In Sheskin's Handbook of Parametric and Nonparametric Statistical Procedures, for each categorical association coefficient (e.g., Pearson's C, Pearson's phi, Cramer's V), the Author also takes into account Cohen's effect size measure.
For example, for Cramer's V, Cohen's measure of effect size (namely, w) can be calculated as:
w=Cramer'sV * squareroot(k-1)
where k is the number of rows or columns, whichever is smaller.
My question:
if Cramer's V can be considered measures of effect size, why should we derive another such measure from it (i.e., w in this case)?
Thank you in advance for any insight into the problem.
Best,
Gm
https://www.amazon.it/Handbook-Parametric-Nonparametric-Statistical-Procedures/dp/1439858012
It looks like Cohen's w is on the same scale as r so it is presumably a comparability issue. However, I would caution against doing this as Cramer's V has multiple df and thus Cohen's w is really akin to R the multiple correlation coefficient and hence fairly useless as an effect size measure. Rather it
Two studies can have identical V or w values and completely different patterns of effect so the statistic isn't really of any use when comparing studies. It might have some use in a power calculation but only if you are interested in nonspecific hypotheses. However, if you are doing a power calculation you should have specific hypotheses to test and therefore use a single df effect size metric to capture those hypotheses.
I'm struggling to get a handle on this issue myself. I think the solution is going to entail buying Cohen's book.
Here's what I figured out so far.
Cramer's V ranges from 0 to 1, which is a desirable property for an effect size. In a 2 x 2 table, Cramer's V reduces to phi, which is good.
Cohen's w is not bound to 1 on the upper end.* It seems that its advantage is that it sticks to the 0.1 = small; 0.3 = medium; and 0.5 = large effect rule.** , that is used with r (https://books.google.com/books?id=5obZnfK5pbsC&pg=PA41).
This is in contrast to Cramer's V, where tables of "small, medium, large" require considering the dimensions of the table. (https://www.researchgate.net/post/How_can_I_intepret_the_effect_sizes_of_Cramers_V_when_DF_3).
In a 2 x 2 table, Cohen's w reduces to phi, which is good. Cohen's w can also be used for a goodness-of-fit test, but I think only if the expected values are equal for each category.
If the smallest dimension of the table is 2, Cramer's V and Cohen's w are the same values anyway.
As these ideas coalesce, I will add them here: http://rcompanion.org/handbook/H_10.html
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* This is obvious from the formula in the question, and also obvious from the calculations for Cramer's V, which is essentially Cohen's w scaled so that the maximum will be 1. Cohen's w can simply be calculated as the square root of (the chi-square value divided by the sum of counts).
** Obviously, such guidelines are depend upon the field of study and other factors, but as Thom Baguley points out, this gives values for Cohen's w the same interpretation as r.
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