1
rc
2001
Abstract
This research sets out to investigate the two-dimensional contingency table r  c
[Table (1)] used in the representation of classed data . Also, Log linear models
describing these data in the light of certain hypotheses for analysis have been set up.
The research also comprises an application in the medical field :
The application is : studying the effect of age on those who suffer from diabetes
and asthma in Baghdad in 2001. the data show that it represented by a twodimensional contingency table .
Log-Linear Models
11
1
2
3
2
(Everitt, B.S. 1977 ) Contingency Tables
1
(Two-Dimensional Contingency Tables)
1211
r
c
(James, T. Mcclave and Frank , H. Dietrich 1989 ) 1
1
2
j
c
1
x11
x12
x1j
x1c
x1.
2
x21
x22
x2j
x2c
x2.
i
xi1
xi2
xij
xic
xi.
r
xr1
xr2
xrj
xrc
xr.
x.1
x.2
x.j
x.c
x.. N
1
(i, j)
, j  1, 2, …. , c
i
xi.  xi1 + xi2 + …. + xic
xij
i  1, 2, …. , r
xi.
c

x
ij
j1
j
x.j
x.j  x1j + x2j + …. + xrj
r

xi j

i 1
N  x..
r

x..
c
 
i1
j1
x
i j
1,2,2
“In Complete Contingency Tables”
B1
Complete Contingency Tables”
( 2001
B2
Configurations
xjk
1,2,3
c
c23
(Bishop, Y.M.M.
“Log-Linear Model”
1,2,4
,Fienberg ,S.E. , and Holland , P. W. 1975 )
1,2 4،1
:”Saturated Model
1
(2.1)
“Unsaturated Model”
2
(Fienberg ,S . E . 1980 ) “The Hierarchy Principle”
1،2،5
u
1
u
2
rc
1،3
rc
“Independence Model”
“Dependence Model”
r
N
xij
c
(2.1)
mij i, j
Log mij  u + u1i + u2j + u12ij ...................................................................... (2.1)
mij
u
(i)
u1i
(j)
u2j
u12ij
u
 u2j  0
0
1i
i
,
u
12 ij
 u12ij  0
0
i
1
û 
rc
û 1i 
,
r
j
c
 Log m
1
c
j
i 1 j1
ij
........................................................................................ (2.2)
c
 Log m
j1
ij
 û
.................................................................................... (2.3)
û 2 j
1

r
û 12ij
r
Log m ij  û

i 1
1
 Log m ij
r
r

i 1
................................................................................... (2.4)
1
Log m ij 
c
c
 Log m
ij
 û
j1
.................................... (2.5)
Log mij  u + u1i + u2j ................................................................................. (2.6)
u12  0
u12 u2 0
……....(2.7)
u12  u1 0
.….…... (2.8)
Log mij  u + u1i
Log mij  u + u2j
rc
1،4
u
u
u1
u2
u12
1
r1
c1
(r  1) (c  1)
Rc
( Fienberg ,S . E . 1980 )
Poisson Distribution
(N)
(P. D. F)
1،5
1،5،1
x

e mij
m ij ij
x ij !
f (xij)  i, j
....................................................................................... (2.55)
(i,j)
xij
Mij
Simple Multinomial Distribution
1،5،2
(N)
(P.D.F)
 m ij 


N

x ij ! i, j 
N!

x ij

f (xij)  i, j
.................................................................... (2.56)
(i,j)
xij
Mij
Product Multinomial Distribution
1،5،3
(x.j)
(c)
(P. D. F)
f (xij) 

j

 x. j !

 xij !
 i
xij 
 mij  
i  x. j   ................................................................. (2.57)

 

Maximum Likelihood Estimates
(Goodman , L . A . 1964 )
(MLE)
(MEL)
1
(Sufficient
u
(Configurations)
Statistics)
(Minimal)
u
( Agresti , A . 1990 ) : Direct Method
1
(Closed Loops)
3
1
2
3
3
2
4
4 1
5
a
b
:Testing Goodness of Fit
1
1993
(Pearson X2)
X2 

1
(observed  Expected ) 2
Expected
.......................................................... (2.69)



X2
(Likelihood-Ration Statistic) (G2)
G2  2
2 2
 observed 
 (observed) Log  Expected  ................................................. (2.70)
G2



(Agresti , A. 1990) :Choosing A best Model
(Goodman)
1
G2
G2
G2 (M1)  G2 (M2)  G2 (M3)  G2 (M4) ................................................. (2.71)
(N  400)
x
1
2
62.5
250
37.5
150
y
14
14
14
0
1
44
15
2
64
45
3
65
4
56
36
15 44
144
25.5
45 64
102
24.5
65
98
2
45-64
-15
15-44
65-
37
93
61
59
19
51
41
39
Maximum Likelihood Estimates
21
Log mij  u + u1i + u2j
j i
2
1
21
x1.  250
x.1 56
m̂ij 
x2.  150
x.2  144
x.3  102
x.4 98
( x i. ) ( x.i )
N
22
3
The Model is : Log mij  u + u1i + u2j
(1, 2)
Formula of Direct Estimate:
m̂ ij 
Expected Values Matrix
35
90
21
54
X2
G2
X2
( x i. ) ( x . j )
N
63.75
38.25
61.25
36.75

2 
( x ij  m ij ) 2
2
m ij
i, j
G2 
2  1.108
3

x ij Log
i, j
x ij
m ij
G2  0.554
G2 2
2
2  7.82
2.4
0.05
2.2
2
û
û  3.82585
1
2
4
û 1i
‫ﻧﻮع اﻟﻤﺮض‬
‫اﻟﻤﺘﻐﯿﺮ‬
0.255
- 0.255
14 , 15 - 44 , 45 - 64 , 65 
5
û 2 j
‫اﻟﻔﺌﺔ اﻟﻌﻤﺮﯾﺔ ﻟﻠﻤﺮﯾﺾ‬
14
45-64
65
0.074
0.034
15-44
‫اﻟﻤﺘﻐﯿﺮ‬
0.526
0.418
15
44
44
15
1993
1995
2
3
2001
Agresti, A. (1990) :” Categorical Data Analysis”. John Wiely & Sons, Inc., New
York.
Everitt, B.S. (1977) :”The Analysis of Contingency Tables”. John Wiley & Sons,
Inc., New York.
Bishop, Y. M. M., Fienberg, S. E., and Holland, P. W. (1975) :” Discrete
Multivariate Analysis”. MIT. Press, Cambridge.
Fienbery, S. E. (1980) :” The Analysis of Cross-Classified Categorical Data”.
James, T. Mcclave and Frank, H. Dietrich (1989): “A First Course in Statistics”.
Goodman, L. A. (1964) : “Simple Methods for Analyzing Three Factor Interaction
in Contingency Tables”. J. Amer. Statist. Assoc. Vol. 59, No. 306, (319-352).