Agriculture Reference
In-Depth Information
From the above calculation we are to take
decision and calculate CD/LSD values required
using the formula discussed during manual cal-
culation. There are certain packages like SPSS
where the pairwise mean comparison also comes
out as output.
Two-Way Analysis of Variance with More
Than One Observations per Cell
Working out the interaction effects of more
than one factor from a single experiment is one of
the major objectives of the two-way analysis of
variance with more than one observation per cell.
To test
m
X
n
X
l
X
2
2
y
ijk
y
000
¼ nl
ð
y
i
00
y
000
Þ
i
j
k
i
2
X
þ ml
y
0
j
0
y
000
j
2
X
X
þ l
y
ij
0
y
i
00
y
0
j
0
þ y
000
i
j
2
X
X
X
þ
y
ijk
y
ij
0
i
j
k
(Other product terms vanish because of the
assumptions as usual.)
or SS(total)
SS(factor A) + SS(factor B) +
SS(factor AB) + SS(error)
or TSS
¼
H
01
: α
1
¼ α
2
¼ α
3
¼α
m
¼
0
;
SS(A) + SS(B) + SS(AB) + ErSS.
The corresponding partitioning of the total
d.f. is as follows:
¼
H
02
: β
1
¼ β
2
¼ β
3
¼β
n
¼
0
;
H
03
: γ
ij
¼
0 for all
i
and
j
,
Lmn
1
¼ m
ð
1
Þþn
ð
1
Þþm
ð
1
Þ
the appropriate model for the purpose would be
n
ð
1
Þþmn l
ð
1
Þ;
X
X
X
1
mnl
y
ijk
¼ μ þ α
i
þ β
j
þ γ
ij
þ e
ijk
;
where
y
000
¼
y
ijk
¼
mean of all
i
j
k
where
i ¼
1, 2,
...
,
m
;
j ¼
1, 2,
...
,
n
;
k ¼
1,
observations
;
X
X
2,
...
,
l
1
nl
y
i
00
¼
y
ijk
¼
i
;
mean of
th level of A
μ ¼
general effect
j
k
α
i
¼
additional effect due to the
i
th group of
X
X
1
ml
factor A
β
j
¼
y
0
j
0
¼
y
ijk
¼
mean of
j
th level of B
;
additional effect due to the
j
th group of
i
k
X
1
l
factor B
γ
ij
¼
y
ij
0
¼
y
ijk
¼
mean of the observations for
interaction effect due to the
i
th group of
k
j
factor A and the
th group of factor B
i
th level of A and
j
th level of B
:
e
ijk
¼
errors that are associated with the
k
th
Dividing the sum of squares by their
corresponding d.f., we have the following:
observation of the
i
th class of factor A and
2
);
the
j
th class of factor B and are i.i.d.
N
(0,
σ
P
α
i
¼
P
j
β
j
¼
P
i
γ
ij
¼
0,
SS
ð
A
Þ
i
Mean sum of squares due to factor A
¼
β
j
m
1
μ ¼ y
000
; α
i
¼ y
i
00
y
000
;
where
¼
MS
ð
A
Þ
,
¼ y
0
j
0
y
000
;
γ
ij
¼ y
ij
0
y
i
00
y
0
j
0
þ y
000
:
and
SS
ð
B
Þ
Mean sum of squares due to factor B
¼
n
1
,
Mean sum of squares due to factor AB
¼
MS
ð
B
Þ
Thus, the linear model becomes
y
ijk
¼ y
000
þ y
i
00
y
000
ð
Þ þ y
0
j
0
y
000
SS
ð
AB
Þ
¼
Þ
¼
MS
ð
AB
Þ
,
:
ðm
1
Þðn
1
þ y
ij
0
y
i
00
y
0
j
0
þ y
000
þ y
ijk
y
ij
0
ErSS
mnð
Mean sum of squares due to error
¼
Þ
l
1
y
000
to the left, squaring both the
sides, and summing over
Transferring
¼
:
ErMS
i
,
j
,
k
, we get
