Sadao Tomizawa

Diagonals-parameter symmetry model for cumulative probabilities in square contingency tables with ordered categories

For the analysis of square contingency tables with ordered categories, Goodman (1979, Biometrika 66, 413-418) considered the diagonals-parameter symmetry (DPS) model, which has a multiplicative form for cell probabilities. This paper proposes another DPS model which has a similar 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. Special cases of the proposed model include conditional symmetry and symmetry. The relationship with a stochastic ordering for marginal distributions is also described. Moreover, the relationships between the proposed model and the (generalized) palindromnic symmetry models are described. An example is given.

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 Tomizawas measures. The proposed measure would be useful for comparing the degree of departure from symmetry in several tables.

Diagonal weighted marginal homogeneity models 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.

Theory & Methods: 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.

A Generalization of the Marginal Homogeneity Model for Square Contingency Tables With Ordered Categories

This article introduces an extended marginal homogeneity model for the analysis of square contingency tables with the same ordinal row and column classifications. The model indicates that the odds that an observation will fall in row category k or below and in column category k + 1 or above, instead of in column category k or below and in row category k + 1 or above, increase (decrease) exponentially as the cut point k increases. An example is given.

Shannon entropy type measure of departure from uniform association in cross-classifications having ordered categories

A measure is proposed to represent the degree of departure from uniform association in cross-classifications having ordered categories. The measure is expressed by using the Shannon entropy or the Kullback-Leibler information, based on the local odds ratios. It would be useful when one wants to use a single summary measure to measure distance of the local odds ratios from uniformity.

Quasi­Diagonals­Parameter Symmetry Model for Square Contingency Tables with Ordered Categories

For the analysis of square contingency tables with ordered categories, this paper introduces the quasidiagonalsparameter symmetry (QDPS) model which is an extension of the quasisymmetry (QS) model. It is shown that the QDPS model is preferable to the QS model for unaided vision data of British women earlier analysed by Stuart (1955). Various extended models of QS are expressed in terms of the local odds ratios and the other odds ratios.

Measures of departure from marginal homogeneity for contingency tables with nominal categories

For square contingency tables with nominal categories, four kinds of measure are proposed to represent the degree of departure from marginal homogeneity (MH). The measures are expressed by using the Shannon entropy or Gini concentration. They are used to measure how far the marginal distributions (or the conditional marginal distributions given that an observation will fall in one of the off-diagonal cells of the table) are distant from those with an MH structure. Two of these measures are also extended to T-way tables (T ≥ 3).

Conditional Difference Asymmetry Model for Square Contingency Tables with Nominal Categories

This paper proposes a model, which is an extension-of-symmetry model, for square contingency tables with the same nominal row and column classifications. The model states that the absolute values of difference between the conditional probability that an observation will fall in cell (i, j) on condition that it falls in cell (i, j) or (j, i) and the conditional probability that it falls in cell (j, i) on the same condition, are constant for every i≠j. The model describes a structure of asymmetry (not symmetry), and it is applied to the data on a nominal scale. An example is given.

Multiplicative models with further restrictions on the usual symmetry model

For square contingency tables with ordered categories, this paper proposes some multiplicative models with further restrictions on the usual symmetry model but they are not so strict as in the double symmetry model considered by Tomizawa (1985a). Each of the proposed models has the structure of symmetry about the main diagonal plus an asymmetry about the reverse diagonal of the square table. Each of the models is unsaturated on the main diagonal. The models are applied to two data sets of unaided distance vision.