Catanova for Two-Way Contingency Tables with Ordinal Variables Using Orthogonal Polynomials

ABSTRACT

The analysis of variance of cross-classified (categorical) data (CATANOVA) is a technique designed to identify the variation between treatments of interest to the researcher. There are well-established links between CATANOVA and the Goodman and Kruskal tau statistic as well as the Light and Margolin R ² for the purposes of the graphical identification of this variation.

The aim of this article is to present a partition of the numerator of the tau statistic, or equivalently, the BSS measure in the CATANOVA framework, into location, dispersion, and higher order components. Even if a CATANOVA identifies an overall lack of variation, by considering this partition and calculations derived from them, it is possible to identify hidden, but statistically significant, sources of variation.

Key Words:

• Goodman and Kruskal tau statistic
• Location, dispersion, and higher-order components
• Orthogonal polynomials

Mathematics Subject Classification:

• Primary 62H25, 62H20
• Secondary 62P10

Acknowledgment

The authors whish to thank the anonymous referees for their helpful remarks and suggestions.