Agriculture Reference
In-Depth Information
X
1
X
5
X
4
X
2
X
3
Fig. 11.1
Path diagram with two dependent and three independent variables
X
1
r
14
r
12
r
24
X
4
X
2
r
13
r
23
r
34
X
3
R
Fig. 11.2
Path diagram of one dependent and three correlated causal variables along with the residual part
The correlation between the response variable
and any component variable (causal variable)
can be looked upon as the additive effect of the
direct and indirect effects of the causal variables.
If
Path analysis is the method of studying the
direct and indirect effects of the variables on the
response variables. But it is not directed towards
discovering the causes of a response.
i 6¼ i
0
¼
a
i
and
b
i
0
ð
1
;
2
; ......; k
Þ
are the direct
and indirect effects of the
i
th variable in a system
11.3.1 Calculation of Path Coefficient
a
i
þ
P
k
k
i6¼i
0
¼
1
b
i
0
¼ r
x
i
y
,
of (
+ 1) variables,
then
p
ij
Generally the path coefficients are denoted by
where
a
i
is the direct effect of the
i
th causal
(
denotes for
the dependent/response variable and
j
for the
independent/causal variable and is defined as
the ratio of the standard deviations due to a
given cause to the total standard deviation of
the response, that is,
a
, in our earlier notation), where
i
variable on the response variable
y
and
b
i
0
s are
the indirect effects of the
th causal variable on
the response variable via the
i
i
0
th causal variable
(other than the
th variable).
In this fashion, each of the
i
number of indi-
vidual correlation coefficients of causal variables
with the response variable can be partitioned into
the additive effect of the direct effect of the
respective variables and the sum of the indirect
effects of the other variables:
k
p
ij
¼
σ
j
σ
i
.
Let us have four variables situation in which
X
1
,
X
3
are the independent variables and
X
4
is the dependent variable. So the variation
X
2
, and
X
4
is supposed to be explained by
X
1
,
X
2
, and
X
3
and
an unexplained portion
R
(Fig.
11.2
).
Thus, we have
X
4
¼ X
1
+
X
2
+
X
3
+
R
.So
r
x
1
y
¼ a
1
þ b
2
þ b
3
þ b
4
þþb
k
r
x
2
y
¼ b
1
þ a
2
þ b
3
þ b
4
þþb
k
r
x
3
y
¼ b
1
þ b
2
þ a
3
þ b
4
þþb
k
:
:
r
x
k
y
¼ b
1
þ b
2
þ b
3
þþa
k
the correlation coefficients of
X
4
with that of
X
1
,
X
2
,
X
3
, and
R
are given as
