# Can Cramér's V be used an effect size measure for McNemar's test?

I'm using McNemar's exact test (with R's "mcnemar.exact" function). I know that giving the odds ratio or proportion would be an adequate measure of effect size (Effect size of McNemar's Test), but I was explictly asked to report Cramér's V. Is Cramér's V adequate in this (repeated measures) situation?

Pointers to R code would also be greatly appreciated.

If anyone needs more details: The same subject is measured twice, with a binary outcome in both cases. I'm interested in the difference in the probability of "true" between both measurements. Counts are rather small, e.g.:

           trial 2
trial 1  True  False
True      9     10
False     0      6


In the context of a 2x2 table, Cramer's $V$ is equivalent to the phi coefficient. Moreover, phi is equivalent to Pearson's product moment correlation of the two columns of $1$'s and $0$'s when the 2x2 table is disaggregated. That corresponds to a different magnitude than the one that McNemar's test is testing. So, no, I don't think it is a good choice.

With McNemar's test, you are comparing two marginal proportions (in your case, $76\%$ true before, and $36\%$ true after). Presenting the complete table is pretty easy, as it's just four numbers. But it may be psychologically helpful to present those proportions and whatever magnitude derived from them makes the most sense to people in your field (e.g., the odds ratio). You could even explicitly refer to the cells that are used by McNemar's test. For instance:

There were 10 cases where people changed their minds; in all such cases, people switched from true to false.

Cramér's $V$ doesn't correspond well to what is tested by McNemar's test.

Edit: Disclosure: the webpage and R package cited below are mine.

Probably the most common effect size statistic for McNemar's test is odds ratio, though Cohen's $g$ could be used. Cohen (1988) also uses a statistic he calls $P$.

For defintions, quoting from here:

Considering a 2 x 2 table, with $a$ and $d$ being the concordant cells and $b$ and $c$ being the discordant cells, the odds ratio is simply the greater of $(b/c)$ or $(c/b)$, and $P$ is the greater of $(b/(b+c))$ or $(c/(b+c))$. Cohen’s $g$ is $P – 0.5$.

Cohen (1988) also gives interpretations ("small", "medium", "large") for his $g$ statistic. Because $g$ is monotonically related to odds ratio, these interpretations can be extended to the odds ratio (same link). Edit: Interpretations of effect size statistics are always relative to the field of study, specific experiment, and practical considerations. Cohen's interpretations should not be considered universal.

Obviously, not much coding is needed to calculate any of these statistics for the 2 x 2 case. However, if they are extended to larger tables, the math can get tricky. In R, the cohenG() function in the rcompanion package makes this relatively easy.