# What is the definition of Cohen's f in ANOVA

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.

• 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? – Sextus Empiricus Feb 4 at 7:48
• Criticism: Explanations like "SSTr: SS Treatment" are a bit cryptical. What do you mean exactly by 'SS Treatment'? – Sextus Empiricus Feb 4 at 7:50
• @SextusEmpiricus, I updated the question. – Royalblue Feb 4 at 7:55
• How many squares do you sum to get SSTr (treatment sum of square)? Could you express it as a formula? – Sextus Empiricus Feb 4 at 8:02
• If we denote $J$ as the number of observations in each group, that will be $IJ$ squares. – Royalblue Feb 4 at 8:17

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)

• 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. – Royalblue Feb 5 at 0:15