Measure of departure from conditional marginal homogeneity for square contingency tables with ordered categories

Abstract

Square contingency tables are a special case commonly used in various fields to analyze categorical data. Although several analysis methods have been developed to examine marginal homogeneity (MH) in these tables, existing measures are single-summary ones. To date, a visualization approach has yet to be proposed to intuitively depict the results of MH analysis. Current measures used to assess the degree of departure from MH are based on entropy such as the Kullback-Leibler divergence and do not satisfy distance postulates. Hence, the current measures are not conducive to visualization. Herein we present a measure utilizing the Matusita distance and introduce a visualization technique that employs sub-measures of categorical data. Through multiple examples, we demonstrate the meaningfulness of our visualization approach and validate its usefulness to provide insightful interpretations.

A Bivariate Index for Visually Measuring Marginal Inhomogeneity in Square Tables

“…For square tables with ordered categories, some indexes that measure the degree of deviation from marginal homogeneity have been proposed other than indexes ϕ (λ) and ψ (λ) . For example, Tahata, Tajima and Tomizawa (2006), Tahata, Iwashita and Tomizawa (2006), and Tahata, Iwashita and Tomizawa (2008). Under the extended marginal homogeneity model or the marginal logit model, these indexes cannot be simply (as a function of parameter τ or θ) expressed such as the indexes ϕ (λ) and ψ (λ) .…”

Asymmetry Index on Marginal Homogeneity for Square Contingency Tables with Ordered Categories

“…Example 2: Consider the unaided vision data in Table 1(b), taken from [9]. We see from Copyright © 2012 SciRes.…”

“…ity (i.e., the r-lef he -r 8. Discussion [6,7] considered the measures to represent the degree of departure from MH. The present paper has considered two types of maximum marginal inhomogene lowe t-marginal inhomogeneity and t upper ightmarginal inhomogeneity).…”

“…The present paper has considered two types of maximum marginal inhomogene lowe t-marginal inhomogeneity and t upper ightmarginal inhomogeneity). The measures in [6,7] take the value 1 in two types of maximum marginal inhomogeneity. The measures  and  in the present paper can distinguish these two kinds of maximum marginal inhomogeneity by the values -1 or 1 although the measures in [6,7] cannot distinguish them.…”

“…We are also interested in considering the other structure of weak MH. The measures to represent the degree of departure from MH are given by, for example, [6,7]. When the structure of weak MH does not hold, we are interested in measuring what degree the departure from weak MH is.…”