Method document

Sample size for a trend in event rates across ordered groups

How many participants are needed in each of three or more ordered groups, such as doses or exposure levels, to show that the rate of a counted event rises or falls across them, from each group’s expected rate. Poisson or over-dispersed negative binomial counts. Covers one- or two-sided tests, clustering and dropout.

Engine version 1.0.0 · Reviewed 9 October 2026

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https://statwisely.com/methods/rates-trend/

When to use it

A dose–response question for a count outcome, analysed with a trend term in Poisson or negative binomial regression.

Not for

Groups with no natural order. Use the test across several groups.
A response that rises and then falls, which a single trend does not describe.

Notation

$n$: Participants in each group
$g$: Number of groups
$\lambda_j$: Expected event rate per unit of time in group j
$t$: Follow-up time per participant
$k$: Negative binomial dispersion: variance μ + kμ²; 0 is Poisson
$u_j$: Information per participant about group j’s log rate, 1/(1/(λⱼt) + k)
$\alpha$: 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
$s_j$: Score of group j; 0, 1, 2, … unless given
$\beta_s$: The u-weighted slope of the log rates on the scores: the log rate ratio per score unit

Formulas

Participants in each group

n = \left\lceil \frac{(z_{1-\alpha/\tau} + z_{1-\beta})^{2}}{\beta_s^{2} \sum_j u_j (s_j - \bar{s})^{2}} \right\rceil, \qquad \beta_s = \frac{\sum_j u_j (s_j - \bar{s})(\ln \lambda_j - \bar{\theta})}{\sum_j u_j (s_j - \bar{s})^{2}}

The Wald test of a log-linear trend, the count counterpart of the Cochran–Armitage test. When the rates are not exactly log-linear in the scores, βₛ is the trend a weighted fit would find; s̄ and θ̄ are u-weighted means [1, 2].

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 [3, 4].

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 [5].

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

Expected rate in each group, in order $\lambda_1, \dots, \lambda_g$

Values: At least three, each above 0, not all equal

Events per participant per unit of time: the smallest pattern of differences worth detecting.

Follow-up per participant $t$

Values: Above 0, in the rates’ time unit

Longer follow-up yields more events, and so more information, per participant.

Dispersion (negative binomial k) $k$ Optional

Values: 0 or more; 0 (the default) for Poisson counts

Clinical counts usually vary between people more than a Poisson process allows; take k from earlier data and try plausible values.

Group scores $s_j$ Optional

Values: One number per group; 0, 1, 2, … by default

The value each group stands for, such as the dose. Only their spacing matters.

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 [6]; 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 [4].

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

The groups are independent and of equal size, every participant is followed for the same time, and each group’s rate is constant over it.
Counts are Poisson, or negative binomial with the stated dispersion, the same in every group.
The log rate changes roughly linearly with the scores.
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 [3].
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

Three doses, Poisson counts

Rates of 1, 1.25 and 1.5 a year over one year, scored 0, 1, 2: βₛ = 0.2013 (a rate ratio of 1.22 per step) and Σuⱼ(sⱼ − s̄)² = 2.433, so n = (2.80)² / (0.2013² × 2.433) = 79.6, rounded up to 80 per group.

80 per group · Total: 240

Protocol statement
To detect rates of 1, 1.25, 1.5 per unit of time with 80% power at a two-sided significance level of 0.05, using a test of a log-linear trend in the event rate across 3 ordered groups, and assuming follow-up of 1 per participant and group scores of 0, 1, 2, the required sample size is 80 per group (240 in total, across 3 groups). Calculated with StatWisely (statwisely.com) using: n per group = (z_(1-α/τ)+z_(1-β))²/(b²·Σuⱼ(sⱼ-s̄)²), b the weighted log-linear trend. These are planning estimates; confirm the final design with a qualified statistician.

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Over-dispersed counts

With dispersion k = 0.5, Σuⱼ(sⱼ − s̄)² falls to 1.508: 128 per group.

128 per group · Total: 384

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One-sided

Testing only for a rise: (2.49)² / (0.2013² × 2.433) = 62.7, so 63 per group.

63 per group · Total: 189

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Unequally spaced doses

Scoring the groups 0, 1, 3 spreads the scores but weakens the trend those rates show: 85 per group.

85 per group · Total: 255

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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.
Simulated Poisson and negative binomial counts at the planned sizes, analysed with the Wald test of a score term in generalised linear models, gave power of 0.79 to 0.83 against a target of 0.80, with the dispersion known or estimated.

How to cite

StatWisely. Sample size for a trend in event rates across ordered groups. Method document, engine version 1.0.0, reviewed 2026-10-09. https://statwisely.com/methods/rates-trend/

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

References

1.Agresti A. Categorical Data Analysis. 3rd ed. Hoboken, NJ: John Wiley & Sons; 2013.
2.Zhu H, Lakkis H. Sample size calculation for comparing two negative binomial rates. Statistics in Medicine. 2014;33(3):376–387.
3.Donner A, Klar N. Design and Analysis of Cluster Randomization Trials in Health Research. London: Arnold; 2000.
4.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
5.Hayes RJ, Moulton LH. Cluster Randomised Trials. 2nd ed. Boca Raton: CRC Press; 2017.
6.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 each group

Clustering

Few clusters

All adjustments together

Dropout

Clusters to recruit

Parameters

Expected rate in each group, in orderλ1,…,λg\lambda_1, \dots, \lambda_gλ1​,…,λg​

Follow-up per participantttt

Dispersion (negative binomial k)kkkOptional

Group scoressjs_jsj​Optional

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

Assumptions

Worked examples

Three doses, Poisson counts

Over-dispersed counts

One-sided

Unequally spaced doses

Validation

How to cite

References

Expected rate in each group, in order $\lambda_1, \dots, \lambda_g$

Follow-up per participant $t$

Dispersion (negative binomial k) $k$ Optional

Group scores $s_j$ Optional

Significance level $\alpha$

Power $1-\beta$

One- or two-sided $\tau$

Expected dropout $q$ Optional

Intracluster correlation (ICC) $\rho$ Optional

Average cluster size $m$ Optional

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

Expected loss of whole clusters $\ell$ Optional

Correction for few clusters $F_t$ Optional