Databases Reference
In-Depth Information
⎧
⎨
3
, dependent
2
, contextual independent
1
, statistical independent
rank
=
⎩
It is easy to see that this discussion can be extended into 3
×
n
contingency
tables.
6.3 Independence of
m × n
Contingency Table
Finally, the relation between rank and independence in a multiway contin-
gency table is obtained from Theorem 4.
Theorem 8.
Let the corresponding matrix of a given contingency table be a
m
n matrix. If the rank of the corresponding matrix is 1, then two attributes
in a given contingency table are statistically independent. If the rank of the
corresponding matrix is
min(
m,n
)
, then two attributes in a given contingency
table are dependent. Otherwise, two attributes are contextual dependent, which
means that several conditional probabilities can be represented by a linear com-
bination of conditional probabilities. Thus,
×
⎧
⎨
min(
m,n
)
dependent
2
,
,
min(
m,n
)
···
rank
=
⎩
−
1
contextual independent
1
statistical independent
7 Pseudo-Statistical Independence: Example
The next step is to investigate the characteristics of linear independence in a
contingency matrix. In other words, a
m
n
contingency table whose rank is
not equal to min(
m,n
). Since two-way matrix (2
×
2) gives a simple equation
whose rank is equal to 1 or 2, let us start our discussion from 3
×
×
3-matrix,
whose rank is equal to 2, first.
7.1 Contingency Table (3
×
3, Rank: 2)
Let
M
(
m,n
) denote a contingency matrix whose row and column are equal
to
m
and
n
, respectively. Then, a three-way contingency table is defined as:
⎛
⎞
x
11
x
12
x
13
x
21
x
22
x
23
x
31
x
32
x
33
⎝
⎠
M
(3
,
3) =
When its rank is equal to 2, it can be assumed that the third row is represented
by the first and second row:
