Databases Reference
In-Depth Information
Definition 4.
A corresponding matrix C
T
a,b
is defined as a matrix the element
of which are equal to the value of the corresponding contingency table T
a,b
of
two attributes a and b, except for marginal values.
Definition 5.
The rank of a table is defined as the rank of its corresponding
matrix. The maximum value of the rank is equal to the size of (square) matrix,
denoted by r.
The contingency matrix of Table 2(
T
(
R
1
,R
2
)) is defined as
C
T
R
1
,R
2
as below:
⎛
⎞
x
11
x
12
···
x
1
n
⎝
⎠
x
21
x
22
···
x
2
n
.
.
.
.
.
.
x
m
1
x
m
2
···
x
mn
6.1 Independence of 2
×
2 Contingency Table
The results in Sect. 3 corresponds to the degree of independence in matrix
theory. Let us assume that a contingency table is given as Table 1. Then the
corresponding matrix (
C
T
R
1
,R
2
) is given as:
x
11
x
12
x
21
x
22
,
Then,
Proposition 1.
The determinant of det
(
C
T
R
1
,R
2
)
is equal to x
11
x
22
−
x
12
x
21
.
Proposition 2.
The rank will be:
rank
=
2
, fd
(
C
T
R
1
,R
2
)
=0
1
, fd
(
C
T
R
1
,R
2
)=0
From Theorem 1,
Theorem 5.
If the rank of the corresponding matrix of a
2
2
contingency
table is 1, then two attributes in a given contingency table are statistically
independent. Thus,
×
rank
=
2
, dependent
1
, statistical independent
This discussion can be extended into 2
×
n
tables. According to Theorem 3,
the following theorem is obtained.
Theorem 6.
If the rank of the corresponding matrix of a
2
n contingency
table is 1, then two attributes in a given contingency table are statistically
independent. Thus,
×
2
, dependent
1
, statistical independent
rank
=
