Achieved power for a rate ratio

When to use it

Assess the sensitivity of a comparison of event rates, such as exacerbations or falls per year, between two arms.

Not for

Clinical counts usually vary more than Poisson allows; if so, the quasi-Poisson or negative-binomial tab is the honest choice.

Notation

$n_1$: Group 1 size; group 2 has r n₁.
$\lambda_1, \lambda_2$: Event rates per unit of time in groups 1 and 2.
$t$: Follow-up time per participant, in the rates’ unit.
$r$: Allocation ratio n₂/n₁.

Formulas

Standardized log rate ratio

z=|\ln(\lambda_1/\lambda_2)|\sqrt{\frac{n_1 t}{1/\lambda_1+1/(r\lambda_2)}}

The variance of the log rate ratio is evaluated at the alternative rates [1].

Power from the normal rejection region

\text{Power}=\Phi(z-z_{1-\alpha/\text{tails}})+I_{\text{tails}=2}\Phi(-z-z_{1-\alpha/\text{tails}})

z₁₋α/tails is the standard-normal critical value.

Parameters

Group 1 size $n_1$

Values: Integer, at least 2

Participants available in group 1; group 2 follows from the allocation ratio.

Allocation ratio $r$ Optional

Values: Positive; 1 for equal groups

Group 2 has n₂ = ⌈r n₁⌉ participants.

Rate in group 1 $\lambda_1$

Values: Positive

Expected events per unit of time.

Rate in group 2 $\lambda_2$

Values: Positive, not equal to λ₁

Expected events per unit of time in the comparator.

Follow-up per participant $t$

Values: Positive

In the same time unit as the rates; doubling it buys about as much as doubling the participants.

Significance level $\alpha$

Values: Between 0 and 1

The type I error rate for the specified test.

Test direction $s$

Values: 1 or 2

One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Assumptions

Events in each participant follow a Poisson process with the stated rate, independently of other participants.
One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Worked examples

A worked input set

With 40 per group followed for 2 years, rates of 0.8 and 1.2 per year at α = 0.05: z = |ln(0.8/1.2)|√(80/(1.25 + 0.83)) = 2.51.

Needed for analysis: 40 · To approach: 80

Open this example in the calculator

Validation

At the size the rate-ratio calculator returns, this power reaches that calculator’s target, and one fewer falls short; it is checked against an independent calculation in R.

How to cite

StatWisely. Achieved power for a rate ratio. Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-rate-ratio/

Cite the original references for the formulas your design uses as well.

References

1.Gu K, Ng HKT, Tang ML, Schucany WR. Testing the ratio of two Poisson rates. Biometrical Journal. 2008;50(2):283–298.