Agriculture Reference
In-Depth Information
Example 10.8.
Five treatments were used to test
the superiority measured in terms of harvest
index. The following information is noted. Find
out the best treatment using DMRT.
Treatment means are as follows: 2.6, 2.1, 2.5,
1.0, and 1.2
ErMS
10.7
Completely Randomized
Design (CRD)
Among all experimental designs, completely
randomized design (CRD) is the simplest one
where only two principles of design of experi-
ments, that is, replication and randomization,
have been used. The principle of local control is
not used in this design. CRD is being analyzed as
per the model of one-way ANOVA. The basic
characteristic of this design is that the whole
experimental area (1) should be homogeneous
in nature and (2) should be divided into as many
numbers of experimental units as the sum of the
number of replications of all the treatments. Let us
suppose there are five treatments A, B, C, D, and
E replicated 5, 4, 3, 3, and 5 times, respectively,
then according to this design, we require the
whole experimental area to be divided into 20
experimental units of equal size. Under labora-
tory condition, completely randomized design is
the most accepted and widely used design.
Analysis
Let there be
¼
0.925,
error
d.f.
¼
21,
no.
of
observations per mean is 5.
Solution. Five treatment means are 2.6, 2.1, 2.5,
1.0, and 1.2 with error mean square
0.925;
replication, 5; and error degrees of freedom
¼
21.
At the first instance, arrange the mean in
descending order,
¼
that
is, 2.6, 2.5, 2.1, 1.2,
and 1.0.
r
ErMS
5
r
0
:
925
5
So the SE
m
¼
¼
¼
0
:
43
:
The
R
p
values for different
p
values at error
degrees of freedom
21 are read from the
studentized range table and are as follows:
¼
p
R
p
(0.05)
R
p
2
2.94
2.94
0.43
¼
1.26
t
number of treatments with
r
1
,
r
2
,
3
3.09
3.09
0.43
¼
1.329
r
3
,
r
t
number of replications, respectively, in
a completely randomized design. So the model
for the experiment will be
...
,
3.18
0.43
¼
1.367
4
3.18
5
3.24
3.24
0.43
¼
1.39
y
ij
¼ μ þ α
i
þ e
ij
,
i ¼
1, 2, 3,
...
,
t
;
j ¼
1, 2,
...
,
r
i
,
From the largest mean (
t
1
) 2.6, we subtract the
where
y
ij
¼
R
p
value for the largest
p ¼
5, that is, 1.39, to get
response corresponding to the
j
th observa-
2.6
1.21.
Declare all the mean values which are less
than this difference value as significantly differ-
ent from the largest mean. In this case
1.39
¼
tion of the
i
th treatment
μ ¼
general effect
α
i
¼
additional effect due to the
i
th treatment
and
P
r
i
α
i
¼
t
4
and
t
5
0
are significantly different from
t
1
.
The difference between treatments
e
ij
¼
errors that are associated with the
j
th obser-
t
1
and
t
2
is
2
)
vation of the
i
th treatment and are i.i.d.
N
(0,
σ
2.6
0.5. Since there are three treatment
means not declared significantly different from
t
1,
we are to compare the above difference with
R
3
value, that is,
2.1
¼
Assumption of the Model
The above model is based on the assumptions
that the effects are additive in nature and the
error components are identically, independently
distributed as normal variate with mean zero and
constant variance.
Hypothesis to Be Tested
H
0
: α
1
¼ α
2
¼ α
3
¼¼α
i
¼¼α
t
¼
R
3
¼
1.329. The difference is
R
3
value. So these treatments are not
different significantly. Thus, the conclusion is
that
less than the
t
1
,
t
2,
and
t
3
are statistically at par while
t
4
and
t
5
are also statistically at par. The largest
treatment mean is for treatment
0
t
1
. In the next
few sections, some basic and complex experi-
mental designs have been discussed.
against the alternative hypothesis
α
0
s
H
1
:
All
are not equal
