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.


Introduction
Consider an R R × square contingency table with the same row and column ordinal classifications.Let ij p denote the probability that an observation will fall in the ith row and jth column of the table ( 1 1 i R j R =, , ; =, , ).The sym- metry (S) model is defined by see Bowker [1] and Bishop et al. [2].This model indicates a structure of symmetry of the probabilities with respect to the main diagonal of the table.Agresti [3] considered the linear diagonals-parameter symmetry (LDPS) model defined by ( ) This indicates that the probability that an observation will fall in the ( ) i j , th cell, i j < , is j i θ − times higher than the probability that it falls in the ( ) , j i th cell.A special case of the LDPS model obtained by putting 1 θ = is the S model.Furthermore, Tomizawa [4] proposed an extended linear diagonals-parameter symmetry (ELDPS) model defined by ( ) This indicates that the probability that an observation will fall in the ( ) times higher than the probability that it falls in the ( ) Consider random variables X and Y having a joint bivariate normal distribution with means ( ) Var Y σ = and covariance ( ) , where ρ is the correlation coefficient ( ) Corr X Y , .The joint bivariate normal density ( ) , where ( ) where A special case of this model obtained by putting 0 m η = is the S model.The TS(m) 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 R R × table.The TS(m) model may be appropriate if it is reasonable to assume an underlying bivariate t-distribution with equal marginal variances having m degrees of freedom (see Iki et al. [6]).Now, we are interested in considering a new model, which is appropriate if it is reasonable to assume an underlying bivariate t-distribution with different marginal variances.
The purpose of present paper is to introduce an extended TS(m) model.Section 2 proposes an extended TS(m) model and describes the properties of the new model.Section 3 describes the relationships between the extended TS(m) model and t-distribution by the simulation study.Section 4 illustrates the use of our new model with father's and son's occupational structure data.Section 5 provides some concluding remarks.

Extended Bivariate T-Distribution Type Symmetry Model
Consider random variables X and Y having a joint bivariate t-distribution with ( ) Var 2 Y m m σ = − , and correlation coefficient ( ) The probability density function ( ) where ( ) See, e.g., Muirhead [7].Another form of the probability density function where We note that ( ) ( ) , .Also, the probability density function ( ) where ( ) ( ) For continuous bivariate data, when we make the R R × square contingency table formed using ( ) 1 R − 's cut points for each of row and column variables, we are interested in the structure of asymmetry of bivariate discrete probabilities For any fixed constant m, we propose a model defined by ( ) ( ) ( ) ) , where ( ) ( ) The ETS(m) model is also expressed as where A special case of this model obtained by putting 0 m γ = is the TS(m) model.
The maximum likelihood estimates of expected frequencies under the ETS(m) model could be obtained using the Newton-Raphson method in the log-likelihood equation (see Appendix 2).For the ETS(m) model, { } ij p are determined by ( ) ψ , , 1 of m γ and 1 of m η , thus a total of ( ) . Therefore, the number of degrees of freedom (df) for the ETS(m) is , which is one less than that for the TS(m) model, and equal to that for the ELDPS model.

Simulation Study
As described in Section 2, the ETS(m) model may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate t-distribution with different marginal variances having m degrees of freedom.We shall consider the simulation study based on bivariate t-distribution.Consider random variables X and Y having a bivariate t-distribution with ( ) freedom, and means ( ) . (Note that it is possible to take various values of ρ.)Such random numbers are obtained by using normal random number and chi-square random number with m degree of freedom.Suppose that there is an underlying bivariate t-distribution with some conditions, namely, × tables on each conditions.From Table 1, we see that the ETS(m) model gives good fit, however, the TS(m) model gives poor fit.Thus, from the result of simulation for comparison the ETS(m) and TS(m) models, we obtain that if it is reasonable to assume the underlying bivariate t-distribution with different marginal variances and the ratio of different marginal variances (i.e.,  σ σ and m de- grees of freedom. (a) For the ETS(m) model

Example
The data in Table 2 is taken from Tominaga [8].These data describe the cross-classification of father's and son's occupational status categories in Japan, which were examined in 1955.From Table 3, we see that the TS(m) ( 5 20 50 100 m = , , , ) models fit these data poorly, however, the ETS(m) ( 5 20 50 100 m = , , , ) and the ELDPS models fit these data well.We obtain the similar result without depending the value of degree of freedom m, although the detail is omitted.We also see that the values of the likelihood ratio chi-squared statistic G 2 for the ETS(m) model approach the value of G 2 for the ELDPS model as   , respectively.
Therefore, the difference between the probability raised to the power 2 2 m − + [=−0.286] that the occupational status category of the father in pair is i and that of his son is ( ) j i j < and the probability raised to the power that the occupational status category of the father in pair is j and that of his son is i, is . For example, the difference between the probability raised to the power that the occupational status category of the father in pair is (1) "upper non-manual" and that of his son is (2) "lower non-manual" and the probability raised to the power that the occupational status category of the father in pair is (2) "lower non-manual" and that of his son is (1) "upper non-manual", is estimated to be -0.400[ 0 335 3 1 405 1 = .× − .× ].Namely, the probability that the occupational status category of the father in pair is (1) "upper non-manual" and that of his son is (2) "lower non-manual" is estimated to be greater than the probability that the occupational status category of the father in pair is (2) "lower non-manual" and that of his son is (1) "upper non-manual".Also, the probability that the occupational status category of the father in pair is (1) and that of his son is (3) is estimated to be greater than the probability that the occupational status category of the father in pair is (3) and that of his son is (1).However, for the other symmetric two probabilities with respect to the main diagonal, the probability that the occupational status category of the father in pair is j and that of his son is ( ) < is estimated to be greater than the probability that the occupational status category of the father in pair is i and that of his son is j.

Concluding Remarks
From the result of simulation studies, the ETS(m) model would be appropriate for a square contingency table if it is reasonable to assume an underlying bivariate t-distribution with different marginal variances, although, the TS(m) model may be appropriate for a square contingency table if it is reasonable to assume an underlying bivariate t-distribution with equal marginal variances having m degrees of freedom.
shall refer to this model as an extended t-distribution type symmetry (ETS(m)) model.The ETS(m) model may be appropriate if it is reasonable to assume an underlying bivariate t-distribution with different marginal variances having m degrees of freedom.Under the ETS(m) model, setting

and 1 . 7 , and a 4 4 ×
table of sample size 1000 is formed using cut points for each variable at 1 shall count the frequencies of acceptance (at the 0.05 significance level) based on the likelihood ratio chi-squared statistic for testing the hypothesis that the ETS(m) model or the TS(m) model with the corresponding m degrees of freedom of underlying t-distribution holds per 10000 times for 4 4

2 2 2 1 σ
σ ) being large, the corresponding ETS(m) model rather than the TS(m) model would fit the data well without depending the value of degrees of freedom m.
For the TS(m) model

Table 1 .
The frequencies of acceptance (at the 0.05 significance level) based on the likelihood ratio chi-squared statistic for testing the hypothesis that the ETS(m) model or the TS(m) model with the corresponding degrees of freedom of underlying t-distribution holds per 10,000 times for4 4

Table 2 .
[8] cross-classification of father's and son's occupational status categories in Japan, which were examined in 1955 from Tominaga[8].(The parenthesized values are the maximum likelihood estimates of expected frequencies under the ETS(5) model.)

Table 3 .
Likelihood ratio chi-squared values G 2 for models applied to Table2.