Means: Many Groups (ANOVA)
Sample size for a one-way within-subjects repeated-measures ANOVA under a constant-correlation covariance structure. Enter the expected means at each measurement, the common standard deviation, the correlation between repeated measurements, alpha, and desired power. The calculation uses the univariate F-test formulation used by nQuery’s MOT2 procedure.
This calculator implements the univariate one-way repeated measures ANOVA with constant correlation, corresponding to the nQuery MOT2 procedure. The design has one group of subjects measured at M repeated time points or conditions. The same subjects contribute observations at every level.
The expected means are summarized by their variance around the average expected mean:
When the common standard deviation is σ and the correlation among repeated measurements is ρ, the nQuery constant-correlation formulation uses the standardized effect size:
For a proposed sample size n, the repeated-measures ANOVA is evaluated with a noncentral F distribution. The numerator and denominator degrees of freedom are:
The noncentrality parameter is:
The calculator finds the critical F value from the central F distribution at 1 − α, then computes power from the corresponding noncentral F distribution. It searches upward over integer values of n and returns the first sample size that achieves the requested power.
The constant-correlation formulation corresponds to a compound-symmetry-type covariance structure. If the covariance structure is expected to violate the assumptions underlying the ordinary univariate test, a Greenhouse–Geisser or other nonsphericity-adjusted approach should be planned instead. PASS’s broader Repeated Measures Analysis procedure supports both univariate tests with Geisser–Greenhouse correction and multivariate approaches.
A published clinical study protocol used nQuery Advisor 7.0’s MOT2 procedure for a four-time-point repeated-measures ANOVA. The planned means were 60, 55, 50, and 45 mm; the standard deviation at each time point was 25 mm; the correlation between visits was 0.50; α was 0.05; and the desired power was 80%.
The published protocol reports that n = 29 provides at least 80% power under these exact assumptions. Using the implementation on this page, n = 28 gives approximately 79.06% power and n = 29 gives approximately 80.67% power, so the calculator returns 29 subjects.
Elashoff, J. D. (2002). nQuery Advisor Version 5.0 User’s Guide. Statistical Solutions Ltd., Los Angeles, CA. The nQuery documentation identifies MOT2 as the univariate one-way repeated-measures ANOVA with constant correlation and specifies the central/noncentral F calculation, with df1 = M − 1, df2 = (M − 1)(n − 1), and noncentrality parameter nM times the MOT2 effect size.
Muller, K. E., & Barton, C. N. (1989). Approximate Power for Repeated-Measures ANOVA Lacking Sphericity. Journal of the American Statistical Association, 84(406), 549–555. DOI: 10.1080/01621459.1989.10478802.
NCSS. PASS Sample Size Software: Repeated Measures Analysis, Chapter 570. The PASS documentation describes its repeated-measures framework, including univariate F tests, Geisser–Greenhouse corrections, and multivariate alternatives.
Clinical Study Protocol 13DUR503, Amendment 02. The protocol reports a four-time-point nQuery Advisor 7.0 MOT2 calculation with means 60, 55, 50, and 45 mm, SD 25 mm, correlation 0.5, α 0.05, 80% power, and required sample size 29.