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I am using partial eta squared to calculate effect sizes for ANOVAs and Aligned-rank transformation (ART) tests. I understand that the recommended values to interpret the results are small (0.01), medium (0.06), and large (0.14). However, how are in-between values interpreted (i.e., above or below these thresholds)? Is it correct to assume small would be 0.01 - 0.05, medium 0.06 - 0.13, and large 0.14 and above?

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You can read about the origin of these thresholds in Statistical Power Analysis for the Behavioral Sciences by Jacob Cohen (1988). For $\eta^2$ and the directly related effect size $f$, see pp. 284-288 for a detailed explanation of these thresholds.

I didn't look very hard, but I didn't find any part where he addresses explicitly your question –I suspect it's on purpose. If it can help, at one occasion in his book, in an illustrative example, he uses the expression "small to medium" to qualify an effect size that happens to be between his small and medium thresholds.

Consider this: if we don't consider effect sizes as a continuum, and instead apply strict thresholds like you suggest, e.g. with anything < 0.06 as small, anything between 0.06 and 0.14 as medium, and anything > 0.14 as large, it would imply that an effect size of 0.059 would have somehow more in common with an effect size of 0.011 than with an effect size of 0.061.

I don't think that using inflexible criteria like that would be very informative or useful - unless, maybe, if you're using some other thresholds designed specifically for your study, because they have some important practical implications.

In any case, you shouldn't use thresholds established by other people without looking why these thresholds have been defined like that in the first place. Their indiscriminate use is controversial. Even Cohen advised multiple times against using the thresholds he suggested, unless you have no way of determining if an effect size is practically important or not. Even then, keep in mind that he was writing primarily for psychologists, and that he established those thresholds based on his experience of this discipline. About the effect size $f$ (hence about $\eta^2$), he says:

Briefly, we note here that these qualitative adjectives are relative, and, being general, may not be reasonably descriptive in any specific area. Thus, what a sociologist may consider a small effect size may well be appraised as medium by a clinical psychologist.

He suggested that analysts could change his thresholds in whatever way they find fit, for the context of their study.

If you have something to say about the practical relevance of the effect size you observed, or if there are other similar studies you can compare the effect size to, it can be a good idea to report that kind of information to show the actual importance of the effect size – rather than focusing on thresholds designed by someone else for a context that may be not relevant to your study.

Finally, you may be interested by the fact that partial eta squared has been criticized as a measure of effect size, so you might want to consider using some alternative instead.

References

Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd edition). Routledge.

Lakens, D. (2015, June 9). The 20% Statistician: Why you should use omega-squared instead of eta-squared. The 20% Statistician. http://daniellakens.blogspot.com/2015/06/why-you-should-use-omega-squared.html

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