# [R] Basis of fisher.test

(Ted Harding) Ted.Harding at nessie.mcc.ac.uk
Thu Jan 12 21:19:06 CET 2006

```I want to ascertain the basis of the table ranking,
i.e. the meaning of "extreme", in Fisher's Exact Test
as implemented in 'fisher.test', when applied to RxC
tables which are larger than 2x2.

One can summarise a strategy for the test as

1) For each table compatible with the margins
of the observed table, compute the probability
of this table conditional on the marginal totals.

2) Rank the possible tables in order of a measure
of discrepancy between the table and the null
hypothesis of "no association".

3) Locate the observed table, and compute the sum
of the probabilties, computed in (1), for this
table and more "extreme" tables in the sense of
the ranking in (2).

The question is: what "measure of discrepancy" is
used in 'fisher.test' corresponding to stage (2)?

(There are in principle several possibilities, e.g.
value of a Pearson chi-squared, large values being
discrepant; the probability calculated in (2),
small values being discrepant; ... )

"?fisher.test" says only:

In the one-sided 2 by 2 cases, p-values are obtained
directly using the hypergeometric distribution.
Otherwise, computations are based on a C version of
the FORTRAN subroutine FEXACT which implements the
network developed by Mehta and Patel (1986) and
improved by Clarkson, Fan & Joe (1993). The FORTRAN
code can be obtained from
<URL: http://www.netlib.org/toms/643>.

I have had a look at this FORTRAN code, and cannot ascertain
it from the code itself. However, there is a Comment to the
effect:

c     PRE    - Table p-value.  (Output)
c              PRE is the probability of a more extreme table, where
c              'extreme' is in a probabilistic sense.

which suggests that the tables are ranked in order of their
probabilities as computed in (2).

Can anyone confirm definitively what goes on?

With thanks,
Ted.

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Date: 12-Jan-06                                       Time: 20:19:02
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