Sample size for repeated measures: comparing average levels across follow-up

When to use it

An outcome measured repeatedly to reduce noise, such as blood pressure at several visits, compared between groups.

Not for

Questions about change over time. Use the rates of change.

Notation

$n_1,\ n_2$: Participants in groups 1 and 2
$M$: Measurements per participant
$\sigma$: Standard deviation of a single measurement
$\rho_m$: Correlation between any two measurements on a participant
$\Delta$: Difference to detect between the groups
$r_a$: Allocation ratio n₂ / n₁
$\alpha$: 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

Variance of the difference in average levels

\operatorname{Var}(\hat{\Delta}) = \sigma^{2}\,\frac{1 + (M-1)\rho_m}{M}\left(\frac{1}{n_1} + \frac{1}{n_2}\right)

Averaging M correlated measurements reduces the variance by the factor [1 + (M − 1)ρ]/M, which is never above 1: measuring repeatedly is never worse than measuring once. The more strongly the measurements are correlated, the less each extra one adds [1]. n₁ is solved on the noncentral t distribution.

Dropout

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

Applied last, to the participants actually recruited.

Parameters

Difference in average level $\Delta$

Values: Not zero

The difference between the groups’ average levels over the follow-up.

Standard deviation $\sigma$

Values: Above 0

Of a single measurement, from previous data.

Measurements per participant $M$

Values: At least 1

Planned visits, equally spaced.

Correlation between measurements $\rho_m$

Values: At least 0 and below 1

How strongly a participant’s measurements track each other, from previous data.

Allocation ratio $r_a$ Optional

Values: Above 0; 1 for equal groups

The planned size of group 2 divided by group 1.

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

Compound symmetry: every pair of measurements on a participant is equally correlated. Other covariance structures, such as correlation that fades with time, can change power in either direction.
Every participant completes every visit; dropout is allowed for by recruiting more, not by modelling partial data.
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

Five measurements each

A difference of 1 with SD 2, 5 measurements correlated 0.5: the factor is (1 + 4 × 0.5) / 5 = 0.6, so 39 per group, against 64 with a single measurement.

Group 1: 39 · Group 2: 39 · Total: 78

Protocol statement
To detect a difference of 1 with 80% power at a two-sided significance level of 0.05, using a comparison of average levels over 5 measurements per participant, and assuming a standard deviation of 2 and a correlation of 0.5 between measurements, the required sample size is 39 per group (78 in total). Calculated with StatWisely (statwisely.com) using: n₁: Var(Δmean)=σ²[1+(m-1)r]/m × (1/n₁+1/n₂), solved on noncentral t. These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

More strongly correlated measurements

With a correlation of 0.8 the factor is 0.8400000000000001: 54 per group. Repetition helps less when measurements agree.

Group 1: 54 · Group 2: 54 · Total: 108

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 repeated measures: comparing average levels across follow-up. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/longitudinal-mean-level/

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

References

1.Diggle PJ, Heagerty P, Liang KY, Zeger SL. Analysis of Longitudinal Data. 2nd ed. Oxford: Oxford University Press; 2002.