Convert probit regression coefficient into correlation

Most coefficients (e.g., odd ratios, mean differences, etc.) can be converted into a (Pearson-like) correlation coefficient.

Q: Is there a conversion from probit coefficient --> correlation coefficient?

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This blog post by A Gelman discusses the approx. relationship between probit and logit coefficients "Take logit coefficients and divide by approximately 1.6 to get probit coefficients". So, this is one way to go: probit $\rightarrow$ logit $\rightarrow$ r –  Bernd Weiss Dec 16 '11 at 13:48
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1 Answer

Thanks to Bernd's comment, I developed following approach: probit coefficient --> logit coefficient --> odd ratio --> correlation. Beware: in many steps are approximations involved!

logit <- 1.7 * probit
OR <- 2.71828^logit

# convert odd ratio to pearson correlation
# Bonett, D. G. (2007). Transforming odds ratios into correlations for meta-analytic research American Psychologist, 62(3), 254–255. doi:10.1037/0003-066X.62.3.254
# Two different approximations: Pearson and Digby
OR2cor <- function(OR) {
return(list(
pearson=cos(pi/(1 + sqrt(OR))),
digby=(OR^.75 - 1)/(OR^.75 + 1)
))
}
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