Agriculture Reference
In-Depth Information
Table 10.5
ANOVA table for two-way classified data with
ml
(
>
1) observations per cell
SOV
d.f.
SS
MS
F
Factor A
m
1
SS(A)
AMS
¼
SS(A)/(
m
1)
AMS/ErMS
Factor B
n
1
SS(B)
BMS
¼
SS(B)/(
n
1)
BMS/ErMS
Interaction (A
B)
(
m
1)(
n
1)
SS(AB)
ABMS
SS(AB)/
(
m
1)(
n
1)
¼
ABMS/ErMS
By subtraction
¼ mn
(
l
1)
ErMS
¼
ErSS/
mn
(
l
1)
Error
ErSS
Total
mnl
1
TSS
The mean sum of squares due to error (ErMS)
always provides an unbiased estimate of
Sum of squares due to A
¼
SS
ð
A
Þ
"
#
2
, but
MS(A), MS(B), and MS(AB) provide unbiased
estimates of
σ
X
X
2
2
i
00
my
000
2
¼ nl
ðy
i
00
y
000
Þ
¼ nl
y
2
under
i
i
H
03,
respec-
tively. They are also independent. The test statis-
tics under
σ
H
01
,
H
02
, and
0
@
1
A
P
j
P
k
2
y
ijk
X
X
1
nl
2
i
H
01
,
H
02
, and
H
03
are given by
¼ nl
CF
¼
y
CF
;
nl
00
i
i
MS
Þ
ErMS
F
m
1
;mnðl
1
Þ
;
ð
A
F
A
¼
Sum of squares due to B
¼
SS
ð
B
Þ
X
X
1
ml
MS
ð
B
Þ
ErMS
F
n
1
;mnðl
1
Þ
;
2
2
0
j
0
¼ ml
ðy
0
j
0
y
000
Þ
¼
y
CF
;
F
B
¼
j
i
MS
Þ
ErMS
F
ðm
1
Þðn
1
Þ;mnðl
1
Þ
;
ð
AB
F
AB
¼
respectively
:
Sum of squares due to
ð
AB
Þ
Thus,
H
03
is rejected at
α
level of significance
m
X
n
y
2
ij
0
l
if
F
AB
> F
α;ðm
1
Þðn
1
Þ;mnðl
1
Þ
; otherwise,
it
is
¼
CF
SS
ð
A
Þ
SS
ð
B
Þ;
accepted. If
H
03
is accepted, that is, interaction
i
j
H
01
and
H
02
can be
is absent,
the tests for
ErSS
¼
TSS
SS
ð
A
Þ
SS
ð
B
Þ
SS
ð
AB
Þ;
;
performed.
H
01
is rejected
if
F
A
> F
α;m
1
;mnðl
1
Þ
:
Similarly
H
02
is rejected
y
i
00
¼
i
where
sum of the observations for
th
if
F
B
> F
α;n
1
;mnðl
1
Þ
:
level of A,
y
0
j
0
¼
sum of the observations for
j
th
H
03
is rejected, that is, interaction is present,
then we shall have to compare for each level of B
at the different levels of A and for each level
of A at the different levels of B (Table
10.5
).
For practical purposes, different sums of squares
are calculated using the following formulae:
If
level of B,
y
ij
0
¼
sum of the observations for
i
th
level of A and
j
th level of B
:
Example 10.3.
The following table gives the
yield (q/ha) of four different varieties of paddy
in response to three different doses of nitrogen.
Analyze the data to show whether:
(a) There exist significant differences in yield of
four rice varieties.
(b) Three doses of nitrogen are significantly
different with respect to production.
(c) The varieties have performed equally under
different doses of nitrogen or not.
m
X
n
X
l
Grand total
¼ G ¼
y
ijk
;
i
j
k
2
mnl
;
Treatment sum of squares
¼
TrSS
¼
G
Correction factor
¼
CF
m
X
n
X
l
2
¼
ðy
ijk
y
000
Þ
i
j
k
m
X
n
X
l
2
¼
ðy
ijk
Þ
CF
;
i
j
k
