How to Report R², Adjusted R² and Cohen's f² in APA 7

Your regression output hands you two numbers in a row that look like the same thing, R Square and Adjusted R Square, a couple of hundredths apart, with no hint as to which one belongs in the paper. In another tab a reviewer has asked for an effect size and told you that R² is not one, and somewhere in a textbook you have met three figures — .02, .15, .35 — that nobody explained the origin of. The worst part is still ahead: in the Discussion you are about to write that your model explains more variance than last year's study, which is the least defensible comparison in the whole section. Here is what each number is, which ones you report, how to work out f² by hand in ten seconds, and what can genuinely be compared. If you are starting from the beginning, the home of all this is how to report a regression in APA 7.

Which numbers go in

Three figures and a test. The R² with the model's F and both degrees of freedom, the adjusted R² beside it, and Cohen's f² as the standardized effect size. The full template is R² = .34, adjusted R² = .32, F(5, 174) = 17.93, p < .001, f² = 0.52. Those degrees of freedom are not decoration: the first is the number of predictors and the second is N − k − 1, so 180 participants and five predictors give F(5, 174). A model summary with a bare R² and no F is a number nobody can check.

Italics and leading zeros follow the usual rule, letter by letter. R and f are Latin letters and are italicized; the superscript 2 stays upright. The leading zero depends on whether the statistic can pass 1: R² runs from 0 to 1, cannot, and drops it (R² = .34); f² has no upper bound, can, and keeps it (f² = 0.52). The adjusted figure is the odd one in the family: it is written without a leading zero, like R², but it can fall below zero, and then it takes the sign and still no zero, −.13.

Its place is the start of the Results, before the coefficients: first how much the whole model accounts for, then what each predictor does inside it. APA 7 sets no symbol for the adjusted figure, so spell it out in the text (“adjusted R²”) and label it in the table note, but say which one it is, once and unambiguously. What does not work is the free-floating sentence “the model explained 34% of the variance” with no F, no degrees of freedom and no N: that is a percentage the reader can neither place nor reproduce.

If you are building the regression table by hand, the APA regression table generator lays out the coefficient rows and the note that carries the R². The other half of the output — the B, the standard error, the standardized beta and its t — has rules of its own and is covered separately.

When the adjusted figure is the honest one

R² has one property that ought to worry you: it never goes down when you add a predictor. Feed it a column of random numbers and it edges up. That is not a flaw in the software, it is what least squares does — more columns, less residual. The adjusted version charges rent for each one, adjusted R² = 1 − (1 − R²)(N − 1)/(N − k − 1), and the penalty grows with the number of predictors and thins out with sample size.

Two cases show the contrast. With N = 180 and five predictors, R² = .34 lands on an adjusted .32: two points, and either number tells the same story. With N = 40 and eight predictors, an R² of .28 — which reads as publishable in an abstract — falls to an adjusted .09, and the model F confirms it: F(8, 31) = 1.51, p = .195. That model is not distinguishable from noise, and of the two figures only the adjusted one was warning you.

It can come out negative, and that is not a calculation error. With N = 30, six predictors and R² = .10, the adjusted value is −.13, alongside F(6, 23) = 0.43, p = .854. A negative adjusted R² is the estimator telling you that, once those six predictors are paid for, the model would do worse than the outcome's own mean. Report it as it stands, −.13; do not round it to .00 and do not drop the row.

Report both, always, and interpret the adjusted one when the number of predictors is large relative to your sample, or when you are talking about what the model would account for in new people. With a caveat plenty of papers miss: the adjusted figure is not “the variance explained in your sample” — that is precisely R² — but a bias-corrected estimate of the population value, which is why it can land outside the range 0 to 1. It is also not the same thing as a cross-validated R², which is worked out by predicting cases the model has never seen.

What f² is, and how it comes out of R²

Cohen's f² is variance accounted for divided by variance left over: f² = R²/(1 − R²). For the model above, .34/.66 = 0.52. For the whole model it tells you nothing R² had not already said — it is a monotone transformation, rising and falling with it — but it is the language power analysis is conducted in: it is the quantity G*Power asks for in the multiple regression family, and the one you need to justify the sample size of the next study.

Where f² earns its place is the incremental version, for a block or a single predictor inside the model: f² = (R²_full − R²_reduced)/(1 − R²_full). Look at the denominator, which belongs to the full model. In the worked example, block 1 with age and gender gives R² = .06, F(2, 177) = 5.65, p = .004; adding rumination, emotion regulation difficulties and perceived stress takes the model to R² = .34, so ΔR² = .28, F(3, 174) = 24.61, p < .001 and f² = .28/.66 = 0.42. How that gets laid out step by step is in how to report a hierarchical regression in APA 7.

For a single predictor, the ΔR² of entering it last is its squared semipartial correlation, so f² = sr²/(1 − R²). With emotion regulation difficulties entered last, ΔR² = .05, F(1, 174) = 13.18, p < .001, f² = 0.08. Notice what the denominator implies: the same predictor, with the same unique contribution, gets a larger f² inside a model that explains little than inside one that explains a lot. f² is local, not absolute.

Two mistakes recur. The first is dividing by (1 − R²_reduced), which inflates f² exactly when the new block contributes most. The second is attaching a p value to f²: it has no test of its own, the ΔF does, and for a single predictor that ΔF is the square of the coefficient's t (3.63² = 13.18, a three-second check). And say where you took it from: Cohen defined f² on R², not on the adjusted value, so anyone computing it from the adjusted figure has to write that down.

Cohen's thresholds are a convention, not a law

The three figures in circulation are .02 small, .15 medium and .35 large (Cohen, 1988). Translated into variance accounted for — the step back is R² = f²/(1 + f²) — they correspond to R² values of .02, .13 and .26. Cohen offered them for the case where there is nothing better to compare against, and wrote as much: reference points for orientation, not a scale on which a model passes or fails.

The field moves the whole scale. In a lab task where you manipulate one variable and hold the rest still, an f² above .35 can be routine. In an observational model of depressive symptoms with five self-reported predictors, a full model above .35 usually means one of two things: a genuinely powerful predictor, or two scales measuring nearly the same construct under different names. The same “useful convention, not law” logic applies across the family, and it is worked out in the piece on interpreting Cohen's d.

Size is also not significance. An R² of .02 with a single predictor gives F(1, 798) = 16.29, p < .001 in 800 people, and that very same .02 gives F(1, 148) = 3.02, p = .084 in 150. Identical effect, opposite verdicts, which is why an effect size travels with a p value rather than in place of one.

The comparison that carries weight is against the effects reported in your own literature, and better still with a confidence interval than with a label. Writing “a large effect” because f² cleared .35 outsources to a table from 1988 the job of knowing what is typical in your field.

Why R² does not travel between samples

R² is a ratio, and its denominator is the variance of your outcome in the people you measured. Recruit only patients above a cut-off, or only first-year undergraduates, and the denominator shrinks while the relationships stay exactly as they were: the same model, applied to a narrower range, returns a lower R². When your R² falls short of the study you are citing, the first thing to check is not your model but who each of you sampled.

The reliability of your instruments does the same thing by another route. Under the classical attenuation formula, two measures with an alpha of .70 turn a true correlation of .50 into an observed .35, and there R² goes from .25 to .12 without anyone touching the phenomenon. An R² is partly a report on your scales: swap the PHQ-9 for a four-item measure and it drops.

Then there is sampling variability, which almost nobody reports. R² is an upward-biased estimator and its interval is wide: R² = .34 with N = 180 and five predictors comes with a 95% CI of [.21, .42], and the small case from earlier, R² = .28 with N = 40 and eight predictors, with a [.00, .36] that spans nothing to quite a lot. With intervals that wide, two studies ten points apart in R² can be perfectly compatible. When your software gives you one (the Steiger and Fouladi noncentral method, ci.R2 in the MBESS package in R), report it.

What can be compared is nested models within the same sample: there ΔR² and its ΔF mean something, because the denominator is the same and so are the people. That is exactly what hierarchical regression exists for. Moving to f² fixes nothing either, since it is a transformation of R² and inherits its problems. One last temptation worth naming: piling in correlated predictors lifts R² by a few thousandths while the coefficients turn unstable, and that shows up in the VIF, not in R².

The whole thing, written out

Here is the complete example with nothing left implicit: “A multiple regression model was fitted to depressive symptoms (N = 180). The covariate block (age and gender) accounted for a small share of the variance, R² = .06, F(2, 177) = 5.65, p = .004. Adding rumination, emotion regulation difficulties and perceived stress increased the variance accounted for significantly, ΔR² = .28, F(3, 174) = 24.61, p < .001, f² = 0.42. The final model accounted for 34% of the variance, R² = .34, 95% CI [.21, .42], adjusted R² = .32, F(5, 174) = 17.93, p < .001. The unique contribution of emotion regulation difficulties was significant but modest, ΔR² = .05, F(1, 174) = 13.18, p < .001, f² = 0.08.” Five sentences, and no number the reader has to guess at.

In a table, R² and its adjusted twin are not one more row among the coefficients: they go in a summary row at the foot of each block, or in the note, with the F, the degrees of freedom and the N. If the paper compares steps, the table also needs a ΔR² column with its ΔF. Delta is Greek and stays upright; R, F and p are Latin and are italicized.

Four things get this section sent back, in descending order of frequency. A free-floating percentage of variance, with no F and no degrees of freedom. An R² of .28 squeezed out of eight predictors and 40 people, presented without the adjusted .09 sitting behind it. An incremental f² divided by the reduced model instead of the full one. And the Discussion sentence that compares your R² with one from another sample as though they measured the same thing. None of the four takes more than a minute once you know where to look.

Already drafted it? Paste your Results section into Review one section before it goes to your supervisor or the journal. You get the first thing a reviewer would object to, with your own sentence quoted.

For a power analysis, G*Power wants f², not R²: convert R squared to Cohen’s f².

Frequently asked questions

How do you report R² in APA 7?

With the model's F and both degrees of freedom next to it, never on its own: R² = .34, adjusted R² = .32, F(5, 174) = 17.93, p < .001. The R is italicized and the superscript 2 is upright, and it drops the leading zero because it cannot exceed 1. The degrees of freedom are the number of predictors and N − k − 1, so 180 participants with five predictors give (5, 174). If your software provides a confidence interval for R², include it.

Should I report R² or adjusted R²?

Both, and interpret one of them in a sentence. R² is the variance accounted for in your sample; the adjusted figure is a bias-corrected estimate of the population value that charges for every predictor: 1 − (1 − R²)(N − 1)/(N − k − 1). With a comfortable sample they barely separate (.34 and .32 with N = 180 and five predictors) and it makes little difference which you discuss. When the sample is not comfortable, the gap is the news: .28 and .09 with N = 40 and eight predictors, and there the honest number is the adjusted one.

Can adjusted R² be negative?

Yes, and it is not a calculation error. With N = 30, six predictors and R² = .10, the adjusted value is −.13, and the model F keeps it company, F(6, 23) = 0.43, p = .854. It means that, once those predictors are paid for, the model would predict worse than the outcome's own mean. Write it as it stands, with the sign and no leading zero; do not round it to .00 and do not hide the row.

How do you calculate Cohen's f² from R²?

For the whole model, f² = R²/(1 − R²): with R² = .34 that is .34/.66 = 0.52. For a block or a single predictor, f² = (R²_full − R²_reduced)/(1 − R²_full), and the denominator always belongs to the full model: a ΔR² of .28 inside a model of .34 gives .28/.66 = 0.42. f² keeps its leading zero because it has no upper bound, and it carries no p value of its own — the test is the ΔF.

What counts as a small, medium or large f²?

Cohen's (1988) convention is .02 small, .15 medium and .35 large, which in variance terms correspond to R² values of .02, .13 and .26. They are reference points for when you have nothing to compare against, not a grading scale: in a well-controlled lab manipulation .35 can be ordinary, while in an observational model with self-reported predictors it often signals that two scales are measuring nearly the same thing. Compare against the effects that are typical in your own literature and, where you can, with a confidence interval rather than a label.

Can I compare my R² with another study's?

Not directly, and it is the commonest Discussion error with this statistic. R² depends on how much variance the outcome had in your sample (restrict the range and it falls), on the reliability of your scales (two measures with an alpha of .70 pull an R² of .25 down to .12) and on sampling luck: R² = .34 with N = 180 carries a 95% CI of [.21, .42]. What can be compared is nested models within the same sample, using ΔR² and its ΔF.

With the R², its adjusted twin, the model F with both degrees of freedom and an f² for each block, your regression is reported the way APA 7 asks and with nothing left for a reviewer to request later. When the question gets harder — non-nested models, an R² flattened by range restriction, or predictors so correlated that the whole model stops meaning what it appears to mean — that is what my statistical consulting is for.