Repeated measures sample size calculator with correlation

Start from the single-measurement sample size for two parallel groups: n₁ = 2σ² (z₁₋α/2 + z₁₋β)² / δ², the formula on the two-means sample size calculator. For δ = 5, σ = 12, two-sided alpha 0.05 and 80% power, n₁ = 2 × 144 × 7.85 / 25 = 90.4, so 91 per group by the normal formula. Now suppose every participant is measured m times after randomisation and the primary analysis uses each participant’s mean (or, equivalently for this purpose, an average of the repeated measures). Under compound symmetry every measurement has variance σ² and every pair has correlation ρ. The variance of the mean of m measurements is then σ² (1 + (m − 1)ρ) / m. The sample size scales with variance, so n = n₁ × (1 + (m − 1)ρ) / m.

The limits are instructive. If ρ = 0, the measurements are independent and the factor is 1/m: three measurements need a third of the participants. If ρ = 1, every repeat is a copy of the first and the factor is 1: measuring again adds nothing. Real clinical data sit between the two. With ρ = 0.5 and m = 3 the factor is (1 + 2 × 0.5) / 3 = 0.667, so n = 90.4 × 0.667 = 60.3, which rounds up to 61 per group. A simulation of the two-sample t-test at 61 per group gives 79.5% power and at 62 gives 80.2%, so plan for 62 to allow for estimating the SD.

Baseline adjustment works through the correlation between the baseline measurement and the outcome. Fitting an ANCOVA with the baseline as a covariate reduces the residual variance of a single follow-up measurement to σ² (1 − ρ²) (Borm et al., J Clin Epidemiol 2007), so n = n₁ × (1 − ρ²). With ρ = 0.5 that is 90.4 × 0.75 = 67.8, so 68 per group. When the outcome is the mean of m follow-ups and a baseline is in the model, the variance is the variance of the mean minus the part explained by the baseline: σ² [ (1 + (m − 1)ρ) / m − ρ² ] (Frison and Pocock, Stat Med 1992). For m = 3 and ρ = 0.5 that is 0.667 − 0.25 = 0.417, so n = 90.4 × 0.417 = 37.7, or 38 per group. Both formulas were checked here against matrix algebra and a 200,000-run simulation of the variances.

These are planning approximations, and the assumptions matter more than the arithmetic. The first is compound symmetry: one correlation ρ for every pair of measurements, regardless of how far apart in time they are. In practice correlation usually decays with the gap between visits. Take adjacent-visit ρ = 0.5 that decays as an autoregressive AR(1) process. With four measurements the variance factor for the mean is 0.516, lower than the 0.625 that compound symmetry predicts, but the correlation between distant visits is then much lower than 0.5. Neither number is wrong; the point is that ρ must be the average correlation among the measurements you will actually use, taken from data on the same endpoint with a similar visit spacing, not a single guessed value.

The second is a constant treatment effect across the m time points, which is what makes the mean a sensible summary. If the effect grows over time, the mean understates the effect at the last visit and the trial may be over- or underpowered for a final-visit claim. The third is complete data: dropout and missed visits remove information, and the factors above give evaluable participants, so divide by (1 − dropout) to get enrolment using the dropout-adjusted sample size calculator. The fourth is that the SD σ is the SD of a single measurement at the scale of the outcome (and, with baseline adjustment, that baseline and outcome are measured the same way).

Finally, the primary analysis must match the calculation. If the SAP specifies a mixed model for repeated measures (MMRM) with an unstructured covariance and a test at a particular visit, the power comes from that model, not from the mean-of-m shortcut; the factors here are a starting point, and the final number should come from a published MMRM formula or a simulation done by the trial statistician. If the analysis is an ANCOVA of the mean of the repeats, the factors apply directly. Group-randomised designs need a different design effect again, covered on the cluster-randomised sample size calculator.

A justification that names the assumptions reads: “Assuming a common SD of 12 mmHg for a single measurement, a correlation of 0.5 among the three post-baseline measurements and between each of them and baseline, and analysis by ANCOVA of the mean of the three measurements with baseline as a covariate, 38 evaluable participants per group provide 80% power to detect a difference of 5 mmHg at the two-sided 5% level. Allowing for 10% dropout, 43 per group will be enrolled.” (38 / 0.9 = 42.2, rounded up to 43.) The correlation should come with a source, such as a previous trial of the same endpoint, and the protocol should state that the SD and ρ will be reviewed at the blinded interim if one is planned.

The number then drives the build. In Capture the repeated measurements are the same eCRF form scheduled at several visits, each with a visit window so the spacing the correlation assumes is respected. The baseline and follow-up values are numeric fields with units and range edit checks, calculated fields can derive the mean of the visits for review during entry, and the monitoring dashboard shows how many participants have reached each visit. Data export as CSV or Excel with a data dictionary for the statistician, and the field-level audit trail records old value, new value, user, time and reason for each change. The power analysis calculator shows what a fixed enrolment can detect, and the crossover sample size calculator is the within-participant design where each person receives both treatments. See the sample size calculation glossary entry for the basic terms.