poisson.exact {exactci}

Exact Poisson tests with Matching Confidence Intervals

Description

Performs an exact test of a simple null hypothesis about the rate parameter in Poisson distribution, or for the ratio between two rate parameters. This is different from poisson.test in that 3 different types of exact two-sided tests (and the matching confidence intervals) are offered. The one-sided tests are the same as in poisson.test.

Usage

poisson.exact(x, T = 1, r = 1,
    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 events. A vector of length one or two.

T

time base for event count. A vector of length one or two.

r

hypothesized rate or rate ratio

alternative

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

tsmethod

character giving two-sided method, one of "central", "minlike" or "blaker", ignored if alternative not equal "two.sided"

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, plot p-value function? For finer control on plot see exactpoissonPlot

midp

logical, use mid-p p-values? Only allowed for two-sided tests if tsmethod='central' (see details)

Details

Confidence intervals are computed similarly to those of binom.exact in the one-sample case, in that there are three two-sided options depending on the tsmethod. For the one-sample case the default intervals use tsmethod="central" giving the Garwood (1936) exact central confidence intervals. For the two-sample case we condition on the total counts and then use binomial methods, see Lehmann and Romano (2005) for that motivation and vignette("exactci") for description of the three different two-sided methods for calculating p-values and confidence intervals.

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 under the null hypothesis, then use midp=TRUE. If you want to ensure bounding of the type I errror rate for all parameter values use midp=FALSE. For a comprehensive discussions of exact inferences including mid-p values see Hirji (2006). Mid-p confidence intervals are calculated by inverting the mid-p-value function. For discussion of mid-p confidence intervals for Poisson see Cohen and Yang (1994).

The mid-p p-values and confidence intervals have not been programmed for the 'blaker' and 'minlike' tsmethods.

Value

A list with class "htest" containing the following components:

statistic

the number of events (in the first sample if there are two.)

parameter

the corresponding expected count

p.value

the p-value of the test.

conf.int

a confidence interval for the rate or rate ratio.

estimate

the estimated rate or rate ratio.

null.value

the rate or rate ratio under the null, r.

alternative

a character string describing the alternative hypothesis.

method

the character string "Exact Poisson test" or "Comparison of Poisson rates" as appropriate.

data.name

a character string giving the names of the data.

Note

The rate parameter in Poisson data is often given based on a “time on test” or similar quantity (person-years, population size). This is the role of the T argument.

References

Cohen and Yang (1994). Mid-p Confidence intervals for the Poisson Expectation. Statistics in Medicine. 13: 2189-2203.

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

Garwood, F (1936). Fiducial limits for the Poisson distribution. Biometrika, 437-442.

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

Lehmann, EL, and Romano, JP (2005). Testing Statistical Hypotheses, third edition. Springer:New York.

See Also

poisson.test, exactpoissonPlot,

Examples

### Suppose you have observed rates of  2 out of 17877 in group A 
### and 10 out of 20000 in group B
### poisson.test gives non-matching confidence intervals
### i.e., p-value using  'minlike' criteria but confidence interval using 'central' criteria
poisson.test(c(2,10),c(17877,20000))
### poisson.exact gives matching CI to the p-values
### defaults to 'central' two-sided method
poisson.exact(c(2,10),c(17877,20000))
### other options
poisson.exact(c(2,10),c(17877,20000),tsmethod="minlike")
poisson.exact(c(2,10),c(17877,20000),tsmethod="blaker")


## Mid-p confidence intervals do not guarantee coverage, 
## but are more likely to have on average closer nominal 
## coverage than exact ones (sometimes erroring on the 
## too liberal side). 
##
## To test the software, here is Table I of Cohen and Yang
## values are equal to the first 2 decimal places
yCY<-c(0:20,20+(1:5)*2,30+(1:14)*5)
TableICohenYang<-matrix(NA,length(yCY),6,dimnames=list(yCY,
    c("90pct LL","90pct UL","95pct LL","95pct UL","99pct LL","99pct UL")))

for (i in 1:length(yCY)){
    TableICohenYang[i,1:2]<-poisson.exact(yCY[i],
        midp=TRUE,conf.level=.9)$conf.int
    TableICohenYang[i,3:4]<-poisson.exact(yCY[i],
        midp=TRUE,conf.level=.95)$conf.int
    TableICohenYang[i,5:6]<-poisson.exact(yCY[i],
        midp=TRUE,conf.level=.99)$conf.int
}
TableICohenYang<-round(TableICohenYang,3)
TableICohenYang