Achieved power for a quasi-Poisson rate ratio

When to use it

Assess a rate comparison whose counts are over-dispersed by a roughly constant factor, as estimated from earlier data.

Not for

When the extra variation grows with the mean, the negative-binomial tab models it better.

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.
$\phi$: Dispersion: the variance of the counts divided by their mean.

Formulas

Standardized log rate ratio, scaled

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

φ = 1 is the Poisson case; φ = 2 halves the information [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.

Dispersion (φ) $\phi$

Values: Positive; 1 is Poisson

The variance-to-mean ratio of the counts in earlier data.

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

The variance of the counts is φ times their mean in both groups.
One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Worked examples

A worked input set

With 60 per group followed for 2 years, rates of 0.8 and 1.2 per year, and φ = 1.5 at α = 0.05: the Poisson information is divided by 1.5.

Needed for analysis: 60 · To approach: 120

Open this example in the calculator

Validation

At the size the quasi-Poisson 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 quasi-Poisson rate ratio. Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-quasi-poisson/

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.