Decompositions of Symmetry Using Generalized Linear Diagonals-Parameter Symmetry Model and Orthogonality of Test Statistic for Square Contingency Tables ()
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 