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Introduction to
Permutation Tests
Peng Liu
3/27/2008
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Assumption for a ztest, ttest or Ftest
±
When conducting a ztest or a ttest, we are actually
assuming that the data (or the random errors) follow a
normal distribution.
±
Based on this assumption, we know the distribution of
the test statistic (T.S.) under the null hypothesis.
±
Based on the distribution (zdistribution, tdistribution or
Fdistribution), we get a pvalue for each observed T.S.
.
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This can be referred to as “
parametric approaches
”.
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What if the distributional assumption does not hold?
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If the normal assumption does not hold for the
data and the sample size is small, the results of
ztest, t or Ftest are not reliable.
±
What can we do?
Transformation of data to make the data normal
Choose some tests that do not make such
distributional assumptions – “nonparametric
approaches”
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Permutation test
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Permutation tests (randomization tests) can be
used without the normal assumption for the
distribution of data.
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Permutation test is a nonparametric approach to
establish the null distribution of a test statistic.
±
Permutation tests are attractive to microarray
study because it makes fewer assumptions.
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Permutation
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Permutation is the
rearrangement of objects
or
symbols into distinguishable sequences.
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Each unique ordering is called a permutation.
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For example, for A, B, C, and D, each possible
ordering of all 4 elements without repetitions is
one permutation, such as B, C, D, A.
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Calculation the number of permutations
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Definition: For a positive number
n
,
n!
(read n factorial) is
the product of all the positive integers less than or equal
to
n
. That is:
n! = n x (n1) x (n2) x … x 3 x 2 x 1
e.g.: 4! = 4 x 3 x 2 x 1 = 24
±
The number of permutations with
n
objects is
n!
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 Spring '08
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 Normal Distribution

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