# #StackBounty: #r #regression #logistic #bayesian log-odds and it's standard error as priors in logistic regression

### Bounty: 50

I’m attempting to complete a Bayesian logistic regression with the outcome of whether or not a crash occurred. I have various covariates in my model that are widely used to predict crash occurrence. As such, I’m using informed priors from prior publications that report the odds ratio and it’s 95% C.I for each covariate.

Here’s an example of a prior provided by the model I’m pulling from

crash at night (OR 13.1; 95% CI 5.0 to 31.5) : log-odds (1.12,.20) from $$frac{log(31.5-5)}{3.92}$$

I wanted to apply the log-odds of these results and their standard error in my updated model as priors. My first thought was to apply the log-odds and it’s a standard error on a normal prior. I’m using logic from the sources 1 & 2 listed at the end of the post.

My question, if my assumptions about applying these log-odds and SE’s on a normal prior are correct, can I simply transform the SE of the log odds to variance and implement?

a normal prior:

βk = (μβk2βk)

requires a variance rather than an SE. According to citation 3 log-odds SE and be transformed into log-odds VAR:

$$SE[log(OR)] = sqrt{VAR[log(OR)]} => SE^2 = VAR[log(OR)]$$

therefore, if I square the standard error x then I should be able to apply this as my final prior:

βk = (1.12,.04)

Is this assumption correct or am I way off? Is there a better way of implementing log-odd priors and their SE’s into a logistic regression model?

Thanks!

1. AdamO (https://stats.stackexchange.com/users/8013/adamo), Prior for Bayesian multiple logistic regression, URL (version: 2016-03-16): https://stats.stackexchange.com/q/202046

"Basically, you have the flexibility to parametrize estimation however
you see fit, but using a model which is linear on the log odds scale
makes sense for many reasons. Furthermore, using a normal prior for
log odds ratios should give you very approximately normal posteriors."

1. Sander Greenland, Bayesian perspectives for epidemiological research: I. Foundations and basic methods, International Journal of Epidemiology, Volume 35, Issue 3, June 2006, Pages 765–775, https://doi.org/10.1093/ije/dyi312

"To start, suppose we model these a priori ideas by placing 2:1 odds
on a relative risk (RR) between ½ and 2, and 95% probability on RR
between ¼ and 4. These bets would follow from a normal prior for the
log relative risk ln (RR) that satisfies…"

1. StatsStudent (https://stats.stackexchange.com/users/7962/statsstudent), How do I calculate the standard deviation of the log-odds?, URL (version: 2020-04-19): https://stats.stackexchange.com/q/266116

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