Agriculture Reference
In-Depth Information
Ψ
2
¼ α
1
X
þ α
2
X
þ α
3
X
þ α
4
X
ð Þk
n
1
þn
2
k
1
F
k;n
1
þn
2
k
1
,
n
1
þn
2
2
2
1
2
2
2
3
2
4
þ
assuming
that
the
variance-covariance matrices
of
the
two
þ α
k
1
X
þ α
k
X
2
k
1
2
k
;
populations are identical but unknown.
X
X
1
i
2
i
where
and
are the sample means of
i
th
2
0
1
D
¼
~
~
~
;
variable for group 1 and 2, respectively.
Let us define
d ¼ X
1
X
1
; X
2
X
2
; ... X
0
1
2
1
2
1
k
¼
~
~
X
k
Þ
0
and
2
1
n
1
þn
2
W ¼
2
~
, where
~
¼
~
þ
~
;
X
k
i¼
1
α
i
d
i
;
1
2
¼
~
~
2
are the sum of squares and products
matrices for the first and second with
1
and
n
2
number of observations, respectively, and W is
the pooled dispersion matrix for the
n
1
and
α
i
's are the linear discriminant coefficients
where
k
characters
and
d
i
's are as defined above.
This
D
2
is known as Mahalanobis generalized
based on two samples.
These
2
statistic.
The relationship between Hotelling's
α
i
's reso snt t
D
Variance between
Ψ
1
and
Ψ
2
Total variance with in groups
2
T
sta-
M ¼
is maximized.
2
statistic is
tistic and Mahalanobis generalized
D
n
1
n
2
n
1
þ n
2
D
...α
k
)
0
is given as a solution
Then
α ¼
(
α
1
,
α
2
,
2
.
2
T
¼
. Let
Ψ
1
and
Ψ
2
be the
sample means of the values for both samples and
let
of the equation
WA ¼ d
To test
H
0
we have
Ψ
1
<Ψ
2
:
Then a unit is assigned to the first
F ¼
n
1
þ n
2
k
1
Ψ
1
þΨ
2
2
population if
Ψ
and to the second popu-
k
n
1
n
2
ðn
1
þ n
2
Þðn
1
þ n
2
Ψ
1
þΨ
2
2
lation if
Ψ ¼
.
2
Þ
D
;
with
k; n
1
2
Ψ ¼ α
1
X
1
þ α
2
X
2
þ α
3
X
3
þ α
4
X
4
þ
þα
k
1
X
k
1
þ α
k
X
k
score obtained from any set
of values for a given object on the above
p
The
þ n
2
k
1d
:
f
:
Contribution of different characters towards
group discrimination is worked out as follows:
α
i
d
i
100
D
variables is used to determine the place of the
particular object in group 1 or 2. A critical score
Ψ
* is calculated (generally the average of
Ψ
1
and
Ψ
2
); if the score of the new object lies
between the
2
.
Test for the equality of
p
(
>
2) population
* and
Ψ
1
, then it belongs to group
1; on the other hand, if it lies between the
mean vectors:
In this problem, our objective is to test the null
hypothesis
Ψ
Ψ
* and
Ψ
2
, then it belongs to group 2. The effectiveness
of the discriminant function is measured with the
help of probability of misclassification. The prob-
ability of misclassification is defined as the pro-
portion of objects/individuals that are placed in a
group but are actually belonging to other groups.
Thus, for a two-group problem, the probability of
misclassification is the proportion of objects
belonging to group 1 but placed in group 2 and
similarly the proportion of objects/individuals
belonging to group 2 but placed in group 1.
To test for the equality of two population
mean vector, that is, to test
H
0
: μ
1
¼ μ
2
¼¼μ
k
vectors based
on
k
characters. In this situation, we use
Λ
statistic,
jj
WþB
defined as
Λ ¼
,where
W
and
B
are the sum
of squares and sum of product matrices for within
population and between population. This is known
as Wilk's
Λ
(lambda) criterion, which is asymp-
totically distributed as
j
j
2
distribution
χ
log
e
Λ:
Þ
k þðp
1
Þþ
1
2
χ
kðp
1
Þ
¼ n
1
2
Example 12.5.
The following table gives the
information on 20 yield and yield components
for 37 varieties distributed in four groups. Justify
whether grouping was made correctly on not,
contribution of different yield components in
group discrimination.
H
0
:
~
¼
~
2
,
1
2
analogous to Fisher's
we calculate statistic
T
t statistic, defined as
T
¼
n
1
n
2
0
1
2
n
1
þn
2
~
~
~
. This
is
known
as
