Achieved power for one rate

When to use it

Assess a single-arm study that compares its event rate with a known benchmark rate.

Not for

The reference rate must be fixed and known; if it comes from a concurrent control group, use a two-group tab.

Notation

$n$: Participants.
$\lambda_1$: Expected event rate.
$\lambda_0$: Reference rate under the null.
$t$: Follow-up time per participant.

Formulas

Power, inverting the sample-size formula

\text{Power}=\Phi\!\left(\frac{|\lambda_1-\lambda_0|\sqrt{nt}-z_{1-\alpha/\text{tails}}\sqrt{\lambda_0}}{\sqrt{\lambda_1}}\right)+I_{\text{tails}=2}\Phi\!\left(\frac{-|\lambda_1-\lambda_0|\sqrt{nt}-z_{1-\alpha/\text{tails}}\sqrt{\lambda_0}}{\sqrt{\lambda_1}}\right)

The null variance sets the threshold and the alternative variance the spread, as in the one-rate sample-size formula.

Parameters

Participants $n$

Values: Integer, at least 2

Participants available for follow-up.

Expected rate $\lambda_1$

Values: Positive

Expected events per unit of time.

Reference rate $\lambda_0$

Values: Positive, not equal to λ₁

The benchmark rate the study is compared with.

Follow-up per participant $t$

Values: Positive

In the same time unit as the rates.

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 25 participants followed for 2 years, an expected rate of 0.8 against a reference of 1.2 per year at α = 0.05: |λ₁ − λ₀|√(nt) = 0.4 × √50 = 2.83.

Sample size: 25

Open this example in the calculator

Validation

At the size the one-rate 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 one rate. Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-one-sample-rate/

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