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I am very confused if Cohen's $f^2$ in ANOVA is $\frac{MSTr}{MSE}$ or $\frac{SSTr}{SSE}$ where:

  • $SSTr$: Treatment Sum of Square
  • $SSE$: Error Sum of Square
  • $MSTr$: Mean SSTr
  • $MSE$: Mean SSE

$MSTr = SSTr/(I-1)$ and $MSE = SSE/(n - I)$, where $I$ is the number of groups and $n$ is the total number of observations.

Some information from the Internet says Cohen's $f^2$ is defined as $R^2/(1-R^2)$ for both regression and ANOVA where $R^2=SSTr/SST$. In that case $f^2=SSTr/SSE$ but other sources seems to state that it is $\frac{MSTr}{MSE}$ in ANOVA.

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  • $\begingroup$ Tip: How would you express MST and MSE in terms of SST and SSE? And how would you express, as a consequence, the ratio MST/MSE in terms of SST/SSE? $\endgroup$ – Sextus Empiricus Feb 4 at 7:48
  • $\begingroup$ Criticism: Explanations like "SSTr: SS Treatment" are a bit cryptical. What do you mean exactly by 'SS Treatment'? $\endgroup$ – Sextus Empiricus Feb 4 at 7:50
  • $\begingroup$ @SextusEmpiricus, I updated the question. $\endgroup$ – Royalblue Feb 4 at 7:55
  • $\begingroup$ How many squares do you sum to get SSTr (treatment sum of square)? Could you express it as a formula? $\endgroup$ – Sextus Empiricus Feb 4 at 8:02
  • $\begingroup$ If we denote $J$ as the number of observations in each group, that will be $IJ$ squares. $\endgroup$ – Royalblue Feb 4 at 8:17
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A useful reference is "Statistical power analysis for the behavioral sciences" by Jacob Cohen. Chapter 8, about ANOVA, is available via google https://books.google.ch/books?id=rEe0BQAAQBAJ&&pg=PA273 or other online sources.

The value for Cohen's f resembles Cohen's d

  • Cohen's d is the ratio of the difference between two population means and the population variance:

$$d = \frac{\mu_1 - \mu_2}{\sigma}$$

  • Cohen's f generalizes to more than two populations and is the ratio of the difference between the population means (in terms of their variance) and the variance/deviation of the populations.

$$ f = \frac{\sigma_ \mu}{\sigma}$$


You can relate these variances roughly with the sum of squares.

$$\sigma_\mu^2 \approx SS_{between}/n$$

and

$$\sigma^2 \approx SS_{within}/n$$

Such that

$$f^2 \approx \frac{SS_{between}}{SS_{within}}$$

You do get some discrepancies because the computation of an average square (using the denominator $n$) is not an unbiased estimator for the variances $\sigma_\mu$ and $\sigma$, and instead you could use some correction terms (which becomes complex for unequal groups) but they will not have a large effect.


F-statistic and F-test

You do have a related term the F-statistic and this is computed as

$$F = \frac{SS_{between}/(I-1)}{SS_{within}/(n-I)}$$

where $n$ is the number of observations and $I$ the number of groups.

This is not the same as Cohen's f.

The numerator $SS_{between}/(I-1)$ relates to the random observed sample variance for the group means given a null hypothesis where the effect is null (absent), $\sigma_\mu = 0$. In that case the error in the group means is solely due to a random effect and is distributed with a variance that decreases as $n$ is larger.

Overall you will have that, given the null hypothesis $\sigma_\mu = 0$ is true, this between error is distributed as a chi-squared variable $SS_{between} \sim \chi^2_{I-1}$ independent of the sample size $n$ (as $n$ grows, the increase in the number of term/squares that are summed will cancel with the decreasing size of those terms/squares).

However, in case there is an effect $\sigma_\mu \neq 0$, then the $SS_{between}$ will grow roughly linear with $n$ and so you have to divide by $n$ and use the average $SS_{between}$. (it will follow approximately a noncentral chi-squared distribution and exactly when the populations are normal distributed)

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  • $\begingroup$ Clear and definitive. Thank you for your great teaching! Indeed, as we evaluate a statistic under the assumption of a null hypothesis in order to test(reject) the null hypothesis, it make sense that we should evaluate the statistic under a alternative hypothesis in order to measure the effect size of the alternative. $\endgroup$ – Royalblue Feb 5 at 0:15

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