Correlation and Regression
Sample size for testing one regression coefficient in a multiple linear regression model while adjusting for the other predictors. The calculation uses the noncentral t distribution, with the expected coefficient, predictor variability, residual variability, and predictor collinearity determining the noncentrality parameter.
The target is an individual coefficient test in an ordinary multiple linear regression. The null hypothesis is H0: β1 = β0, versus HA: β1 ≠ β0. The fitted coefficient is tested with a t statistic, with N − k − 1 residual degrees of freedom.
For planning, the coefficient's noncentrality parameter is determined by the expected coefficient difference, the predictor's standard deviation, the amount of predictor variance that remains after adjustment for the other predictors, and the residual standard deviation:
For each candidate N, the calculator uses a two-sided critical value from the central t distribution with N − k − 1 degrees of freedom and evaluates power from the corresponding noncentral t distribution. Equivalently, because T2 follows a noncentral F distribution with 1 and N − k − 1 degrees of freedom, the browser implementation evaluates the upper tail of that noncentral F distribution. The reported sample size is the first integer N meeting the requested power.
R²X1|others is the squared multiple correlation between the predictor being tested and the other predictors. As this value increases, less independent variation remains in X1, increasing the variance of its adjusted coefficient estimate and therefore increasing the required sample size. Setting this value to 0 corresponds to a predictor that is uncorrelated with the other predictors.
A published regression-power example specifies an expected standardized coefficient of β = 0.20, three total predictors, R² = 0.30 for the full outcome model, 80% power, α = 0.05, and two-sided testing. With standardized outcome and predictor SDs of 1, the residual SD implied by R² = 0.30 is √(1 − 0.30) = 0.836660. With no specified predictor collinearity (R²X1|others = 0), the exact noncentral-t calculation gives N = 140.
Vittinghoff, E., Glidden, D.V., Shiboski, S.C., & McCulloch, C.E. (2012). Regression Methods in Biostatistics: Linear, Logistic, Survival, and Repeated Measures Models, 2nd ed. Springer. The Linear Regression chapter includes sample-size, power, and detectable-effect calculations for individual regression coefficients.
Statistical Solutions Ltd. (2010). Advisor Version 5.0 User's Guide, Appendix: References and Formulas. The ROT4 procedure describes multiple linear regression tests for a set of covariates adjusted for prior covariates; with one tested covariate, the F test is equivalent to the squared individual-coefficient t test. The manual cites Gatsonis & Sampson (1989) for the exact conditional multiple-regression power framework.
Gatsonis, C., & Sampson, A.R. (1989). Multiple correlation: Exact power and sample size calculations. Psychological Bulletin, 106, 516–524.
Dupont, W.D., & Plummer, W.D. Jr. (1998). Power and sample size calculations for studies involving linear regression. Controlled Clinical Trials, 19, 589–601.
For the worked validation target, the published pwrss regression vignette gives the single-coefficient example with β = 0.20, three predictors, R² = 0.30, α = 0.05, and 80% power, reporting N = 140.