Confidence Interval for the Ratio of Two Poisson Rates

Estimates the ratio of two independent Poisson event rates and calculates a confidence interval.

Usage

poissonRatioCI(
  x1,
  n1 = 1,
  x2,
  n2 = 1,
  conf.level = 0.95,
  sides = c("two.sided", "left", "right"),
  method = c("exact", "midp", "wald-log")
)

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: "exact", "midp", or "wald-log"; may be abbreviated and defaults to "exact"

Value

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

est: estimated rate ratio
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 ratio $\theta = \lambda_1/\lambda_2$, estimated by $\hat{\theta} = (x_1/n_1)/(x_2/n_2)$.

The available confidence-interval methods are:

"exact": the exact conditional interval obtained by conditioning on $x_1 + x_2$ and transforming a Clopper–Pearson interval for the resulting binomial probability; this is the construction used by poisson.test() for two samples
"midp": the corresponding conditional mid-p interval, which is generally shorter but does not guarantee conservative coverage
"wald-log": the asymptotic Wald interval, symmetric on the logarithmic scale

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 0, the lower limit of the parameter space.

The log-Wald interval cannot be calculated when either count is zero and is then returned as [0, Inf]. If both counts are zero, the rate ratio and its point estimate are undefined; the function returns NA for est and [0, Inf] for the confidence interval for every method. If only x2 is zero, the point estimate is 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

Sahai, H. and Khurshid, A. (1993). Confidence intervals for the ratio of two Poisson means. The Mathematical Scientist, 18, 43–50.

Graham, P. L., Mengersen, K. and Morton, A. P. (2003). Confidence limits for the ratio of two rates based on likelihood scores. Statistics in Medicine, 22(12), 2071–2083.

Examples

# 15 events in 100 person-years compared with
# 6 events in 120 person-years
poissonRatioCI(15, 100, 6, 120)
#>      est      lci      uci 
#> 3.000000 1.099947 9.437411 

# The exact interval agrees with the two-sample conditional test in stats
poisson.test(c(15, 6), c(100, 120))$conf.int
#> [1] 1.099947 9.437411
#> attr(,"conf.level")
#> [1] 0.95

poissonRatioCI(15, 100, 6, 120, method = "midp")
#>      est      lci      uci 
#> 3.000000 1.188883 8.415485 

# Zero counts are handled explicitly
poissonRatioCI(0, 100, 6, 120)
#>      est      lci      uci 
#> 0.000000 0.000000 1.019173