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.