How to calculate the confidence interval of incidence rate under the Poisson distribution
Incidence rate (IR) = # event (N) / person-time at risk (T)
(1- α) % CI of IR = 
If N is large, 95% CI of N = 
95% CI of IR = 
= 
= 
You may use PQRS or use an online calculate like GraphPad QuickCalcs, or Quantile Function Calculator to get a quantile of the chi-square distribution.
The Online OpenEpi, a nice Epidemiologic Calculator, provides different ways to get these confidence intervals. I noticed, if N = 0, the calculator gives N = 0.5 to avoid unrealistic results.
Stata can give exact Poisson CI:
.cii T N, poisson
10 comments:
There is a typo in the CI for the incidence rate for Poisson distribution, second line of IR for "large N": the first formula is correct, but the second is not, the denominator of the half-width is sqrt(T) in your notation, not sqrt(N).
Thank you Birgit for your question, but the SQRT(N) is not a typo. SQRT(N)/T = [SQRT(N)*SQRT(N)]/[T*SQRT(N)] = [N/[T*SQRT(N)] = (N/T)/SQRT(N) = IR/SQRT(N). Thanks again for double-checking my formula.
You are right. Sorry.
useful, thanks for this information.
Can you explain why this formula is different from the Wald interval (Normal approximation) formula which would look like this: IR∓1.96*√(IR/N)?
Thanks!
Does 1.96 * IR / √N represent the standard error of the IR? Thank you!
That's my question as well: shouldn't standard error be √IR / √N with 1.96 being a 95% z-score... Thank you!
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Thanks for the post, it is really helpful! Do we know how to calculate p-value for this?
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