Sample size for comparing two over-dispersed rates: quasi-Poisson

When to use it

Counts whose variance is a roughly constant multiple of their mean, as a quasi-Poisson model will assume.

Not for

Heterogeneity that grows with the rate, which a negative binomial model describes better.

Notation

$n_1,\ n_2$: Participants in groups 1 (treatment) and 2 (comparator)
$\lambda_1,\ \lambda_2$: Event rates per unit of time in groups 1 and 2
$t$: Follow-up time per participant
$r_a$: Allocation ratio n₂ / n₁
$\phi$: Dispersion: variance divided by mean
$\alpha$: Significance level: the chance of a false positive
$1-\beta$: Power: the chance of detecting the effect if it is real
$\tau$: Number of tails: 2 for a two-sided test, 1 for one-sided
$\rho$: Intracluster correlation coefficient (ICC)
$m$: Average number of participants per cluster
$\mathrm{CV}$: Coefficient of variation of cluster size
$C$: Clusters per group
$\ell$: Expected loss of whole clusters
$q$: Expected dropout

Formulas

Participants in group 1

n_1 \approx \phi\;\frac{(z_{1-\alpha/\tau} + z_{1-\beta})^{2}\left[\dfrac{1}{t\lambda_1} + \dfrac{1}{r_a\,t\lambda_2}\right]}{\left(\ln \lambda_1/\lambda_2\right)^{2}}

The Poisson requirement multiplied by the dispersion φ, the ratio of the variance to the mean [1]. The calculator solves the test’s power function exactly for n₁; the formula shown is its large-sample form, which agrees to within a participant or two.

Clustering

\mathrm{DE} = 1 + \left[(\mathrm{CV}^{2} + 1)\,m - 1\right]\rho

When participants are recruited or randomised in clusters, observations within a cluster repeat information, and the sample is multiplied by the design effect. With equal clusters (CV = 0) this is 1 + (m − 1)ρ [2, 3].

Few clusters

F_t = \left(\frac{t_{2(C-1),\,1-\alpha/\tau} + t_{2(C-1),\,1-\beta}}{z_{1-\alpha/\tau} + z_{1-\beta}}\right)^{2}

Optional. A cluster trial is analysed on cluster-level summaries, so its critical values come from t on 2(C − 1) degrees of freedom rather than the normal distribution. With few clusters this costs participants; because C depends on the sample, it is iterated until it settles [4].

All adjustments together

n_{\text{adj}} = \left\lceil n \mathrm{DE}\;F_t \right\rceil

The adjustments multiply, in this order, and the result is rounded up once. Any adjustment not used contributes a factor of 1.

Dropout

n_{\text{recruit}} = \left\lceil \frac{n_{\text{adj}}}{1 - q} \right\rceil

Applied last, to the participants actually recruited.

Clusters to recruit

C = \left\lceil \frac{n_{\text{recruit}}}{m} \right\rceil, \qquad C_{\text{recruit}} = \left\lceil \frac{C}{1 - \ell} \right\rceil

Per group. Losing a whole cluster loses every participant in it, so an expected loss of clusters inflates the clusters recruited rather than the participants.

Parameters

Rate in the treatment group $\lambda_1$

Values: Above 0, events per unit of time

The rate you expect with the intervention, in the same time unit as the follow-up.

Rate in the comparator group $\lambda_2$

Values: Above 0, different from λ₁

From routine data or previous studies.

Follow-up per participant $t$

Values: Above 0, in the rates’ time unit

Doubling the follow-up buys roughly the same information as doubling the participants, for a Poisson outcome.

Allocation ratio $r_a$ Optional

Values: Above 0; 1 for equal groups

The planned size of group 2 divided by group 1.

Dispersion $\phi$

Values: Above 0; 1 is Poisson

From previous data on the same outcome: the variance of the counts divided by their mean.

Significance level $\alpha$

Values: Between 0 and 1; usually 0.05

The false-positive rate you accept. A smaller α, for example to allow for several primary outcomes, needs a larger sample.

Power $1-\beta$

Values: Between 0 and 1; usually 0.80 or 0.90

The chance of a significant result if the assumed effect is real. Funders and ethics committees commonly expect at least 0.80.

One- or two-sided $\tau$

Values: 2 (two-sided) or 1 (one-sided)

Two-sided is the default. A one-sided test is justified only when an effect in the other direction would be treated exactly as no effect, and should be stated in advance.

Expected dropout $q$ Optional

Values: 0% up to, but not including, 100%

The share of participants expected to provide no outcome, from comparable studies. Recruiting more restores the numbers, not the comparability of the groups: if dropout is related to the outcome, the estimate can still be biased.

Intracluster correlation (ICC) $\rho$ Optional

Values: From 0 to 1

For clustered designs only. Published ICCs for health outcomes in primary-care and community clusters are mostly small, often below 0.05 [5]; prefer an estimate from a comparable trial.

Average cluster size $m$ Optional

Values: At least 1

The average number of participants per cluster. Given with the ICC.

Variation in cluster size $\mathrm{CV}$ Optional

Values: 0 or more; 0 for equal clusters

The SD of cluster sizes divided by their mean. Unequal clusters cost power on their own account [3].

Expected loss of whole clusters $\ell$ Optional

Values: 0% up to, but not including, 100%

Clusters, such as practices or schools, expected to withdraw entirely.

Correction for few clusters $F_t$ Optional

Values: On or off

Worth applying when there are few clusters per group, roughly fewer than 15–20.

Assumptions

Participants are independent apart from any clustering specified, each followed for the same time.
The variance of the counts is φ times their mean in both groups.
With clustering: clusters are independent, with the same ICC throughout, and the analysis accounts for the clustering, for example with a mixed model or cluster-level summaries [2].
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

Dispersion 1.5

The Poisson example with φ = 1.5: about 1.5 × 355, solved exactly as 533 per group.

Group 1: 533 · Group 2: 533 · Total: 1066

Protocol statement
To detect a difference between rates of 0.8 and 1 per unit of time with 80% power at a two-sided significance level of 0.05, using a quasi-Poisson comparison of two rates (dispersion 1.5), and assuming follow-up of 1 per participant, the required sample size is 533 per group (1066 in total). Calculated with StatWisely (statwisely.com) using: n₁: solve the log rate-ratio z test for λ₁, λ₂, exposure and allocation; variance scaled by the dispersion coefficient. These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

Dispersion 2

With φ = 2: 710 per group.

Group 1: 710 · Group 2: 710 · Total: 1420

Open this example in the calculator

Validation

Every worked example on this page is recomputed by the StatWisely engine whenever the software is built. If a number or formula here stopped matching the calculator, the release would be stopped until this document was corrected.
An independent implementation in R recomputes this test every night and before each production release, and the results are compared with the engine’s.

How to cite

StatWisely. Sample size for comparing two over-dispersed rates: quasi-Poisson. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/rates-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.
2.Donner A, Klar N. Design and Analysis of Cluster Randomization Trials in Health Research. London: Arnold; 2000.
3.Eldridge SM, Ashby D, Kerry S. Sample size for cluster randomized trials: effect of coefficient of variation of cluster size and analysis method. International Journal of Epidemiology. 2006;35(5):1292–1300. doi:10.1093/ije/dyl129
4.Hayes RJ, Moulton LH. Cluster Randomised Trials. 2nd ed. Boca Raton: CRC Press; 2017.
5.Adams G, Gulliford MC, Ukoumunne OC, Eldridge S, Chinn S, Campbell MJ. Patterns of intra-cluster correlation from primary care research to inform study design and analysis. Journal of Clinical Epidemiology. 2004;57(8):785–794. doi:10.1016/j.jclinepi.2003.12.013

When to use it

Not for

Notation

Formulas

Participants in group 1

Clustering

Few clusters

All adjustments together

Dropout

Clusters to recruit

Parameters

Rate in the treatment groupλ1\lambda_1λ1​

Rate in the comparator groupλ2\lambda_2λ2​

Follow-up per participantttt

Allocation ratiorar_ara​Optional

Dispersionϕ\phiϕ

Significance levelα\alphaα

Power1−β1-\beta1−β

One- or two-sidedτ\tauτ

Expected dropoutqqqOptional

Intracluster correlation (ICC)ρ\rhoρOptional

Average cluster sizemmmOptional

Variation in cluster sizeCV\mathrm{CV}CVOptional

Expected loss of whole clustersℓ\ellℓOptional

Correction for few clustersFtF_tFt​Optional