Agriculture Reference
In-Depth Information
where
y
ij
¼
H
0
: α
1
¼ α
2
¼ α
3
¼¼α
k
¼
0 against the
response due to j
th
observation of i
th
alternative hypothesis
group,
μ ¼
general mean effect,
H
1
:
All
α
's are not equal
:
additional effect due to i
th
class group,
α
i
¼
errors component associated with j
th
obser-
vation of i
th
class group and are i.i.d.N
(0,
e
ij
¼
Let the level of significance chosen be equal
to 0.05.
2
),
σ
X
k
1
n
i
α
i
¼
0
:
y
ij
¼ y
00
þðy
i
0
y
00
Þþðy
ij
y
i
0
Þ;
where
y
i
0
¼
mean of the
i
th group
=
class,
y
00
¼
i¼
We want to test the equality of the population
means, that is,
grand mean,
;
y
ij
y
00
¼ðy
i
0
y
00
Þþðy
ij
y
i
0
Þ:
Squaring and taking sum for both the sides over
i
and
j;
we have
X
X
X
X
2
2
ðy
ij
y
00
Þ
¼
ðy
i
0
y
00
Þþðy
ij
y
i
0
Þ
i
j
i
j
X
X
X
X
2
X
i
X
2
2
ðy
i
0
y
00
Þ
ðy
ij
y
i
0
Þ
ðy
i
0
y
00
Þðy
ij
y
i
0
Þ
¼
þ
þ
i
j
i
j
j
X
X
X
X
2
X
i
X
2
2
¼
ðy
i
0
y
00
Þ
þ
ðy
ij
y
i
0
Þ
þ
ðy
i
0
y
00
Þ
ðy
ij
y
i
0
Þ
i
j
i
j
j
"
#
X
X
X
X
2
2
¼
n
i
ðy
i
0
y
00
Þ
þ
ðy
ij
y
i
0
Þ
ðy
ij
y
i
0
Þ¼
0
,
i
i
j
j
SS
ð
Total
Þ¼
SS
ð
class
=
group
Þþ
SS
ð
error
Þ
,
TSS
¼
CSS
þ
ErSS
:
Thus, the total sum of squares is partitioned
into sum of squares due to classes/groups and
sum of squares due to error.
n
(
n k
) d.f. since it is based on quantities which
are
subjected
to
k
linear
constraints
P
ðy
ij
y
i
Þ¼
0
; i ¼
1
;
2
;
3
; ...; k
.Itistobe
j
1 d.f. is attributed to SS(total) or TSS
because it is computed from the
noted that CSS and ErSS add up to TSS and the
corresponding d.f. is also additive. Dividing SS
by their respective d.f., we will get the respective
mean sum of squares, that is, MSS due to class
and error. Thus,
n
quantities of
the form
ðy
ij
yÞ
which is subjected to one linear
constraint
P
j
ðy
ij
y
i
Þ¼
0
; i ¼
1
;
2
;
3
; ::::; k
.
Similarly, (
k
1) d.f. is attributed to CSS since
P
n
i
ðy
i
0
y
00
Þ¼
0. Finally ErSS will have
i
CSS
k
CMS
¼
1
¼
mean sum of squares due to class
=
group,
ErSS
n k
¼
ErMS
¼
mean sum of squares due to error
:
