ABSTRACT

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.

1. Introduction

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 [1] considered the symmetry (S) model defined by

This model describes the structure of symmetry with respect to the cell probabilities As a model which indicates the structure of asymmetry for Agresti [2] considered the linear diagonals-parameter symmetry (LDPS) model defined by

A special case of this model obtained by putting is the S model. Yamamoto and Tomizawa [3] considered the generalized linear diagonals-parameter symmetry (LDPS(K)) model as follows; for a fixed

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. [4] considered the cumulative linear diagonals-parameter symmetry (CLDPS) model defined by

which indicates a structure of asymmetry for The CLDPS model is different from the LDPS model. Yamamoto and Tomizawa [3] considered the generalized cumulative linear diagonals-parameter symmetry (CLDPS(K)) model as follows; for a fixed

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. [5] gave Theorem 1. The S model holds if and only if both the LDPS and ME models hold.

Yamamoto and Tomizawa [6] gave Theorem 2. The S model holds if and only if both the CLDPS and ME models hold.

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.

2. Decompositions of Symmetry Model

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.

3. Test Statistic and Orthogonality

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 [7] and Lang [8] considered the simultaneous modeling of a model for the joint distribution and a model for the marginal distribution. Aitchison [9] discussed the asymptotic separability, which is equivalent to the orthogonality in Read [10] and the independence in Darroch and Silvey [11], of the test statistic for goodness-of-fit of two models (also see Tomizawa and Tahata [12], Tahata et al. [13], and Tahata and Tomizawa [14]). On the orthogonality of test statistic for models in Theorem 6, we obtain.

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

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

(1)

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 [15], Lang and Agresti [7], and Tahata and Tomizawa [16], we denote the linear space spanned by columns of the matrix by with the dimension Note that where is the vector of 1 elements, and thus Let be an where full column rank matrix such that the linear space is the orthogonal component of the space Thus, where is the zero matrix. Therefore, the LDPS(K) model is expressed as

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 [17], Darroch and Silvey [11], Aitchison [9]), we obtain Theorem 8. The proof is completed.

4. Analysis of Data

Table 1 taken directly from Agresti [18, p. 232] summarizes responses to the questions “How successful is the government in (1) providing health care for the sick? (2) Protecting the environment?”.

Table 2 gives the values of the likelihood ratio test statistic for models applied to these data. The S model does not fit these data so well. Also, each of the ME (i.e., MNE(0)), MNE(K) and the GS models does not fit these data so well. However each of the LDPS(K) models and the CLDPS(K) models fit these data very well. Using Theorems 3 through 7 (including Theorems 1 and 2), we shall consider the reason why the S model fits these data poorly. For the structure of cell probabilities we see from Theorems 3, 5, 6 and 7 that the poor fit of the S model is caused by the influence of the lack of structure of the ME model (the GS model or the MNE(K) model) rather than the LDPS(K) model For the structure of cumulative probabilities we see from Theorem 4 that the poor fit of the S model is caused by the influence of the lack of structure of the ME model rather than the CLDPS(K) model