Agriculture Reference
In-Depth Information
8.5.1 Rank Correlation
according to character B are
y
n
.
Assuming that no two individuals are tied in
either the classification mentioned, wherein both
x
y
1
,
y
2
,
y
3
,
...
,
As discussed above, there are other groups of
variables, which are not arithmetically measur-
able, rather they can be grouped or ranked into
different groups or ranks. Even sometimes some
measurable characters cannot be measured
appropriately because of constraints in time of
measurement, cost involvement (may be in the
form of the cost of the measuring instrument),
etc. In all these cases, we can arrange the
individuals according to their assigned ranks.
For example, in case of tea, the aroma and taste
of different teas can be ranked rather than
measured. Similarly, the susceptibility of the dif-
ferent varieties of scented rice towards pest and
diseases can be ranked; it is very difficult to
measure the susceptibility exactly. Now, if we
want to have a degree of association between
aroma and susceptibility, then one can use the
Spearman's rank correlation coefficient.
Suppose we have
, then
Spearman's rank correlation coefficient is defined
and
y
are all integral values from 1 to
n
6
P
d
2
as
are
the difference between the ranks obtained in two
different characters by the
r
R
¼
1
i
, where
d
i
i ¼
ð
1
;
2
; ...; n
Þ
nn
ð
2
1
Þ
th individual.
If more than one individual have the same rank,
then the above formula is redefined as follows:
i
n
o
6
P
d
i
þ
pðp
1
Þ
12
2
þ
qðq
1
Þ
12
2
2
r
R
¼
1
, where
nn
ð
2
1
Þ
p
are the number of individuals involved in
tied ranks for characters
and
q
A
and
B
, respectively.
Range
of
rank
correlation
coefficient
:
1
r
R
þ
1.
Example 8.28.
Ten different types of teas were
examined for aroma as well as for taste. The
ranks are shown as follows. Find the association
between aroma and pest susceptibility:
n
individuals whose ranks
A
x
1
,
x
2
,
x
3
,
...
x
n
and
according to a character
are
,
Variety
T
1
T
2
T
3
T
4
T
5
T
6
T
7
T
8
T
9
T
10
Aroma:
3
1
6
7
8
4
2
5
9
10
Susceptibility:
6
2
9
10
4
5
1
7
8
3
The differences between the rankings in
aroma and susceptibility are
Thus, there is a low association between
the aroma and the pest susceptibility in scented
rice.
3,
1,
3,
3,
4,
2, 1, and 7.
So,
P
i¼
1
d
1, 1,
2
i
¼
9+1+9+9+16+1+
Example 8.29.
Farmers are ranked for their edu-
cational status and motivation index as follows.
Find the association between the standard of
education and the motivation index of
1+4+1+49
¼
100
6
:
P
10
2
i
10 10
1
1
d
the
6
:
100
10
:
99
¼
1
20
33
¼
13
33
:
i¼
) r
R
¼
1
Þ
¼
1
farmers:
ð
Farmers
F
F
F
F
F
F
F
F
F
F
1
2
3
4
5
6
7
8
9
10
Education
6
10
3.5
5
7
3.5
2
1
9
8
Motivation index
2
9
7
4.5
6
1
4.5
3
8
10
