Abstract
For two-way contingency tables with ordered categories, the present paper gives a theorem that the independence model holds if and only if the logit uniform association model holds and equality of concordance and discordance for all pairs of adjacent rows and all dichotomous collapsing of the columns holds. Using the theorem, we analyze the cross-classification of duodenal ulcer patients according to operation and dumping severity.
Highlights
Consider the r × c contingency tables with ordered categories, let X and Y denote the row and column variables, and let P ( X= i,Y= j=)pij (>0) for i = 1, r and j = c Goodman [1]considered the uniform association (U) model which was defined by= pij μαi β = jθ ij (i 1,=, r; j 1, c).See Agresti ([2], p. 76)
The logit uniform association (logit U) model indicates the constant of the odds ratios for the (r −1)(c −1) 2× 2 tables obtained by taking all pairs of adjacent rows and all dichotomous collapsing of the response (Agresti [2], p. 122)
We see that the rejection of the hypothesis that the I model holds assuming that the logit U model holds is caused by the influence of the lack of structure of the CDE model, because the hypothesis that the I model holds assuming that the logit U model holds is equivalent to the CDE model from Theorem 1
Summary
Consider the r × c contingency tables with ordered categories, let X and Y denote the row and column variables, and let. Miyamoto and Sakurai [3] give the theorem that the I model holds if and only if the Pearson’s correlation coefficient ρ for X and Y equals zero and the U model holds. Miyamoto and Tomizawa [5] give the theorem that the I model holds if and only if the Spearman’s ρs equals zero and the U model holds. The logit U model indicates the constant of the odds ratios for the (r −1)(c −1) 2× 2 tables obtained by taking all pairs of adjacent rows and all dichotomous collapsing of the response { } interested in what structure of probabilities pij is necessary for obtaining the I model, in addition to the logit.
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