Confidence Interval for the Difference Between Two Poisson Rates

Estimates the difference between two independent Poisson event rates and calculates a confidence interval.

Usage

poissonDiffCI(
  x1,
  n1 = 1,
  x2,
  n2 = 1,
  conf.level = 0.95,
  sides = c("two.sided", "left", "right"),
  method = c("mover", "wald")
)

Arguments

x1: non-negative integer event count or vector of counts for the first sample
n1: positive exposure associated with x1, such as observation time, person-time, or population at risk; may be a vector and defaults to 1
x2: non-negative integer event count or vector of counts for the second sample
n2: positive exposure associated with x2; 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: "mover" or "wald"; may be abbreviated and defaults to "mover"

Value

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

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

Otherwise, a data.frame containing these three columns followed by the recycled values of x1, n1, x2, n2, and conf.level.

Details

The function assumes two independent counts $$X_i \sim \mathrm{Poisson}(n_i\lambda_i), \quad i = 1, 2.$$ The parameter of interest is the rate difference $\Delta = \lambda_1 - \lambda_2$, estimated by $\hat{\Delta} = x_1/n_1 - x_2/n_2$.

The available confidence-interval methods are:

"mover": the method of variance estimates recovery (MOVER), which combines separate exact Garwood limits for the two rates
"wald": the normal-approximation interval based on the standard error $\sqrt{x_1/n_1^2 + x_2/n_2^2}$

The Wald interval has zero width when both counts are zero.

For sides = "left", the function returns a lower one-sided confidence bound and sets uci to Inf. For sides = "right", it returns an upper one-sided confidence bound and sets lci to -Inf.

The numeric arguments are recycled to a common length only when their lengths are compatible whole multiples. Incompatible lengths produce an error. sides and method must each identify a single choice.

References

Zou, G. Y. and Donner, A. (2008). Construction of confidence limits about effect measures: a general approach. Statistics in Medicine, 27(10), 1693–1702.

Examples

# 15 events in 100 person-years compared with
# 6 events in 120 person-years
poissonDiffCI(15, 100, 6, 120)
#>        est        lci        uci 
#> 0.10000000 0.01155263 0.20241565 

poissonDiffCI(15, 100, 6, 120, method = "wald")
#>        est        lci        uci 
#> 0.10000000 0.01419326 0.18580674 

# A 95% lower confidence bound for the rate difference
poissonDiffCI(15, 100, 6, 120, sides = "left")
#>        est        lci        uci 
#> 0.10000000 0.02462852        Inf 

# Recycling returns one row per comparison
poissonDiffCI(x1 = c(15, 20), n1 = 100, x2 = 6, n2 = 120)
#>    est        lci       uci x1  n1 x2  n2 conf.level
#> 1 0.10 0.01155263 0.2024156 15 100  6 120       0.95
#> 2 0.15 0.05243411 0.2633907 20 100  6 120       0.95