I have developed a index of drug addiction risk 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 log10(`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 log10(`Literature_var`) and my index are the independent continuous variables in distinct single-term models. In particular, I compared log10(`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 ~ log10(Literature_var)

I expected the same performance but obtained opposite results, with `Index_normalized` being better than log10(`Literature_var`) (much lower AIC), and `Index` being worse (much 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`?

Or maybe should I apply the same normalization formula to the other variable log10(`Literature_var`)?

Thank you