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Would it make sense to apply a measure of association test like Cramér's V on a 1xk table?

I am a bit puzzled because in the literature phi and Cramér V are normally used on 2x2 or bigger contingency tables, but while I was looking for some more info, I found the following statement on Wikipedia:

"Cramér's V may also be applied to goodness of fit chi-squared models when there is a 1×k table (e.g.: r=1). In this case k is taken as the number of optional outcomes and it functions as a measure of tendency towards a single outcome."

No reference is provided though. What do you think about it? Could you please give me some bibliographic reference?

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    $\begingroup$ Since the section you describe has been there since the initial creation of the article by Tal Galili you might try contacting him (that link gives a link to his blog, at which in turn you can find a contact email under 'Contact me') and asking him firstly what it even means to have association within a single variable, how he intends the computation to be done, and for a reference (if there's no citation, it should probably be removed from the article as original research) - generally, if a person says something, ask them what they mean. $\endgroup$
    – Glen_b
    Commented Jun 18, 2014 at 0:33

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Since your question in 2014, the information has been removed from the Wikipedia article. If you read the discussion page, the person who originally added this information mentions that they didn't have a reliable source for it. I assume this addition was made at a time when Wikipedia had more relaxed rules for contributing.

If you are strictly looking for a bibliographic reference, the R handbook mentions it. But my guess is that this reference itself relied on the Wikipedia article or on discussions about it on CrossValidated. If it's the case, it's a bit circular and does not really answer your implied question of where this variant of Cramér's V is originally coming from.

An unknown origin does not automatically mean it doesn't have some merits, but if your question is about using an effect size with a known origin, you have several other options. There is the classic Cohen's $\omega$, which is quite useful for power calculation. It varies between 0 and a maximum value depending on the situation. Otherwise, there is the effect size Fei (Ben-Shachar et al., 2023) and two other effect sizes for the chi-squared and likelihood-ratio tests called $E_{\chi^2}$ and $E_L$ (Johnston et al., 2006). These three effect sizes are always bounded between 0 and 1, contrary to $\omega$.

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