Agriculture Reference
In-Depth Information
X
X
2
R ¼
y
ij
μ α
i
βx
ij
minimum for
μ; α
i
; β
i
j
X
X
2
y
ij
y þ βx y
i
þ y þ β x
i
x
Þ βx
ij
¼
ð
i
j
X
X
X
X
2
XX
2
2
2
y
ij
y
i
β x
ij
x
i
β
¼
¼
y
ij
y
i
x
ij
x
i
i
j
i
j
2
xy
E
xx
;
¼ E
yy
E
since
β ¼
E
xy
E
xx
with
ðn t
1
Þ
d
:
f
:
The residual
sum of
squares under
the
For testing
H
01
:
β ¼
0, we have the test statis-
restricted model when
H
01
holds good, that is,
1
R=ðn t
R
0
=
tic
F ¼
with (1,
n t
1) d.f.
y
ij
¼ μ þ α
i
þ e
ij
,is
Þ
1
If the hypothesis is not rejected, we may not
go for working out the adjusted treatment sum of
squares because in that case, the assumed model
will turn out to be
X
X
2
R
2
¼
y
ij
μ α
i
minimum
μ; α
i
i
j
¼
X
i
X
¼
X
i
X
2
2
y
ij
¼ μ þ α
i
þ e
ij
, the model of
one-way classified data. If the regression coeffi-
cient proves to be significant, we will proceed to
test
y
ij
y y
i
þ y
y
ij
y
i
j
j
¼ E
yy
with
ðn tÞ
d
:
f
:
H
02
.
The residual
Hence, the regression sum of squares will be
R
0
¼ R
2
R ¼
sum of
squares under
the
restricted model when
H
02
holds good, that is,
E
2
xy
E
xx
with 1 d
:
f
:
y
ij
¼ μ þ βx
ij
þ e
ij
,is
X
X
2
R
1
¼
y
ij
μ βx
ij
minimum
μ
,
β
i
j
2
XX
y
ij
y þ βx βx
ij
¼
PP
x
ij
x
y
ij
y
h
i
2
XX
¼
S
xy
S
xx
y
ij
y β x
ij
x
where
β ¼
¼
;
2
PP
x
ij
x
XX
XX
2
2
2
β
¼
y
ij
y
x
ij
x
¼ S
yy
S
2
xy
S
xx
with
ðn
2
Þ
d
:
f
:
Hence,
the
adjusted treatment
sum of
is significant, we conclude that the
treatment means after they are adjusted for
In case
F
x
S
2
xy
S
xx
squares will be
R
00
¼ R
1
R¼ S
yy
differ significantly.
The analysis of covariance table is as follows
(Table
11.1
):
Let
with
2
xy
E
xy
E
E
yy
t
1d
:
f
:
For
testing
H
02
,
we
have
F ¼
y
0
i
be the mean of the
i
th treatment
R
00
=ðt
1
Þ
with
ðt
1,
n t
1
Þ
d
:
f
:
y
x
adjusted for the linear regression of
on
. Then,
R=ðn t
1
Þ