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alternative and loss function and not be limited by the availability of
tables.
We take as our model
X
ij
= m + a
i
+ e
jj
, where we select m so that the
treatment effects a
i
sum to zero;
i
= 1,..., I denotes the treatment, and
j
= 1,...,
n
i
. We assume that the error terms {e
jj
} are independent and
identically distributed.
We consider two loss functions: one in which the losses associated with
overlooking a real treatment effect, a Type II error, are proportional to
the sum of the squares of the treatment effects a
i
2
(LS), the other in
which the losses are proportional to the sum of the absolute values of the
treatment effects, |a
i
| (LAD).
Our hypothesis, a null hypothesis, is that the differential treatment
effects, the {a
i
}, are all zero. We will also consider two alternative
hypotheses:
K
U
that at least one of the differential treatment effects a
i
is
not zero, and
K
O
that
K
U
is true and there is an ordered response such
that a
1
£ a
2
£ ...£ a
I
.
For testing against
K
U
with the LS loss function, Good [2002, p. 126]
recommends the use of the statistic
F
2
=S
i
(S
j
X
ij
)
2
which is equivalent
to the
F
ratio once terms that are invariant under permutations are
eliminated.
For testing against
K
U
with the LAD loss function, Good [2002, p.
126] recommends the use of the statistic
F
1
=S
i
|S
j
X
ij
|.
For testing against
K
0
, Good [2001, p. 46] recommends the use of the
Pitman correlation statistic S
i
f
[
i
]S
j
X
ij
, where
f
[
i
] is a monotone increasing
function of
i
that depends upon the alternative. For example, for testing
for a dose response in animals where
i
denotes the dose, one might use
f
[
i
] = log[
i
+ 1].
A permutation test based on the original observations is appropriate
only if one can assume that under the null hypothesis the observations are
identically distributed in each of the populations from which the samples
are drawn. If we cannot make this assumption, we will need to transform
the observations, throwing away some of the information about them so
that the distributions of the transformed observations are identical.
For example, for testing against
K
0
, Lehmann [1999, p. 372] recom-
mends the use of the Jonckheere-Terpstra statistic, the number of pairs in
which an observation from one group is less than an observation from a
higher-dose group. The penalty we pay for using this statistic and ignoring
the actual values of the observations is a marked reduction in power for
small samples and is a less pronounced loss for larger ones.
If there are just two samples, the test based on the Jonckheere-Terpstra
statistic is identical to the Mann-Whitney test. For very large samples,
with identically distributed observations in both samples, 100 observations
would be needed with this test to obtain the same power as a permutation
