# Rescaling random intercept coefficients from hierarchical logistic regression

My logistic regression model includes an overall intercept, multiple categorical variables + and continuous covariates like so:

$$logit(\mu)$$ = $$\beta_0$$ + $$\alpha_{j}$$ + $$\gamma_k$$ + $$\betaX$$

where $$\beta_0$$ is the overall intercept, $$\alpha$$ and $$\gamma$$ are categorical variables with $$j$$ and $$k$$ levels (essentially variable intercept terms) and $$\beta$$ is a vector of coefficients associated with multiple continuous covariates $$X$$. I have scaled each variable in $$X$$ by substract the mean and dividing by the standard deviation. I understand that rescaled $$\beta_0$$ = $$\beta_0^{*}$$ - $$\sum(\beta^{*}\times\frac{\bar{x}}{sd_x})$$ and rescaled $$\beta=\frac{\beta^*}{sd_x}$$. My question is: Is it necessary to rescale $$\alpha$$ and $$\gamma$$ ? If so, how?

• Is there some reason why you centered and scaled the $X$ values to start with?
– EdM
Sep 28, 2021 at 20:27
• Scaling the covariates drastically improved convergence in JAGS. Sep 28, 2021 at 22:09

If your model is strictly additive as you show, then there's no need to adjust $$\alpha$$ or $$\gamma$$. For each individual observation with the adjustments you show, the net contributions to the linear predictor (the right side of your equation) of $$X$$ values and corresponding $$\beta$$ coefficients for the scaled and back-transformed versions of the model are the same. Thus the contributions of $$\alpha$$ and $$\gamma$$ to the linear predictor must be the same in both forms of the model.