\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)
\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.
\item[\code{hybrid}] a logical indicating whether the exact probabilities
(default) or a hybrid approximation thereof should be computed.
In the hybrid case, asymptotic chi-squared probabilities are only
used provided that the \dQuote{Cochran} conditions are satisfied.
\item[\code{control}] a list with named components for low level algorithm control.
\item[\code{or}] the hypothesized odds ratio.  Only used in the 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 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 2 by 2 case.
\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.

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}.  Note this fails (with an error
message) when the entries of the table are too large.

In the 2 by 2 case, the null of conditional independence is equivalent
to the hypothesis that the odds ratio equals one.  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).
\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 2 by 2 case.
\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 2 by 2 case.
\item[\code{null.value}] the odds ratio under the null, \code{or}.
Only present in the 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
Alan Agresti (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.

Cyrus R. Mehta \& Nitin R. Patel (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.

Douglas B. Clarkson, Yuan-an Fan \& Harry Joe (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.
\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 women'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), Convictions of like-sex twins in criminals
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
\end{ExampleCode}
\end{Examples}

