Confidence interval for a rate

Steve Simon


*Dear Professor Mean


A rate

A rate is different from a proportion where the numerator and denominator are both counts and where the numerator is a subset of the denominator. So the number of children born with birth defects divided by the number of live births is a proportion

The simplest way to get a confidence interval for a single rate is to treat the denominator as a fixed quantity and treat the numerator as a Poisson random variable. Since the standard deviation of a Poisson random variable is simply the square root of the expected value

as a confidence interval for a Poisson count variable equal to x. Then simply divide the limits of the confidence interval by the same quantity that you divided the count by the same denominator. This produces the following formula

If you let R=x/T

Example: In a study of AIDS patients

Source: Antiretroviral therapy at a district hospital in Ethiopia prevents death and tuberculosis in a cohort of HIV patients. Jerene, D.

I’ve written a simple Excel spreadsheet to calculate a confidence interval for a single rate.

If the statistic in question is a proportion rather than a rate

though if the proportion is small and the sample size is large