Means: Many Groups (ANOVA)
Determine the equal per-group sample size needed for a one-factor MANOVA using the Wilks’ Lambda approximate F method documented in PASS. Enter the hypothesized multivariate means and residual variance-covariance matrix, then specify alpha and desired power. Runs entirely in your browser.
This calculator implements the one-factor, equal-allocation special case of the general linear multivariate MANOVA procedure documented in PASS. PASS calculates power for Wilks’ Lambda, Pillai-Bartlett trace, and Hotelling-Lawley trace; this calculator uses Wilks’ Lambda.
Let p be the number of response variables and q the number of groups. The rows of the means matrix correspond to response variables and its columns correspond to groups. The residual covariance matrix describes the joint within-group variability of the response variables.
For a one-factor design, an orthonormal Helmert contrast matrix is used to form the hypothesis matrix. With equal group size n, the hypothesis sum-of-products matrix is formed from the contrast-transformed means, while the error matrix is E = Σ(N − r), where N = qn and r = q.
The Wilks’ Lambda statistic is calculated as the determinant ratio of the error matrix to the total matrix:
PASS converts Wilks’ Lambda to an approximate F statistic using the Muller, LaVange, Ramey, and Ramey approximation. With a = q − 1 and p response variables:
The approximate F statistic is then evaluated against the central F critical value at the selected alpha level. Power is calculated from the corresponding noncentral F distribution with noncentrality parameter λ = df1F. Sample size is searched upward over integer values of n until the requested power is reached.
The resulting sample size is the number of evaluable subjects required per group. The total evaluable sample size is q × n. If a nonzero dropout rate is entered, the enrollment sample size is inflated as nenroll = ceil[n / (1 − dropout)] for each group. The power calculation itself uses the evaluable sample size.
The PASS documentation assumes continuous response variables, multivariate normal residuals with a constant variance-covariance matrix, and independent subjects. This implementation is specifically for a one-factor design with equal group allocation and the Wilks’ Lambda approximate F test. More complicated factorial MANOVA designs require construction of the appropriate contrast matrices for the individual main effects and interactions.
The implementation reproduces the PASS validation example at n = 4: Wilks’ Lambda = 0.79290842, approximate F = 0.49209029, degrees of freedom = (4, 16), noncentrality = 1.96836116, and power = 0.13706319. These values agree with the published PASS validation calculation to the displayed precision.
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.
Muller, K. E., LaVange, L. E., Ramey, S. L., & Ramey, C. T. (1992). Power Calculations for General Linear Multivariate Models Including Repeated Measures Applications. Journal of the American Statistical Association, 87(420), 1209–1226.
NCSS, LLC. PASS Sample Size Software, Chapter 605: Multivariate Analysis of Variance (MANOVA). The PASS documentation specifies the general linear multivariate model, Wilks’ Lambda approximate F calculation, noncentral-F power calculation, worked examples, and validation example used here.