Agriculture Reference
In-Depth Information
Table 11.1
Analysis of covariance table
SOV
d.f.
SS(
x
)
SP(
xy
)
SS(
y
)
SS(Reg.)
SS(y)adj.
d.f.(adj.)
Treatment
t
1
T
xx
T
xy
T
yy
Error
n t
xx
E
xy
E
yy
2
xy
E
xx
¼ R
0
2
xy
E
xx
¼ R
n t
1
E
E
yy
E
Treatment
þ
n
1
T
xx
þ
E
xx
¼ S
xx
T
x
y
þ
E
xy
¼ S
xy
T
yy
þ
E
yy
¼ S
yy
2
xy
S
xx
2
xy
S
xx
¼ R
n
2
S
S
yy
S
error
1
Treatment
(adjusted)
R
1
R ¼ R
00
t
1
information on character
x
. Give the standard error
of the difference between average effects of two
feeds after adjusting the means for variation in
y
0
i
¼ y
i
β x
i
x
ð
Þ;
"
#
2
V β
¼ σ
2
1
n
i
þ
x
i
x
ð
Þ
V y
0
ðÞ¼V yðÞþx
i
x
2
ð
Þ
;
x
.
E
xx
Feed
F1
"
#
2
n
j
þ
x
i
x
j
1
n
i
þ
1
F2
F3
F4
V y
0
i
y
0
j
2
¼ σ
:
E
xx
x
y
x
y
x
y
x
y
11
120
12
145
9
123
10
132
10
135
11
152
11
110
12
130
This
requires
a
separate
calculation
of
2
13
121
9
160
10
120
9
140
x
i
x
j
for each
i
and
j
.D. .Finney
10
130
13
142
12
112
10
128
2
σ
2
(1946) has, however,
suggested to use
n
0
h
1
i
as an average value of
þ
T
xx
=ðt
1
Þ
E
xx
V y
0
i
y
0
j
Solution. From the above analysis, it is clear
that the response variable here is the average
weekly gain in body weight (
y
) and the covariate
is the initial body weight (
n
i
¼ n
j
¼ n
0
.
The estimated variances are given by the
corresponding expressions with
averaged over all
t
(
t
1) pairs when
2
substituted by
σ
x
). The appropriate
covariance model is
2
xy
E
xx
E
E
yy
2
s
y:x
¼
Þ
:
y
ij
¼ μ þ α
i
þ βx
ij
þ e
ij
;
i ¼
1
;
2
;
3
;
4
;
ðn t
1
j ¼
1
;
2
;
3
;
4
;
For testing the hypothesis
H
0
: α
i
¼ α
j
,we
where
y
ij
¼
have the test statistic:
gain in body weight for
j
th calves fed
2
with
i
th feed,
y
0
i
y
0
j
μ ¼
general effect,
h
i
F ¼
with
ð
1
; n t
1
Þ
d
:
f
:
2
s
2
y:x
n
0
þ
T
xx
=ðt
1
Þ
E
xx
α
i
¼
effect due to
i
th feed,
1
β ¼
regression coefficient of
y
on
x
,
It may be noted that the denominator remains
unchanged for any
e
ij
¼
independent normal variate with mean
zero
. Below, we present an
example by Sahu and Das (2010).
i
and
j
2
ð
0
Þ
and variance
; σ
:
We want to test
Example 11.1.
Four different feeds are given to
four calves each; initial age in months (
H
01
: β ¼
;
H
02
: α
1
¼ α
2
¼ α
3
¼ α
4
¼
0
x
) and
average weekly gain (
) in body weight of the
calves over a period of 8 weeks are recorded.
Analyze the data of gain in body weight using the
y
0
:
