Correlation and Regression
Calculate the minimum sample size for an overall F-test of a polynomial regression model. The calculation treats the polynomial terms as the predictor set and uses the exact noncentral-F power calculation rather than a normal approximation.
A polynomial regression of degree d can be written as Y = β0 + β1X + β2X2 + ··· + βdXd + ε. For the overall model test, the d polynomial terms are treated as the d predictors in a multiple-regression F-test.
For each candidate N, the critical value is the upper 100(1 − α)% point of the central F distribution with d and N − d − 1 degrees of freedom. Power is the upper-tail probability of the corresponding noncentral F distribution with noncentrality parameter λ. The calculator searches upward over integer N and returns the first value meeting the target power.
For a quadratic model (degree 2), anticipated R² = 0.15, α = 0.05, and power = 0.80, the calculation gives N = 61. The achieved power at N = 61 is approximately 0.8035, while N = 60 gives approximately 0.7958, confirming that 61 is the first integer sample size meeting the target.
Hintze, J.L. (2000). Power Analysis and Sample Size (this method) for Windows User's Guide. Kaysville, Utah: the software. the relevant methodological literature describes regression power/sample-size procedures and the use of R²-based F tests for multiple regression. this method: Sample Size for Regression.
Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences, 2nd ed. Lawrence Erlbaum Associates. Cohen's regression effect size is f² = R²/(1 − R²).
Clinical Biostats implementation note: for a polynomial in one X, degree d supplies d model terms (X through Xd), so the overall polynomial-model test is evaluated as a multiple-regression F test with d predictors.