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
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