Achieved power for a linear regression (R²)

When to use it

Assess whether a planned multiple regression has enough participants to detect that its predictors together explain the outcome.

Not for

This is the test of the whole model. Power for one coefficient, adjusted for the others, is usually much lower and needs the single-coefficient calculation.

Notation

$N$: Participants.
$p$: Number of predictors.
$R^2$: Expected proportion of outcome variance the predictors explain.
$f^2$: Cohen’s f², R²/(1 − R²).

Formulas

Effect size, degrees of freedom and noncentrality

f^2=\frac{R^2}{1-R^2}, \qquad \nu_1=p, \qquad \nu_2=N-p-1, \qquad \lambda=f^2 N

This is G*Power’s noncentrality for the R² deviation-from-zero test [1, 2].

Power of the overall F test

\text{Power}=P\left(F_{\nu_1,\nu_2,\lambda} > F_{1-\alpha;\,\nu_1,\nu_2}\right)

F and chi-square tests reject only in the upper tail, so there is no one- or two-sided choice.

Parameters

Sample size $N$

Values: Integer, more than predictors + 1

Enter participants with complete data on the outcome and every predictor.

Number of predictors $p$

Values: Integer, at least 1

Count every term in the model: a categorical variable with c levels adds c − 1.

Expected R² $R^2$

Values: Between 0 and 1

From comparable published models; f² = 0.02, 0.15, and 0.35 (R² ≈ 0.02, 0.13, and 0.26) are small, medium, and large.

Significance level $\alpha$

Values: Between 0 and 1

The type I error rate for the specified test.

Assumptions

Observations are independent, the residuals are approximately normal with constant variance, and the predictors are treated as fixed.
F and chi-square tests reject only in the upper tail, so there is no one- or two-sided choice.

Worked examples

A worked input set

For 100 participants, 3 predictors, and an expected R² of 0.10 at α = 0.05: f² = 0.111, ν₁ = 3, ν₂ = 96, and λ = 11.1.

Sample size: 100

Open this example in the calculator

Validation

Power is the exact noncentral-F tail probability. It agrees with pwr::pwr.f2.test in R and with G*Power, and at the size the linear-regression calculator returns it reaches that calculator’s target power.

How to cite

StatWisely. Achieved power for a linear regression (R²). Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-linear-regression/

Cite the original references for the formulas your design uses as well.

References

1.Cohen J. Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Hillsdale, NJ: Lawrence Erlbaum Associates; 1988.
2.Faul F, Erdfelder E, Lang AG, Buchner A. G*Power 3: a flexible statistical power analysis program for the social, behavioral, and biomedical sciences. Behavior Research Methods. 2007;39(2):175–191.