Can AIC/BIC be used to compare models that differ by transformation of response variable? [duplicate]

Question

I fit two models: one with a log-transformed response variable, and one without. All else is the same between them. Can I use AIC (or BIC or other, similar criteria) to choose which approach best fits the data?

Answer 1

No it should not be used.

The Akaike Information Criterion (AIC) and the Bayes Information Criterion (BIC) are derived from the log-likelihood $\cal{L}$ of the model:

$$\text{AIC}= -2\cal{L}+ 2p$$ $$\text{BIC}= -2\cal{L} + p\log(n)$$

where p is number of predictors in model (degrees of freedom), n the number of observations. The -2 loglikelihood (also known as the deviance) term above is normally dependent on the dependent variable

For example in linear regression models, it is $n\log(\text{RSS}/n)$, where $\text{RSS}$ is the sum squared of residuals:

$$\text{RSS}=\sum_{i=1}^n (y_i-\hat{y_i})^2$$

We can look at an example, in R, where below, both logged response and original scale give, i would say, equally good fit, but differs in BIC. If one uses BIC or AIC it will be misleading

fit = lm(carb ~.,data=mtcars)
fit_log = lm(log(carb) ~.,data=mtcars)

BIC(fit)
[1] 95.80717
BIC(fit_log)
[1] 30.53078

par(mfrow=c(1,2))
plot(fitted(fit),mtcars$carb)
mtext(paste("cor=",round(cor(fitted(fit),mtcars$carb),digits=3)))
plot(fitted(fit_log),log(mtcars$carb))
mtext(paste("cor=",round(cor(fitted(fit_log),log(mtcars$carb)),digits=3)))