# Comparing AIC of a model and its log-transformed version

The essence of my question is this:

Let $Y \in \mathbb{R}^n$ be a multivariate normal random variable with mean $\mu$ and covariance matrix $\Sigma$. Let $Z := \log(Y)$, i.e. $Z_i = \log(Y_i), i \in \{1,\ldots,n\}$. How do I compare the AIC of a model fit to observed realizations of $Y$ versus a model fit to observed realizations of $Z$?

My initial and slightly longer question:

Let $Y \sim \mathcal{N}(\mu,\Sigma)$ be a multivariate normal random variable. If I want to compare a model fit to $Y$ versus a model fit to $\log(Y)$, I could look at their log-likelihoods. However, since these models aren't nested, I can't compare the log-likelihoods (and stuff like AIC, etc.) directly, but I have to transform them.

I know that if $X_1,\ldots,X_n$ are random variables with joint pdf $g(x_1,\ldots,x_n)$ and if $Y_i = t_i(X_1,\ldots,X_n)$ for one-to-one transformations $t_i$ and $i \in \{1,\ldots,n\}$, then the pdf of $Y_1,\ldots,Y_n$ is given by $$f(y_1,\ldots,y_n)=g(t_1^{-1}(y),\ldots,t_n^{-1}(y))\det(J)$$ where $J$ is the Jacobian associated with the transformation.

Do I simply have to use the transformation rule to compare

$$l(Y) = \log(\prod_{i=1}^{n}\phi(y_i;\mu,\Sigma))$$ to $$l(\log(Y))=\log(\prod_{i=1}^{n}\phi(\log(y_i);\mu,\Sigma))$$

or is there something else I can do?

 Forgot to place logarithms in the last two expressions.

• You also seem to have lost the Jacobian in the last expression. – whuber Jun 10 '13 at 16:21
• I don't understand the $\log$ transformation. How can you take $\log Y$ when $Y$ is negative? – semibruin Jul 29 '13 at 22:19

You cannot compare the AIC or BIC when fitting to two different data sets i.e. $Y$ and $Z$. You only can compare two models based on AIC or BIC just when fitting to the same data set. Have a look at Model Selection and Multi-model Inference: A Practical Information-theoretic Approach (Burnham and Anderson, 2004). They mentioned my answer on page 81 (section 2.11.3 Transformations of the Response Variable):
Akaike (1978, pg. 224) describes how the AIC can be adjusted in the presence of a transformed outcome variable to enable model comparison. He states: “the effect of transforming the variable is represented simply by the multiplication of the likelihood by the corresponding Jacobian to the AIC ... for the case of $log\{y(n)+1\}$, it is −2 $\cdot \sum log\{y(n)+1\}$, where the summation extends over $n = 1,2, ..., N$.”