Convert probit regression coefficient into correlation

Question

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?

Answer 1

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)
    ))
}