Rob:   Fisher's exact test is conceptually possible for any r x c 
contingency table problem and uses the observed multinomial table 
probability as the test statistic.   Other tests for r x c contingency 
tables use a different test statistic (Chi-squared, likelihood ratio, 
Zelterman's).  It is possible that the probabilities for any of these 
procedures may differ slightly for the same table configuration even if 
the probabilities for each test are calculated by enumerating all possible 
permutations (hypergeometric) under the null hypothesis.   See Mielke and 
Berry 2007 (Permutation Methods:  A distance function approach) Chps 6 
and7.   Mielke has provided efficient Fortran algorithms for enumerating 
the exact probabilities for 2x2, 3x2, 4x2, 5x2, 6x2 ,3x3,and even 2x2x2 
tables for Fisher's exact and Chi-square statistics.   I don't remember 
whether Cyrus Meta's algorithms for Fisher's exact can do more.    But the 
important point to keep in mind is that it is possible to use different 
statistics for evaluating the same null hypothesis for r x c tables 
(Fisher's exact uses one form, Chi-square uses another, etc.) and the 
probabilities can be computed by exact enumeration of all permutations 
(what people expect Fisher's exact to do but also possible for Chi-square 
statistic) or by some approximation (asymptotic distribution, Monte Carlo 
resampling).  The complete enumeration of test statistics under the null 
becomes computationally intractable for large dimension r x c problems 
whether using the observed table probability (like Fisher's exact) as a 
test statistic or other like Chi-square statistic.

So in short, yes you can use Fisher's exact on your 4 x 2 problem, and the 
result might differ from using a Chi-square statistic even if you compute 
the P-value for the Chi-square test by complete enumeration.   Note that 
the minimum expected cell size for the Chi-square test is related to 
whether the Chi-square distributional approximation (an asymptotic 
argument) for evaluating the Chi-square statistic will be reasonable and 
is irrelevant if  you calculate your probabilities by exact enumeration of 
all permutations.

Brian
 

Brian S. Cade, PhD

U. S. Geological Survey
Fort Collins Science Center
2150 Centre Ave., Bldg. C
Fort Collins, CO  80526-8818

email:  brian_cade@usgs.gov
tel:  970 226-9326



From:
viostorm <rob.schutt@gmail.com>
To:
r-help@r-project.org
Date:
04/29/2011 01:23 PM
Subject:
Re: [R] fisher exact for > 2x2 table
Sent by:
r-help-bounces@r-project.org




After I shared comments form the forum yesterday with the biostatistician 
he
indicated this:

"Fisher's exact test is the non-parametric analog for the Chi-square 
test for 2x2 comparisons. A version (or extension) of the Fisher's Exact 
test, known as the Freeman-Halton test applies to comparisons for tables 
greater than 2x2. SAS can calculate both statistics using the following 
instructions.

  proc freq; tables a * b / fisher;"

Do people here still stand by position fisher exact test can be used for 
RxC
contingency tables ?  Sorry to both you all so much it is just important 
for
a paper I am writing and planning to submit soon. ( I have a 4x2 table but
does not meet expected frequencies requirements for chi-squared.)

I guess people here have suggested R implements, the following, which
unfortunately are unavailable at least easily at my library but  at least 
by
the titles indicates it is extending it to RxC 

Mehta CR, Patel NR. A network algorithm for performing Fisher's exact test
in r c contingency tables. Journal of the American Statistical Association
1983;78:427-34.
 
Mehta CR, Patel NR. Algorithm 643: FEXACT: A FORTRAN subroutine for 
Fisher's
exact test on unordered r x c contingency tables. ACM Transactions on
Mathematical Software 1986;12:154-61.

The only reason I ask again is he is exceptionally clear on this point.

Thanks again, 

-Rob



viostorm wrote:
> 
> Thank you all very kindly for your help.
> 
> -Rob
> 
> --------------------------------
> Robert Schutt III, MD, MCS 
> Resident - Department of Internal Medicine
> University of Virginia, Charlottesville, Virginia
> 

viostorm wrote:
> 
> Thank you all very kindly for your help.
> 
> -Rob
> 
> --------------------------------
> Robert Schutt III, MD, MCS 
> Resident - Department of Internal Medicine
> University of Virginia, Charlottesville, Virginia
> 


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