Search

Summary For square contingency tables with ordered categories, Iki, Tahata and Tomizawa (2012) considered a measure to represent the degree of departure from marginal homogeneity. However, the maximum value of this measure cannot distinguish two kinds of marginal inhomogeneity. The present paper proposes a measure which can distinguish two kinds of marginal inhomogeneity. In particular, the proposed measure is useful for representing the degree of departure from marginal homogeneity when the marginal cumulative logistic model holds.

For square contingency tables, a double symmetry model having a matrix structure that combines both symmetry and point symmetry was proposed. Also, an index which represents the degree of departure from double symmetry was proposed. However, this index cannot simultaneously characterize the degree of departure from symmetry and the degree of departure from point symmetry. For measuring the degree of departure from double symmetry, the present paper proposes a bivariate index vector that can simultaneously characterize the degree of departure from symmetry and the degree of departure from point symmetry.

The purpose of this paper is to propose a new model of asymmetry for square contingency tables with ordered categories. The new model may be appropriate for a square contingency table if it is reasonable to assume an underlying bivariate t-distribution with different marginal variances having any degrees of freedom. As the degrees of freedom becomes larger, the proposed model approaches the extended linear diagonals-parameter symmetry model, which may be appropriate for a square table if it is reasonable to assume an underlying bivariate normal distribution. The simulation study based on bivariate t-distribution is given. An example is given.

For square contingency tables with ordered categories, the present article proposes a two-dimensional measure vector with existing indexes to the elements that can simultaneously analyze the degree of asymmetry and the directionality for two kinds of asymmetries. The proposed measure vector would be useful to measure the degree of asymmetry, while distinguishing the directionality for two kinds of asymmetries. Also, the proposed measure vector would be useful for visually comparing the degrees of asymmetry in several square contingency tables using confidence region.

For the analysis of two-way contingency tables with ordered categories, Yamamoto, Tahata, Suzuki, and Tomizawa (2011) considered a measure to represent the degree of departure from marginal point-symmetry. The maximum value of the measure cannot distinguish two kinds of marginal complete asymmetry with respect to the midpoint. A measure is proposed which can distinguish two kinds of marginal asymmetry with respect to the midpoint. It also gives large-sample confidence interval for the proposed measure.

This study proposes original symmetry models with a structure wherein the local odds ratios increase exponentially as the row number of the square contingency table increases in each sub-table. The proposed models provide a better fit than the existing models.

For square contingency tables, the present paper newly considers the partial symmetry model which indicates that there is a symmetric structure of probabilities for at least one of pairs of symmetric cells. It also proposes the measure to express the degree of departure from the partial symmetry model. Examples are given.

For multi-way tables with ordered categories, the present paper gives a decomposition of the point-symmetry model into the ordinal quasi point-symmetry and equality of point-symmetric marginal moments. The ordinal quasi point-symmetry model indicates asymmetry for cell probabilities with respect to the center point in the table.

For square contingency tables with ordered categories, there would be some situations that one would like to analyze them by using collapsed 3×3 tables combining some adjacent categories in the original table. This paper considers the marginal homogeneity for collapsed tables and proposes a measure which represents the degree of departure from the marginal homogeneity. The proposed measure lies between 0 and 1, and it takes zero when the marginal homogeneity holds. Examples are given.

For square contingency tables, Caussinus (1965) considered the quasi-symmetry (QS) model. Kateri and Agresti (2007) considered the ordinal quasi-symmetry (OQS[ f ]) model based on f -divergence. The present paper gives a decomposition of the symmetry (S) model into the OQS[ f ] and marginal mean equality models. It also shows that the test statistic for goodness-of-fit of the S model is asymptotically equivalent to the sum of those for the decomposed models.

For square contingency tables with ordered categories, this paper gives the theorem that the symmetry model holds if and only if the following three models hold: (1) extended palindromic symmetry, (2) equality of marginal moments, and (3) subsymmetry of cumulative probabilities from the upper right and lower left corners of the table.

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.

For square contingency tables with ordered categories, there may be some cases that one wants to analyze them by considering collapsed tables with some adjacent categories combined in the original table. This paper considers the symmetry model for collapsed square contingency tables and proposes a measure to represent the degree of departure from symmetry. The proposed measure is defined as the arithmetic mean of submeasures each of which represents the degree of departure from symmetry for each collapsed 3×3 table. Each submeasure also represents the mean of power-divergence or diversity index for each collapsed table. Examples are given.

For square contingency tables, we propose a quasi-symmetry model with an exponential form along subdiagonal and give the theorem that Tomizawa’s (1992) diagonal exponent symmetry model holds if and only if the proposed model and marginal means equality model hold with the orthogonality of test statistics.

For the analysis of square contingency tables with the same row and column ordinal classifications, McCullagh (1978) considered the palindromic symmetry (PS) model, which has a multiplicative form for cumulative probabilities that an observation will fall in row (column) category i or below and column (row) category j (> i) or above. The present paper proposes a modified PS model which indicates that (1) the symmetric odds for cumulative probabilities with distance 1 from main diagonal of the table are constant and (2) there is the structure of quasi-symmetry for them with distance k (≥ 2). Also the present paper gives the decomposition of the symmetry model using the proposed model. Examples are given.

For square contingency tables, this paper considers the local symmetry model which indicates that there is a symmetric structure of probabilities for only one of pairs of symmetric cells. Also it proposes the measure to express the degree of departure from the local symmetry model. The measure is expressed as the weighted harmonic mean of the diversity index including the Shannon entropy. Examples are given.

For square contingency tables with ordered categories, there may be some cases that one wants to analyze them by considering collapsed tables with some adjacent categories combined in the original table. This paper proposes three kinds of new models which have the structure of point-symmetry (PS), quasi point-symmetry and marginal point-symmetry for collapsed square tables. This paper also gives a decomposition of the PS model for collapsed square tables. The father's and his daughter's occupational mobility data are analyzed using new models.

For square contingency tables with ordered categories, the present paper gives several theorems that the symmetry model holds if and only if the generalized linear diagonals-parameter symmetry model for cell probabilities and for cumulative probabilities and the mean nonequality model of row and column variables hold. It also shows the orthogonality of statistic for testing goodness-of-fit of the symmetry model. An example is given.

For square contingency tables with ordered categories, we decompose the symmetry model into three models; i.e., the palindromic symmetry, the marginal means equality, and the cumulative subsymmetry models. The palindromic symmetry model is also decomposed into the generalized palindromic symmetry and the extended marginal homogeneity models. The decompositions are applied to the unaided vision data.

For square contingency tables with ordered categories, the present paper proposes an asymmetry model with m-additional parameters, which indicates (1) the generalized marginal homogeneity and (2) the structure of quasi-symmetry for cumulative probabilities. The proposed model includes a modified palindromic symmetry model by Iki, Oda and Tomizawa [7]. Also the present paper gives the decomposition of the symmetry model using the proposed model. Examples are given.

Abtsrcat For the square contingency tables with ordered categories, there may be some cases that one wants to analyze them by considering collapsed tables with some adjacent categories combined in the original table. This paper considers some symmetry models for collapsed square contingency tables and proposes two new models which have the structure of global symmetry and that of quasi-symmetry for the collapsed square tables. Also this paper gives two kinds of decompositions of the symmetry model for collapsed square tables. The new models and decompositions are useful for analyzing the collapsed square tables. Examples are given.

For square contingency tables with ordered categories, this paper proposes a measure to represent the degree of departure from the marginal homogeneity model. It is expressed as the weighted sum of the power-divergence or Patil–Taillie diversity index, and is a function of marginal log odds ratios. The measure represents the degree of departure from the equality of the log odds that the row variable is i or below instead of i+1 or above and the log odds that the column variable is i or below instead of i+1 or above for every i. The measure is also extended to multi-way tables. Examples are given.

Problem statement: First, this study considers how the structure of symmetry for probabilities is decomposed into two structures. Secondly, this study infers the structure of unknown probabilities which indicates how the right eye is better (or worse) than the left eye for three kinds of data on unaided distance vision of (1) women in Britain, (2) students in an university of Japan and (3) pupils in elementary schools in Tokyo, Japan. This study proposes a new decomposition of symmetry model for probabilities and analyzes these vision data using the decomposition. Approach: This study considers a new decomposition theorem that for the probabilities the symmetry model (indicates that the right eye vision is symmetric to the left eye vision) holds. Also this study analyzes the vision data using this decomposition. Results: From the statistical approach, we can see that (1) for the vision data of women, the right eye is better than the left eye and the mean of right eye is not equal to the mean of left eye, (2) for the vision data of students, the right eye is worse than the left eye and the mean of right eye is not equal to the mean of left eye and (3) for the vision data of pupils, the right eye is symmetric to the left eye and the mean of right eye is equal to the mean of left eye. Conclusion: When the symmetry model fits the data poorly, this new decomposition is useful for seeing which of decomposed two models influences stronger. We can see the structure of asymmetry for vision data in more details.

For square contingency tables with ordered categories, the present paper considers two kinds of weak marginal homogeneity and gives measures to represent the degree of departure from weak marginal homogeneity. The proposed measures lie between –1 to 1. When the marginal cumulative logistic model or the extended marginal homogeneity model holds, the proposed measures represent the degree of departure from marginal homogeneity. Using these measures, three kinds of unaided distance vision data are analyzed.

For square contingency tables with ordered categories, Iki, Tahata and Tomizawa (12) considered the ridit score type quasi-symmetry model. The present paper proposes an extension of that model having one more parameter. Using the proposed model, the global symmetry model and the marginal mean ridits equality model, this paper also gives a theorem such that the symmetry model holds if and only if these three models hold. Examples are given.

For three-way contingency tables with two ordinal and one nominal variables, this paper shows that (i) the conditional independence model holds if and only if conditional covariance between ordinal variables at each level of the nominal variable equals zero and the heterogeneous linear-by-linear association model holds, (ii) the conditional independence model holds if and only if an average conditional covariance between ordinal variables across levels of the nominal variable equals zero and the homogeneous linear-by-linear association model holds, and (iii) the likelihood ratio test statistic for the conditional independence model is asymptotically equivalent to the sum of the likelihood ratio test statistic for the decomposed models.

Problem statement: For square contingency tables with ordered categories, this study considers four kinds of extensions of the marginal homogeneity model and gives decompositions for the ordinal quasi-symmetry model. The decompositions are extensions of some existing decompositions. Approach: This study gives a decomposition theorem that the ordinal quasi-symmetry model holds if and only if the quasi-symmetry model and the proposed weighted marginal homogeneity model hold. An example is given. Results: For the data of cross-classification of father's and his son's occupational status in Denmark, the decomposition of the ordinal quasi-symmetry model is applied and the detailed analysis is given. Conclusion: When the ordinal quasi-symmetry model fits the data poorly, this decomposition is useful for seeing which of decomposed two models influences stronger.

For two-way contingency tables with a nominal explanatory and an ordinal response variables, the present paper proposes a proportional reduction in error measure, which describes the relative decrease in the probability of making an error in predicting the value of the response variable when the value of the explanatory variable is known, as opposed to when it is not known. The measure depends on the order of listing the response categories. Examples are shown.

For multi-way tables with same categories, Bhapkar and Darroch [4] and Agresti [2], p. 440, considered the general order quasi-symmetry model and the ordinal quasi-symmetry model, respectively. Yamamoto, Iwashita and Tomizawa [16] considered the three kinds of linear ordinal quasi-symmetry models, which are special cases of the first and second order quasi-symmetry models. For multi-way tables with same ordinal categories, the present paper proposes the general linear ordinal quasi-symmetry models, which are special cases of the general order quasi-symmetry model, and gives new decomposition of the symmetry model. Moreover it is shown that the likelihood ratio statistic for testing goodness-of-fit of the symmetry model is asymptotically equivalent to the sum of those for testing goodness-of-fit of the decomposed models.