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Say I have a chi-squared test that compares 3 different groups:

$\chi^2_2 = 9.81$, $p=.012$, Cramer's $V=.33$

How would I go about converting this to a $d$-statistic?

Is the proper approach in this case:

$\sqrt{(4\times 9.81)/N}$, with $N$ = number of participants?

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  • $\begingroup$ Where did that formula come from? In what circumstances does it make sense? $\endgroup$
    – Sal Mangiafico
    Commented Jul 14, 2018 at 14:58
  • $\begingroup$ How could you have a single value for Cohen's d --- which compares two groups --- when you say you have three groups to begin with? $\endgroup$
    – Sal Mangiafico
    Commented Oct 17, 2018 at 17:37

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There is no general method to convert Cramér's V to Cohen's d.

Both statistics are considered effect size statistics, but they convey different kinds of information.

Cohen's d compares values from a continuous variable between two groups, as might be analyzed with a t-test. It is essentially the difference in means between two groups divided by the pooled standard deviation. The absolute value of Cohen's d ranges from 0 to infinity.

Cramér's V is used for contingency tables of counts, for tables larger than 2 x 2. It expresses how far the counts are from expected values. It can be thought as the chi-square value normalized for sample size. It ranges from 0 to 1.

Perhaps certain conditions could be specified where a conversion between Cramér's V and Cohen's d makes sense. For example, where it is known that a continuous variable had a certain distribution and was dichotomized according to a certain rule. But this would not hold for cases outside of these pre-determined conditions.

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Cramer's $V$ is an effect size from the 'correlation family' of effect sizes. Therefore, you could convert using the formula for converting $r$ to $d$:

$$\text{Cohen's }d=\frac{2r}{\sqrt{1-r^2}}$$

However, note that this '$d$' is quite different from the regular Cohen's $d$, which represents the difference between two means expressed in standard deviations.

Note

If you're conducting a meta-analysis and the variable that's "trichotomized" in this case, but dichotomized in all other cases, this can be ok if the variable is the same variable (although you'll have to caution when interpretating the result). If the 'trichotomized' variable in reality is continuous, but happens to be operationalized as a categorical variable in this case, I'd suggest instead converting all effect sizes to $r$. Whether these conversions are appropriate at all depends very much on the context and nature of the variables.

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  • $\begingroup$ What is r here ? What is this Cohen's d that "is quite different from the regular Cohen's d" ? $\endgroup$
    – Sal Mangiafico
    Commented Jul 14, 2018 at 14:31
  • $\begingroup$ $r$ is Pearson's correlation; and (Cohen's) $d$ normally represents the difference between two means. However, when converting from a metric that is not based on one dichotomous and one continuous variable (e.g. two continuous variables, or one categorical variable with three levels and one with four levels, such as you may encounter when Cramer's V is reported), this interpretation makes no sense. $\endgroup$
    – Matherion
    Commented Jul 15, 2018 at 14:29
  • $\begingroup$ Okay. So this formula works when there are two groups, and r = cor(Y, Group) and d = cohensD(Y ~ Group). It looks like this matches the form of Cohen's d where the denominator is (n) not (n - 2). $\endgroup$
    – Sal Mangiafico
    Commented Jul 15, 2018 at 15:55
  • $\begingroup$ I still have no idea how this would relate to Cramer's V. $\endgroup$
    – Sal Mangiafico
    Commented Jul 15, 2018 at 15:56
  • $\begingroup$ What is your question exactly? Or what is the problem you need to solve? $\endgroup$
    – Matherion
    Commented Jul 16, 2018 at 7:11

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