Fisher's Exact Test for Count Data

Performs Fisher's exact test for testing the null of independence of rows and columns in a contingency table with fixed marginals, or with a formula expression.

Usage

fisher.test(
  x,
  y,
  workspace,
  hybrid,
  control,
  or,
  alternative,
  conf.int,
  conf.level,
  simulate.p.value,
  B
)

# S3 method for default
fisher.test(x, ...)

# S3 method for formula
fisher.test(
  x,
  y = NULL,
  workspace = 200000,
  hybrid = F,
  control = list(),
  or = 1,
  alternative = "two.sided",
  conf.int = T,
  conf.level = 0.95,
  simulate.p.value = F,
  B = 2000
)

Arguments

x: either a two-dimensional contingency table in matrix form, a factor object, or a formula of the form lhs ~ rhs where lhs and rhs are factors.
y: a factor object; ignored if x is a matrix or a formula.
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 \(2 \times 2\) tables. Since R version 3.5.0, this also increases the internal stack size which allows larger problems to be solved, however sometimes needing hours. In such cases, simulate.p.values=TRUE may be more reasonable.
hybrid: a logical. Only used for larger than \(2 \times 2\) tables, in which cases it indicates whether the exact probabilities (default) or a hybrid approximation thereof should be computed.
control: a list with named components for low level algorithm control. At present the only one used is "mult", a positive integer \(\ge 2\) with default 30 used only for larger than \(2 \times 2\) tables. This says how many times as much space should be allocated to paths as to keys: see file fexact.c in the sources of this package.
or: the hypothesized odds ratio. Only used in the \(2 \times 2\) case.
alternative: indicates the alternative hypothesis and must be one of "two.sided", "greater" or "less". You can specify just the initial letter. Only used in the \(2 \times 2\) case.
conf.int: logical indicating if a confidence interval for the odds ratio in a \(2 \times 2\) table should be computed (and returned).
conf.level: confidence level for the returned confidence interval. Only used in the \(2 \times 2\) case and if conf.int = TRUE.
simulate.p.value: a logical indicating whether to compute p-values by Monte Carlo simulation, in larger than \(2 \times 2\) tables.
B: an integer specifying the number of replicates used in the Monte Carlo test.
...: additional params to feed to original fisher.test

Value

A list with class "htest" containing the following components: p.value: the p-value of the test. conf.int: a confidence interval for the odds ratio. Only present in the 2 by 2 case and if argument conf.int = TRUE. estimate: an estimate of the odds ratio. Note that the _conditional_ Maximum Likelihood Estimate (MLE) rather than the unconditional MLE (the sample odds ratio) is used. Only present in the 2 by 2 case. null.value: the odds ratio under the null, or. Only present in the 2 by 2 case. alternative: a character string describing the alternative hypothesis. method: the character string "Fisher's Exact Test for Count Data". data.name: a character string giving the names of the data.

Details

If x is a matrix, it is taken as a two-dimensional contingency table, and hence its entries should be nonnegative integers. Otherwise, both x and 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 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 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 2^31 - 1.)

For 2 by 2 tables, 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). The alternative for a one-sided test is based on the odds ratio, so alternative = "greater" is a test of the odds ratio being bigger than or.

Two-sided tests are based on the probabilities of the tables, and take as 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 2 by 2 tables and hybrid = TRUE, asymptotic chi-squared probabilities are only used if the ‘Cochran conditions’ are satisfied, that is if no cell has count zero, and more than 80 exact calculation is used.

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.)

References

Agresti, A. (1990) _Categorical data analysis_. New York: Wiley. Pages 59-66.

Agresti, A. (2002) _Categorical data analysis_. Second edition. New York: Wiley. Pages 91-101.

Fisher, R. A. (1935) The logic of inductive inference. _Journal of the Royal Statistical Society Series A_ *98*, 39-54.

Fisher, R. A. (1962) Confidence limits for a cross-product ratio. _Australian Journal of Statistics_ *4*, 41.

Fisher, R. A. (1970) _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 r*c contingency tables. _ACM Transactions on Mathematical Software_, *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 r x c Contingency Tables. _ACM Transactions on Mathematical Software_, *19*, 484-488.

Patefield, W. M. (1981) Algorithm AS159. An efficient method of generating r x c tables with given row and column totals. _Applied Statistics_ *30*, 91-97.

Examples