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Context

Suppose I am performing a meta-analysis on the relationship between two variables (say depression and life satisfaction). Unfortunately, many studies in the prior literature do not report simple correlations between depression and life satisfaction, but instead categorize depression into high versus low.

So far, no problem (ish). One can easily convert a standardized difference (i.e., Cohen's d) into a correlation coefficient using the following formula:

$\frac{d}{\sqrt{d^2 + a}}$

where

$a = \frac{(n_1 + n_2)^2}{n_1 n_2}$

However, the correlation between depression and life satisfaction will probably be higher (in absolute value) than the correlation between categorized depression and life satisfaction (because dichotomizing a variable introduces unreliability, which attenuates validity).

Problem 1: Dichotomizing a variable introduces unreliability, so the conversion from d to r is expected to be smaller than the correlation between the continuous depression and life satisfaction.

Furthermore, suppose these studies are only interested in those really high or really low in depression. In other words, they throw out the middle group. (I know, it's a dumb thing to do, but I can't change the way others have published their data). Now, converting from d to r using the above formula doesn't make sense. The equation wasn't designed to be used in those sorts of situations.

Problem 2: Using d to r wasn't designed to be used in situations where one has categorized based on the extremes of the continuous variable.

Question: Is there a formula out that converts a d to r when the d was obtained from extreme high versus extreme low categorizations?

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1) You should compute the biserial correlation coefficient. See:

Jacobs, P., & Viechtbauer, W. (2017). Estimation of the biserial correlation and its sampling variance for use in meta-analysis. Research Synthesis Methods, 8(2), 161-180.

2) To deal with dichotomization based on extreme groups, see:

Pustejovsky, J. E. (2014). Converting from d to r to z when the design uses extreme groups, dichotomization, or experimental control. Psychological Methods, 19(1), 92-112.

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