Orthogonal decomposition of point-symmetry for multiway tables

For multiway contingency tables, Wall and Lienert (Biom. J. 18:259–264, 1976) considered the point-symmetry model. For square contingency tables, Tomizawa (Biom. J. 27:895–905, 1985) gave a theorem that the point-symmetry model holds if and only if both the quasi point-symmetry and the marginal point-symmetry models hold. This paper proposes some quasi point-symmetry models and marginal point-symmetry models for multiway tables, and extends Tomizawa’s (Biom. J. 27:895–905, 1985) theorem into multiway tables. We also show that for multiway tables the likelihood ratio statistic for testing goodness of fit of the point-symmetry model is asymptotically equivalent to the sum of those for testing the quasi point-symmetry model with some order and the marginal point-symmetry model with the corresponding order. An example is given.     

Measure of departure from symmetry for the analysis of collapsed square contingency tables with ordered categories

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.     

Quasi local odds symmetry model for square contingency table with ordinal categories

This paper proposes a new model for square contingency tables. The proposed model tests the equality of local odds ratios between the one side of the main diagonal and corresponding other side and it represents the non-symmetric structure of the square contingency table. The proposed model is compared with twenty-five models introduced for analysing the square contingency tables for both symmetric and non-symmetric structures. The results show that the proposed model provides best fit performance than other existing models for square contingency tables.     

Measure of Asymmetry for Square Contingency Tables Having Ordered Categories

For the analysis of square contingency tables with nominal categories, Tomizawa and coworkers have considered measures that represent the degree of departure from symmetry. This paper proposes a measure that represents the degree of asymmetry for square contingency tables with ordered categories (instead of those with nominal categories). The measure proposed is expressed using the Cressie–Read power-divergence or Patil–Taillie diversity index, defined for the cumulative probabilities that an observation falls in row (column) category i or below and column (row) category j (> i ) or above. The measure depends on the order of listing the categories. It should be useful for comparing the degree of asymmetry in several tables with ordered categories. The relationship between the measure and the normal distribution is shown.     

Power-divergence-type measure of departure from diagonals-parameter symmetry for square contingency tables with ordered categories

For the analysis of square contingency tables with ordered categories, Goodman considered the diagonals-parameter symmetry (DPS) model. This paper proposes a measure to represent the degree of departure from the DPS model. The proposed measure is expressed by applying Read and Cressie’s power-divergence or Patil and Taillie’s diversity index. The measure would be useful for comparing the degree of departure from the DPS model in several tables. Examples are given.     

Diagonal,weighted marginal homogeneity model and decompositions for linear diagonals-parameter symmetry model

For square contingency tables rith ordered categories, this paper proposes two kinds of extensions of marginal homogeneity model and gives decompositions for the Liseer diagonals-parameter symmetry model considered by Agresti (1983a) using the proposed models- The proposed models are also applied to an unaided vision data.     

A model of uniform association plus two-diagonalsparameter and its application to occupational mobility table data

For the analysis of square contingency tables with ordered categories, Agresti (1988) introduced a model having the structure of uniform association plus a main-diagonal parameter. This paper extends that model. The extended model has the structure of uniform association plus two-diagonals-parameter, and it is a special case of the quasi-uniform association model introduced by Goodman (1979). The Danish occupational mobility table data are analyzed using the models introduced here.     

Modeling symmetry models in square contingency tables with ordered categories

This paper provides alternative methods for fitting symmetry and diagonal-parameters symmetry models to square tables having ordered categories. We demonstrate here the implementation of the class of models discussed in Goodman (1979c) using GEN-MOD in SAS. We also provide procedures for testing hypotheses involving model parameters. The methodology provided here can readily be used to fit the class of models discussed in Lawal and Upton (1995). If desired, composite models can be fitted. Two data sets, the 4 × 4 unaided distance vision of 4746 Japanese students Tomizawa (1985) and the 5 × 5 British social mobility data Glass (1954) are employed to demonstrate the fitting of these models. Results obtained are consistent with those from Goodman (1972, 1979c, 1986) and Tomizawa (1985, 1987).     

New critical values for the generalized t and generalized rank-sum procedures

Student's t test as well as Wilcoxon's rank-sum test may be inefficient in situations where treatments bring about changes in both location and scale. In order to rectify this situation, O'Brien (1988, Journal of the American Statistical Association 83, 52-61) has proposed two new statistics, the generalized t and generalized rank-sum procedures, which may be much more powerful than their traditional counterparts in such situations. Recently, however, Blair and Morel (1991, Statistics in Medicine in press) have shown that referencing these new statistics to standard F tables as recommended by O'Brien results in inflations of Type I errors. This paper provides tables of critical values which do not produce such inflations. Use of these new critical values results in Type I error rates near nominal levels for the generalized t statistic and slightly conservative rates for the generalized rank-sum test. In addition to the critical values, some new power results are given for the generalized tests.     

DECOMPOSITIONS FOR CONDITIONAL SYMMETRY MODEL INTO PALINDROMIC SYMMETRY AND MODIFIED MARGINAL HOMOGENEITY MODELS

For square contingency tables with ordered categories, the conditional symmetry model is decomposed into the palindromic symmetry and the modified marginal homogeneity models, and moreover into the generalized palindromic symmetry model and two other models. The data on unaided vision first analysed by Stuart (1953, 1955) is analysed again by using the decompositions.     

Marginal inhomogeneity models for square contingency tables with nominal categories

For the analysis of square contingency tables with nominal categories, this paper proposes two kinds of models that indicate the structure of marginal inhomogeneity. One model states that the absolute values of log odds of the row marginal probability to the corresponding column marginal probability for each category i are constant for every i. The other model states that, on the condition that an observation falls in one of the off-diagonal cells in the square table, the absolute values of log odds of the conditional row marginal probability to the corresponding conditional column marginal probability for each category i are constant for every i. These models are used when the marginal homogeneity model does not hold, and the values of parameters in the models are useful for seeing the degree of departure from marginal homogeneity for the data on a nominal scale. Examples are given.     

A class of exact UMP unbiased tests for conditional symmetry in small-sample square contingency tables

Testing conditional symmetry against various alternative diagonals-parameter symmetry models often provides a point of departure in studies of square contingency tables with ordered categories. Typically, chi-square or likelihood-ratio tests are used for such purposes. Since these tests depend on the validity of asymptotic approximation, they may be inappropriate in small-sample situations where exact tests are required. In this paper, we apply the theory of UMP unbiased tests to develop a class of exact tests for conditional symmetry in small samples. Oesophageal cancer and longitudinal income data are used to illustrate the approach.     

Power-divergence-type measure of departure from symmetry for square contingency tables that have nominal categories

For square contingency tables that have nominal categories, Tomizawa considered two kinds of measure to represent the degree of departure from symmetry. This paper proposes a generalization of those measures. The proposed measure is expressed by using the average of the power divergence of Cressie and Read, or the average of the diversity index of Patil and Taillie. Special cases of the proposed measure include Tomizawa's measures. The proposed measure would be useful for comparing the degree of departure from symmetry in several tables.     

Diagonal uniform association symmetry modelsfor cumulative probabilities in square tables

For the analysis of square contingency tables with ordered categories, Tomizawa (1991) considered the diagonal uniform association symmetry (DUS) model, which has a multiplicative form for cell probabilities and has the structure of uniform association in the tables constructed using two diagonals that are equidistant from the main diagonal. This paper proposes another DUS model which has a similar multiplicative form for cumulative probabilities. The model indicates that the odds that an observation will fall in row category i or below and column category i+k or above, instead of in column category i or below and row category i+k or above, increase (decrease) exponentially as the cutpoint i increases for a fixed k. Examples are given.     

Tables for the Friedman rank test

This note presents tables for Friedman's test for two-way analysis of variance by ranks. These tables are more accurate than those that are presented in the literature. After intensive simulations, we have found for particular critical values some discrepancies with tables published earlier. The tables are also more extensive than those previously available.     

KENDALL'S TAU AND CONTINGENCY TABLES

Kendall's tau is a coefficient of concordance between two rankings of n objects. Its definition and large sample normal approximation are easily extended to the case where one of the rankings contains ties. In this paper, definition and normal approximation are extended further to the case where both rankings contain ties. The results are applied to give a fully distribution-free test for two-way contingency tables with ordered categories.     

The iterative proportional fitting algorithm for adjusted agreement in a non‐inferiority diagnostic clinical trial

In this paper, we describe a method of comparing agreement between two diagnostic contingency tables after adjustment to more clinically relevant marginal distributions using the iterative proportional fitting algorithm. When the categories of a contingency table represent mild, moderate, and severe outcomes, the majority of patients often are in the mild category. Because it is often of more interest to evaluate agreement when patients are uniformly distributed among categories, we present the primary results of two clinical trials with adjustment to this structure. We also describe the relationship between the sponsor's pre‐specified agreement measure for the observed contingency table and kappa for the adjusted table; and by either criterion, we then show that the agreement of the new diagnostic tool with the standard diagnostic tool is comparably non‐inferior to the agreement of the standard diagnostic tool with itself. Copyright © 2014 John Wiley & Sons, Ltd.     

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

For the analysis of square contingency tables with ordered categories, Tomizawa et al. (S. Tomizawa, N. Miyamoto, and N. Ashihara, Measure of departure from marginal homogeneity for square contingency tables having ordered categories, Behaviormetrika 30 (2003), pp. 173–193.) and Tahata et al. (K. Tahata, T. Iwashita, and S. Tomizawa, Measure of departure from symmetry of cumulative marginal probabilities for square contingency tables with ordered categories, SUT J. Math., 42 (2006), pp. 7–29.) considered the measures which represent the degree of departure from the marginal homogeneity (MH) model. The present paper proposes a measure that represents the degree of departure from the conditional MH, given that an observation will fall in one of the off-diagonal cells of the table. The measure proposed is expressed by using the Cressie–Read power-divergence or the Patil–Taillie diversity index, which is applied for the conditional cumulative marginal probabilities given that an observation will fall in one of the off-diagonal cells of the table. When the MH model does not hold, the measure is useful for seeing how far the conditional cumulative marginal probabilities are from those with an MH structure and for comparing the degree of departure from MH in several tables. Examples are given.