# Transformation of Chi-Square value from different test types into effect sizes

While working on a meta-analysis, I found different types of Chi-Square tests in the literature. I wonder if all these different types can be handled the same way when converting χ² to Pearson correlation r or Cohens' D.

Type A: The Chi-square test according to Pearson (Pearson's chi-square statistic) includes the squared difference between the observed and expected frequencies.
Type B: The Likelihood ratio chi-squared statistic is based on the quotient of observed and expected frequencies. (source: https://support.minitab.com/minitab/21/help-and-how-to/statistics/tables/how-to/chi-square-test-for-association/before-you-start/example/)

According to literature, following convertion formula is recommended for Chi-Square values: Pearson correlation r = SQRT(χ² / sample size n) (see Lipsey, Wilson 2001, Practical Meta-analysis, p. 201).

Does this formula above apply to any of the two Chi-Square tests? Or only to one of those, and if so, to which? Apparently, they are a bit different from each other because one uses the squared difference between observed and expected frequencies and the other the quotient.

I would be very grateful, if somebody could help me.

• Hi Milewa. Lipsey and Wilson recommend using this effect size, but what do they recommend to use it for? Many people don't have a copy of the book, can you quote what it says exactly? It might help answering you. Commented Nov 2, 2023 at 17:31
• Note that it's possible for correlations to be negative, but $\sqrt{\chi^2/n}$ is necessarily non-negative. $\chi^2/n$ is the square of the $\phi$-coefficient ("phi-coefficient") in a $2\times 2$ contingency table, and $\phi$ is indeed a form of correlation. See en.wikipedia.org/wiki/Phi_coefficient . When the binary values are not considered ordinal, of course the sign of $\phi$ would not be of consequence. This is in relation to the Pearson $\chi^2$ but the G-test $\chi^2$ will be equivalent to it in sufficiently large samples Commented Nov 2, 2023 at 17:32