EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 6, No. 3, 2013, 299-306
ISSN 1307-5543 – www.ejpam.com

Decomposition of Symmetry Model into Three Models for
Cumulative Probabilities in Square Contingency Tables

Kouji Tahata1,∗, Kouji Yamamoto2, Sadao Tomizawa1

1 Department of Information Sciences, Faculty of Science and Technology, Tokyo University of
Science, Noda City, Chiba, 278-8510, Japan
2 Department of Medical Innovation, Osaka University Hospital, 2-15, Yamadaoka, Suita, Osaka,
565-0871, Japan

Abstract. For square contingency tables with ordered categories, we decompose the symmetry model
into three models for cumulative probabilities. Three models are the cumulative two ratios-parameter
symmetry, the global symmetry, and the marginal means equality models. An example is given.

2010 Mathematics Subject Classifications: 62H17

Key Words and Phrases: Decomposition, Global symmetry, Marginal mean, Square contingency table,
Symmetry

1. Introduction

For an r × r square contingency table with the same row and column classifications, let
pi j denote the probability that an observation will fall in the ith row and jth column of the
table (i = 1, . . . , r; j = 1, . . . , r). Bowker [3] considered the symmetry (S) model defined by

pi j = p ji (i 6= j);

see [2, p.282]. Caussinus [4] considered the quasi-symmetry (QS) model defined by

pi j = µαiβ jψi j (i = 1, . . . , r; j = 1, . . . , r),

where ψi j = ψ ji . A special case of QS model obtained by putting {αi = βi} is the S model.
The marginal homogeneity (MH) model is defined by

pi· = p·i (i = 1, . . . , r),

∗Corresponding author.

Email addresses: kouji_tahata@is.noda.tus.ac.jp (K. Tahata),
yamamoto-k@hp-crc.med.osaka-u.ac.jp (K. Yamamoto), tomizawa@is.noda.tus.ac.jp (S.
Tomizawa)

http://www.ejpam.com 299 c© 2013 EJPAM All rights reserved.



K. Tahata, K. Yamamoto, S. Tomizawa / Eur. J. Pure Appl. Math, 6 (2013), 299-306 300

where

pi· =
r
∑

t=1

pi t , p·i =
r
∑

s=1

psi;

see [8]. Caussinus [4] gave the following theorem.

Theorem 1. The S model holds if and only if both the QS and MH models hold.

Tomizawa [11] considered the two ratios-parameter symmetry (2RPS) model defined by

pi j

p ji
= γφ j−i (i < j).

Special cases of 2RPS model obtained by putting φ = 1 and γ = 1 are McCullagh’s [6]
conditional symmetry (CS) and Agresti’s [1] linear diagonals-parameter symmetry (LDPS)
models, respectively.

Define the global symmetry (GS) model by

δU = δL ,

where
δU =
∑∑

i< j

pi j , δL =
∑∑

i> j

pi j .

Let X and Y denote the row and column variables, respectively. Define the marginal means
equality (ME) model by

E(X ) = E(Y ),

where

E(X ) =
r
∑

i=1

ipi·, E(Y ) =
r
∑

i=1

ip·i .

Yamamoto, Iwashita and Tomizawa [15], and Tahata, Yamamoto and Tomizawa [10] gave
the following theorem.

Theorem 2. The S model holds if and only if both the LDPS and ME models hold.

Tahata and Tomizawa [9] gave the theorem as follows.

Theorem 3. The S model holds if and only if all the 2RPS, GS and ME models hold.

Let

Gi j =
i
∑

s=1

r
∑

t= j

pst (i < j),

and

Gi j =
r
∑

s=i

j
∑

t=1

pst (i > j).



K. Tahata, K. Yamamoto, S. Tomizawa / Eur. J. Pure Appl. Math, 6 (2013), 299-306 301

Note that the S model may be expressed as

Gi j = G ji (i 6= j).

The MH model may be expressed as

Gi,i+1 = Gi+1,i (i = 1, . . . , r − 1).

Miyamoto, Ohtsuka and Tomizawa [7] considered the cumulative quasi-symmetry (CQS)
model defined by

Gi j = µξiη jΨi j (i 6= j),

where Ψi j =Ψ ji . Yamamoto, Ando and Tomizawa [16] gave the following theorem.

Theorem 4. The S model holds if and only if both the CQS and MH models hold.

Miyamoto et al. [7] also considered the cumulative linear diagonals-parameter symmetry
(CLDPS) model defined by

Gi j

G ji
=Θ j−i (i < j).

Yamamoto and Tomizawa [17] gave the theorem as follows.

Theorem 5. The S model holds if and only if both the CLDPS and ME models hold.

Tomizawa, Miyamoto, Yamamoto and Sugiyama [13] considered the cumulative two ratios-
parameter symmetry (C2RPS) model defined by

Gi j

G ji
= ΓΘ j−i (i < j).

We are now interested in whether or not Theorem 3 with the 2RPS model replaced by the
C2RPS model holds.

The purpose of this paper is to decompose the S model into three models, i.e., the C2RPS,
the GS, and the ME models.

2. New Decomposition of Symmetry

We can obtain a new decomposition of the symmetry model as follows.

Theorem 6. The S model holds if and only if all the C2RPS, GS, and ME models hold.

Proof. If the S model holds, then all the C2RPS, GS, and ME models hold. Assume that the
C2RPS, GS, and ME models hold, and then we shall show that the S model holds. We see

E(X ) =
r
∑

i=1

ipi·



K. Tahata, K. Yamamoto, S. Tomizawa / Eur. J. Pure Appl. Math, 6 (2013), 299-306 302

=
r
∑

s=1

r
∑

t=s
pt·

=
r
∑

s=1

(1− F X
s−1)

=r −
r−1
∑

i=1

F X
i ,

where F X
i = P(X ≤ i). Similarly we see

E(Y ) = r −
r−1
∑

i=1

F Y
i ,

where F Y
i = P(Y ≤ i). Thus we see

E(Y )− E(X ) =
r−1
∑

i=1

F X
i −

r−1
∑

i=1

F Y
i

=
r−1
∑

i=1

Gi,i+1−
r−1
∑

i=1

Gi+1,i .

From the ME model, we see
r−1
∑

i=1

Gi,i+1 =
r−1
∑

i=1

Gi+1,i . (1)

From the C2RPS model, we obtain

r−1
∑

i=1

Gi,i+1 = ΓΘ
r−1
∑

i=1

Gi+1,i .

From (1) we see Γ = Θ−1. Thus

Gi j

G ji
=Θ j−i−1 (i < j). (2)

We can see that
r−1
∑

i=1

Gi,i+1 =
r−1
∑

i=1

r
∑

j=i+1

( j− i)pi j ,

and
r−1
∑

i=1

Gi+1,i =
r−1
∑

i=1

r
∑

j=i+1

( j− i)p ji .



K. Tahata, K. Yamamoto, S. Tomizawa / Eur. J. Pure Appl. Math, 6 (2013), 299-306 303

Also we can see that
r−2
∑

i=1

Gi,i+2 =
r−2
∑

i=1

r
∑

j=i+2

( j− i− 1)pi j ,

and
r−2
∑

i=1

Gi+2,i =
r−2
∑

i=1

r
∑

j=i+2

( j− i− 1)p ji .

Therefore we can see that

δU =
r−1
∑

i=1

Gi,i+1−
r−2
∑

i=1

Gi,i+2,

and

δL =
r−1
∑

i=1

Gi+1,i −
r−2
∑

i=1

Gi+2,i .

From the GS model (i.e., δU = δL) and from (1), we can obtain

r−2
∑

i=1

Gi,i+2 =
r−2
∑

i=1

Gi+2,i .

From (2) we obtain
r−2
∑

i=1

ΘGi+2,i =
r−2
∑

i=1

Gi+2,i .

Thus Θ= 1, i.e., the S model holds. The proof is completed.

3. Goodness-of-fit Test

Let x i j denote the observed frequency in the ith row and jth column of the r × r table
(i = 1, . . . , r; j = 1, . . . , r), with N =

∑∑

x i j . Let mi j denote the corresponding expected
frequency. Assuming that {x i j} have a multinomial distribution, the maximum likelihood esti-
mates of expected frequencies {mi j} under each model could be obtained, for example, using
the Newton-Raphson method to the log-likelihood equations. The goodness-of-fit of each
model can be tested by, e.g., the likelihood ratio chi-squared statistic G2 with the correspond-
ing degrees of freedom, defined by

G2 = 2
r
∑

i=1

r
∑

j=1

x i j log

�

x i j

m̂i j

�

,

where m̂i j is the maximum likelihood estimate of mi j under the model. The numbers of
degrees of freedom for each model are omitted here; however, when r = 4, those are given in
Table 2.



K. Tahata, K. Yamamoto, S. Tomizawa / Eur. J. Pure Appl. Math, 6 (2013), 299-306 304

4. Example

Consider the vision data in Table 1. The row variable X is the right eye grade and the
column variable Y is the left eye grade. The categories are ordered from best (1) to worst (4).
These data have been analyzed by many statisticians, including Stuart [8], Bishop et al. [2,
p.284], McCullagh [6], Goodman [5], Tomizawa [12], Miyamoto et al. [7], and Tomizawa
and Tahata [14].

Table 1: Unaided distance vision of 7, 477 women aged 30-39 employed in Royal Ordnance factories in
Britain from 1943 to 1946; from [8].

Right eye Left eye grade
grade Best (1) Second (2) Third (3) Worst (4) Total

Best (1) 1520 266 124 66 1976
Second (2) 234 1512 432 78 2256

Third (3) 117 362 1772 205 2456
Worst (4) 36 82 179 492 789

Total 1907 2222 2507 841 7477

Table 2 gives the values of the likelihood ratio chi-squared statistic G2 for each model. The
S model fits the data in Table 1 poorly. We can see from Theorem 1 that the poor fit of the S
model is caused by the influence of the lack of structure of the MH model rather than that of
the QS model.

Table 2: Likelihood ratio chi-square values for models applied to the data in Table 1. (∗ means significant
at the 0.05 level.)

Applied Degrees of Likelihood ratio
models freedom chi-square

S 6 19.25∗

QS 3 7.27
MH 3 11.99∗

CS 5 7.35
LDPS 5 7.28
2RPS 4 6.83
GS 1 11.90∗

ME 1 11.98∗

CLDPS 5 8.63
C2RPS 4 6.26
CQS 3 8.43∗

We can also see from Theorem 2 (Theorem 3) that the poor fit of the S model is caused
by the influence of the lack of structure of the ME model (the GS and ME models) rather than
that of the LDPS (the 2RPS) model.



REFERENCES 305

The S model also indicates the structure of symmetry of cumulative probabilities {Gi j},
i 6= j, instead of the cell probabilities {pi j}, i 6= j. Therefore we shall next consider the reason
why the S model fits the data in Table 1 poorly using the models which describe 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 structures of both the CQS and MH models.

We also see from Theorem 5 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 model.

In more details, we can see from Theorem 6 that the poor fit of the S model is caused by
the influence of the lack of structures of the GS and ME models rather than the C2RPS model.

5. Concluding Remarks

Theorems 1 through 6 would be useful for seeing the reason for poor fit of the S model
when the S model fits the data poorly.

When we are interested in the structure of symmetry of cumulative probabilities {Gi j},
i 6= j, instead of the cell probabilities {pi j}, i 6= j, Theorems 4, 5 and 6 would be useful.
Especially, Theorem 6 rather than Theorem 5 would be useful for seeing in more details the
reason for poor fit of the S model when the S model fits the data poorly.

References

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[3] A H Bowker. A test for symmetry in contingency tables. Journal of the American Statis-
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[4] H Caussinus. Contribution à l’analyse statistique des tableaux de corrélation. Annales
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[5] L A Goodman. Multiplicative models for square contingency tables with ordered cate-
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[6] P McCullagh. A class of parametric models for the analysis of square contingency tables
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[8] A Stuart. A test for homogeneity of the marginal distributions in a two-way classification.
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