Databases Reference
In-Depth Information
Theorem 12.
For a contingency table with size m
×
n:
∆
(
u,v
)=
⎧
⎨
m−s
m−s
(1 +
k
ri
)
⎧
⎨
⎫
⎬
r
=1
i
=1
n
n
×
x
uv
(
x
ij
)
−
x
iv
(
x
uj
)
⎩
⎭
j
=1
j
=1
(1
≤
u
≤
m
−
s,
1
≤
v
≤
m
)
(23)
⎩
m−s
n
m−s
x
r
1
x
ij
r
=1
i
=1
j
=1
×{
(
k
ur
− k
ui
)
m−s
m−s
+
k
ur
k
ri
− k
ui
k
rq
r
=1
r
=1
(
n
−
s
+1
≤
u
≤
m,
1
≤
v
≤
m
)
Thus, from the above theorem, if and only if
∆
(
u,v
) = 0 for all
v
, then the
u
-th row will satisfy the condition of statistically independence. Especially,
the following theorem is obtained.
Theorem 13.
If the following equation holds for all v
(1
n
)
, then the
condition of statistical independence will hold for the u-th row in a contingency
table.
≤
v
≤
= 0
m−s
m−s
r
=1
{
m−s
m−s
(
k
ur
−
k
ui
)+
k
ur
k
ri
−
k
ui
k
rq
(24)
i
=1
r
=1
r
=1
It is notable that the above equations give diophatine equations which can
check whether each row (column) will satisfy the condition of statistical inde-
pendence. As a corollary,
Corollary 2.
If k
ui
is equal for all i
=1
,
···
,n
−
s , then the u-th satisfies
the condition of statistical independence.
The converse is not true.
Example 3.
Let us consider the following matrix:
⎛
⎝
⎞
⎠
112
223
445
x
41
x
42
x
43
x
51
x
52
x
53
F
=
,
