Reisensburg 1996: Abstract J. Röhmel
**Statistical Computing '96 - Schloß Reisensburg**

**The Permutation Distribution of the Friedman Test
**

**
Joachim Röhmel
**

Bundesinstitut für Arzneimittel und Medizinprodukte, Berlin
During the last ten years, many algorithms for calculating exact
distributions of discrete
random variables have been developed, particularly for the independent 2-sample
case. Exact distributions for quadratic statistics such as the Friedman test
or the
Kruscal Wallis test are still lacking. The Friedman test is part of most of
the major
statistical packages, though usually not very well documented, and of course
only
applicable in its approximate form. I develop an algorithm for calculating
the exact
distribution of the Friedman test which is feasible with respect to time and
memory
contraints on commonly available PC's.

For establishing this algorithms, some properties of the symmetric group G

will be used. In addition, two operators on k-dimensional cubes are
introduced, the take
operator and the rotate operator which prove to be very helpful in
describing the
algorithm as well as in realizing it in the computer language APL.
The Friedman statistic is a quadratic statistic. It is useful in situations
where one has no
idea about the direction of the deviations between the treatments under the
alternative
hypothesis. In the proposed development we will make use of particular
properties of
the usual Friedman statistic only in the final part. In fact, for the
derivation of
distributions of other often used tests one can apply similar ideas.
Examples are
provided by all linear functions (i.e. contrasts between the treatments).
Among others,
the exact distribution of the Page test falls into this class, which is of
use, when dose
response is investigated, i.e. when the treatments t1, t2, ..., tk
represent increasing
doses of the same substance. More examples can be constructed, when
a particular
(partial) order is specified under the alternative hypthesis. The
computation of exact
distributions of linear statistics is straight forward and can be achieved
by convolution.
Only quadratic statistics need special attention.

The complexity of the algorithms will be discussed, and APL programms will be
provided. Some further simplifications of the algorithms by means of
combinatorial
properties will be proposed.

**Literatur:**

- Ralphs V.W., Zimmermann H. (1995). Nonparametric methods in standard statistical packages:
The case of the Friedman Analysis. The Statistical Software Newsletter 20, 101-110.

Statistical Computing '96 auf
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