Modeling and Measuring Association for Ordinal Data
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We propose the modified Pearson goodness-of-fit statistic and consider it in the cumulative contingency tables. For the non-cumulative tables, we provide the scoring systems and the variance minimization correction term to Pearson goodness-of-fit statistic. We also put forward the average of individual odds ratios as the common odds ratio and modify the Mantel-Haenszel estimator of common odds ratio in a K x K ordinal contingency table where the category numbers of the variables are the same. The M[superscript2] statistic with scoring systems is also introduced. From the simulation results, we figure out that the chi-square statistics do not work except for those without the ordinal information. However, the M[superscript2] statistic accurately detects the association between ordinal variables in a contingency table no matter if it is sparse or not. The scoring systems can be considered together with the M[superscript2] statistic. The estimators of common odds ratio are appropriate as well. However, the estimators of common odds ratio are more appropriate for the lightly sparse tables because the test errors are relatively larger for the extremely sparse tables. The Mantel-Haenszel estimator of common odds ratio works better than the average of individual odds ratios in the extremely sparse tables.