Two-Sample Poisson Rates Sample Size

Methodology

This calculator implements the large-sample Z-test for the difference between two independent Poisson event rates described in this method Chapter 436, which cites Paul Mathews (2010), Sample Size Calculations: Practical Methods for Engineers and Scientists. this method defines the event-rate difference as λ₂ − λ₁ and describes the two-sided large-sample test as appropriate when the observed event counts are sufficiently large.

Sample Size Formula

For equal sample sizes and a common observation time t per subject, the required number of evaluable subjects per group is

For a one-sided test, z_1−α/2 is replaced by z_1−α. The formula follows directly from the large-sample power expression documented in the statistical literature: the standard error of the estimated difference is √[λ₁/(N₁t) + λ₂/(N₂t)], which reduces to √[(λ₁ + λ₂)/(Nt)] under equal allocation.

What the Inputs Mean

• λ₁ and λ₂: expected Poisson event rates for Groups 1 and 2, expressed per unit of observation time.
• Target power: probability of rejecting the null hypothesis when the specified alternative rates are true.
• Alpha: Type I error probability. For a two-sided test, α is divided between the two tails.
• Observation time: fixed exposure or follow-up time contributed by each subject. this method sets this time to 1 without loss of generality; entering a larger common observation time therefore reduces the number of subjects required for the same amount of information.

Worked Validation Example

this method Chapter 436, Example 1, uses λ₁ = 1, λ₂ = 1.1, 90% power, α = 0.05, a two-sided large-sample test, and equal allocation. This yields N₁ = 2,207 and N₂ = 2,207, for a total of 4,414 subjects.

The calculator reproduces this result before the displayed sample size is rounded upward. The achieved power at the rounded sample size is also reported in the results panel.

Assumptions and Limitations

The method assumes independent Poisson counts, fixed observation time, and a sufficiently large-sample approximation. Note that its large-sample Z-test is appropriate when the event counts are large (it specifically states X_i > 30 for this test). If expected event counts are small, overdispersion is present, follow-up times differ materially between subjects, or clustering or repeated observations are important, a different sample-size method may be needed.

References

Mathews, P. (2010). Sample Size Calculations: Practical Methods for Engineers and Scientists. Mathews Malnar and Bailey, Fairport Harbor, Ohio. This is the source cited by this method for the large-sample Poisson-rate difference method.

the software, LLC. this method Sample Size Software: Tests for the Difference Between Two Poisson Rates, Chapter 436. This method uses the large-sample Z-test, its power equation, and the validation example used above.

Statsols. . provides sample-size and power procedures for Poisson/incidence-rate designs, including two-rate comparisons and normal-approximation approaches.