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

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

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?

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

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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.

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