Is standardize always necessary when predictors having very different scales?

Question

I am running a two-level mixed model, where individual economic status and GDP per capita (PPP) are predictors, and subjective well-being (SWB) is outcome.

Two predictors (economic status, PPP) are centered before entering the model. But they have very different scales.

> summary(CPPP) #PPP
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 -23918  -16514   -8705       0   11722   75858 
> summary(Ceco) #economic status
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
-2.7684 -0.7684  0.2316  0.0000  1.2316  2.2316     360 

When running the model (B001:country, who5: SWB), I received the warning message: Warning: Some predictor variables are on very different scales: consider rescaling. But the model has no converge issue. Does standardize necessary in this case? Without rescaling, the result seems easier to interpret (?)

Here is the model summary output:

model1.2 <- lmer(who5 ~ Ceco + CPPP + (1|B001), data = md)


The results of the model is: 
Linear mixed model fit by REML ['lmerMod']
Formula: who5 ~ Ceco + CPPP + (1 | B001)
   Data: md

REML criterion at convergence: 207560

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.5456 -0.7194  0.0753  0.7646  3.2508 

Random effects:
 Groups   Name        Variance Std.Dev.
 B001     (Intercept)   7.614   2.759  
 Residual             387.660  19.689  
Number of obs: 23581, groups:  B001, 27

Fixed effects:
              Estimate Std. Error t value
(Intercept)  5.382e+01  5.496e-01  97.919
Ceco         5.218e+00  9.299e-02  56.117
CPPP        -3.148e-05  2.496e-05  -1.261

Correlation of Fixed Effects:
     (Intr) Ceco  
Ceco -0.003       
CPPP -0.008 -0.052
fit warnings:
Some predictor variables are on very different scales: consider rescaling

Thank you in advance!

Answer 1

In principle linear regression (including the mixed effects model you run) is scale equivariant, meaning that in theory there is no need to standardise. However vastly different scales may cause numerical issues in the algorithm, therefore the recommendation. These issues do not necessarily lead to warnings about convergence. Obviously you can compute with and without rescaling and see how much of a difference you get. There are certainly situations in which you get that warning but it doesn't make a difference that carries any meaning (which may well be so in your case), but without trying out it's hard to know.

Surely if scales differ by a factor of less than 10 or even 100, there should be no reason to worry.