\HeaderA{fisher.test}{Fisher's Exact Test for Count Data}{fisher.test}
\keyword{htest}{fisher.test}
\begin{Description}\relax
Performs Fisher's exact test for testing the null of independence of
rows and columns in a contingency table with fixed marginals.
\end{Description}
\begin{Usage}
\begin{verbatim}
fisher.test(x, y = NULL, workspace = 200000, hybrid = FALSE,
            control = list(), or = 1, alternative = "two.sided",
            conf.int = TRUE, conf.level = 0.95,
            simulate.p.value = FALSE, B = 2000)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] either a two-dimensional contingency table in matrix form,
or a factor object.
\item[\code{y}] a factor object; ignored if \code{x} is a matrix.
\item[\code{workspace}] an integer specifying the size of the workspace
used in the network algorithm.  In units of 4 bytes.  Only used for
non-simulated p-values larger than \eqn{2 \times 2}{2 by 2} tables.
\item[\code{hybrid}] a logical. Only used for larger than \eqn{2 \times 2}{2 by 2}
tables, in which cases it indicated whether the exact probabilities
(default) or a hybrid approximation thereof should be computed.
See Details.
\item[\code{control}] a list with named components for low level algorithm
control.  At present the only one used is \code{"mult"}, a positive
integer \eqn{>= 2}{} with default 30 used only for larger than
\eqn{2 \times 2}{2 by 2} tables.  This says how many times as much
space should be allocated to paths as to keys: see file
\file{fexact.c} in the sources of this package.
\item[\code{or}] the hypothesized odds ratio.  Only used in the
\eqn{2 \times 2}{2 by 2} case.
\item[\code{alternative}] indicates the alternative hypothesis and must be
one of \code{"two.sided"}, \code{"greater"} or \code{"less"}.
You can specify just the initial letter.  Only used in the
\eqn{2 \times 2}{2 by 2} case.
\item[\code{conf.int}] logical indicating if a confidence interval should be
computed (and returned).
\item[\code{conf.level}] confidence level for the returned confidence
interval.  Only used in the \eqn{2 \times 2}{2 by 2} case if
\code{conf.int = TRUE}.
\item[\code{simulate.p.value}] a logical indicating whether to compute
p-values by Monte Carlo simulation, in larger than \eqn{2 \times
2}{2 by 2} tables.
\item[\code{B}] an integer specifying the number of replicates used in the
Monte Carlo test.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
If \code{x} is a matrix, it is taken as a two-dimensional contingency
table, and hence its entries should be nonnegative integers.
Otherwise, both \code{x} and \code{y} must be vectors of the same
length.  Incomplete cases are removed, the vectors are coerced into
factor objects, and the contingency table is computed from these.

For \eqn{2 \times 2}{2 by 2} cases, p-values are obtained directly
using the (central or non-central) 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 and Joe (1993).
The FORTRAN code can be obtained from
\url{http://www.netlib.org/toms/643}.  Note this fails (with an error
message) when the entries of the table are too large.  (It transposes
the table if necessary so it has no more rows than columns.  One
constraint is that the product of the row marginals be less than
\eqn{2^{31} - 1}{2^31 - 1}.)

For \eqn{2 \times 2}{2 by 2} tables, the null of conditional
independence is equivalent to the hypothesis that the odds ratio
equals one.  \sQuote{Exact} inference can be based on observing that in
general, given all marginal totals fixed, the first element of the
contingency table has a non-central hypergeometric distribution with
non-centrality parameter given by the odds ratio (Fisher, 1935).  The
alternative for a one-sided test is based on the odds ratio, so
\code{alternative = "greater"} is a test of the odds ratio being bigger
than \code{or}.

Two-sided tests are based on the probabilities of the tables, and take
as \sQuote{more extreme} all tables with probabilities less than or
equal to that of the observed table, the p-value being the sum of such
probabilities.

For larger than  \eqn{2 \times 2}{2 by 2} tables and \code{hybrid =
    TRUE}, asymptotic chi-squared probabilities are only used if the
\dQuote{Cochran conditions} are satisfied, that is if no cell has
count zero, and more than 80\% of the cells have counts at least 5.

Simulation is done conditional on the row and column marginals, and
works only if the marginals are strictly positive.  (A C translation
of the algorithm of Patefield (1981) is used.)
\end{Details}
\begin{Value}
A list with class \code{"htest"} containing the following components:
\begin{ldescription}
\item[\code{p.value}] the p-value of the test.
\item[\code{conf.int}] a confidence interval for the odds ratio.
Only present in the \eqn{2 \times 2}{2 by 2} case if argument
\code{conf.int = TRUE}.
\item[\code{estimate}] an estimate of the odds ratio.  Note that the
\emph{conditional} Maximum Likelihood Estimate (MLE) rather than the
unconditional MLE (the sample odds ratio) is used.
Only present in the \eqn{2 \times 2}{2 by 2} case.
\item[\code{null.value}] the odds ratio under the null, \code{or}.
Only present in the \eqn{2 \times 2}{2 by 2} case.
\item[\code{alternative}] a character string describing the alternative
hypothesis.
\item[\code{method}] the character string
\code{"Fisher's Exact Test for Count Data"}.
\item[\code{data.name}] a character string giving the names of the data.
\end{ldescription}
\end{Value}
\begin{References}\relax
Agresti, A. (1990)
\emph{Categorical data analysis}.
New York: Wiley.
Pages 59--66.

Fisher, R. A. (1935)
The logic of inductive inference.
\emph{Journal of the Royal Statistical Society Series A} \bold{98},
39--54.

Fisher, R. A. (1962)
Confidence limits for a cross-product ratio.
\emph{Australian Journal of Statistics} \bold{4}, 41.

Fisher, R. A. (1970)
\emph{Statistical Methods for Research Workers.}  Oliver \& Boyd.

Mehta, C. R. and Patel, N. R. (1986)
Algorithm 643. FEXACT: A Fortran subroutine for Fisher's exact test
on unordered \eqn{r*c}{} contingency tables.
\emph{ACM Transactions on Mathematical Software}, \bold{12},
154--161.

Clarkson, D. B., Fan, Y. and Joe, H. (1993)
A Remark on Algorithm 643: FEXACT: An Algorithm for Performing
Fisher's Exact Test in \eqn{r \times c}{r x c} Contingency Tables.
\emph{ACM Transactions on Mathematical Software}, \bold{19},
484--488.

Patefield, W. M. (1981)
Algorithm AS159.  An efficient method of generating r x c tables
with given row and column totals.
\emph{Applied Statistics} \bold{30}, 91--97.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{chisq.test}{chisq.test}}
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
## Agresti (1990), p. 61f, Fisher's Tea Drinker
## A British woman claimed to be able to distinguish whether milk or
##  tea was added to the cup first.  To test, she was given 8 cups of
##  tea, in four of which milk was added first.  The null hypothesis
##  is that there is no association between the true order of pouring
##  and the woman's guess, the alternative that there is a positive
##  association (that the odds ratio is greater than 1).
TeaTasting <-
matrix(c(3, 1, 1, 3),
       nr = 2,
       dimnames = list(Guess = c("Milk", "Tea"),
                       Truth = c("Milk", "Tea")))
fisher.test(TeaTasting, alternative = "greater")
## => p=0.2429, association could not be established

## Fisher (1962, 1970), Criminal convictions of like-sex twins
Convictions <-
matrix(c(2, 10, 15, 3),
       nr = 2,
       dimnames =
       list(c("Dizygotic", "Monozygotic"),
            c("Convicted", "Not convicted")))
Convictions
fisher.test(Convictions, alternative = "less")
fisher.test(Convictions, conf.int = FALSE)
fisher.test(Convictions, conf.level = 0.95)$conf.int
fisher.test(Convictions, conf.level = 0.99)$conf.int

## A r x c table  Agresti (2002, p. 57) Job Satisfaction
Job <- matrix(c(1,2,1,0, 3,3,6,1, 10,10,14,9, 6,7,12,11), 4, 4,
dimnames = list(income=c("< 15k", "15-25k", "25-40k", "> 40k"),
                satisfaction=c("VeryD", "LittleD", "ModerateS", "VeryS")))
fisher.test(Job)
fisher.test(Job, simulate=TRUE, B=1e5)
\end{ExampleCode}
\end{Examples}

