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): Investigators should be sure that all hypotheses are modeled using the same response variable (e.g., if the whole set of models were based on log(y), no problem would be created; it is the mixing of response variables that is incorrect). 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 ⋅∑log{𝑦(𝑛)+1},