Confidence Interval for a Poisson Rate

Estimates a Poisson event rate and calculates an exact or approximate confidence interval.

Usage

poissonCI(
  x,
  n = 1,
  conf.level = 0.95,
  sides = c("two.sided", "left", "right"),
  method = c("exact", "score", "wald", "byar")
)

Arguments

x: non-negative integer event count or vector of counts
n: positive exposure associated with x, such as observation time, person-time, or population at risk; may be a vector and defaults to 1
conf.level: numeric confidence level between 0 and 1; defaults to 0.95
sides: type of confidence interval: "two.sided", "left", or "right"; may be abbreviated
method: method used to calculate the confidence interval: "exact", "score", "wald", or "byar"; may be abbreviated and may contain several methods; defaults to "exact"

Value

If all arguments identify a single result, a named numeric vector with elements:

est: estimated event rate
lci: lower confidence bound
uci: upper confidence bound

Otherwise, a data.frame containing these three columns and the recycled argument values that identify each result.

Details

The function assumes $$X \sim \mathrm{Poisson}(n\lambda),$$ where x is the observed event count, n is the exposure, and $\lambda$ is the event rate. The point estimate is $\hat{\lambda} = x/n$.

The available confidence-interval methods are:

"exact": the exact Poisson interval calculated by poisson.test(), equivalent to the Garwood interval
"score": the interval obtained by inverting the Poisson score test
"wald": the normal-approximation interval centred at $\hat{\lambda}$
"byar": Byar's cube-root normal approximation

The lower bound is restricted to the parameter space $[0, \infty)$. For sides = "left", the function returns a lower one-sided confidence bound and sets uci to Inf. This corresponds to the alternative "greater" in a hypothesis test. For sides = "right", it returns an upper one-sided confidence bound and sets lci to 0.

Compatible vector lengths are recycled. Supplying several methods therefore provides the corresponding intervals in a single data.frame.

References

Garwood, F. (1936). Fiducial limits for the Poisson distribution. Biometrika, 28, 437–442.

Rothman, K. J. and Boice, J. D. Jr. (1979). Epidemiologic Analysis with a Programmable Calculator. NIH Publication No. 79-1649. Washington, DC: US Government Printing Office.

Examples

# Deaths from horse kicks in 280 Prussian army corps-years
count <- 0:4
corpsYears <- c(144, 91, 32, 11, 2)

x <- sum(count * corpsYears)
n <- sum(corpsYears)

poissonCI(x, n)
#>       est       lci       uci 
#> 0.7000000 0.6054271 0.8051570 
poissonCI(x, n, method = c("exact", "score", "wald", "byar"))
#>   est       lci       uci   x   n conf.level     sides method
#> 1 0.7 0.6054271 0.8051570 196 280       0.95 two.sided  exact
#> 2 0.7 0.6086218 0.8050977 196 280       0.95 two.sided  score
#> 3 0.7 0.6020018 0.7979982 196 280       0.95 two.sided   wald
#> 4 0.7 0.6070833 0.8032497 196 280       0.95 two.sided   byar

# A 95% lower confidence bound for the event rate
poissonCI(x, n, sides = "left")
#>       est       lci       uci 
#> 0.7000000 0.6198372       Inf 

# SMR for Welsh nickel workers
poissonCI(x = 137, n = 24.19893)
#>      est      lci      uci 
#> 5.661407 4.753125 6.692709