# Small/Medium/Large Cramér's V effect size tables available for degrees of freedom > 5?

I've come across numerous tables published online categorizing Cramér's V effect sizes for categorical tables with degrees of freedom $$\leq$$ 5, for example:

$$\begin{array}{c|lcr} \text{Degrees of freedom} & \text{Negligible} & \text{Small} & \text{Medium} & \text{Large}\\ \hline \text{1} & \lt 0.10 & [0.10, 0.30) & [0.30, 0.50) & 0.50+\\ \text{2} & \lt 0.07 & [0.07, 0.21) & [0.21, 0.35) & 0.35+\\ \text{3} & \lt 0.06 & [0.06, 0.17) & [0.17, 0.29) & 0.29+\\ \text{4} & \lt 0.05 & [0.05, 0.15) & [0.15, 0.25) & 0.25+\\ \text{5} & \lt 0.05 & [0.05, 0.13) & [0.13, 0.22) & 0.22+\\ \end{array}$$

What are the "Negligible/Small/Medium/Large" rules of thumb for interpreting Cramér's V effect sizes for degrees of freedom > 5?

The table you reproduced relies on Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed).

If you want to follow Cohen's rule of thumb (which may or may be not a good idea) for any given degrees of freedom, first you have to convert Cramér's V to Cohen's omega ($$\omega$$).

You do that by multiplying $$V$$ by the square root of the table's degrees of freedom. Cohen gives this formula in his book (formula 7.2.7, page 223):

$$\omega = \phi ' \sqrt{r-1}$$

where $$\phi '$$ is Cramér's V, and $$(r-1)$$ is the degrees of freedom ($$r$$ being the contingency table's smallest dimension, not necessarily the number of rows, contrary to what the letter "r" might lead one to believe).

Then, you can use the rule of thumb he describes pp. 224-227 for interpreting $$\omega$$, which is:

$$\omega$$ Effect Size
.10 small
.30 medium
.50 large

As an example: if you have a $$9 \times 8$$ table with $$V = 0.2$$, you get $$\omega = 0.2 \times \sqrt{8-1} \approx 0.53$$, which is a large effect size according to Cohen.

The table you mention in your question is essentialy an aid to avoid having to calculate $$\omega$$ yourself (which isn't really complicated if you have a computer at your disposal). It seems that the table is constructed from the table 7.2.3 on page 222 of Cohen's book, where Cohen shows various equivalents of $$\omega$$ in terms of Cramer's V, up to 5 degrees of freedom (I guess that saving ink and space was the main rationale for limiting the examples to 5 degrees of freedom).

You can easily check that the table you copied relies on Cohen's formula, for example if you take the line "4 degrees of freedom": $$0.15 \times \sqrt{4} = 0.3$$ (which is a medium effect for $$\omega$$), or if you take the line "5 degrees of freedom" with $$0.22 \times \sqrt{5} \approx 0.5$$ (large effect for $$\omega$$).

Note that contrary to Cramér's V, Cohen's $$\omega$$ does not have a upper bound, so it's entirely possible to end up with results where $$\omega > 1$$.

Also note that Cohen gives a word of caution about using his rule of thumb, page 224 of his book:

The best guide here, as always, is the development of some sense of magnitude ad hoc, for a particular problem or a particular field.

and:

The investigator is best advised to use the conventional definitions as a general frame of reference for ES and not to take them too literally.

• Wouldn't this mean that Cohen's w is the same as applying the Phi coefficient to a larger than 2x2 contingency table, since it is just removing the square root degree of freedom term from the denominator of the Cramer's V equation? Mar 8 at 15:18
• @stat_is_quo In his book (1988), Cohen mentions the identity between $\omega$ and Phi ($\phi$) in the case of $2\times2$ tables, but he does not describe $\omega$ as a generalization of Phi ($\phi$) for larger tables. Additionally, he explicitly says that Cramér's V is a generalization of $\phi$ for larger tables. Page 222 of his book, he gives a table listing equivalent values of $\omega$ in terms of C (contingency coefficient), $\phi$, and Cramér's V. I think it's generally better to stick to definitions given by authors, in order to avoid confusion. Mar 8 at 17:47
• sure, distinct concepts and whatnot. But to clarify phi and Cohen's omega are mathematically equivalent, correct? The square root of the quantity chi-square divided by the sample size. Mar 8 at 20:16
• Following up to note that yes, in fact, in this table in Cramer's 1998 book the phi and Cohen's omega values are identical. Mar 8 at 21:17
• @stat_is_quo Yes when $\phi$ is calculated that way. Note that there's an alternative formula for $\phi$ in $2\times2$ tables, where phi is directional and is bounded between -1 and 1. In this scenario $w$ and $\phi$ are not the same thing. (I have to admit that I'm not sure why $\phi$ is used for denoting two different formulas). Mar 9 at 7:43