The Confidence Interval around an Event Count or Rate
There are many approximate formulas for the CIs (confidence intervals) around an observed event count or rate (also called a Poisson CI). Suppose that there were 36 fatal highway accidents in your county in the last three months.
If that’s the only safety data you have to go on, then your best estimate of the monthly fatal accident rate is simply the observed count (N), divided by the length of time (T) during which the N counts were observed: 36/3, or 12.0 fatal accidents per month. What is the 95 percent CI around that estimate?
The simplest method is based on approximating the Poisson distribution by a normal distribution. It should be used only when N is large (at least 50). You first calculate the SE of the event rate. The Poisson distribution tells us that the SE of the total observed number of counts (N) is simply the square root of N, so the SE of the event rate is given by:
Using these numbers, N = 36 and T=3, the SE for the event rate is
Then you use the normal-based formulas, which say that the CI around the observed rate is equal to the observed rate ± k×SE.
k is 1.96 for 95 percent CLs. So CLL = 12.0 – 1.96 x 1.67 and CLU = 12.0 + 1.96 x 1.67, which works out to 95 percent confidence limits of 8.73 and 15.27. You report your result this way: “The fatal accident rate was 12.0, 95%CI = 8.7–15.3 fatal accidents per month.”
If you wanted to calculate the CI around the total 3-month accident count itself (rather than around the monthly rate), you would estimate the SE of the count N as
So the SE of the 36 observed fatal accidents in a three-month period is simply
which equals 6.0. Then you would calculate the CI around the observed count, using the normal-based formulas. So CLL = 36.0 – 1.96 x 6.0 and CLH = 36.0 + 1.96 x 6.0, which works out to a 95 percent CI of 24.2 to 47.8 accidents in a three-month period.
Many other approximate formulas for CIs around observed event counts and rates are available, most of which are more reliable when N is small. There are also several exact methods. They’re too complicated to attempt by hand, involving evaluating the Poisson distribution repeatedly to find values for the true mean event count that are consistent with (that is, not significantly different from) the count you actually observed.
Fortunately, many statistical packages can do these calculations for you.
You can also go to the “Poisson Confidence Intervals” section of the online web calculator at StatPages.info. Enter the observed count (36) and press the Compute button. The page calculates the exact 95 percent CI for the 3-month total accident count as (25.2 – 49.8). You could then calculate the exact 95% CI around the average monthly accident rate by dividing these lower and upper confidence limits by 3 months, giving (8.4 – 6.6) accidents per month.
For this example, the normal-based CI is only a rough approximation to the exact CI, mainly because the total event count was only 36 accidents. For small samples, you should report exact confidence limits, and not normal-based confidence limits.