Sample size for the Cochran–Armitage test for trend

When to use it

A dose-response study or an ordered exposure (none, low, medium, high) with a yes/no outcome, where the question is whether the outcome becomes more or less likely as the exposure increases.

Not for

Two groups. Compare the two proportions.
Unordered groups, or any difference between groups rather than a trend. Use the chi-square test of independence.
A pattern expected to rise and then fall. A linear trend test can miss it entirely.

Notation

$k$: Number of ordered groups
$n_i$: Participants in group i
$p_i$: Outcome probability in group i
$d_i$: Score of group i, such as its dose
$\bar{d},\ \bar{p}$: Size-weighted mean score and outcome probability
$U$: The trend statistic, Σ xᵢ(dᵢ − d̄), where xᵢ is the number of outcomes in group i
$\alpha$: 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

The trend statistic under the null and the alternative

E(U) = \sum_{i=1}^{k} n_i p_i (d_i - \bar{d}), \quad V_0 = \bar{p}(1-\bar{p}) \sum_{i=1}^{k} n_i (d_i - \bar{d})^{2}, \quad V_1 = \sum_{i=1}^{k} n_i p_i (1-p_i)(d_i - \bar{d})^{2}

V₀ is the variance of U when every group shares one probability, V₁ its variance under the expected proportions [2, 1].

The smallest group sizes with enough power

1-\beta = \Phi\!\left(\frac{|E(U)| - z_{1-\alpha/s}\sqrt{V_0}}{\sqrt{V_1}}\right) + \left[s = 2\right]\Phi\!\left(\frac{-|E(U)| - z_{1-\alpha/2}\sqrt{V_0}}{\sqrt{V_1}}\right)

s is 1 for a one-sided test and 2 for a two-sided one. The first group gets the smallest whole number of participants reaching the power, and every other group that number times its relative size. No continuity correction is applied [1].

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 [4, 5].

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 [6].

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 proportion in each group $p_1, \dots, p_k$

Values: At least three, each between 0 and 1, from the lowest group to the highest

The smallest trend worth detecting, written as the proportion expected in each ordered group.

Group scores $d_1, \dots, d_k$ Optional

Values: One per group, increasing; equally spaced if omitted

The dose or exposure level of each group. Only the spacing matters, so 1, 2, 3 and 10, 20, 30 give the same size.

Relative group sizes $n_i / n_1$ Optional

Values: One per group, each above 0; equal if omitted

How many participants each group gets relative to the others.

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 [7]; 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 [5].

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 in exactly one group.
The proportions change roughly monotonically across the groups; the test has power against a linear trend in 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 [4].
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

Three doses

Success rates of 80%, 85% and 90% at one, two and three doses a day; two-sided α = 0.05, 80% power: 199 per group, N = 597.

199 per group · Total: 597

Protocol statement
To detect outcome proportions of 80%, 85%, 90% across the groups with 80% power at a two-sided significance level of 0.05, using a Cochran–Armitage test for trend across 3 ordered groups, the required sample size is 199 per group (597 in total, across 3 groups). Calculated with StatWisely (statwisely.com) using: Nam: solve power of U=Σxᵢ(dᵢ-d̄), Var₀=p̄q̄Σnᵢ(dᵢ-d̄)², Var₁=Σnᵢpᵢqᵢ(dᵢ-d̄)². These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

Only an increasing trend matters

The same rates with a one-sided test: 157 per group, N = 471.

157 per group · Total: 471

Open this example in the calculator

Twice as many at the lowest dose

Allocating 2 : 1 : 1: 300 at one dose a day and 150 at each of the others, N = 600.

group1: 300 · group2: 150 · group3: 150 · Total: 600

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.
StatWisely reproduces the four worked examples of Stata’s power trend command, which implements the same method: 597 for the three doses, 471 one-sided, 600 with 2 : 1 : 1 allocation, and 1,030 for five unequally spaced exposures [3]. Simulating the test itself at those sizes gives power of 0.80 to 0.81.

How to cite

StatWisely. Sample size for the Cochran–Armitage test for trend. Method document, engine version 1.0.0, reviewed 2026-10-04. https://statwisely.com/methods/proportions-trend/

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

References

1.Nam JM. A simple approximation for calculating sample sizes for detecting linear trend in proportions. Biometrics. 1987;43(3):701–705. doi:10.2307/2531969
2.Armitage P. Tests for linear trends in proportions and frequencies. Biometrics. 1955;11(3):375–386. doi:10.2307/3001775
3.StataCorp. Stata Power, Precision, and Sample-Size Reference Manual, Release 19: power trend. College Station, TX: Stata Press; 2025.
4.Donner A, Klar N. Design and Analysis of Cluster Randomization Trials in Health Research. London: Arnold; 2000.
5.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
6.Hayes RJ, Moulton LH. Cluster Randomised Trials. 2nd ed. Boca Raton: CRC Press; 2017.
7.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

The trend statistic under the null and the alternative

The smallest group sizes with enough power

Clustering

Few clusters

All adjustments together

Dropout

Clusters to recruit

Parameters

Expected proportion in each groupp1,…,pkp_1, \dots, p_kp1​,…,pk​

Group scoresd1,…,dkd_1, \dots, d_kd1​,…,dk​Optional

Relative group sizesni/n1n_i / n_1ni​/n1​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

Only an increasing trend matters

Twice as many at the lowest dose

Validation

How to cite

References

Expected proportion in each group $p_1, \dots, p_k$

Group scores $d_1, \dots, d_k$ Optional

Relative group sizes $n_i / n_1$ 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