Interpretation of Cohen's $f^2$ for effect size in multiple regression I am struggling with the interpretation of the effect size of a multiple regression model measured by Cohen's $f^2$. I know the guideline to determine if the effect is small, moderate or high, but what I am looking for is a simple explanation (may be in lay language) of the effect size for a multiple regression model. I looked into some resources but did not get any satisfactory answer. I have calculated the $f^2$=0.04 by the Gpower software for a given $\alpha$, power, and sample size. Can anybody explain what does this effect size mean, or how to interpret the $f^2$=0.04? Thank you very much in advance!
 A: From a simple Google search, I found this useful paper. I recommend reading it before continuing with your work, A Practical Guide to Calculating Cohen’s f2, a Measure of Local Effect Size, from PROC MIXED. Albeit the authors use SAS, you should be fine interpreting your results in whatever software you choose to use.
From the paper, it reads

According to Cohen’s (1988) guidelines, $f^2$≥ 0.02, $f^2$≥ 0.15, and  $f^2$ ≥ 0.35 represent small, medium, and large effect sizes, respectively.

To answer the question of what meaning $f^2$, the paper reads

However, the variation of Cohen’s $f^2$ measuring local effect size is much more relevant to the research question:
$f^2$ = $\frac{R^2_{AB} − R^2_{A}}{1−R^2_{AB}}$ 
  (2) where B is the variable of interest (i.e.,
  either smoking quantity or nicotine dependence score), A is the set of
  all other variables (i.e., gender and depending on what B is at the
  moment, nicotine dependence score or smoking quantity), $R^2_{AB}$ is the
  proportion of variance accounted for by A and B together (relative to
  a model with no regressors), and R2A is the proportion of variance
  accounted for by A (relative to a model with no regressors). Thus, the
  numerator of (2) reflects the proportion of variance uniquely accounted for by B, over and above that of all other variables (Cohen, 1988).

So in simple terms, the statistic shows the marginal effect size of including covariate B in the variable set, A, already in the model.
