Beta estimates outside of -3 to 3 I am performing logistic regression with glmmTMB with random intercepts and random slopes in R, with 5 fixed effects. I have 9 different datasets that I am running through the same model. Before running my model I have centered and scaled each dataset. If I understand correctly, my beta estimates should be between -3 and 3 because of the standard deviations associated with scaled and centered data. Well, in 4 of my datasets, the beta estimates for one particular covariate range from -3.13 to -5.89. I have double checked these datasets and everything appears to be fine. Depending on the dataset, my sample size is 28-32 animals.
So my questions are:

*

*Is my understanding correct that beta estimates should be between -3 and 3 with scaled and centered data?

*If this is true, what could be causing my beta estimate to be outside of this range?

*Could it be due to too much (or too little) variation for this particular covariate? If so, what can I do about this?


I'll give an example using a vanilla logistic regression (no random effects) to show why this assertion that the coefficients must between $-3$ and $3$ need not hold.
set.seed(2022)
N <- 1000
x1 <- rnorm(N, 0, 1)
x2 <- rnorm(N, 0, 1)
z <- 4*x1 - 4*x2 # Linear combination of the x1 and x2 features
pr <- 1/(1 + exp(-z)) # Transform through the inverse logit
y <- rbinom(N, 1, pr) # Draw from the conditional binomial distributions
x1s <- (x1 - mean(x1))/sd(x1) # Standardize x1
x2s <- (x2 - mean(x2))/sd(x2) # Standardize x2
Ls <- glm(y ~ x1s + x2s, family = binomial) # Fit with standardized features 
summary(Ls)

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)   0.1534     0.1230   1.246    0.213    
x1s           3.7805     0.2878  13.138   <2e-16 ***
x2s          -4.3728     0.3215 -13.599   <2e-16 ***

Interestingly, the coefficients are not that different when we fit to the original features.
L <- glm( y ~ x1  + x2 , family = binomial) # Fit with original features
summary(L)

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)   0.1328     0.1228   1.081     0.28    
x1            3.7872     0.2883  13.138   <2e-16 ***
x2           -4.2710     0.3141 -13.599   <2e-16 ***

