Decompositions for Ordinal Quasi-Symmetry Model in Square Contingency Tables with Ordered Categories

Problem statement: For square contingency tables with ordered categor i s, this study considers four kinds of extensions of the marginal homogeneity model and gives decompositions for the ordinal quasi-symmetry model. The decomposition s are extensions of some existing decompositions. Approach: This study gives a decomposition theorem that the o rdinal quasisymmetry model holds if and only if the quasi-symme try model and the proposed weighted marginal homogeneity model hold. An example is given. Results: For the data of cross-classification of father's and his son's occupational status in Denmark, the d ecomposition of the ordinal quasi-symmetry model is applied and the detailed analysis is given. Conclusion: When the ordinal quasi-symmetry model fits the data poorly, this decomposition is useful for s eeing which of decomposed two models influences stronger.


INTRODUCTION
Consider an R×R square contingency tables with the same row and column classifications. Let p ij denote the probability that an observation will fall in the ith row and jth column of the table (i =1, ··· ,R; j =1, ··· ,R). The Quasi-Symmetry (QS) model (Caussinus, 1965)  where, ψ ij = ψ ji . A special case of this model with {α i = β i }is the usual Symmetry (S) model (Bowker, 1948). The QS model can be also expressed in a log-linear form as (Bishop et al., 1978): ∑ ∑ (Stuart, 1955). Caussinus (1965) gave the decomposition of the S model as follows: Theorem 1: The S model holds if and only if both the QS and MH models hold.
Each of the S, QS and MH models is invariant under arbitrary permutations of same row and column categories. So, these models are suitable for analyzing the data on a nominal scale, but they are possible for analyzing the data on an ordinal scale because they only require a categorical scale.
The Ordinal Quasi-Symmetry (OQS) model (Agresti, 2002), which is a special case of the QS model, is often useful when the categories are ordered. Let u 1 <···<u R denote the ordered known scores assigned for both the rows and columns of same classifications. The OQS model is defined by: where, ψ ij = ψ ji . The OQS model with scores {i} instead of {u i } is the Linear Diagonals-Parameter Symmetry (LDPS) model (Agresti, 1983). Thus, the LDPS model is useful for analyzing the data with positive integer scores or equal-interval scores. While the OQS model is useful for analyzing the data with any increasing (or decreasing) scores. Tomizawa (1987) introduced two kinds of Diagonal Weighted Marginal homogeneity (DWM) models and gave the decompositions of the LDPS model as follows.
Theorem 2: The LDPS model holds if and only if both the QS and DWM models hold.
Consider the data in Table 1, taken directly from Bishop et al. (1978), which represents father's and his son's occupational status in Denmark. Each category means that (1) is High professionals, (2) White-collar employees of higher education, (3) White-collar employees of less high education, (4) Upper working class and (5) Unskilled workers. We treat the classifications as ordinal.
First, we apply the LDPS model to these data. For testing goodness-of-fit of the LDPS model, the value of likelihood ratio statistic (denoted by G 2 ) is 19.054 with 9 degrees of freedom (p = 0.025). Thus, the LDPS model does not fit these data well. This may be because we assign the equal-interval scores for the categories.
Next we apply the OQS model to the data in Table  1. We shall assign the same ordered scores for both the rows and columns of same classifications by using empirical ridit. Then, we use the average of ridit for a row category and that for the corresponding column category. Therefore, we assign the scores u 1 = 0.014, u 2 = 0.089, u 3 = 0.293, u 4 = 0.604 and u 5 = 0.886 for both the rows and columns of same classifications. Then the G 2 value for the OQS model is 16.654 with 9 degrees of freedom (p = 0.054). Thus, the OQS model fits these data moderately.  Also, we apply the QS model to these data. Since the value of G 2 is 6.468 with 6 degrees of freedom (p = 0.373), the QS model also fits these data well. Note that each of the OQS and LDPS models is a special case of the QS model.
Then, since the OQS model implies the QS model and these models fit the data in Table 1 well, we consider which of the models fits these data much better. For testing the goodness-of-fit of the OQS model under the assumption that the QS model holds true, the difference between the G 2 values for the OQS and QS models is 10.186 with 3 degrees of freedom (p = 0.017). Therefore, we see that the OQS model does not fit these data well, assuming that the QS model holds true.
Therefore, we shall consider a decomposition of the OQS model, to investigate the reason that the OQS model has a poor fit for the data in Table 1.
The purpose of this study is to give a decomposition of the OQS model, introducing four kinds of new models.

MATERIALS AND METHODS
To consider decompositions of the OQS model, we shall introduce four kinds of new models. First, consider a model defined by: This model indicates that the row marginal totals summed by multiplying the probabilities p ij for the cells with a distance i-j (>0) below main diagonal in the table by the weight i j u u 1 are equal to the column marginal totals summed by the same way. We will refer to this model as the weighed marginal homogeneity 1 (WMH-1) model.
Next, consider the following model defined by: This model states that the row marginal totals summed by multiplying the probabilities p ij for the cells with a distance j-i (> 0) above main diagonal in the table by the weight j i u u 2 ( 0) − δ > are equal to the column marginal totals summed by the same way. This is contrast to the WMH-1 model. We will refer to this model as the weighed marginal homogeneity 2 (WMH-2) model.
Thirdly, we introduce a model defined by: δ . This is contrast to the WMH-3 model. We will refer to this model as the weighted marginal homogeneity 4 (WMH-4) model.
We obtain the decompositions of the OQS model as follows.
Theorem 3: For t fixed (t = 1, 2, 3, 4), the OQS model holds if and only if both the QS and WMH-t models hold. For t fixed (t = 1, 2, 3, 4), if the OQS model holds, then both the QS and WMH-t models hold.

Proof:
Conversely, assuming that the QS and WMH-t models hold and then we shall show that the OQS model holds.
First, consider the case of t = 1. From the assumption that the QS model holds, we obtain Eq. 1: Where: Also we obtain Eq. 2: Where: Since Eq. 1 is equal to Eq. 2 in terms of the assumption that the WMH-1 model holds, we obtain: and "t" denotes the transpose and the (i, m)th element of the R×R matrix W is: All elements of W are positive and satisfy WJ R = J R where J R = (1, ··· ,1) t is a vector of order R whose components are all unity. Thus from Eq. 4 and the restriction Σ k v 1(k) = Σ k v 2(k) = 0, we get f = J R . Namely we obtain: Thus noting that v 12(ij) = v 12(ji) , we see for i<j: Thus the OQS model holds. The case of t = 2 can also be proved in the similar way.
Next consider the case of t = 3. From the assumption that the QS model holds, we obtain: Where: with v 12(im) = v 12(mi) . All elements of W * are positive and satisfy W * J R = J R . Thus from Eq. 8 and the restriction Σ k v 1(k) = Σ k v 2(k) = 0, we get f * = J R . Namely we get: Thus the OQS model holds. The case of t = 4 can also be proved in the similar way. Thus, the proof is completed.
For a full multinomial sampling, the maximum likelihood estimates of expected frequencies and the parameter δ t under the WMH-t (t = 1, 2, 3, 4) model can be obtained using the Newton-Raphson method to the log-likelihood equations. Each WMH-t model can be tested for goodness-of-fit by likelihood ratio statistic G 2 with R-2 degrees of freedom, which is only one less than the number of degrees of freedom for the MH model. The OQS and QS models have (R+1)(R-2)/2 and (R-1)(R-2)/2, degrees of freedom, respectively. Table 1 again. Table 2 gives the values of likelihood ratio test statistic G 2 for each model.

Consider the data in
As described in the section of INTRODUCTION, the OQS and QS models fit these data well. However, according to the test based on the difference between the G 2 values for the OQS and QS models (since the G 2 value is significant at the 0.05 level under the assumption that the QS model holds true), the OQS model does not fit the data well. Also, we see from Table 2 that the WMH-t (t = 1, 2, 3, 4) model has a poor fit for the data in Table 1.

DISCUSSION
We see from Theorem 3 that the poor fit of the OQS model is caused by the influence of the lack of structure of the WMH-t (t = 1, 2, 3, 4) model rather than the QS model.

CONCLUSION
We have proposed the WMH-t (t = 1, 2, 3, 4) models and have given Theorem 3, which is the decompositions of