Point-symmetry models and decomposition for collapsed square contingency tables

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.     

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.     

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.     

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.     

Marginal asymmetry model for square contingency tables with ordered categories

For the analysis of square contingency tables with ordered categories, this paper proposes a model which indicates the structure of marginal asymmetry. The model states that the absolute values of logarithm of ratio of the cumulative probability that an observation will fall in row category i or below and column category i+1 or above to the corresponding cumulative probability that the observation falls in column category i or below and row category i+1 or above are constant for every i. We deal with the estimation problem for the model parameter and goodness-of-fit tests. Also we discuss the relationships between the model and a measure which represents the degree of departure from marginal homogeneity. Examples are given.     

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 marginal homogeneity using marginal odds for multi-way tables with ordered categories

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.     

Quasi-asymmetry model for square tables with nominal categories

For an R×R square contingency table with nominal categories, the present paper proposes a model which indicates that the absolute values of log odds of the odds ratio for rows i and j and columns j and R to the corresponding symmetric odds ratio for rows j and R and columns i and j are constant for every i<j<R. The model is an extension of the quasi-symmetry model and states a structure of asymmetry of odds ratios. An example is given.     

Double linear diagonals-parameter symmetry and decomposition of double symmetry for square tables

For square contingency tables with ordered category, the present paper proposes the double linear diagonals-parameter symmetry (D-LDPS) model which implies the structure of both asymmetry with respect to the main diagonal and with respect to the reverse diagonal in the table. The D-LDPS model may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate normal distribution with equal marginal variances. The present paper also gives the orthogonal decomposition of the double symmetry model into the D-LDPS model and the double marginal mean equality model. An example is given.