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Cohen’s f² is the effect-size measure built for regression, not the one built for group comparisons. Cohen’s d answers “how far apart are two group means, in standard-deviation units?” Cohen’s f² answers a different question: “how much variance does this predictor, or this block of predictors, actually add to a regression model?” It is derived directly from R² (or from the change in R² when a predictor is added to an existing model), which makes it the natural effect size for multiple regression, hierarchical/sequential regression, and the power-analysis software researchers actually use to plan sample size for those designs. This guide covers the formula for a single predictor or block, Cohen’s (1988) small/medium/large conventions, a fully reproducible worked example, and exactly how f² differs from Cohen’s d.
What Cohen’s f² measures
For a regression model as a whole, Cohen’s f² expresses the proportion of variance explained relative to the variance not explained:
f² = R² / (1 − R²)This is the effect size for the omnibus test of whether a model’s R² is significantly different from zero — it rescales R² so that equal-sized increments feel more like equal-sized increments in explanatory power, which R² itself does not do (going from R² = 0.10 to 0.20 is a much bigger jump in f² terms than going from 0.50 to 0.60).
Calculating the local f² for a single predictor or block
Most of the time, a researcher does not want the effect size for an entire model — they want the effect size for one predictor, or one theoretically meaningful block of predictors, controlling for everything else already in the model. This is the “local” or “partial” f², and it is what a sequential (hierarchical) regression is built to test. Fit the model twice:
- Model A — the base model, without the predictor(s) of interest.
- Model AB — the same model with the predictor(s) of interest added.
Then:
f² = (R²AB − R²A) / (1 − R²AB)The numerator is the familiar R² change (ΔR²) reported in a sequential regression table. The denominator rescales that change against the variance the full model still leaves unexplained. This is the exact quantity G*Power’s “F tests — Linear multiple regression: special (R² increase)” module asks for as its effect size input, because it is the effect size that determines statistical power for testing whether adding a specific predictor or block improves the model.
Cohen’s small/medium/large conventions
Cohen (1988) proposed benchmark values for f² in multiple regression, distinct from the conventions he separately proposed for Cohen’s d (group comparisons) and for Cohen’s f (the ANOVA-family effect size based on eta-squared). The three are not interchangeable and do not convert into one another by simple arithmetic — each was calibrated to its own test family.
| Effect size | f² value | Rough R² equivalent (single predictor) |
|---|---|---|
| Small | f² = 0.02 | R² ≈ 0.02 |
| Medium | f² = 0.15 | R² ≈ 0.13 |
| Large | f² = 0.35 | R² ≈ 0.26 |
As with Cohen’s d benchmarks, treat these as a fallback for when a field has no better-established yardstick of its own — not as a universal pass/fail line. A predictor with a “small” f² can still be the theoretically decisive one in a model; a “large” f² on a trivial control variable is not automatically the finding worth leading a paper with.
A worked example
The numbers below come from a seeded simulation (n = 150, fixed random seed, reproducible), not an invented case study. A base model predicts a continuous research-productivity outcome from two predictors — years of continuous funding and team size. A second model adds a third predictor, mentorship hours, to test whether it explains additional variance once the first two are already in the model.
| Model | Predictors | R² |
|---|---|---|
| A (base) | years funded, team size | 0.1470 |
| AB (full) | years funded, team size, mentorship hours | 0.1996 |
ΔR² = 0.1996 − 0.1470 = 0.0526. Applying the local-f² formula:
f² = (0.1996 − 0.1470) / (1 − 0.1996) = 0.0526 / 0.8004 ≈ 0.066An f² of about 0.066 sits between Cohen’s small (0.02) and medium (0.15) benchmarks — mentorship hours adds a real but modest amount of explanatory power once the other two predictors are already accounted for. The overall model’s f² (all three predictors against a null model) works out to R² / (1 − R²) = 0.1996 / 0.8004 ≈ 0.249, which is close to Cohen’s large benchmark for the model as a whole — illustrating why the omnibus f² and a single predictor’s local f² routinely tell different stories about the same model, and why reporting only one of them can be misleading.
How f² differs from Cohen’s d
| Cohen’s d | Cohen’s f² | |
|---|---|---|
| Question it answers | How far apart are two group means, in SD units? | How much variance does a predictor (or block) add to a regression model? |
| Design it fits | Two-group comparisons (t-test family) | Multiple/hierarchical regression, and by extension the general linear model |
| Built from | Mean difference divided by pooled SD | R², or the change in R² between two nested models |
| Small/medium/large | 0.2 / 0.5 / 0.8 | 0.02 / 0.15 / 0.35 |
| What it can’t do | Doesn’t apply once there are three or more predictors in play at once | Doesn’t tell you which specific group differs from which; it’s about variance explained, not group location |
A regression model can be converted to a two-group comparison in special cases (a single binary predictor with no covariates reduces to a t-test, and f² and d become mathematically related through R²), but the two indices exist because they answer different questions for different designs — use whichever one matches the actual analysis being reported, rather than converting one into the other for the sake of a single “effect size” figure in a results section.
Using f² in a power analysis
Because f² is scaled specifically for regression, it is the effect-size input G*Power and similar tools expect for two related but distinct questions: whether an entire model’s R² differs from zero, and whether adding a specific predictor or block improves an already-specified model. Both require an a priori f² estimate — drawn from a pilot study, closely related published work, or (only as a last resort) one of Cohen’s benchmark values — entered alongside the number of predictors, the number of predictors already in the base model for a local test, and the desired power (conventionally 0.80) and alpha (conventionally 0.05).
Frequently asked questions
Is Cohen’s f² the same as Cohen’s f?
No. Cohen’s f is a related but separate index Cohen proposed for ANOVA-family designs, built from eta-squared, with its own small/medium/large benchmarks (0.10 / 0.25 / 0.40). Cohen’s f² is the regression-specific index built from R², with the benchmarks 0.02 / 0.15 / 0.35 given above. The similar names and the fact both come from the same 1988 source are the main reason the two get confused; they are not interchangeable and their benchmark values do not convert into one another.
Can f² be negative?
The formula itself cannot produce a negative value from a valid R² (which is bounded between 0 and 1 in ordinary least squares regression), but a local f² computed from R²AB − R²A can come out at or effectively at zero if the added predictor(s) contribute nothing once the base model is already fit — that is a legitimate, reportable result, not a calculation error.
What sample size does a given f² require?
That depends on the number of predictors in the test and the desired power, not on f² alone — the same f² needs a larger sample to detect reliably as the number of predictors in the model grows. Run the actual numbers through a power-analysis tool such as G*Power rather than estimating from the f² value in isolation.
Does f² work for logistic or other non-OLS regression?
Cohen’s original f² was defined for ordinary least squares multiple regression, where it is derived directly from R². For logistic and other generalized-linear-model regressions, R² itself is defined multiple, non-equivalent ways (Cox & Snell, Nagelkerke, McFadden’s pseudo-R²), so an f² computed from one of those pseudo-R² values is not directly comparable to an OLS f² or to Cohen’s original benchmarks — treat any such figure as a rough approximation, and say explicitly which pseudo-R² it was built from.
See also: How to Interpret R-Squared, How to Interpret Cohen’s d, Cohen’s d Effect Size: Calculation & Interpretation Guide, Effect Size: Choosing, Reporting and Interpreting It, Multiple Regression in SPSS, Statistical Power Analysis & G*Power Sample Size Guide, Degrees of Freedom in Statistics, and the Research Methods hub.
