I am a PhD student using logistic regression to investigate mental health epidemiology. Since participants in my cohort study have a diagnosis or not (coded 1 or 0) - I’m using logistic regression to estimate the assocation of mental health disorders with some *categorical* exposures

Reading Gelman et al. (2008), I understand one approach to Bayesian logistic regression (not hierarchical) is to standardize the input variables. They say, scale variables before setting priors by doing the following:

- “Binary inputs shifted to have a mean of 0 and to differ by 1 in their lower and upper conditions. (For example, if a population is 10% African-American and 90% other, we would define the centered “African-American” variable to take on the values 0.9 and −0.1.)”
- “Other inputs are shifted to have a mean of 0 and scaled to have a standard deviation of 0.5. This scaling puts continuous variables on the same scale as symmetric binary inputs (which, taking on the values ±0.5, have standard deviation 0.5).”

Once data is scaled in this way, Gelman et al. (2008) and Gelman again in Stan Prior Choice Guidance recomend using a (scaled) Student’s t distribution with 3<\nu<7 as a weakly informative prior for the coefficients in the predictor.

**BUT** what if you have a multi-category variable: say ethnicity? Let me illustrate with an example:

What if I have the following ethnicities (UK-context, and by no means a full list of ethnicities!):

- White
- Black-Caribbean
- Black-African
- South-Asian
- East-Asian
- Other

Now say I wanted to look at ethnicity as a predictor of the presence a mental health disorder (a dichotomous variable: 1 or 0). I would normally fit a model using dummy variables for

log(\frac{p_i}{1-p_i}) = \beta_0 + \beta_1x_{1i} + \beta_2x_{2i} + \beta_3x_{3i} + \beta_4x_{4i} + \beta_5x_{5i}

where x_1 is whether you are Black-Caribbean (1) or not (0) … up to x_5 (East-Asian or not) and x_6 other or not WITH the references category being white. This makes sense in terms of the research question as we are comparing mental health of *minority* ethnicities with the dominant ethnicity.

This is where I get stuck. If I change my binary/dummy variables for each ethnicity by shifting the variables (90% ones and 10% zeros would be shifted to take on the values 0.9 and −0.1). When I have done this scaling - this changes the value of the intercept. Previous the intercept was the logit (logit(.)) of the probability p that white subjects had a mental health disorder. Since the output coefficient estimates variables can only be interpreted as the ratio of log-odds as compared to white participants - I am confused about what the benefit is of scaling? Below is the STAN code for the simplest possible version of this model. There are K predictors which represents P-1 categories.

To summarize:

Scaling the variables by shifting them as Gelman et al. (2008) descibes changes ths intercept. *How is this intercept now interpreted?*

*Should this method forscaling variables be used multi-category variables (so that a weakly informative prior can be set on them all)?*

*If this is not suitable, are there any other guides/references on how to set weakly informative priors for categories with more than two factors?*

*Any ideas about how weakly informative priors are extended to hierarchical logistic regression models?*

```
data {
int<lower=0> N;
int<lower=0> K; //number of fixed effect predictors (inc intercept)
matrix[N, K] X_mat;
int<lower=0,upper=1> y[N];
}
parameters {
//FE coeffiecents in mean function for y_repeat
vector[K] beta;
}
model {
//priors
beta ~ student_t(5, 0, 1);
//binomial likelihood
y ~ bernoulli_logit(X_mat * beta);
}
****
```