Agriculture Reference
In-Depth Information
analysis of variance, canonical analysis, etc.,
while the other group consists of factor analysis,
cluster analysis, etc.
have the following correlation matrix for the
variables involved in the regression analysis:
2
3
r
00
r
01
r
02
r
0
j
... r
0
k
r
10
r
11
r
12
r
1
j
... r
1
k
........................
r
k
0
4
5
;
<¼
12.2
Regression Analysis
r
k
1
r
k
2
r
kj
:::: r
kk
The main task of the researcher is to work out
the actual relationship between or among the
variables under study. In agricultural and other
experiments, mainly three types of variables are
recoded: (a) the treatments or factors such as
variety, insecticide, doses or type of fertilizers,
different chemical treatments, and different man-
agement practices; (b) environmental parameters
like rainfall, temperature, humidity, sunshine
hours, and wind speed; and (c) various responses
in the form of different growth and yield
parameters, qualitative changes, etc. Regression
analysis is one of the most important statistical
tools used in this endeavor. Regression analysis
is a technique by virtue of which one can study
the relationship of the ultimate variables, say,
adoption index, awareness, empowerment status,
etc., within different fields of studies (different
demographic, social, economical, educational,
and other parameters).
In regression analysis, the dependent variable
where
r
01
,
r
02
...
,
r
kj
's are the correlation
X
J
and
the diagonal elements are 1. If we denote
Y
X
1
,
Y
X
2
... X
K
and
between
and
and
R
for
the corresponding determinant and
R
ij
for the
cofactor of
r
ij
element in
, then one can have
ℜ
b
j
¼
s
0
:
R
0
j
R
00
Þ
j
1
Þ
jþ
2
s
j
ð
1
ð
1
¼
s
0
s
j
:
R
0
j
R
00
;
j ¼
1
;
2
; ...k
and
S
0
and
S
j
are
the standard deviations of
Y
and
X
j
;
respectively,
and
X
n
j¼
1
b
j
X
j
¼ Y þ
X
k
s
0
s
j
:
R
0
j
b
0
¼ Y
R
00
X
j
:
j¼
1
Thus, multiple regression equation of
y
on
X
1
,
X
2
,
...X
k
becomes
y
c
¼ y
s
0
s
1
:
R
01
Þ
s
0
s
2
:
R
02
Y
(
) is the function of one or more independent
variables (
R
00
X
1
X
1
R
00
X
2
X
2
ð
ð
Þ
X
's) and the error term (
u
's), which
s
0
s
k
:
R
0
k
can be represented in the form of
Y ¼ f
(
X
i
,
u
i
).
R
00
X
k
X
k
ð
Þ:
Let us suppose that
other variates
which are denoted by
X
1
,
X
2
,
...X
k
.
These need
not be independent. The usual problem is to find
the best linear predicting equation for
Y
depends on
k
The coefficient
b
j
is known as partial regres-
sion coefficient of
Y
of the
Y
on
X
j
for fixed
form
+
b
k
X
k
for the true regression equation in the popu-
lation
Y
c
¼ b
0
+
b
1
X
1
+
b
2
X
2
+
b
3
X
3
+
X
1
; X
2
; ...X
j
1
; X
jþ
1
; ...X
k
and is written in
the form
b
yj:
12
...ðj
1
Þðjþ
1
Þ...k
¼ b
0
j:
12
...ðj
1
Þðjþ
1
Þ...
Y ¼ β
0
þ β
1
x
1
þ β
2
x
2
þþβ
k
x
k
.We
present here three methods of getting such
relationships below. Readers are free to select
any one of these as per the convenience:
1. Now, using the procedure adopted (Chap.
8
)
for calculation of simple correlation
coefficients among the variables, one can
R
0
j
R
00
:
2. The objective is to find out the relationship of
the
k ¼
s
0
s
j
form
Y
i
¼ β
1
X
1
i
þ β
2
X
2
i
þ β
1
X
3
i
þ
þβ
k
X
ki
þ u
i
:
For
number of observations, the regres-
sion equation can be written as:
n
