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

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.


Introduction
Consider an square contingency table with the same row and column classifications.Let ij 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   .  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 , i j The S model may be expressed as Thus the S model also has the structure of symmetry with respect to the cumulative probabilities   , ij G .i j  Miyamoto et al. [4] considered the cumulative linear diagonals-parameter symmetry (CLDPS) model defined by 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 X and Y denote the row and column variables, respectively.We consider the mean equality (ME) model as where and and 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.

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 1 1 From the LDPS(K) model, we see  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

 
This model indicates that the difference between the means of X and Y is K times higher than the difference between the global symmetric probabilities.When 0, K  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.
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 0 K  0 L  Theorem 7 is identical to Theorem 3; and 3) Theorem 7 is an extension of Theorem 6 because when K L  Theorem 7 is identical to Theorem 6.

Test Statistic and Orthogonality
 where is the maximum likelihood estimate of ij 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 where .

S X
with the dimension Note that where 1 is the vector of 1 elements, and thus full column rank matrix such that the linear space is the orthogonal component of the space Thus, The MNE(K) model may be expressed as From Theorem 6, the S model may be expressed as are the numbers of degrees of freedom for testing goodness-of-fit of the LDPS(K), MNE(K) and S models, respectively.Let     Thus we obtain From the asymptotic equivalence of the Wald statistic and the likelihood ratio statistic (Rao [17], Darro d Si om Agresti [18, p. 232] sumquestions "How successful is ch an lvey [11], Aitchison [9]), we obtain Theorem 8.The proof is completed.

Analysis of Data
Table 1 taken directly fr marizes responses to the 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 2  G for models applied to these data.The S model does not fit these data so well.Also, each of the ME (i.e.NE(0)), MNE(K)   1, 2, , 5 K   and the GS models does not fit these data so well.However each of the LDPS(K) models , M  

Concluding Remarks
e kinds of decompositions of 4 are extensions of Theheorem 5 is another deting goodness-of-fit of the S m s significant We have given the new fiv the S model.Theorems 3 and orems 1 and 2, respectively.T composition of the S model.Theorem 6 is another extension of Theorem 1, and Theorem 7 is an extension of Theorem 3.These theorems may be useful for seeing in more details the reason for the poor fit when the S model fits the data poorly.
From the orthogonality of test statistic given by Theorem 8, we point out that for instance, the likelihood ratio chi-squared statistic for tes odel assuming that the LDPS(K) model holds true is and this is asymptotically equivalent to the likelihood ratio chi-squared statistic for testing goodness-of-fit of the MNE(K) model, i.e.,    

A
special case of this model obtained by putting 1  

2 G
Let ij denote the observed frequency in the ith row and jth column of the R distribution.The maximum likelihood estimates of expected frequencies ha   ij under each model could be obtained, for example, using the Newton-Raphson method to the log-likelihood equations.Each model (say, model m M ) can be tested for goodness-of-fit by the likelihood ratio chi-squared statistic   M with the corresponding degrees of freedom, defined by