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
![](https://www.scirp.org/html/2-1240249\e5dcb4aa-73ac-444d-8c85-75f2dd063de3.jpg)
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
![](https://www.scirp.org/html/2-1240249\604cd0ec-d41e-4944-85f8-d66b4ca77f52.jpg)
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 ![](https://www.scirp.org/html/2-1240249\472007a1-94e0-4c48-a34e-c9126c20ff44.jpg)
![](https://www.scirp.org/html/2-1240249\ef2a1d76-ec70-451f-87b1-9ce77a2e256e.jpg)
Especially the LDPS(0) model is equivalent to the LDPS model.
Let for ![](https://www.scirp.org/html/2-1240249\f4566300-1aae-4cb6-a7c9-0fb4de81463a.jpg)
and ![](https://www.scirp.org/html/2-1240249\cdcd3042-50e9-4f87-bf67-6a642a765497.jpg)
The S model may be expressed as
![](https://www.scirp.org/html/2-1240249\99b168c0-2fb6-4ce7-97fe-86f105986c1e.jpg)
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
![](https://www.scirp.org/html/2-1240249\fb467248-e359-427e-9aa4-ef2268fadfee.jpg)
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 ![](https://www.scirp.org/html/2-1240249\67a17601-18fe-4f1e-96de-1636f87c8ad4.jpg)
![](https://www.scirp.org/html/2-1240249\73cdd4e2-b267-4d4c-9995-a83d75eff580.jpg)
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
![](https://www.scirp.org/html/2-1240249\0f0a32ef-5e55-40aa-b139-bfa14662da9c.jpg)
where
and
and ![](https://www.scirp.org/html/2-1240249\8fb9be16-fad9-438b-9f5e-8c44b7ba41c2.jpg)
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
![](https://www.scirp.org/html/2-1240249\76151bb3-b821-4b11-a089-d75250c37703.jpg)
From the LDPS(K) model, we see
![](https://www.scirp.org/html/2-1240249\14620a78-c1a8-42fa-ae0c-a8c7d51de4cd.jpg)
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
![](https://www.scirp.org/html/2-1240249\2d64526f-e50c-47df-9bb8-55ec606baa8e.jpg)
namely
![](https://www.scirp.org/html/2-1240249\aeb33d78-6252-452f-80e8-62f3d2920c05.jpg)
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:
![](https://www.scirp.org/html/2-1240249\d96d712a-3659-4e79-9b9a-7b96e88254d3.jpg)
which is
![](https://www.scirp.org/html/2-1240249\fc2abbce-6ecc-4b75-b257-fd013821d944.jpg)
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
![](https://www.scirp.org/html/2-1240249\440b6bc7-9e03-4504-99cd-f6ecb6b42bd1.jpg)
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 ![](https://www.scirp.org/html/2-1240249\7401ff17-ec5b-4cd1-ac8b-97e18b684f55.jpg)
Proof. The LDPS(K) model may be expressed as
(1)
where
Let
![](https://www.scirp.org/html/2-1240249\214e7be6-bc4b-49e8-8897-64bda888582d.jpg)
![](https://www.scirp.org/html/2-1240249\55131025-51fb-43a4-85d3-11b032b02607.jpg)
where “t” denotes the transpose, and
![](https://www.scirp.org/html/2-1240249\998991f2-47b2-4d17-bf9e-41f7629413b2.jpg)
is the
vector. The LDPS(K) model is expressed as
![](https://www.scirp.org/html/2-1240249\a9bf0c53-764f-4ca6-a810-d15e5e315cfb.jpg)
where
is the
matrix with
and
is the
vector with
![](https://www.scirp.org/html/2-1240249\55714a73-7bed-4b1b-a64a-8f9352fbd869.jpg)
where
![](https://www.scirp.org/html/2-1240249\a309ef51-0a9a-462e-87e1-16d15dbb9949.jpg)
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
![](https://www.scirp.org/html/2-1240249\6b2b2da8-974a-4129-ac90-2e2138781f19.jpg)
where
is the
zero matrix, and
![](https://www.scirp.org/html/2-1240249\f661b3a9-564a-4cec-8b70-2d9bf9b006b2.jpg)
The MNE(K) model may be expressed as
![](https://www.scirp.org/html/2-1240249\d1f1553d-dbed-4b7e-92d8-1469f58f1e02.jpg)
where ![](https://www.scirp.org/html/2-1240249\87cbd870-823d-4be3-8d2d-dd9bfa072e9b.jpg)
![](https://www.scirp.org/html/2-1240249\c95507ac-b8e3-4dd5-9aad-f400735e436e.jpg)
Note that
From Theorem 6, the S model may be expressed as
![](https://www.scirp.org/html/2-1240249\884b8a33-2597-425f-957f-1189b78065b6.jpg)
where ![](https://www.scirp.org/html/2-1240249\a0d25e68-2aa0-4965-9e36-7ca56b696d50.jpg)
![](https://www.scirp.org/html/2-1240249\df15ae6a-db3b-4d3d-9f97-f91544eabe10.jpg)
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
![](https://www.scirp.org/html/2-1240249\3cd41800-6619-4b6b-bb1e-b3166f176cd2.jpg)
because
and that
![](https://www.scirp.org/html/2-1240249\26f0d77f-4092-45df-a060-166430a91437.jpg)
![](https://www.scirp.org/html/2-1240249\42788b13-3643-45bd-be83-3b11867254a8.jpg)
Thus we obtain
![](https://www.scirp.org/html/2-1240249\7648285c-e664-4b77-865a-414f3ad25dfb.jpg)
Therefore we obtain
where
![](https://www.scirp.org/html/2-1240249\cb400bce-d323-4873-b039-b9975584d60e.jpg)
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 ![](https://www.scirp.org/html/2-1240249\b46f43d9-c1e3-4e82-bb51-9903c897b01d.jpg)