# Extremely large difference in AICs between two models

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)

[1] 1050033

AIC(modelc)+2*sum(log(data\$Migration),na.rm=TRUE)

[1] 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?

So even if the two models are equally complex in terms of number of parameters, the AIC values will completely different since one response is Migration and the other log(Migration).