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For square contingency tables with ordered categories, the present paper gives several theorems that the symmetry model holds if and only if the generalized linear diagonals-parameter symmetry model for cell probabilities and for cumulative probabilities and the mean nonequality model of row and column variables hold. It also shows the orthogonality of statistic for testing goodness-of-fit of the symmetry model. An example is given.

Consider an square contingency table with the same row and column classifications. Let denote the probability that an observation will fall in the ith row and jth column of the table Bowker [

This model describes the structure of symmetry with respect to the cell probabilities As a model which indicates the structure of asymmetry for Agresti [

A special case of this model obtained by putting is the S model. Yamamoto and Tomizawa [

Especially the LDPS(0) model is equivalent to the LDPS model.

Let for

and

The S model may be expressed as

Thus the S model also has the structure of symmetry with respect to the cumulative probabilities Miyamoto et al. [

which indicates a structure of asymmetry for The CLDPS model is different from the LDPS model. Yamamoto and Tomizawa [

Especially the CLDPS(0) model is equivalent to the CLDPS model.

Let and denote the row and column variables, respectively. We consider the mean equality (ME) model as

where and and

Yamamoto et al. [

Yamamoto and Tomizawa [

The present paper gives several decompositions of the S model using the LDPS(K) and CLDPS(K) models. It also proposes the mean nonequality model, and gives the orthogonal decomposition for testing goodness-of-fit of the S model. An example is given.

We shall give five kinds of decompositions of the S model using the LDPS(K) and CLDPS(K) models.

Theorem 3. For a fixed the S model holds if and only if both the LDPS(K) and ME models hold.

Proof. If the S model holds, then both the LDPS(K) and ME models hold. Conversely, assuming that the LDPS(K) and ME models hold and then we shall show that the S model holds. The ME model may be expressed as

From the LDPS(K) model, we see

Therefore we obtain. Namely the S model holds. The proof is completed.

Theorem 4. For a fixed the S model holds if and only if both the CLDPS(K) and ME models hold.

Considering the global symmetry (GS) model as

namely

we obtain Theorem 5. For a fixed the S model holds if and only if both the LDPS(K) and GS models hold.

We shall omit the proofs of Theorems 4 and 5 because these are obtained in a similar manner to the proof of Theorem 3.

For a fixed consider the mean nonequality (MNE(K)) model as follows:

which is

This model indicates that the difference between the means of and is times higher than the difference between the global symmetric probabilities. When the MNE(0) model is identical to the ME model. We obtain Theorem 6. For a fixed the S model holds if and only if both the LDPS(K) and MNE(K) models hold.

Theorem 7. For a fixed and for a fixed the S model holds if and only if both the LDPS(K) and MNE(L) models hold.

We shall omit the proofs of Theorems 6 and 7 because there are obtained in a similar manner to the proof of Theorem 3. Note that: 1) Theorem 6 is an extension of Theorem 1 because when Theorem 6 is identical to Theorem 1; 2) Theorem 7 is an extension of Theorem 3 because when Theorem 7 is identical to Theorem 3; and 3) Theorem 7 is an extension of Theorem 6 because when Theorem 7 is identical to Theorem 6.

Let denote the observed frequency in the ith row and jth column of the table with and let denote the corresponding expected frequency. Assume that has a multinomial distribution. The maximum likelihood estimates of expected frequencies under each model could be obtained, for example, using the Newton-Raphson method to the log-likelihood equations. Each model (say, model) can be tested for goodness-of-fit by the likelihood ratio chi-squared statistic with the corresponding degrees of freedom, defined by

where is the maximum likelihood estimate of under the model. The number of degrees of freedom for the S model is and that for each of the LDPS(K) and CLDPS(K) models is (being one less than that for the S model). That for each of ME, GS, and MNE(K) models is 1. Note that the number of degrees of freedom for the S model is equal to the sum of those for the decomposed models.

Lang and Agresti [

Theorem 8. For a fixed test statistic is asymptotically equivalent to the sum of and

Proof. The LDPS(K) model may be expressed as

where Let

where “t” denotes the transpose, and

is the vector. The LDPS(K) model is expressed as

where is the matrix with and is the vector with

where

and is matrix of 0 or 1 elements determined from (1). The matrix is full column rank which is In a similar manner to Haber [

where is the zero matrix, and

The MNE(K) model may be expressed as

where

Note that From Theorem 6, the S model may be expressed as

where

Note that are the numbers of degrees of freedom for testing goodness-of-fit of the LDPS(K), MNE(K) and S models, respectively.

Let denote the matrix of partial derivatives of with respect to i.e., Let where denotes a diagonal matrix with ith component of as ith diagonal component. We see that

because and that

Thus we obtain

Therefore we obtain where

From the asymptotic equivalence of the Wald statistic and the likelihood ratio statistic (Rao [