exact2x2 {exact2x2}R Documentation

Exact Conditional Tests for 2 by 2 Tables of Count Data

Description

Performs either Fisher's exact test or Blaker's exact test for testing hypotheses about the odds ratio from a two by two contingency table with fixed marginals. The commands follow the style of fisher.test, the difference is that the confidence intervals are matching ones (see details).

Usage

exact2x2(x, y = NULL, or = 1, alternative = "two.sided", tsmethod = "minlike", conf.int = TRUE, conf.level = 0.95, tol = 0.00001, conditional = TRUE)
fisher.exact(x, y = NULL, or = 1, alternative = "two.sided", tsmethod = "minlike", conf.int = TRUE, conf.level = 0.95, tol = 0.00001)
blaker.exact(x, y = NULL, or = 1, alternative = "two.sided", conf.int = TRUE, conf.level = 0.95, tol = 0.00001)

Arguments

x either a two-dimensional contingency table in matrix form, or a factor object.
y a factor object; ignored if x is a matrix.
or the hypothesized odds ratio. Must be a single numeric.
alternative indicates the alternative hypothesis and must be one of "two.sided", "greater" or "less". if "two.sided" uses method defined by tsmethod.
tsmethod one of "minlike","central", or "blaker". Defines type of two-sided method (see details). Ignored if alternative="less" or "greater".
conf.int logical indicating if a confidence interval should be computed.
conf.level confidence level for the returned confidence interval. Only used if conf.int = TRUE.
tol tolerance for confidence interval estimation.
conditional TRUE. Unconditional exact tests not supported at this time.

Details

The motivation for this package is to match the different two-sided conditional exact tests for 2x2 tables with the appropriate confidence intervals.

There are three ways to calculate the two-sided conditional exact tests, motivated by three different ways to define the p-value. The usual two-sided Fisher's exact test defines the p-value as the sum of probability of tables with smaller likelihood than the observed table (tsmethod="minlike"). The central Fisher's exact test defines the p-value as twice the one-sided p-values (but with a maximum p-value of 1). Blaker's (2000) exact test defines the p-value as the sum of the tail probibility in the observed tail plus the largest tail probability in the opposite tail that is not greater than the observed tail probability.

In fisher.test the p-value uses the two-sample method associated with tsmethod="minlike", but the confidence interval method associated with tsmethod="central". The probability that the lower central confidence limit is less than the true odds ratio is bounded by 1-(1-conf.level)/2 for the central intervals, but not for the other two two-sided methods. The confidence intervals in for exact2x2 match the test associated with alternative. In other words, the confidence interval is the smallest interval that contains the confidence set that is the inversion of the associated test (see Fay, 2009). The functions fisher.exact and blaker.exact are just wrappers for certain options in exact2x2.

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.

P-values are obtained directly using the (central or non-central) hypergeometric distribution.

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.

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
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.
null.value the odds ratio under the null, or.
alternative a character string describing the alternative hypothesis
method a character string, changes depending on alternative and tsmethod
data.name a character string giving the names of the data

Author(s)

Michael Fay

References

Blaker, H. (2000) Confidence curves and improved exact confidence intervals for discrete distributions. Canadian Journal of Statistics 28: 783-798.

Fay, M. P. (2009). Confidence intervals for Fisher's exact and Blaker's exact tests. (unpublished manuscript, see pdf in doc directory of this package).

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

See Also

fisher.test

Examples

## In example 1, notice how fisher.test rejects the null at the 5 percent level, 
## but the 95 percent confidence interval on the odds ratio contains 1 
## The intervals do not match the p-value.
## In fisher.exact you get p-values and the matching confidence intervals 
example1<-matrix(c(6,12,12,5),2,2,dimnames=list(c("Group A","Group B"),c("Event","No Event")))
example1
fisher.test(example1)
fisher.exact(example1,tsmethod="minlike")
fisher.exact(example1,tsmethod="central")
blaker.exact(example1)
## In example 2, this same thing happens, in some examples... this cannot be avoided because of the 
## holes in the confidence set.
##  
example2<-matrix(c(7,255,30,464),2,2,dimnames=list(c("Group A","Group B"),c("Event","No Event")))
example2
fisher.test(example2)
exact2x2(example2,tsmethod="minlike")
exact2x2(example2,tsmethod="central")
exact2x2(example2,tsmethod="blaker")

[Package exact2x2 version 0.9-3.1 Index]