Asymmetry Index on Marginal Homogeneity for Square
Contingency Tables with Ordered Categories
Kouji Tahata, Kanau Kawasaki, Sadao Tomizawa
Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Chiba, Japan
Email: {kouji_tahata, tomizawa}@is.noda.tus.ac.jp, [email protected]
Received December 27, 2011; revised January 25, 2012; accepted February 10, 2012
ABSTRACT
For square contingency tables with ordered categories, the present paper considers two kinds of weak marginal homo- geneity and gives measures to represent the degree of departure from weak marginal homogeneity. The proposed meas- ures lie between –1 to 1. When the marginal cumulative logistic model or the extended marginal homogeneity model holds, the proposed measures represent the degree of departure from marginal homogeneity. Using these measures, three kinds of unaided distance vision data are analyzed.
Keywords: Marginal Homogeneity; Marginal Cumulative Logistic Model; Measure; Square Contingency Table
1. Introduction
Consider an square contingency table with or- dered categories. Let ij denote the probability that an observation will fall in the ith row and jth column of the
table ( ; ). Also let
R R
p
1, ,
i R j1, , R X and denote the row and column variables, respectively. The marginal homogeneity (MH) model ([1]) is defined by
Y
X Y
i i
F F for i1, , R1
1
i k
k
where
i X
F
p 1
i Y
i k
k
, F p
1
k kt
t
1
R
k sk
s
p p
1 1
1
R
X Y
i i
i
with
R
p
p , .When the MH model does not hold, we are interested in applying the model that has weaker restriction than the MH model. As such a model, for example, there are the marginal cumulative logistic (ML) model ([2]) and the extended marginal homogeneity (EMH) model ([3-5]). We are also interested in considering the other structure of weak MH. The measures to represent the degree of departure from MH are given by, for example, [6,7]. When the structure of weak MH does not hold, we are interested in measuring what degree the departure from weak MH is.
The present paper considers two kinds of structures of weak MH and proposes the measures to represent the
degree of departure from weak MH.
2. Weak Marginal Homogeneity I and
Measure
2.1. Submeasure I
Let
F F
,
and
* 1
1
X i i
F
F
,
* 2
1
Y i i
F
F , for .
i1, , R1
1 * *
1 2
1 1
R
i i
i F F
Note that
. Assuming that
X Y 0
i i
F F , consider the submeasure defined by
1
1
* *
1 1 2
1
4 π
π 4
R
i
i i
i
F F
,where
1 1
2 2
sin
Y i i
X Y
i i
F
F F
.
Noting that 01 π 2 1 1
1 1
i , we see that 1) 1 ,
2) Y 0
i
F FX 0
1, ,R 1
if and only if and i
(i ), and 3) 11 if and only if FX 0 i
and Y 0 (i i
F 1, , R1 1
2.2. Submeasure II
Let
1
i i
SX FX SY 1 FY 1, , 1
X Y
S S 1, , 1
, i i for i R . The MH model may be expressed as
i i for i R
1 1 R X Y i i i S S
. Let 2 1, , 1
i R
,
and for ,
2 sin1
i 2 2 X i X Y i i S S S , * 1 2 i i S X S , * 2 2 Y i i S S
1 * *
1 2 1
R i i S S 2 .
Note that
i1 . Assuming that
X Y 0
i i
S S , we shall define the submeasure as
follows;
1 2 4 π 4 R ii i
* *
2 1 2
1
πi
S S
.Noting that 0i2 π 2
2 1
X 0
S , we see that 1)
2 ; 2) if and only if i
1 1
and
( ); and 3) if and only if
and (i ). When the MH
model holds, equals zero.
0
Y i
S i1,
0
Y i
S Si
,R1 1
X 1,
X Y 0
F F X Y 0
i i
S S
2
,R 1
0
2
2.3. Complete Measure
Assume that i i and
. Con-sider a measure defined by
1 2
2 1 1 1
X
F SX FY 0 SY 1
1
.
We see that 1) , 2) if and only if
i 1 (then i 0) and i (then i ) for
all i1, , R1, and 3) 1 if and only if X 0 i
F
(then ) and (then ) for all . Thus, indicates that 1R
1
X i
S
1, , 1
i R
1
Y i
F
0
Y i S
1
p 1
and the other cell probabilities are zero (say, upper-right- marginal inhomogeneity), and indicates that
1
R and the other cell probabilities are zero (say, lower-left-marginal inhomogeneity). When , we shall refer to this structure as the weak marginal homo- geneity I (WMH-I). We note that if the MH model holds then the structure of WMH-I holds, but the converse does not hold. 1 1 p 0
Therefore, using the measure , we can see whether
the structure of WMH-I departs toward the upper-right- marginal inhomogeneity or toward the lower-left-margi- nal inhomogeneity. As the measure approaches –1, the departure from WMH-I becomes greater toward the upper-right-marginal inhomogeneity. While as the approaches 1, it becomes greater toward the lower-left- marginal inhomogeneity.
3. Weak Marginal Homogeneity II and
Measure
Let
1 Pr i X c i k kT X i X Y p
,
1 Pr i Y c i k kT Y i X Y p
1, , 1
i R
,
, where for
1 c
k k kk
p p p
,
1 c
k k kk
p p p
, st
s t p
X Y i iT T i1, , R1
1
1
.
The MH model may be expressed by
for .
We shall consider the submeasure which is de- replaced
Xi
F and
fined by the submeasure
Yi
F by
TiX and
Y i T 1 X X i iU T UiY 1 TiY i1, , R1
X Y
i i
U U i1, , R1
2
2
, respectively. Let
, for .
The MH model may be expressed by
for .
We shall consider the submeasure which is de- replaced
X
andi
S
fined by the submeasure
Yi
S by
UiX and
Y iU , respectively.
X Y 0
Ti Ti
0
X Y
i i
U U
and
Assume that .
Consider a measure defined by
1 2
12
1 1
.
We see that . Let c
pij pij (i j). In a similar way to , 1 c 1
p indicates that 1R
and
the other pijc are zero (i j) (say, conditional upper- right-marginal inhomogeneity), and indicates that
1 R 1 1 c
p and the other ij are zero ( ) (say, condi- tional lower-left-marginal inhomogeneity). When
c
p i j
X Y
L L 1, , 1
4. Relationships between Measures and
Models
We shall consider the relationship between the measure (or ) and the ML model. The ML model is given by
i i for i R ,
where log 1 X i X i F F X i
L , log
1 Y Y i i Y i F L F 0
.A special case of this model obtained by putting is the MH model. The ML model may also be expressed as
exp 1 exp i X i F i
, exp
1 exp i Y i i F 1 R 0
X Y
i i
,
for i1, , . Therefore, when the ML model holds, 1) if and only if F F , 2) 0 if and
only if
X Y
i i
F F
Y i, and 3) 0 if and only if
Xi
F F
0
0 0
0
0 0
0 0
0
1i 2i
G G
. We obtain the following theorem.
Theorem 1. When the ML model holds,
1) if and only if ( ), 2) if and only if ( ),
3) (i.e., the MH model holds) if and only if
( ).
Next, we shall consider the relationship between the measure (or ) and the EMH model, defined by
i1, , R1
1
1 1
i R
for ,
where i st
s t i
G p
2 1 1
R i
,
i sts i t
G p
1 . A special case of this model obtained by putting is the MH model. Noting that 1 2 X Yi i
i i
G G F F 1, ,R 1
(i
1
), we obtain the following theorem.
Theorem 2. When the EMH model holds,
if and only if 0
1) 0
1
( ), if and only if
2) 0 0
1
( ),
(i.e., the MH model holds) if and only if
3) 0
( 0).
Thus, when the ML (EMH) model holds, the measures and are adequate to represent the degree of departure from MH.
5. Approximate Confidence Interval for
Measures
Let ij denote the observed frequency in the ith row and
jth column of the table ( ; ). As-
suming that a multinomial distribution applies to the
n
1, ,
i R j1, , R
R R table, we shall consider an approximate standard error and large-sample confidence interval for the meas- ure , using the delta method, as described by [8]. The sample version of , i.e., ˆ , is given by with
pˆij , where
pij replaced by pˆijnij nn n
and
ij . Using the delta method, we obtain the fol-lowing theorem.
Theorem 3. n ˆ
n
2 ˆ
has asymptotically (as ) a normal distribution with mean zero and vari-ance , where
22 1 1 1 ˆ 4 R R
ij ij ij i j
a b p
,with
1 1 1 2 2 1 2 14 R X Y
Y X
k k
ij k k k
X Y
k k
R i j
F F
a I j k I i k F I j k F
F F 1 1 π k
I i k
,
1 2 2 2 1 2 2 2 1 4 π X Y R Y X k kij k k k
X Y
k
k k
i j
S S
b I i k I j k I i k S I j k S
S S
2 ,and I I
1 ˆ
p
is the indicator function, if true, 0 if not.
Also, the sample version of , i.e., , is given by
with
ij replaced by
pˆij . We obtain the fol-lowing theorem.Theorem 4. n
ˆ
n
2 ˆ
has asymptotically (as
) a normal distribution with mean zero and vari- , where
ance
22 1 1 1 ˆ 4 R R
ij ij ij i j
j i
c d p
, with
1 3 2 2 1 3 1 1 3 4 π π 4 1 , X Y RX Y k k X Y Y X
ij k k k k k k k
X Y k k k R X Y k k k T T
c I i k T I j k T I i k T T I j k T T
T T
I i k T I j k T
1
4
2 2
1 4
1
2 1
4
4 π
π 4
1
,
X Y
R
X Y k k X Y
ij k k k k k
X Y
k
k k
R
X Y
k k
k
U U
d I i k U I j k U I i k U U I j
U U
I i k U I j k U
Y X
k k
k U U
i
1
1
X Y
i i
U U
,
1
1
R
X Y
i i
T T
, 4 Ri
3
3 sin 1
Y i i
T
2 2
X Y
i i
T T
,
4 sin1
i 2 2
X i
X Y
i i
U
U U
.
Let ˆ2 denote
ˆ
2 ˆ with
pij replaced by
pˆij . Then ˆ n is estimated approximateˆ
, and
ˆ
ror for
standard er ˆ zp2ˆ ˆ n is ap-
e 100 1
p
percent confidence interval for proximat, where zp2 from
ibution co o p. We a
les
m
7477 women aged 30 to 39 employed in Royal Ordnance factories in Britain from 1943 to 1946.
ble 2 that for the data in Table 1(a), the
e measure is –0.0130 and all is the percentage point the stan- dard mal distr rresponding to a two-tail prob y equal t lso obtain the similar result for measure .
nor abilit
6. Examp
Example 1: Consider the unaided distance vision data in
Table 1(a) taken fro [1]. There are data on unaided
distance vision of
We see from Ta
estimated value of th
values in confidence interval for are negative. There- fore, the structure of WMH-I for a woman’s right and left eyes departs toward the upper-right-marginal inhomoge- neity. Also we see from Table 3 that for the data in Ta-
ble 1(a), the estimated value of the measure is
–0.0436 and all values in confiden e interval for c are negative. Therefore, the structure of WMH-II for a woman’s right and left eyes departs toward the condi- tional upper-right-marginal inhomogeneity.
Table 4 gives the values of likelihood ratio chi-squared
statistic for testing goodness-of-fit of each of MH, L, and EMH models. We see from Table 4 that each of ML
and EMH models fits these data well. Thus the measures and would indicate the degree of departure from MH. We can see from these measures that the de
M
gree of de
inhomogeneity which indicates that the grade of right eye for arbitrary woman is “Best” and the grade of her left eye is “Worst”.
Example 2: Consider the unaided vision data in Table
1(b), taken from [9]. We see from Table 2 that for the
Br m
parture from MH for the vision data in Table 1(a) is
estimated to be 1.30 (4.36) percent of the maximum departure toward the (conditional) upper-right-marginal
Table 1. Unaided distance vision data of (a) 7477 women in itain fro [1]; (b) 3242 men in Britain from [9] and (c) 4746 students in Japan from [3].
(a) Women in Britain
Right eye Left eye grade
grade (1) (2) (3) (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
(b) Men in Britain Right eye Left eye grade
grade (1 1)
) (4) Total
Best ( 821 35 1053
(2) (3)
112 85
Second ( 116 494 145 27
T 72 151 583 87
W 43 106 331
791 919 480 3242
2) 782
hird (3) 893
orst (4)
Total 1052
34 514
(c) Stude apa Right eye
nts in J n Left eye grade
grade (1) (2 (4) Total
Be 1291 130 4
) (3)
st (1) 0 22 1483
Second ( 149 221 114
T 64 124 660
W 20 25 249 1429
1524 500 1063
2) 23 507
hird (3) 185 1033
orst (4) 1723
Total 1659 4746
Table 2. Estimates of , estimated approximate standard errors for ˆ , and
for
approximate 95% confidence intervals , applied to Table
ˆ
1.
Table S. E. C. I.
1(a) –0.0130 0037 0. (–0.0203,–0.0056)
1(b) 0.0055 0.0064 (–0.0071, 0.01
0040 (0.0 0.020
81)
Table 3. Estim p
ates of timate prox standard
erro ˆ
for lie
, es d ap imate
rs fo , ap
r , and approximate 95% d to Table
confidence intervals 1.
Table ˆ S. E. C. I.
1(a) –0.0436 0.0126 (–0.0683, –0.0190)
1( 0.0201 (–0.0222, 0.0566)
0.0 7 3 (0.0198, .0836)
b) 0.0172
1(c) 51 0.016 0
Table 4. Values of likelihood i-sq
the M L, an mod lied t
ratio ch els app
uared statistic for o Table 1.
H, M d EMH
Applied Degrees of Table
models freedom 1(a) 1(b) 1(c) H 3
M 11.99* 3.68 11.18*
ML 2 0.39 3.16 .41
E 0 2
1
MH 2 .005 .94 0.56
*Mean ant at t vel.
data in Table 1(b) the estimated value of measure
s signific he 0.05 le
is
H-I in the data in Table 1(b) se Table 3 that for
th Table 1(b), estima of measure
72 a o r includes
zero. So this m indicat there structure of WMH-II in the data in Tabl ).
E e 3: C nsider t ta in le 1( en
om [3,10]. We see from that for the data in value of measure 0.0055 and the confidence interval for includes zero. So this may indicate that there is a structure of WM
. Also we the nfide
e from ted value terval fo e data in
is 0.01 nd the c nce in
ay e that is a
e 1(b
xampl o he da
Table 2
Tab c) tak
fr
Table 1(c), the estimated the is
0.0125 and all values in confidence interval for
f
are positive. Therefore, the structure of WMH-I for a stu- dent’s right and left eyes departs toward the lower-left- marginal inhomogeneity. Also we see from Table 3 that
for the data in Table 1(c), the estimated value of the
measure is 0.0517 and all values in con idence in- terval for are positive. Therefore, the structure of WMH-II for a student’s right and left eyes departs to- ward the conditional lower-left-marginal inhomogeneity. We see from Table 4 that each of ML and EMH
mod-els fits these data well. Thus the measures and would indicate the degree of departure from MH. We can see from these measures that the degree of departure from MH for the vision data in Table 1(c) is estimated to
be 1.25 (5.17) percent of the maximum departure toward the (conditional) lower-left-marginal inhomogeneity which indicates t at the grade of right eye for arbitrary student is “Worst” a d the grade of his/her left eye is “Best”.
7. Concluding Remarks
hn
Fo
t H
r the analysis of square contingency tables with or- dered categories, when the ML model, or he EM model, or other asymmetry models, for example, [11]’s
conditional symmetry model (defined by pij pji for i j) holds, the proposed measures and are adequate to represent the degree of departure from the MH model toward two maximum departures, i.e., toward
the (conditional) lower-left-marginal inhomogeneity or toward the (conditional) upper-right-marginal inhomoge- neity.
ity (i.e., the
r-lef he -r
8. Discussion
[6,7] considered the measures to represent the degree of departure from MH. The present paper has considered two types of maximum marginal inhomogene
lowe t-marginal inhomogeneity and t upper ight- marginal inhomogeneity). The measures in [6,7] take the value 1 in two types of maximum marginal inhomoge- neity. The measures and in the present paper can distinguish these two kinds of maximum marginal inhomogeneity by the values –1 or 1 although the meas- ures in [6,7] cannot distinguish them. Also the proposed esent the degree of departure from MH
to ss measures can repr
when the ML or the EMH models, or the other asym- metry models hold. Therefore for the ordinal data, the proposed measures rather than those in [6,7] may be useful to represent the degree of departure from MH.
9. Acknowledgements
The authors would like expre their sincere thanks to the editor and a referee for their helpful comments.
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