# Solved – Is the squared Cohen’s $f$ the same as Cohen’s $f^2$

cohens-dinteractionregressionstatistical-powerterminology

Question

Is the squared version of Cohen's $$f$$ the same as Cohen's $$f^2$$? Or do they only share the same letter by coincidence?

Explanation

At first, I wondered if one can simply square Cohen's $$f$$ to obtain the Cohen's $$f^2$$. For example, papers often report Cohen's $$d$$, but power analyses (for regression) often require $$f^2$$ as estimate of the effect size. According to this source, it is possible to convert Cohen's $$d$$ to $$f$$ with the following formula :

$$f = d/2$$

Therefore, if one were to square the resulting $$f$$, would one be able to use this value as estimated effect size in a power analysis?

I would think that you cannot do that and that they are not related because they are part of different families and use different calculations ($$f^2$$ is part of the correlation family, while $$f$$ is part of the difference family). I was speaking with a statistician friend recently who was unsure himself because they're both related to regression under the hood. So I just wanted to confirm this with the community?

The effect size index $$f^2$$ is defined by Cohen as $$\frac{R^2}{1-R^2}$$ in a regression model where $$R^2$$ is the coefficient of determination (the % of variance explained by the model). Cohen emphasized the fact that is fundamentally identical to the $$f$$ index, hence you can take the square root of $$f^2$$ to get your estimate for $$f$$ (Statistical Power Analysis for the behavioral Sciences, 2nd ed., Lawrence Erlbaum Associates, 1988; §9.2, p. 410). It is just easier to work with a squared index when you already work with squared correlation values.
Sometimes, it is also computed as $$\frac{\eta^2}{1-\eta^2}$$ or $$\frac{\eta_\text{partial}^2}{1-\eta_\text{partial}^2}$$, especially when the authors focus on ANOVA. This is discussed later in \$9.2.1, but you will find the same explanation on the UCLA website.