Correspondence Analysis: Theory, Practice and New Strategies

Chapter 2Pearson's Chi-Squared Statistic

2.1 Introduction

There are a great number of books and articles that describe in varying details the historical development of procedures designed to analyse the association between categorical variables. Here we will not give a comprehensive history, although it is important to highlight the importance of Pearson's chi-squared statistic, analogous measures, and their role in the correspondence analysis of a contingency table. As we shall see in the coming chapters, it is also the central measure of association around which the classical approach to correspondence analysis is based. In this chapter, we shall describe some of the key contributions to the development of association for categorical data that have arisen. Many of these measures will be considered for the adaptation of correspondence analysis. There are many excellent reviews on the history of the measure of association in contingency tables. In particular, the reader is invited to consider those of Goodman and Kruskal (1959), and Agresti (2002, Chapter 16). Yule and Kendall (1950), Bishop et al. (1975) and Liebetrau (1983) also provided excellent discussions of a large number of measures of association for contingency tables.

2.2 Pearson's Chi-Squared Statistic

2.2.1 Notation

Consider two categorical variables X and Y where X consists of I categories and Y consists of J categories. We shall denote the I categories of X by X₁, X₂, ..., X_I and the J categories of Y by ...