Agriculture Reference
In-Depth Information
11.3.2 Calculation of Residual
B
1
can be worked out fol-
lowing the method of matrix inversion. The
inversion of matrix is available in the commonly
used MS Excel subprogram of the MS Office
software also. In Example 1, we shall demon-
strate how, with the help of the MS Excel soft-
ware, path analysis can be done.
The inverse matrix
Once the path coefficient values corresponding to
different independent variables are worked out,
the next problem lies in estimating the residual
(
), that is, the unexplained part of the model.
We have assumed that
R
X
4
¼ X
1
+
X
2
+
X
3
+
R
4
2
or
σ
¼ σ
ð
X
1
þ X
2
þ X
3
þ R
Þ
2
1
2
2
2
3
2
R
þ
2
1
2
2
2
3
2
R
¼ σ
þ σ
þ σ
þ σ
2
σ
12
þ
2
σ
13
þ
2
σ
23
,where
; σ
; σ
; σ
;
and
σ
are the variance of the
X
1
; X
2
; X
3
variables and residual and
σ
ij
ði 6¼ j ¼
1,2,3
Þ
is the covariance between
X
i
and
X
j
ð
covariances between the variables and the residual are assumed to be zero
Þ
1
2
3
R
¼ σ
þ σ
þ σ
þ σ
þ
2
r
12
σ
1
σ
2
þ
2
r
13
σ
1
σ
3
þ
2
r
23
σ
2
σ
3
;
4
1
2
3
2
R
þ
or
σ
¼ σ
þ σ
þ σ
þ σ
2
r
12
σ
1
σ
2
þ
2
r
13
σ
1
σ
3
þ
2
r
23
σ
2
σ
3
;
2
4
2
2
2
2
σ
4
¼
σ
1
þ σ
2
þ σ
3
þ σ
R
þ
2
r
12
σ
1
σ
2
þ
2
r
13
σ
1
σ
3
þ
2
r
23
σ
2
σ
3
;
4
σ
σ
¼
σ
1
4
þ
σ
2
4
þ
σ
3
4
þ
σ
2
R
2
r
12
σ
1
σ
2
σ
2
r
13
σ
1
σ
3
σ
2
r
23
σ
2
σ
3
σ
1
4
þ
þ
þ
4
4
4
σ
σ
σ
σ
2
2
2
2
¼ p
41
þ p
42
þ p
43
þ p
4
R
þ
2
r
12
p
41
p
42
þ
2
r
13
p
41
p
43
þ
2
r
23
p
24
p
34
:
2
2
2
2
;
p
4
R
¼
1
p
41
þ p
42
þ p
43
þ
2
r
12
p
41
p
42
þ
2
r
13
p
41
p
43
þ
2
r
23
p
42
p
43
number of independent variables, the
generalized formula for getting residual will be
For the
k
X
k
j¼
1
p
2
X
k
X
k
j<j
0
2
2
p
yR
¼
1
yj
r
jj
0
p
yj
p
yj
0
;
j
0
th variables
where
p
yj
; p
yj
0
are the path coefficients of
j
th and
;
respectively,
r
jj
0
0
s are the correlation coefficients among the independent variables
:
and
11.3.3 Types of Path Coefficients
phenotypic, genotypic, and environmental levels
gives an indication about the genetic architecture
and their interrelationships among the characters.
Using the variances and covariances at the phe-
notypic, genotypic, and environmental levels, the
correlation coefficients among the variables at
the respective levels can also be worked out.
From the correlation coefficients, one can work
out the path coefficients at the phenotypic, geno-
typic, and environmental levels also. The analy-
sis of correlation and path coefficients at the
Interpretation of Path Coefficients
1.
Value of the residual
: If the value of the resid-
ual is substantial, then the question is whether
or not the variables/characters under consid-
eration are sufficient to explain the variations
in the dependent variable/character. If one
