Extended Diagonal Exponent Symmetry Model and Its Orthogonal Decomposition in Square Contingency Tables with Ordered Categories ()
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
. The symmetry (S) model is defined by
![](//html.scirp.org/file/4-1240496x8.png)
where
see Bowker [2] . Caussinus [3] considered the quasi-symmetry (QS) model defined by
![](//html.scirp.org/file/4-1240496x10.png)
where
The marginal homogeneity (MH) model is defined by
![](//html.scirp.org/file/4-1240496x12.png)
where
and
see Stuart [4] . Caussinus [3] gave the theorem that the S model holds if and only if both the QS and MH models hold.
Tomizawa [1] considered the diagonal exponent symmetry (DES) model defined by
![](//html.scirp.org/file/4-1240496x15.png)
By putting
and
this model is also expressed as
![](//html.scirp.org/file/4-1240496x18.png)
Note that the DES model implies the S model; thus the DES model implies the QS (MH) model. The DES model states that
is
times higher than
; in other words, for fixed distance k
from the main diagonal of the table,
increase (decrease) exponentially along every subdiagonal of the table as the value i increase
.
Iki, Yamamoto and Tomizawa [5] considered the quasi-diagonal exponent symmetry (QDES) model defined by
![]()
A special case of the QDES model obtained by putting
is the DES model. Note that the QDES model implies the QS model. Let X and Y denote the row and column variables, respectively. We define the mean equality (ME) model as
. Iki et al. [5] gave the theorem that the DES model holds if and only if both the QDES and ME models hold.
Iki et al. [5] described the relationship between the QDES model and a joint bivariate normal distribution, and showed that the QDES model may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate normal distribution with equal marginal variances. We are interested in considering the new model which is appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate normal distribution without equal marginal variances, and a decomposition using the proposed models.
The present paper proposes two models, and gives the decomposition using the proposed models. Also it shows the orthogonality of the test statistics for decomposed model.
2. New Models
Consider a model defined by
![]()
A special case of this model obtained by putting
is the DES model. Thus we shall refer to this model as the extended diagonal exponent symmetry (EDES) model. The EDES model states that
is
times higher than
; in other words, for fixed distance from the main diagonal of the table, the ratio of
to
increases (decreases) exponentially along every subdiagonal of the table. Note that the EDES model implies the S model.
Next, consider a model defined by
![]()
A special case of this model obtained by putting
is the QDES model. Thus we shall refer to this model as the extended quasi-diagonal exponent symmetry (EQDES) model. A special case of the EQDES model obtained by putting
and
is the EDES model. The EQDES model states that
is
times higher than
; in other words, for fixed distance from the main diagonal of the table, the ratio of
to
increases (decreases) exponentially along every subdiagonal of the table. Note that the EQDES model implies the QS model.
Under the EQDES model, we can see
![]()
where
. This indicates that the odds that an observation will fall in the
th cell, instead of the
th cell is
times higher than the odds that the observation will fall in the
th cell, instead of the
th cell. Also we can see
![]()
where
and
. If
, for corresponding i and j, the structure of
holds. Also if
, the structure of
holds.
In Figure 1, we show the relationships among models. In figure,
indicates that model A implies model B.
3. Decomposition
Refer to model of equality of marginal means and variances, i.e.,
and
as the MVE model. This model is also expressed as
and
We obtain the decomposition of the EDES model as follows:
Theorem 1. The EDES model holds if and only if the EQDES and MVE models hold.
Proof. If the EDES model holds, then the EQDES and MVE models hold. Assuming that both the EQDES and MVE models hold, then we shall show that the EDES model holds. Let
denote the cell probabilities which satisfy both the EQDES and MVE models. Since the EQDES model holds, we see
(1)
Let
with
We denote that
with
Then, since
satisfy the EQDES and MVE models, we see
(2)
and
(3)
where
with
and
.
Then, we denote
by
and
by ![]()
Consider the arbitrary cell probabilities
satisfying
(4)
where
and ![]()
From (2), (3) and (4), we see
(5)
Using the Equation (5), we obtain
![]()
where
![]()
and
is the Kullback-Leibler information between
and
. Since
being a func-
tion of
is fixed, we see
![]()
and then
uniquely minimizes
(see Bhapkar and Darroch [6] ).
Let
for
Then
(6)
Noting that
the Equation (6) is also expressed as
(7)
From (3), (4) and (7), we see
(8)
Using the Equation (8), we obtain
![]()
Since
being a function of
is fixed, we see
![]()
and then
uniquely minimizes
Therefore, we see
Thus, ![]()
From (1) and (6), for
, we see
![]()
Thus, we obtain
and
Namely, the EDES model holds. The proof is completed.
4. Orthogonality of Test Statistics
Let nij denote the observed frequency in the (i, j)th cell of the table
with
, and let
denote the corresponding expected frequency. Assume that
have a multinomial distribution. The maximum likelihood estimates (MLEs) of
under the EDES and EQDES models could be obtained using iterative procedures; for example, see Darroch and Ratcliff [7] . The MLEs of
under the MVE model could be obtained using Newton-Raphson method to the log-likelihood equations.
Let
denote the likelihood ratio chi-squared statistic for testing goodness-of-fit model M. The numbers of degrees of freedom (df) for the EDES and EQDES models are
and
, respectively.
The orthogonality (asymptotic separability or independence) of the test statistics for goodness-of-fit of two models is discussed by, e.g., Darroch and Silvey [8] and Read [9] . We obtain as follow:
Theorem 2. The test statistic
is asymptotically equivalent to the sum of
and
.
Proof. The EQDES model is expressed as
(9)
where
Let
![]()
where “t” denotes the transpose, and
![]()
is the
vector. The EQDES model is expressed as
![]()
where X is the
matrix with
(the R2 × 1 vector),
(the R2 × 1 vector),
(the R2 × 1 vector),
(the R2 × 1 vector), and
is the
matrix of 1 or 0 elements determined from (9),
is the
vector of 1 elements, ![]()
and
denotes the Kronecker product. The matrix X is full column rank which is K. In a similar manner to Haber [10] , we denote the linear space spanned by the columns of the matrix X by
with the dimension K.
Let U be an
, where
, full column rank matrix such that
is the orthogonal complement of
. Thus,
, where
is the s × t zero matrix. Therefore the EQDES model is expressed as
![]()
where
is the
zero vector, and
. The MVE model is expressed as
![]()
where
, and
, with
being the
matrix. Namely,
. Thus
belongs to
Hence
From Theorem 1, the EDES model is expressed as
![]()
where
and
.
Let
denote the
matrix of partial derivative of
with respect to p, i.e.,
Let
, where
denotes a diagonal matrix with ith component of p as ith diagonal component. Let
denote p with
replaced by
. Then
has asymptotically a normal distribution with mean
and covariance matrix
. Using the delta method,
has asymptotically a normal distribution with mean
and covariance matrix
![]()
Note that
belongs to
because
. Thus
. Since
and
, we see
![]()
Thus, we obtain
where
(10)
Under each
, the Wald statistic
has asymptotically a chi-squared distribution with
degrees of freedom. From (10), we see that
From the asymptotic equivalence of the Wald statistic and likelihood ratio statistic, we obtain Theorem 2.
5. Examples
Example 1. Consider the data in Table 1, taken from Bishop, Fienberg and Holland [11] , which describe the cross-classification of father’s and son’s occupational status categories in Denmark. The row is the father’s status category and column is the son’s status category. The categories are ordered from (1) to (5) (high to low). These data have also been analyzed by some statisticians; see for example, Kullback [12] , Haberman [13] , Goodman [14] , and Yamamoto, Tahata and Tomizawa [15] .
![]()
Table 1. Occupational status for Danish father-son pairs; from Bishop et al. [11] . (The parenthesized values are MLEs of expected frequencies under the EQDES model.)
Note: Status (1) is high professionals, (2) white-collar employees of higher education, (3) white-collar employees of less high education, (4) upper working class, and (5) unskilled workers.
We see from Table 3 that the EQDES and QS models fit these data well, although the other models fit poorly. The EQDES model is a special case of the QS model. We shall test the hypothesis that the EQDES model holds assuming that the QS model holds for these data. Since
with 6 df being the difference between the numbers of df for the EQDES and the QS models, this hypothesis is accepted at the 0.05 significance level. Therefore, the EQDES model would be preferable to the QS model.
Under the EQDES model, the MLEs of
and
are
and
respectively. Therefore the probability that a father’s and his son’s status categories are
and
, respectively, is estimated to be
times higher than the probability that those are i and j, respectively
. Since the values of
for
and
are greater than 1 and it for
is less than 1 (see Table 2), the probability that a father’s and his son’s status categories are
and
, respectively, is estimated to be greater than the probability that those are i and j, respectively
.
Also the MLEs of
are
,
,
,
,
, ![]()
and
, respectively. Therefore, it is estimated that there is the structure of
for
with
and
for
with
and 4.
We see from Table 3 that the poor fit of the EDES model is caused by the influence of the lack of structure of the MVE model rather than the EQDES model.
Example 2. Consider the data in Table 4 taken from Tomizawa [16] . These data are an unaided distance vision of 3168 pupils comprising nearly equal number of boys and girls aged 6 - 12 at elementary schools in Tokyo, Japan, examined in June 1984. These data have also been analyzed by Tomizawa [1] , Tahata and Tomizawa [17] , and Iki et al. [5] . The row is the right eye grade and column is the left eye grade.
We see from Table 3 that the EDES and EQDES models fit these data well, although the MVE model fits poorly. The EDES model is a special case of the EQDES model. We shall test the hypothesis that the EDES
![]()
Table 2. Values of
,
, under the EQDES model applied to Table 1.
![]()
Table 3. Likelihood ratio chi-squared values
for models applied to Table 1 and Table 4.
*means significant at the 0.05 level.
model holds assuming that the EQDES model holds for these data. Since
![]()
with 2 df being the difference between the numbers of df for the EDES and the EQDES models, this hypothesis is rejected at the 0.05 significance level. Therefore, the EQDES model would be preferable to the EDES model.
Under the EQDES model, the MLEs of
and
are
and
respectively. Therefore the probability that a pupil’s right eye grade and his or her left eye grade are
and
, respectively, is estimated to be
times higher than the probability that those are i and j, respectively
. Since all values of
,
, are less than 1 (see Table 5), the probability that a pupil’s right eye grade and his or her left eye grade are
and
, respectively, is estimated to be less than the probability that those are i and j, respectively
.
Also the MLEs of
are
,
,
,
and
, respectively. Therefore, it is estimated that there is the structure of
for
with
and 4 and
for
with
and 7.
6. Simulation Studies
Under the QDES model, we see the structure of
which is the structure of Agresti’s [18]
![]()
Table 4. Unaided distance vision of 3168 pupils comprising nearly equal number of boys and girls aged 6 - 12 at elementary schools in Tokyo, Japan, examined in June 1984; from Tomizawa [16] . (Upper and lower parenthesized values are MLEs of expected frequencies under the EDES and EQDES models, respectively.)
![]()
Table 5. Values of
,
, under the EQDES model applied to Table 4.
linear diagonals-parameter symmetry model, and under the EQDES model, we see the structure of
which is the structure of Tomizawa’s [19] extended linear diagonals-parameter symmetry model. Also under the DES and EDES models, we see the structure of
for
.
Consider now random variables U and V having a joint bivariate normal distribution with means
and
variances
and
and correlation
Then the joint bivariate normal density function
satisfies
![]()
Namely,
has the form
for constant
and
. Agresti [18] described relationship between the linear diagonals-parameter symmetry model and the joint bivariate normal distribution (see also Tomizawa [19] ). We now consider the relationship between the QDES (DES) and EQDES (EDES) models and the joint bivariate normal distribution in terms of simulation studies.
Table 6 gives the
tables of sample size 5000 formed by using cut points for each variable at
,
, for underlying bivariate normal distribution with the conditions
, and
and
(Table 6(a)),
and
(Table 6(b)),
and
(Table 6(c)) and
and
(Table 6(d)).
(c) ![]()
(d) ![]()
![]()
Table 7. Likelihood ratio chi-squared values
for models applied to Tables 6(a)-6(d).
*means significant at the 0.05 level.
We see from Table 7 that the EQDES model fits well for each of Tables 6(a)-6(d), although the QDES model fits well for each of Table 6(a) and Table 6(b), and fits poorly for each of Table 6(c) and Table 6(d). The DES and EDES models fit well for Table 6(a) and fit poorly for each of Tables 6(b)-6(d). Thus the EQDES model may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate normal distribution (without the equality of marginal variances), although the QDES model may be appropriate if it is reasonable to assume it with equal marginal variances, and the DES and EDES models may be appropriate if it is reasonable to assume it with both equal marginal means and equal marginal variances.
7. Concluding Remarks
Theorem 1 may be useful for seeing the reason for the poor fit when the EDES model fits the data poorly; in fact, see from Example 1, a poor fit of the EDES model would be caused by a poor fit of the MVE model rather than the EQDES model.
From Theorem 2, we point out that the
can be easily calculated using the
and
; in fact, see from Table 3, the value of
is very close to the value of the sum of
and ![]()
From Simulation studies, the EQDES model may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate normal distribution without equal marginal means and equal marginal variances; although the QDES model may be appropriate if it is reasonable to assume it with equal marginal variances.
Acknowledgements
The authors would like to thank the editor and the referee for theirhelpful comments.