How would we go about scaling your priors by the number of predictors in logistic regression? In a linear regression, @andrewgelman, Hill, and @avehtari (2020: 208) suggest scaling your predictors and then giving each of them a prior SD equal to

\sqrt{R^2_*/p}\times\text{sd}(y),

where R^2_* is your best guess at maximum R^2 achievable with these predictors, p is the number of predictors. They also say to give \sigma an exponential prior with mean \sqrt{1-R^2_*}\times \text{sd}(y).

But what formulas would I use in (multilevel) logistic regression with a Bernoulli outcome? I donâ€™t know what to plug in for sd(y) when thereâ€™s a link function in between. Supposedly the standard logistic distribution has a variance of \pi^2/3, but somehow I doubt if thatâ€™s the right quantity.

Iâ€™m also unsure which pseudo-R^2 metric to base the speculative maximum on, given that many different alternatives exist (McFadden, Cox&Snell, Nagelkerke, Nagakawa&Schielzethâ€¦). To be honest, Iâ€™d rather base the educated guess on something like the ROC AUC, given that I have a better sense of what kind of values to expect (0.90 or so).

Gelman, Andrew, Jennifer Hill, and Aki Vehtari. 2021. *Regression and Other Stories*. Cambridge New York, NY Port Melbourne, VIC New Delhi Singapore: Cambridge University Press.