Agriculture Reference
In-Depth Information
and
For practical purposes the various sums of
squares are calculated using the following
formulae:
μ ¼ y
00
;
α
i
¼ y
i
0
y
00
;
β
j
¼ y
0
j
y
00
;
m
X
n
X
X
X
G ¼
grand total
¼
1
y
ij
;
1
mn
1
n
1
m
with
y
00
¼
y
ij
; y
i
0
¼
y
ij
; y
0
j
¼
y
ij
:
i¼
1
j¼
i;j
j
i
þ
2
mn
;
Þ¼
G
ð
Correction factor
CF
Thus,
y
ij
¼ y
00
þ y
i
0
y
00
ð
Þ þ y
0
j
y
00
X
X
. (The product terms van-
ish because of assumption of independence.)
X
2
Total sumof squares
ð
TSS
Þ¼
ðy
ij
y
00
Þ
y
ij
y
i
0
y
0
j
þ y
00
i
j
X
X
2
¼
y
ij
CF
;
X
X
2
2
i
j
y
ij
y
00
¼ n
ð
y
i
0
y
00
Þ
Sum of squares
ð
A
Þ¼
SS
ð
A
Þ
i
j
i
X
"
#
þ m
y
0
j
y
00
X
X
2
2
i
2
00
¼ n
ðy
i
0
y
00
Þ
¼ n
y
my
j
2
X
X
0
i
i
þ
y
ij
y
i
0
y
0
j
þ y
00
0
1
n
2
i
j
1
y
ij
n
X
X
@
A
or SS total
ð
Þ¼
SS factor A
ð
Þ þ
SS factor B
ð
Þ þ
SS error
ð
Þ
1
n
j¼
2
00
2
i
¼ n
nmy
¼
y
CF,
0
or TSS
¼
SS
ðÞþ
SS
ðÞþ
ErSS
:
i
i
The corresponding partitioning of the total d.f.
is as follows:
X
n
j¼
1
y
ij
where
; y
i
0
¼
is the sum of observations
mn
1
¼ m
ð
1
Þþn
ð
1
Þþm
ð
1
Þ n
ð
1
Þ:
for
i
th level of factor A
:
Hence, the test statistic under
H
01
is given by
Sum of squares
ð
B
Þ¼
SS
ð
B
Þ
SS
ð
A
Þ
1
m
ErSS
σ
1
2
X
X
1
m
F ¼
:
:
0
¼ m
y
0
j
y
00
¼
y
j
CF,
σ
2
1
2
ðm
1
Þðn
1
Þ
j
j
MS
Þ
ErMS
ð
A
¼
m
i¼
1
y
ij
where
y
0
j
¼
is the sum of observations
which follows
F
distribution with [(
m
1),
th
for
j
level of factor B
;
(
m
1)(
n
1)] d.f.
ErSS
¼
TSS
SS
ð
A
Þ
SS
ð
B
Þ
Thus,
the null hypothesis
H
01
is rejected
at
α
level of significance if
Dividing this SSs by their respective degrees
of freedom, we will get the corresponding mean
sum of squares, that is, mean sum of squares due
to classes and error mean sum of squares.
We have Table
10.4
as the ANOVA table for
two-way classification with one observation per
cell.
MS
Þ
ErMS
> F
α; ðm
ð
A
F ¼
d
:
f
:
1
Þ; ðm
1
Þðn
1
Þ
Similarly the test statistics under
H
02
is given by
MS
Þ
ErMS
ð
B
F ¼
with
α; ðm
1
Þ; ðm
1
Þðn
1
Þ
d
:
f
:
