Estimate complete-observation sample size for the fixed-model omnibus multiple-regression F-test from effect size, alpha, target power and predictor count.
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Regression Sample Size Formula
This calculator plans the fixed-model omnibus test that all non-intercept slopes are zero (population R² = 0). Its four inputs are expected full-model f², upper-tail alpha, target power and predictor count. It assumes a full-rank fixed predictor design, independent normal errors and constant error variance. It does not plan an individual coefficient or an added block of predictors. The GPower manual, section 13.5, describes this model.
- N = required sample size
- Fcrit = central-F critical value with upper-tail probability α
- 1−β = target power of the omnibus regression test
- f² = Cohen’s regression effect size
- k = number of predictor terms in the model, excluding the intercept
The calculator evaluates the noncentral-F power for each candidate N and searches for the smallest eligible integer that meets the target. Alpha is not split between two normal tails. Smaller effects and stricter power or alpha requirements generally increase the required sample. The R pwr package vignette explains the corresponding numerator and residual degrees of freedom.
What the Inputs Mean
| Input | What it Represents | Typical Planning Choice |
|---|---|---|
| Effect size (f²) | How strong the regression signal is expected to be | Estimated from prior studies, pilot data, or expected R² |
| Alpha (α) | Probability of a Type I error | 0.05 is a common overall alpha; the F rejection region is upper-tail |
| Power (1−β) | Probability of detecting the effect if it truly exists | 0.80 or 0.90 are common targets |
| Predictors (k) | Total number of estimated predictor terms | Count each dummy variable, interaction, and nonlinear term separately |
How to Estimate Effect Size
If you know the expected model R² instead of f², convert it before using the calculator.
Common interpretation ranges for regression effect size are shown below.
| f² | General Interpretation | Practical Meaning |
|---|---|---|
| 0.02 | Small | Subtle effect; often needs a much larger sample |
| 0.15 | Medium | Moderate signal; often used in planning examples |
| 0.35 | Large | Strong relationship; usually needs fewer observations |
Because f² controls the noncentrality λ = Nf², a smaller expected effect generally requires more observations to achieve the same power. The small/medium/large conventions are broad references, not universal practical-importance thresholds.
How to Use the Calculator
- Estimate the expected effect size using prior evidence, pilot data, or an anticipated R².
- Select the alpha level for the hypothesis test.
- Choose the desired statistical power.
- Count the number of predictor terms in the planned regression model.
- Enter f² and the predictor count, select alpha and target power, then choose Calculate. This calculator solves only the required N.
- Round the resulting sample size up to the next whole number.
Example
Suppose f² = 0.15, alpha = 0.05, target power = 0.80 and k = 3. The fixed-model omnibus F calculation requires 77 complete observations:
Under these assumptions, plan for at least 77 complete observations. This minimum does not automatically allow for missing data or unusable records. Plan additional recruitment separately.
If Your Sample Size Is Fixed
When N is already fixed, achieved omnibus power can be evaluated with the same central/noncentral-F distributions. Solving for a detectable f² requires numerical inversion; it is not a rearrangement of a normal-z expression. This widget solves N only.
Use a method that matches the intended hypothesis. An incremental R² test also needs the number of tested predictor columns; power for a single slope can depend on the predictor correlations. Those analyses are outside this calculator.
Planning Tips for Better Regression Studies
- Count terms, not just variables. If a categorical variable uses 3 dummy variables, all 3 count toward k.
- Include interactions and polynomial terms. Each added coefficient increases the number of predictors.
- Round up, then add a buffer. A small oversample helps preserve power after exclusions and missing values.
- Use realistic effect sizes. Overly optimistic assumptions make required sample sizes look smaller than they should be.
- Watch for multicollinearity. Strong correlation among predictors can make estimates unstable even when total N appears adequate.
- Match the method to the model. This calculator is specifically for the fixed-model omnibus multiple-linear-regression F-test. Individual slopes, added blocks, mixed models, penalized regression and clustered designs need other methods.
Common Questions
- Does k include the intercept?
- No. k counts the tested slopes. The intercept uses one degree of freedom, so the residual degrees are N − k − 1.
- Should interaction terms count as predictors?
- Yes. Any estimated coefficient beyond the intercept should generally be counted in k, including interactions, spline terms, and polynomial terms.
- Why is the required sample size sometimes much larger than expected?
- Small effects, higher desired power, stricter alpha levels, and larger models all increase the required sample quickly. In many cases, effect size is the biggest driver.
- Can this be used for logistic regression?
- Not directly. Logistic regression and other non-linear models usually require model-specific sample size approaches.
When This Calculator Is Most Useful
Use this model-based calculation when the fixed-design omnibus hypothesis and assumptions match the study. It does not establish representative sampling, coefficient precision, prediction quality or adequate model selection. Supported software limits are 1–100 predictor terms, f² from 0.000001 to 10, and a search up to 1,000,000 complete observations; a larger requirement is reported as a limit, not a fabricated answer.
