Information Technology Reference
In-Depth Information
Lest we keep you in suspense, a distribution-free exact and more power-
ful test for comparing variances can be derived based on the permutation
distribution of Aly's statistice.
This statistic proposed by Aly [1990] is
m
-
1
Â
im
(
)
(
)
d =
-
i X
-
X
(
)
( )
i
+
1
i
i
=
1
where
X
(1)
£
X
(2)
£ ...£
X
(
m
)
are the order statistics of the first sample.
Suppose we have two sets of measurements, 121, 123, 126, 128.5, 129
and in a second sample, 153, 154, 155, 156, 158. We replace these with
the deviations
z
1
i
=
X
(
i
+ 1)
-
X
(
i
)
or 2, 3, 2.5, .5 for the first sample and
z
2
i
= 1, 1, 1, 2 for the second.
The original value of the test statistic is 8 + 18 + 15 + 2 = 43. Under
the hypothesis of equal dispersions in the two populations, we can
exchange labels between
z
1
i
and
z
2
i
for any or all of the values of
i
. One
possible rearrangement of the labels on the deviations puts {2, 1, 1, 2} in
the first sample, which yields a value of 8 + 6 + 6 + 8 = 28.
There are 2
4
= 16 rearrangements of the labels in all, of which only one
{2, 3, 2.5, 2} yields a larger value of Aly's statistic than the original obser-
vations. A one-sided test would have two out of 16 rearrangements as or
more extreme than the original, and a two-sided test would have four. In
either case, we would accept the null hypothesis, though the wiser course
would be to defer judgment until we have taken more observations.
If our second sample is larger than the first, we have to resample in two
stages. First, we select a subset of
m
values at random without replacement
from the
n
observations in the second, larger sample and compute the
order statistics and their differences. Last, we examine all possible values of
Aly's measure of dispersion for permutations of the combined sample as
we did when the two samples were equal in size and compare Aly's
measure for the original observations with this distribution. We repeat this
procedure several times to check for consistency.
COMPARING THE MEANS OF
K
SAMPLES
The traditional one-way analysis of variance based on the
F
ratio
Â
I
2
(
)
(
)
nX
-
X
I
-
1
i
i
.
..
i
=
1
Â
I
Â
n
2
(
)
(
)
i
XX NI
-
-
ij
i
.
i
=
1
j
=
1
has at least three major limitations:
