Theory & Methods: Partitioning Pearson’s Chi‐Squared Statistic for a Completely Ordered Three‐Way Contingency Table

Abstract

The paper presents a partition of the Pearson chi-squared statistic for triply ordered three-way contingency tables. The partition invokes orthogonal polynomials and identifies three-way association terms as well as each combination of two-way associations. This partition provides information about the structure of each variable by identifying important bivariate and trivariate associations in terms of location (linear), dispersion (quadratic) and higher order components. The significance of each term in the partition, and each association within each term can also be determined.The paper compares the chi-squared partition with the log-linear models of Agresti (1994) for multi-way contingency tables with ordinal categories, by generalizing the model proposed by Haberman (1974).

A Non-Iterative Alternative to Ordinal Log-Linear Models

“…The polynomials used as those considered by Beh [3,4], Beh and Davy [6,7], and Rayner and Best [24].…”

“…See for example Gilula [14], Gilula and Haberman [15,16], and Gilula and Ritov [17]. In this section we consider the correlation models for two-way ordinal cross -classifications as seen in Beh [3], Beh and Davy [6,7], Rayner and Best [24] and Best and Rayner [8].…”

“…For the analysis of three-way ordinal contingency tables, Beh and Davy [6,7] presented some models which are extensions of the bivariate models presented in the previous section. These trivariate models can be used to estimate the parameter values from a log-linear model of the data.…”

“…This paper presents a method which relies on orthogonal polynomials based on those in Emerson [12] and described in Beh [4] and Rayner and Best [24]. We review the alternative approach introduced in Beh and Davy [6,7] which is applicable for ordinal multi-way contingency tables. Parameter estimation using the technique described in this paper is also discussed in Beh [5] where the Pearson chi-squared statistic for a singly ordered two-way contingency table is partitioned using orthogonal polynomials for the ordered variable and generalised basic vectors for the nonordered variable.…”

Approximating Exact Test of Mutual Independence in Multiway Contingency Tables via Stochastic Approximation Monte Carlo

“…This example considers the 3 ×4×5 three-way contingency table with the ordering of the happiness, schooling and sibling variables to identify the important bivariate and trivariate moments and identify important location, dispersion and higher order components (Beh and Davy, 1998). The data classifies 1517 people according to their reported happiness, number of completed years of schooling, and number of siblings.…”