Compare Independent Poisson Rates

Match the event and exposure definitions

A rate comparison begins with two event counts and the exposure over which each count could occur. Record whether events are first occurrences or recurrent episodes, when exposure begins and ends, and which periods are excluded. A count of affected people and a count of repeated episodes answer different questions even when they share the same denominator.

Use identical exposure units in both groups. If one source reports person-months and the other person-years, convert explicitly before calculation and preserve the original values. The statsmodels rate-ratio documentation distinguishes several interval methods, including the log-Wald method used here.

Recompute a fictional comparison

Suppose group A has 40 events during 200 person-years and group B has 25 events during 250 person-years. Their rates are 0.20 and 0.10 events per person-year, so the rate ratio A/B is 2.00. Multiplying both rates by 100 changes the display to 20 and 10 events per 100 person-years but leaves the ratio unchanged.

For independent Poisson counts, the approximate SE of the log rate ratio is sqrt(1/40+1/25), approximately 0.25495. A two-sided 95% log-Wald interval is exp(log(2) ± 1.959964×0.25495), approximately 1.213 to 3.296. Exponentiate both endpoints; do not add the log-scale margin directly to the ratio.

The Poisson rate comparison tool implements this specific approximation. It requires positive event counts and positive exposures. Zero events need a method that handles the boundary; entering an arbitrary small count changes the data and the method.

Check whether the approximation fits the data structure

Independence concerns both groups and the underlying event-generating process. Multiple events from one person, center-level clustering, or strong variation in individual event propensity can make the simple count-based variance inadequate. Large aggregate counts do not automatically remove these problems.

Investigate sparse counts rather than treating a finite interval as evidence of reliable coverage. Exact conditional and other methods are different procedures; the R exact Poisson test documentation describes conditioning on the total count for a two-sample comparison. Its output should not be labeled as a log-Wald interval.

Preserve the interpretation

Use the rate comparison worksheet to record event definitions, exposure units, independence assumptions, and the selected interval method. State the direction of comparison in every table. Reversing A/B to B/A reciprocates the estimate and reverses the reciprocal endpoints.

Finish with both absolute rates and their denominators. A rate ratio is not a risk ratio over a specified follow-up period, and this unadjusted calculation does not address confounding. Keep the observational comparison separate from a causal claim about changing exposure or treatment.

Sources checked on 5 November 2026; all numerical examples are fictional and independently recomputed.