Extended Diagonal Exponent Symmetry Model and Its Orthogonal Decomposition in Square Contingency Tables with Ordered Categories

For square contingency tables with ordered categories, this article proposes new models, which are the extension of Tomizawa’s [1] diagonal exponent symmetry model. Also it gives the decomposition of proposed model, and shows the orthogonality of the test statistics for decomposed models. Examples are given and the simulation studies based on the bivariate normal distribution are also given.


Introduction
Consider an R R × square contingency table with the same row and column classifications. Let ij p denote the probability that an observation will fall in the ith row and jth column of the ij ji ψ ψ = see Bowker [2]. Caussinus [3] considered the quasi-symmetry (QS) model defined by = ∑ see Stuart [4]. Caussinus [3] gave the theorem that the S model holds if and only if both the QS and MH models hold.
Tomizawa [1] considered the diagonal exponent symmetry (DES) model defined by is the DES model. Note that the QDES model implies the QS model. Let X and Y denote the row and column variables, respectively. We define the mean equality (ME) model as ( ) ( ) Iki et al. [5] gave the theorem that the DES model holds if and only if both the QDES and ME models hold.
Iki et al. [5] described the relationship between the QDES model and a joint bivariate normal distribution, and showed that the QDES model may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate normal distribution with equal marginal variances. We are interested in considering the new model which is appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate normal distribution without equal marginal variances, and a decomposition using the proposed models.
The present paper proposes two models, and gives the decomposition using the proposed models. Also it shows the orthogonality of the test statistics for decomposed model.

Orthogonality of Test Statistics
Let n ij denote the observed frequency in the (i, j)th cell of the G M denote the likelihood ratio chi-squared statistic for testing goodness-of-fit model M. The numbers of degrees of freedom (df) for the EDES and EQDES models are 2 The orthogonality (asymptotic separability or independence) of the test statistics for goodness-of-fit of two models is discussed by, e.g., Darroch and Silvey [8] and Read [9]. We obtain as follow: where "t" denotes the transpose, and ( ) 5  1  2  1 11 22 , , , , , , , ,

Examples
Example 1. Consider the data in Table 1, taken from Bishop, Fienberg and Holland [11], which describe the cross-classification of father's and son's occupational status categories in Denmark. The row is the father's status category and column is the son's status category. The categories are ordered from (1) to (5) (high to low). These data have also been analyzed by some statisticians; see for example, Kullback [12], Haberman [13], Goodman [14], and Yamamoto, Tahata and Tomizawa [15]. We see from Table 3 that the EQDES and QS models fit these data well, although the other models fit poorly. The EQDES model is a special case of the QS model. We shall test the hypothesis that the EQDES model holds assuming that the QS model holds for these data. Since  Table 2), the probability that a father's and his son's status categories are 1 i + and 1 j + , respectively, is estimated to be greater than the probability that those are i and j, respectively , 5 1 03 ξ = . , 6 1 06 ξ = . , 7 1 10 ξ = . , 8 Table 3 that the poor fit of the EDES model is caused by the influence of the lack of structure of the MVE model rather than the EQDES model. Example 2. Consider the data in Table 4 taken from Tomizawa [16]. These data are an unaided distance vision of 3168 pupils comprising nearly equal number of boys and girls aged 6 -12 at elementary schools in Tokyo, Japan, examined in June 1984. These data have also been analyzed by Tomizawa [1], Tahata and Tomizawa [17], and Iki et al. [5]. The row is the right eye grade and column is the left eye grade.
We see from Table 3 that the EDES and EQDES models fit these data well, although the MVE model fits poorly. The EDES model is a special case of the EQDES model. We shall test the hypothesis that the EDES  Table 3. Likelihood ratio chi-squared values 2 G for models applied to Table 1 and Table 4.
Applied with 2 df being the difference between the numbers of df for the EDES and the EQDES models, this hypothesis is rejected at the 0.05 significance level. Therefore, the EQDES model would be preferable to the EDES model. Under the EQDES model, the MLEs of 1 α , 2 α , 1 β and 2 β are 1 0 34 α = . , 2 1 13 α = . , 1 0 72 β = . and 2 0 97 β = . , respectively. Therefore the probability that a pupil's right eye grade and his or her left eye grade are 1 i + and 1 j + , respectively, is estimated to be ( ) ( ) , i j ≠ , are less than 1 (see Table 5), the probability that a pupil's right eye grade and his or her left eye grade are 1 i + and 1 j + , respectively, is estimated to be less than the probability that those are i and j, respectively
Applied 2 G models df We see from Table 7 that the EQDES model fits well for each of Tables 6(a)-6(d), although the QDES model fits well for each of Table 6(a) and Table 6(b), and fits poorly for each of Table 6(c) and Table 6(d). The DES and EDES models fit well for Table 6(a) and fit poorly for each of Tables 6(b)-6(d). Thus the EQDES model may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate normal distribution (without the equality of marginal variances), although the QDES model may be appropriate if it is reasonable to assume it with equal marginal variances, and the DES and EDES models may be appropriate if it is reasonable to assume it with both equal marginal means and equal marginal variances.

Concluding Remarks
Theorem 1 may be useful for seeing the reason for the poor fit when the EDES model fits the data poorly; in fact, see from Example 1, a poor fit of the EDES model would be caused by a poor fit of the MVE model rather than the EQDES model. From