Kiyotaka Iki

An Extended Bivariate T-Distribution Type Symmetry Model for Square Contingency Tables

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.

Decomposition of Parsimonious Independence Model Using Pearson, Kendall and Spearman's Correlations for Two-Way Contingency Tables

For two-way contingency tables with ordered categories, Tomizawa (1992) considered the parsimonious Linear-by-Linear association model. This model can be described in terms of fewer parameters than the Linear-by-Linear association model (Agresti, 1983). The purpose of this paper is (i) to define the parsimonious independence model, (ii) to show the parsimonious independence model holds if and only if the parsimonious Linear-by-Linear association model holds and the each one of various correlation coefficients is equal to zero, and (iii) show the statistic for testing the parsimonious independence model is asymptotically equivalent to the sum of test statistics for the decomposed models.

A generalized asymmetry model for square contingency tables with ordered categories

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.

Point-Symmetric Multivariate Density Function and Its Decomposition

For a -variate density function, the present paper defines the point-symmetry, quasi-point-symmetry of order (), and the marginal point-symmetry of order and gives the theorem that the density function is -variate point-symmetric if and only if it is quasi-point-symmetric and marginal point-symmetric of order . The theorem is illustrated for the multivariate normal density function.

Bivariate t-distribution type symmetry model for square contingency tables with ordered categories

For the analysis of square contingency tables with the same row and column ordinal classifications, this article pro- poses a new model, which may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate t-distribution with equal marginal variances having any degrees of freedom. The proposed model indicates that the difference between the symmetric two probabilities raised to the power is proportional to the distance from the main diagonal of the table. The model is an extension of the symmetry model. An example is given. The simulation studies based on bivariate t-distribution are 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.

On partitioning conditional independence model for three-way contingency tables

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.