I have developed a index of drug addiction risk (Index) whose formula is Index = 1/log10(a_given_variable). The raw values of the calculated Index range from -4 to 0. Since I wanted the Index to be expressed on a scale of 1 (lowest risk) to 10 (highest risk), I normalized the Index with the following min-max formula:

Index_normalized = xi - min(x) / max(x) - min(x)

Now I want to know if my index is a good predictor of drug addiction risk compared to Literature_var, which is a good predictor know from literature. To this end, I want to use logistic regression models, using a dataset where Addicted was the dependent variable (0 = non-addicted and 1 = addicted) and Literature_var and my index are the independent continuous variables in distinct single-term models. In particular, I compared Literature_var with both the raw and normalized version of my index, i.e. with both Index and Index_normalized, using the AIC to identify the most parsimonious model.

Model 1a: Addicted ~ Index_normalized

Model 1b: Addicted ~ Index

versus

Model 2: Addicted ~ Literature_var

I expected the same performance but obtained opposite results, with Index_normalized being better than Literature_var (lower AIC), and Index being a worse (higher AIC).

My doubt is about which of the two proposed models (Model 1a or Model 1b) is the correct one. That is, should I compare Literature_var with the raw values of Index, or with the normalized values of the desired 1-to-10 Index_normalized?

Thank you