagreements) or not, this chapter defines six difference functions to weigh these coincidences according to the ordering of phenomena to be recorded. It provides the standardized definitions for each and exemplifies their weights by means of 10-by-10 difference matrices: for nominal (unordered) data, for ordinal (rank ordered) data, for interval (linearly ordered) scales, for ratio data (proportions with one absolute zero value), for polar opposite data (being constrained by two extremes), and for circular data (recursions that come back to their origins).
It discusses their strengths in terms of what the αs for particular metrics include or omit respectively and it visualizes the extremes of the interval α and ratio α relative to the nominal α to demonstrate their comparability. This is followed by numerical examples to show how they differ.
The chapter also defines the criteria for using less formalized metrics, whether obtained by experiments (e.g., networks), measures of configurations (maps of terrains), or in conformity with theories.
