binom.exact {exactci}R Documentation

Exact tests with matching confidence intervals for single binomial parameter

Description

Calculates exact p-values and confidence intervals for a single binomial parmeter. This is different from binom.test only when alternative='two.sided', in which case binom.exact gives three choices for tests based on the 'tsmethod' option. The resulting p-values and confidence intervals will match.

Usage

binom.exact(x, n, p = 0.5, 
   alternative = c("two.sided", "less", "greater"), 
   tsmethod = c("central", "minlike", "blaker"), 
   conf.level = 0.95, 
   control=binomControl(),plot=FALSE, midp=FALSE)

Arguments

x

number of successes, or a vector of length 2 giving the numbers of successes and failures, respectively.

n

number of trials, ignored if x has length 2.

p

hypothesized probability of success.

alternative

indicates the alternative hypothesis and must be one of "two.sided", "greater" or "less". You can specify just the initial letter.

tsmethod

indicates the method for a two-sided alternative hypothesis and must be one of "minlike", "central" or "blaker". You can specify just the initial letter.

conf.level

confidence level for the returned confidence interval.

control

list with settings to avoid problems with ties, etc, should not need to change this for normal use, see binomControl

plot

logical, do basic plot of p-value function by null hypothesis value, see exactbinomPlot for more plot options

midp

logical, use mid-p for p-values and confidence intervals? midp Confidence intervals are not available for tsmethod='minlike' and 'blaker'

Details

Traditionally, hypothesis tests and confidence intervals are treated separately. A more unified approach suggested by Hirji (2006) is to use the same p-value function to create confidence intervals. There is essentially only one way to calculate one-sided p-values and confidence intervals so these methods are the same in binom.test and binom.exact. However, there are three main ways that binom.exact allows for defining two-sided p-values.

minlike: sum probabilities of all likelihoods equal or less than observed
central: double minimum one-sided p-value
blaker: combine smaller observed tail probability with opposite tail not greater than observed tail

The 'minlike' method is the p-value that has been used in binom.test, and 'blaker' is described in Blaker (2000) or Hirji (2006), where it is called the 'combined tails' method. Once the p-value function is defined we can invert the test to create 'matching' confidence intervals defined as the smallest interval that contains all parameter values for which the two-sided hypothesis test does not reject. There are some calculation issues for the 'minlike' and 'blaker' methods which are the same as for exact tests for 2x2 tables (see Fay, 2010).

All of the above traditional p-values can be thought of as estimating Pr[X=xobs or X is more extreme than xobs] under the null hypothesis, where more extreme is defined differently for different methods. The mid-p-value replaces this with 0.5*Pr[X=xobs]+ Pr[X is more extreme than xobs]. The mid-p p-values are not valid. In other words, for all parameter values under the null hypothesis we are not guaranteed to bound the type I error rate. However, the usual exact methods that guarantee the type I error rate are typically conservative for most parameter values in order to bound the type I error rate for all parameter values. So if you are interested in rejecting approximately on average about 5 percent of the time for arbitrary parameter values and n values under the null hypothesis, then use midp=TRUE. If you want to ensure bounding of the type I errror rate for all n and all parameter values use midp=FALSE. (See for example, Vollset, 1993, or Hirji, 2006).

The associated midp confidence intervals have not been programmed for tsmethod='blaker' and 'minlike'.

Value

An object of class 'htest': a list with items

p.value

p-value

conf.int

confidence interval, see attributes 'conf.level' and perhaps 'conf.limit.prec'

statistic

number of successes

parameter

number of trials

estimate

observed proportion of success

null.value

null hypothesis probability of success, 'p'

alternative

a character string describing alternative hypothesis

method

a character string describing method

data.name

a character string giving the names of the data

Note

The 'central' method gives the Clopper-Pearson intervals, and the 'minlike' method gives confidence intervals proposed by Stern (1954) (see Blaker, 2000). The 'blaker' method is guaranteed to be more powerful than the 'central' method (see Blaker, 2000, Corollary 1), but both the 'blaker' method and 'minlike' method may have some undesireable properties. For example, there are cases where adding an additional Bernoulli observation REGARDLESS OF THE RESPONSE will increase the p-value, see Vos and Hudson (2008). The 'central' method does not have those undesireable properties.

The Blyth-Still-Casella intervals given in StatXact (and not by binom.exact) are the shortest possible intervals, but those intervals are not nested. This means that the Blyth-Still-Casella intervals are not guaranteed to have the 95 percent interval contain the 90 percent interval. See Blaker (2000) Theorem 2.

Author(s)

M.P. 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. (2010). Confidence intervals that Match Fisher's exact and Blaker's exact tests. Biostatistics. 11:373-374.

Fay, M.P. (2010). Two-sided Exact Tests and Matching Confidence Intervals for Discrete Data. R Journal 2(1): 53-58.

Hirji K. F. (2006). Exact analysis of discrete data. Chapman and Hall/CRC. New York.

Stern, T (1954). Some remarks on confidence and fiducial limits. Biometrika, 275-278.

Vollset, S. E. (1993). Confidence intervals for a binomial proportion. Statistics in medicine, 12(9), 809-824.

Vos, P.W. and Hudson, S. (2008). Problems with binomial two-sided tests and the associated confidence interals. Aust. N.Z. J. Stat. 50: 81-89.

See Also

binom.test, for two-sample exact binomial tests see exact2x2

Examples

## Notice how binom.test p-value is given by tsmethod='minlike'
## but the confidence interval is given by tsmethod='central'
## in binom.exact p-values and confidence intervals match
binom.test(10,12,p=20000/37877)
binom.exact(10,12,p=20000/37877,tsmethod="minlike")
binom.exact(10,12,p=20000/37877,tsmethod="central")
binom.exact(10,12,p=20000/37877,tsmethod="blaker")
## one-sided methods are also available
## as in binom.test


[Package exactci version 1.4-4 Index]