Agriculture Reference
In-Depth Information
If
a
ij
¼
1 for a particular value of
j
and for
functional relationship between the mean and
the variance may help us in identifying whether
a distribution is normal or non-normal. Homo-
geneity of error variance is one of the important
assumptions in the analysis of variance. In prac-
tice we may not get exactly equal (homosce-
dastic) variances. The variances can be tested
for homogeneity through
all
is termed general mean or general
effect. A linear model in which all the
i
, then
γ
j
γ
j
's are
unknown constants (known as parameters) is
termed
fixed effect model
. On the other hand,
a linear model in which
γ
j
's are random variables
excepting the general mean or general effect is
known as
Hartley's
test
or
random
effect model
. A linear model in which at least one
γ
j
variance component model
or
Bartlett's
test.
γ
j
is a
constant (other than general effect or general
mean) is called a
is a random variable and at least one
10.2 One-Way ANOVA
.
Assumptions in Analysis Variance
The analysis of variance is based on following
assumptions:
1. The effects are additive in nature.
2. The observations are independent.
3. The variable concerned must be normally
distributed.
4. Variances of all populations from which
samples have been drawn must be the same.
In other words, all samples should be drawn
from normal populations having common
variance with the same or different means.
The interpretation of analysis of variance is
valid only when the assumptions are met. A larger
deviation from these assumptions affects the
level of significance and the sensitivity of
mixed effect model
In one-way analysis of variance, different
samples for only one factor are considered. That
means the whole data set comprised several
groups with many observations based on one
factor as shown below:
Let
y
i
1
; y
i
2
; y
i
3
; ...y
in
i
be a random sample
from an
N μ
i
; σ
ð
2
Þ
population and
i ¼
1, 2, 3, 4,
...
,
k
. The random samples are independent.
n ¼
P
k
Suppose there are
i¼
1
n
i
observations
y
ij
(
i ¼
1, 2, 3,
...
.,
k
;
j ¼
1, 2, 3, 4,
...
,
n
i
),
which are grouped into
k
classes in the following
way (Table
10.1
):
If we consider the fixed effect model, then it
can be written as
F
-and
y
ij
¼ μ
i
þ e
ij
, where
μ
i
is a fixed
t
Two independent factors are said to be
additive in nature if the effect of one factor
remains constant over the levels of the other fac-
tor. On the contrary when the effects of one factor
remain constant by a certain percentage over the
levels of other factors, then the factors are multi-
plicative or nonadditive in nature
. The models
for additive and multiplicative effects may be
presented as follows:
-test.
i
e
ij
s
effect due to
th class and
are independently
2
N
μ
i
can be regarded as the sum of
two components, namely,
ð
0
; σ
Þ
.This
, the overall mean,
and a component due to the specific class. Thus,
we can write
μ
μ
i
¼ μ þ α
i
:
Thus
the mathematical model will be
y
ij
¼ μ þ α
i
þ e
ij
;
;
y
ij
¼ μ þ α
i
þ β
j
þ e
ij
ð
additive model
Þ
y
ij
¼ μ þ α
i
þ β
j
þðαβÞ
ij
þ e
ij
ð
multiplicative model
Þ
Table 10.1
One-way classification of data
The second assumption is related to the error
associated with the model. Precisely, this
assumption means that the errors should be inde-
pendently and identically distributed with mean
0 and constant (
1
2
.........
..
i.........
.
k
y
y
y
i
1
y
k
1
11
21
y
y
y
i
2
y
k
2
12
22
:
:
:
:
j
:
y
ij
y
jk
2
) variance.
Several normality test procedures are avail-
able in the literature. However, the nature of the
σ
:
:
:
:
y
n
y
n
y
n
y
k
n
k
1
1
2
2
i
i
