Abstract
For the analysis of square contingency tables, Tomizawa (1994), Tomizawa, Seo, and Yamamoto (1998), and Tomizawa, Miyamoto, and Hatanaka (2001) considered measures to represent the degree of departure from symmetry. However, the maximum value of these measures cannot distinguish two kinds of complete asymmetry (say, complete-upper-asymmetry and complete-lower-asymmetry). The present paper proposes a measure which can distinguish two kinds of complete asymmetry for square tables with ordered categories. Especially the proposed measure is useful for representingthe degree of departure from symmetry when the conditional symmetry model holds. Examples are given.
Highlights
Consider an R × R square contingency table
The purpose of this paper is to propose such a measure which can distinguish two kinds of complete asymmetry for square contingency tables with ordered categories
For comparisons between several tables, if it can be estimated that there is a structure of conditional symmetry in each table, the measure φ would be adequate for representing and comparing the degree of departure from the symmetry toward the complete-upper-asymmetry and toward the complete-lower-asymmetry
Summary
Consider an R × R square contingency table. Let pij denote the probability that an observation will fall in the ith row and jth column of the table (i = 1, . . . , R, j = 1, . . . , R). For square contingency tables with ordered categories, Tomizawa et al (2001) proposed another power-divergence-type measure to represent the degree of departure from symmetry the detail is omitted. Using the measure Φ(λ) (and using the Tomizawa et al, 2001 measure), we cannot distinguish two kinds of complete asymmetry, namely, that the complete asymmetry is which of (i) pij = 0 ( pji > 0) for all i < j, or (ii) pji = 0 ( pij > 0) for all i < j (i.e., which of (i) all observations concentrate in the lower left triangle cells, or (ii) those concentrate only in the upper right triangle cells) Since these two kinds of complete asymmetry indicate the opposite different maximum departures from symmetry, we are interested in proposing a measure which can take the different values for them. The purpose of this paper is to propose such a measure which can distinguish two kinds of complete asymmetry for square contingency tables with ordered categories
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