I am currently fitting a mixed model where I analye longitudinal trends in migration between country pairs (68335 observations nested in 6442 groups). One of the first questions I wanted to have answered was whether I should estimate time as linear or exponential. Based on theory and heuristic examination of the plots I have good reason to believe that it should be exponential, but I also wanted to compare the AIC of the linear and exponential models to see which fit the data better. When I tried this, however, I found the difference in AICs to be absurdly high (>480,000). I immediately became suspicious of this but as far as I know, I didn't make any mistakes.
This is the r syntax I used for the models:
modela <- lmer(Migration~(1+Time|Country_number) + Time,REML=FALSE) modelc <- lmer(log(Migration)~(1+Time|Country_number) + Time,REML=FALSE)
You can see that the only difference is that in model
c the dependent variable has been transformed using the natural logarithm.
I then calculated the AICs of both models using the steps outlined HERE
With the following r script:
AIC(modela)  1050033 AIC(modelc)+2*sum(log(data$Migration),na.rm=TRUE)  566092.1
You can see that the difference in AICs is $1,050,033 - 566,092 = 483,940.9$. I have heard it say that a difference that is >10 generally means you have a better model fit. However, this large of a difference seems rather excessive.
Did I mistake somewhere, or is this is a normal occurrence?
Thanks in advance.