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.

  • 1
    $\begingroup$ 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. $\endgroup$
    – J-J-J
    Nov 2, 2023 at 17:31
  • 2
    $\begingroup$ 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 $\endgroup$
    – Glen_b
    Nov 2, 2023 at 17:32

1 Answer 1


For sufficiently large samples the Pearson chi-squared and the likelihood ratio chi-squared are equivalent. The Wikipedia article on the G-test explains more about the issues.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.