Method document

Sample size to detect a change over time in a repeated binary outcome

How many participants, each measured repeatedly, are needed to show that the probability of a yes/no outcome changes over time in one group, from the outcome probability, the odds ratio per unit of time, the visit schedule and the correlation between a participant’s measurements. Uses logistic GEE with an exchangeable correlation. Covers dropout.

Engine version 1.0.0 · Reviewed 4 October 2026

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https://statwisely.com/methods/longitudinal-binary-trend/

When to use it

A cohort with a yes/no outcome recorded at every visit, testing whether it becomes more or less common over follow-up.

Not for

Comparing how two groups change. That needs a group-by-time interaction and its own calculation.
Participant-specific effects from a logistic mixed model, which differ from population-average ones.

Notation

$n$: Number of participants
$p$: Outcome probability at the middle of follow-up
$\mathrm{OR}$: Odds ratio per unit of time to detect
$m,\ T$: Measurements per participant, equally spaced over a follow-up of length T
$\rho$: Correlation between two measurements on the same participant
$V(\beta)$: Per-participant variance of the GEE time slope
$\alpha$: 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
$q$: Expected dropout

Formulas

Wald test of the GEE time slope

n = \left\lceil \frac{(z_{1-\alpha/s} + z_{1-\beta})^{2}\, V(\beta)}{(\ln \mathrm{OR})^{2}} \right\rceil, \qquad V(\beta) = \left[(D^{\top} V^{-1} D)^{-1}\right]_{\beta\beta}

V(β) is the model-based GEE variance over the schedule at the expected odds ratio, with D = A X, V = A^{1/2} R(ρ) A^{1/2} and A = diag(μ(1 − μ)) at each visit, the exchangeable working correlation being taken as the true one [1]. With a small change it is (1 − ρ)/[p(1 − p) m s_t²] [2].

Dropout

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

Applied last, to the participants actually recruited.

Parameters

Outcome probability at the middle of follow-up $p$

Values: Between 0 and 1

From routine data or a previous cohort.

Odds ratio per unit of time $\mathrm{OR}$

Values: Above 0, not 1

In the unit the follow-up is entered in.

Measurements per participant $m$

Values: A whole number of at least 2

Equally spaced over the follow-up.

Follow-up length $T$

Values: Above 0

A longer follow-up spreads the visits and strengthens the test.

Correlation between measurements $\rho$

Values: At least 0 and below 1

On the binary scale, the same for every pair of visits.

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.

Assumptions

Participants are independent; a participant’s measurements share one correlation (exchangeable); visits are equally spaced, every participant attends them all, and the log odds change linearly with time.
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

Four visits over three years

p = 0.3, OR = 1.3 per year, ρ = 0.5, two-sided α = 0.05, 80% power: n = 57.

Sample size: 57

Protocol statement
To detect an odds ratio of 1.3 per unit of time with 80% power at a two-sided significance level of 0.05, using a logistic GEE test of change over time in one group with 4 measurements, and assuming an outcome probability of 30% at the middle of follow-up and a correlation of 0.5 between a participant’s measurements, the required sample size is 57 participants. Calculated with StatWisely (statwisely.com) using: n = (z_(1-α/tails)+z_(1-β))² × V(β) / (ln OR)², V(β) the exchangeable GEE variance over m visits. These are planning estimates; confirm the final design with a qualified statistician.

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More visits over the same period

7 visits over the same three years: n = 41.

Sample size: 41

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Allowing for dropout

Expecting 20% to drop out: 57 / 0.8 = 71.3, so 72 are enrolled.

Needed for analysis: 57 · To recruit: 72

Protocol statement
To detect an odds ratio of 1.3 per unit of time with 80% power at a two-sided significance level of 0.05, using a logistic GEE test of change over time in one group with 4 measurements, and assuming an outcome probability of 30% at the middle of follow-up and a correlation of 0.5 between a participant’s measurements, the required sample size is 57 participants. Allowing for 20% dropout, 72 participants should be recruited. Calculated with StatWisely (statwisely.com) using: n = (z_(1-α/tails)+z_(1-β))² × V(β) / (ln OR)², V(β) the exchangeable GEE variance over m visits. These are planning estimates; confirm the final design with a qualified statistician.

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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.
Simulating 1,000 cohorts of 57 participants in R, with correlated binary outcomes from a Gaussian copula calibrated to ρ = 0.5 and analysed with geepack::geeglm, gave power of 0.812 against a target of 0.80. An independent R computation of the GEE variance reproduces the engine’s sizes.

How to cite

StatWisely. Sample size to detect a change over time in a repeated binary outcome. Method document, engine version 1.0.0, reviewed 2026-10-04. https://statwisely.com/methods/longitudinal-binary-trend/

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

References

1.Liu G, Liang KY. Sample size calculations for studies with correlated observations. Biometrics. 1997;53(3):937–947. doi:10.2307/2533554
2.Diggle PJ, Heagerty P, Liang KY, Zeger SL. Analysis of Longitudinal Data. 2nd ed. Oxford: Oxford University Press; 2002.

When to use it

Not for

Notation

Formulas

Wald test of the GEE time slope

Dropout

Parameters

Outcome probability at the middle of follow-upppp

Odds ratio per unit of timeOR\mathrm{OR}OR

Measurements per participantmmm

Follow-up lengthTTT

Correlation between measurementsρ\rhoρ

Significance levelα\alphaα

Power1−β1-\beta1−β

One- or two-sidedτ\tauτ

Expected dropoutqqqOptional

Assumptions

Worked examples

Four visits over three years

More visits over the same period

Allowing for dropout

Validation

How to cite

References

Outcome probability at the middle of follow-up $p$

Odds ratio per unit of time $\mathrm{OR}$

Measurements per participant $m$

Follow-up length $T$

Correlation between measurements $\rho$

Significance level $\alpha$

Power $1-\beta$

One- or two-sided $\tau$

Expected dropout $q$ Optional