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I'm wondering why I should or should not use Cohen's $\omega$ (omega) for effect size over Cramer's V for a chi-square goodness-of-fit test?

What would be the advantage or disadvantage of favoring Cohen's $\omega$?

I am using the rcompanion package with the cohenW() and cramerV() functions.

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  • $\begingroup$ You are using a chi-square goodness-of-fit test , not a chi-square test of association ? $\endgroup$ Mar 1, 2023 at 16:43
  • $\begingroup$ Yes, I am using chi-square goodness-of-fit test in this case. $\endgroup$
    – Chris_F
    Mar 1, 2023 at 17:46
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    $\begingroup$ Why would Cramer's V make sense in goodness of fit? It's for measuring association in 2x2 tables. $\endgroup$
    – Glen_b
    Mar 1, 2023 at 23:26
  • $\begingroup$ @glen_b you discussed this almost 10 years ago here :) stats.stackexchange.com/questions/110600/… This "variant of Cramér's V" seems to come from the Wikipedia article in its original 2011 version (after looking around, I couldn't find any reference older than that). The only implementation of this variant I'm aware of is in the R library rcompanion. I guess that calling it "Cramér's V" may be considered a misuse of language; as far as I can tell, it isn't Cramér who proposed this adjustement to the V coefficient for the case of one-dimensional tables. $\endgroup$
    – J-J-J
    Mar 2, 2023 at 5:04
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    $\begingroup$ Yes, as discussed there, that's not actually Cramer's V, it's just somewhat related/analogous to it. I agree with your assessment of the circumstances. $\endgroup$
    – Glen_b
    Mar 2, 2023 at 18:50

2 Answers 2

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As you mention that you're using the variant* of Cramér's V in a goodness-of-fit scenario, in the case of equally-distributed expected frequencies, one advantage of the $V$ variant over Cohen's $\omega$ is its upper bound of 1. So you can see how far the effect size is from its maximum possible value. There's the same advantage in the case of a contingency table, where $V$ is a measure the association between two variables. As noted in Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.) (in the case of $\omega$ used as a measure of association in contingency tables):

Although $\omega$ is a useful ES index in the power analysis of contingency tables, as a measure of association it lacks familiarity and convenience

I think that the same remark holds for Cohen's $\omega$ in the case of a one-dimensional table, not just contingency tables.

Note that Cohen's omega has also an upper bound of 1 in the case of two categories (which implies $\omega=V$, see the penultimate paragraph below) with expected proportions of 0.5. But in other cases, $\omega$ maximum possible value exceeds 1, so it might be difficult to interpret.

On the other hand, Cohen's $\omega$ allows you to directly use it for power calculation - again, see Cohen (1988).

In cases other than equally-distributed expected frequencies (always in the situation of a one-dimensional table), as the variant of V can exceed 1, it loses its only advantage. You might have some context-dependent advantages (e.g. if you're more familiar with one coefficient than the other, or if you're trying to compare studies using the same coefficient) but I don't think this is generally a massive advantage – though I'd be happy to be corrected here. As Cohen's $\omega$ is generally the effect size that software ask you as an input in the context of power analysis (see the pwr library for example), for any pratical purposes it's probably more relevant to use it than the $V$ variant in unequally-distributed expected frequencies.

As a side note, you might be interested by the fact that $\omega = V\times\sqrt{k-1}$, where $k$ is the number of categories used in the Cramér's V formula, so converting one to the other is quite straightforward.

You might be also interested by the Pearson contingency coefficient (C) and by the Fei coefficient (פ), that are bounded between 0 and 1. If you refer to the R effectsize library, they can be used as an effect size measure in the case of a one-dimensional table. You can find formulas for these two coefficients in the documentation of the effectsize library, page 5. The Fei coefficient is also explained more thoroughly in this 2023 article by Ben-Shachar et al.

The effectsize library authors recommend using Fei over Cohen's $\omega$ and Pearson's C for reasons explained here (i.e. they claim that Cohen's $\omega$ and Pearson's C are inflated, but I'm not sure what they mean exactly by this, and the article by Ben-Shachar et al. doesn't seem to tackle this specific point).


* calling it "Cramér's V" is probably a misuse of language, as this variant for one-dimensional table doesn't seem to have been proposed by Cramér. After looking around, the origin of this $V$ variant is unclear to me. After re-reading Cohen's book discussion on $\omega$ and $V$ (pp. 216-224), it seems that Cohen implicitly excludes the use of $V$ for one-dimensional tables. See Cramér's $V$ for one variable for some additional discussion on the use of the $V$ variant on one-dimensional tables.

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As far as I know, neither Cohen's w nor Cramér’s V is routinely used for a goodness-of-fit test. (Edit: see the comments following this answer.)

Cramér’s V can be modified for use in a goodness-of-fit test. If the theoretical proportions for the test are all equal, Cramer's V is relatively easy to interpret. It's more difficult to interpret if the proportions are not equal.

Considering a test of association, Cramér’s V is a popular effect size statistic, and has the advantage of being confined to the range of 0 to 1.

Cohen's w doesn't have this quality, and may have popularity simply because it was presented by Jacob Cohen (Statistical Power Analysis for the Behavioral Sciences, 1988) and is used there in power analysis.

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    $\begingroup$ See also, in the test of association context: www.researchgate.net/post/Whats_the_reason_for_deriving_Cohens_w_from_Cramers_V_if_the_latter_is_already_a_measure_of_effect_size $\endgroup$ Mar 1, 2023 at 18:36
  • $\begingroup$ Thanks. I was mainly going off Table 1 in Cohen (1992): web.mit.edu/hackl/www/lab/turkshop/readings/cohen1992.pdf $\endgroup$
    – Chris_F
    Mar 1, 2023 at 18:47
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    $\begingroup$ Interesting. For readers, the article recommends Cohen's w for a chi-square goodness-of-fit test. ... I think the dispositive difference, then, is that Cramer's V is constrained to the range of 0 to 1, which, to me, is a desirable characteristic. ... It appears that Cohen's w will tend to result in a larger value. $\endgroup$ Mar 1, 2023 at 19:11
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    $\begingroup$ As a software note, in the rcompanion package, it appears that the cohenW() function works fine if the vector of observed counts is passed to it. For Cramer's V, the cramerVFit() function needs be used in the goodness-of-fit case. $\endgroup$ Mar 1, 2023 at 19:14
  • $\begingroup$ Thanks for all your support with this. Really appreciate it. $\endgroup$
    – Chris_F
    Mar 2, 2023 at 14:25

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