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?

Best Answer

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.