Agriculture Reference
In-Depth Information
have the relationship for the research data
using the above package. In example one,
the same has been demonstrated using MS
Excel as well as SPSS software.
1
Y
1
¼ β
1
X
11
þβ
2
X
21
þβ
3
X
31
þβ
4
X
41
þþβ
k
Xk
1
þu
1
2
Y
2
¼ β
1
X
12
þβ
2
X
22
þβ
3
X
32
þβ
4
X
42
þþβ
k
Xk
2
þu
2
Y
3
¼ β
1
X
13
þβ
2
X
23
þβ
3
X
33
þβ
4
X
43
þþβ
k
Xk
3
þu
3
3
nY
n
¼ β
1
X
1
n
þβ
2
X
2
n
þβ
3
X
3
n
þβ
4
X
4
n
þþβ
k
X
kn
þu
n
12.3 Multiple Correlation
0
@
1
A
0
@
1
A
0
@
1
A
0
@
1
A
Y
X
... X
k
1
X
12
X
22
X
32
X
42
... X
k
2
X
X
X
β
1
β
2
u
1
u
2
)
1
Y
2
11
21
31
41
x
1
; x
2
; ...x
k
is
nothing but the simple correlation coefficient
between the observed
y
Multiple correlation of
with
¼
þ
y
c
.
Thus, multiple correlation coefficient is given as
y
and the predicted
Y
n
X
1
n
X
2
n
X
n
X
n
... X
kn
β
k
u
n
3
4
n
1
n k
k
1
n
1
In matrix notation, the above equations can
be written as
~
¼
~
~
þ
~
ðy; y
c
Þ
VðyÞ
cov
p
p
Vðy
c
Þ
R
y:
12
...k
¼
;
R
0
:
12
...k
with
Eð
~
Þ¼
~
~
; Eð
~
Þ¼
0
~
s
Vðy
c
Þ
VðyÞ
1
2
R
R
00
0
Þ¼ σ
2
and
Eð
~ ~
~
;
where
~
is an
n n
unit
¼
; R
0
:
12
...k
¼
:
1
matrix
:
Our objective will be to minimize
0
~
:
L ¼
~
We have
0
0
X
n
i¼
1
y
i
y
X
n
i¼
1
y
i
y
ic
þy
ic
y
0
Let
L ¼
~
~
¼
~
~
~
~
~
~
2
2
TSS
ðyÞ¼
ð
Þ
¼
ð
Þ
;
0
0
0
0
0
¼
~
~
2
~
~
~
þ
~
~
~
~
:
X
n
X
n
2
2
1
y
ic
y
¼
1
y
i
y
ic
ð
Þ
þ
ð
Þ
Let
. Then
using the procedure of ordinary least square,
~
be the least square estimators of
~
i¼
i¼
¼
RSS
þR
g
SS
;
where
y
i
and
y
ic
are the observed and expected
@L
@
~
values of
Y
for
i
th observation, respectively,
¼
0
;
and writing
~
for
~
;
we have
TSS
þ
R
g
SS
2
~
0
~
2
~
0
~ ~
or
~
TSS
TSS
¼
RSS
TSS
¼
RSS
þ R
g
SS
)
;
TSS
0
0
~ ~
¼
~
~
;
RSS
TSS
þ R
RSS
TSS
¼ R
1
2
2
)
1
¼
)
1
:
0
0
or
~
¼
~
~
~
~
;
1
exists
:
~
0
~
provided
2
Clearly, from the above, 0
R
1
2
indicates the proportion of variation of the
dependent variable explained by the line of mul-
tiple regression.
We shall demonstrate the process of getting
regression equation using MS Excel software of
MS Office taking the following example:
R
Once after getting the
for a given set of
observations, one can have the multiple
regression
b
y ¼ b
1
x
1
þ b
2
x
2
þ
þb
k
x
k
, the sample regression equation.
3. Most probably the simplest way of getting the
above multiple regression equation is the use
of data analysis module of MS Excel software
of MS Office mostly available in computers.
In this method, one need not to go into details
about the steps of calculations; rather, one can
equation
Example 12.1.
The following table gives the
information on ten varieties of mulberry. Find
out the regression equation of leaf yield on
other yield attributing characters.
