Correlation and Regression
Find the minimum sample size for an F-test of an R² effect in multiple linear regression. Enter the number of predictors being tested, the covariates already in the model, the expected incremental R², desired power, and alpha. The calculation uses the conditional noncentral-F method documented in the statistical literature and.
This calculator uses the conditional power approach for a multiple-regression F-test. The tested predictors are added to a model containing the control covariates, and the hypothesis is that their incremental R² is zero:
Let K be the number of predictors tested, L the number of covariates already in the model, ΔR² the expected increase in R², and R²(C) the R² from the covariates alone. The full-model R² is R²(C) + ΔR². The corresponding Cohen effect size is:
For a candidate sample size N, the full-versus-reduced-model F-test has numerator degrees of freedom K and denominator degrees of freedom N − K − L − 1. The noncentrality parameter is λ = Nf². The critical value is the upper 1 − α quantile of the corresponding central F distribution, and power is the upper-tail probability of the noncentral F distribution at that critical value.
The calculator searches integer sample sizes and returns the first N that reaches the requested power. This is the conditional method described in the relevant methodological literature and corresponds to ROT3 framework for a multiple linear regression test of R² = 0 when there are no control covariates, with the same noncentral-F formulation.
the Example 1 uses K = 1 tested predictor, L = 4 control covariates, R²(C) = 0.30, ΔR² = 0.10, and α = 0.05. For a target power of 0.80, the documented required sample size is N = 50; This yields achieved power 0.8060 at N = 50. For the 0.90 target, This yields N = 66 with achieved power 0.9037.
Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Lawrence Erlbaum Associates, Hillsdale, NJ.
Gatsonis, C., & Sampson, A. R. (1989). Multiple correlation: Exact power and sample size calculations. Psychological Bulletin, 106(3), 516–524.
the software, LLC. this method Sample Size Software: Multiple Regression, Chapter 858. The documentation gives the conditional noncentral-F calculation, formulas, worked examples, and validation results used here.
Statistical Solutions, Ltd. Advisor User's Guide, ROT3: Multiple linear regression test that R² = 0 for k normally distributed covariates. The guide specifies central/noncentral F with df K and N−K−1 and noncentrality Nf².