Means: Many Groups (ANOVA)
Calculate the sample size required for a one-way ANCOVA F-test using the exact noncentral-F approach. Specify the anticipated group means, response standard deviation, number of covariates, and R-squared between the response and covariates. Runs entirely in your browser.
This calculator follows the current this method Analysis of Covariance (ANCOVA) procedure. it uses the Keppel formulation for ANCOVA power calculations and computes power from the noncentral F distribution. The model contains a categorical group factor and quantitative covariates, with the covariate slopes assumed to be equal across groups.
Let σ2 be the common response variance before adjustment for the covariates and let R2 be the coefficient of multiple determination between the response and the covariates. This calculator uses the residual variance
Thus, increasing the anticipated R2 reduces the residual variance and increases the power available for detecting the specified group means.
If the group sample sizes are n1,.., nG and N = Σni, the weighted mean is
The noncentrality parameter used by the this method formulation is
For G groups and p covariates, the numerator and denominator degrees of freedom are G − 1 and N − G − p, respectively. The critical value is the upper 1 − α quantile of the corresponding central F distribution. Power is the probability that a noncentral F random variable with the same degrees of freedom and noncentrality parameter λ exceeds that critical value.
Because the sample size is an integer, the calculator evaluates candidate designs sequentially and returns the first design whose computed power is at least the requested target. With equal allocation, the group sample sizes are constrained to be equal. With allocation ratios supplied, integer group sizes are constructed to follow those ratios as closely as possible.
This worked example: two groups; means 0 and 0.6; SD = 1.2; one covariate; R2 = 0.25; α = 0.05; target power = 0.80; equal allocation. The exact noncentral-F calculation gives N = 98, or 49 subjects per group, with achieved power approximately 0.80752. this method explicitly notes that its exact calculation gives N = 98, whereas the normal approximation in Borm et al. (2007) gives N = 95.
The implementation below was independently checked against that this method example: with N = 98, σm = 0.3, σe = 1.03923, and the resulting noncentral-F power is approximately 0.80752.
the software / this method. Analysis of Covariance (ANCOVA). this method Sample Size Software, Chapter 591. The documentation identifies the Keppel formulation as the method used by this method and describes the exact noncentral-F power calculation.
Keppel, G. (1991). Design and Analysis: A Researcher's Handbook. 3rd ed. Prentice Hall, Englewood Cliffs, NJ, pp. 323–324.
Borm, G. F., Fransen, J., & Lemmens, W. A. J. G. (2007). A simple sample size formula for analysis of covariance in randomized clinical trials. Journal of Clinical Epidemiology, 60(12), 1234–1238. DOI: 10.1016/j.jclinepi.2007.02.006.
Official the relevant methodological literature: Analysis of Covariance (ANCOVA), the software/this method
Published Borm et al. article: PubMed record