Visualising Departures from Symmetry and Bowker’s <i>X</i>² Statistic

Abstract

Sometimes, the same categorical variable is studied over different time periods or across different cohorts at the same time. One may consider, for example, a study of voting behaviour of different age groups across different elections, or the study of the same variable exposed to a child and a parent. For such studies, it is interesting to investigate how similar, or different, the variable is between the two time points or cohorts and so a study of the departure from symmetry of the variable is important. In this paper, we present a method of visualising any departures from symmetry using correspondence analysis. Typically, correspondence analysis uses Pearson’s chi-squared statistic as the foundation for all of its numerical and visual features. In the case of studying the symmetry of a variable, Bowker’s chi-squared statistic, presented in 1948, provides a simple numerical means of assessing symmetry. Therefore, this paper shall discuss how a correspondence analysis can be performed to study the symmetry (or lack thereof) of a categorical variable when Bowker’s statistic is considered. The technique presented here provides an extension to the approach developed by Michael Greenacre in 2000.

Keywords

• correspondence analysis
• Bowker’s chi-squared statistic
• singular value decomposition
• skew-symmetric matrix