Correlation and Regression
Determine the sample size needed to test whether a set of predictors explains a meaningful increase in R² after controlling for other covariates. Uses the exact conditional noncentral-F method documented in the statistical literature and runs entirely in your browser.
This calculator implements the conditional power and sample size method in the this method Multiple Regression procedure. The method tests whether a specified set of K predictors contributes additional explanatory power after controlling for L other predictors.
The null hypothesis is that all regression coefficients associated with the tested predictors are zero:
The test is a full-versus-reduced-model F-test. The reduced model contains the L covariates, while the full model additionally contains the K predictors being tested.
The key effect-size quantity is the increase in R² obtained by adding the tested predictors:
Here, R²(C) is the expected R² from the covariates alone and R²(T,C) is the expected R² from the full model. Therefore the calculator internally obtains the full-model R² as R²(C) + R²(T|C).
For a candidate sample size N, the numerator and denominator degrees of freedom are:
this method specifies the noncentrality parameter for the conditional calculation as:
The critical value is the upper α quantile of the central F distribution with K and N − K − L − 1 degrees of freedom. Power is then the upper-tail probability of the corresponding noncentral-F distribution with noncentrality parameter λ. The calculator searches upward through integer N values and returns the first N for which the achieved power is at least the requested power.
Important: This is the this method conditional method. It treats the observed/planned predictor values as fixed for the conditional calculation. It is not a prediction-model development sample-size method based on calibration, optimism, shrinkage, or events-per-variable.
This worked example uses power = 0.80, alpha = 0.05, K = 1 tested predictor, L = 4 covariates, incremental R² = 0.10, and covariate R² = 0.30. The documented required sample size is N = 50, with achieved power approximately 0.8060.
If a dropout rate is entered, the calculator inflates the evaluable sample size using N′ = N / (1 − dropout rate), rounding the enrollment target upward. The statistical power calculation itself is based on the evaluable sample size N.
the software, LLC. this method Sample Size Software — Multiple Regression, Chapter 858. the relevant methodological literature. The procedure describes the conditional noncentral-F method, the definition of R²(T|C), the noncentrality parameter, and the published validation examples.
Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences, 2nd ed. Lawrence Erlbaum Associates, Hillsdale, NJ.
Gatsonis, C. and Sampson, A.R. (1989). "Multiple Correlation: Exact Power and Sample Size Calculations." Psychological Bulletin, 106(3), 516–524.