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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. In R:

Model 1a: glm(Addicted ~ Index_normalized, data = df, family = "binomial")

Model 1b: glm(Addicted ~ Index, data = df, family = "binomial")

versus

Model 2: glm(Addicted ~ log10(Literature_var), data = df, family = "binomial")

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

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  • $\begingroup$ I assume you mean (xi - min(x)) / (max(x) - min(x)) ? And what is your model exactly? By "Addicted ~ Index" do you mean "logit-probability for addicted = intercept + regression coefficient * index" or do you mean "logit-probability for addicted = regression coefficient * index"? If it's the latter, then that would explain it. $\endgroup$
    – Björn
    Commented Jan 30 at 13:01
  • $\begingroup$ I corrected the normalization formula, thank you. Regarding the model, I added the R code, I am using a binomial GLM whose output for each model is the intercept + regression coefficient. So, for prediction of addiction risk, for example, I would calculate log-odds with the formula intercept + regression coefficient * Index or Index_normalized or log10(Literature_var), respectively. $\endgroup$
    – M.S.
    Commented Jan 30 at 13:09

1 Answer 1

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You say you know for sure that log10(Literature_var) is a good index. Therefore I think I would start with checking whether you the two forms of the index you calculated are correlated with it.

Then, in order to compare the performance of the 3 potential indexes (literature, index & normalized index) I believe what you want is the amount of variance in "Addicted" that is explained by each of the 3 models you have built. This can be done by comparing the likelihood ratio between each of the 3 models to the null model Model 0: glm(Addicted ~ 1, data = df, family = "binomial") Then you could use lrtest with an R command which would look a bit like : lrtest(Model 0,Model 1a) then lrtest(Model 0,Model 1b) and finally lrtest(Model 0,Model 2)

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  • $\begingroup$ Actually, I am not interested in exploring the correlation with Log10(Literature_var), I am interested in finding statistical evidence for my Index_normalized, where for evidence I mean the AIC of the corresponding model is lower than that of the model with Log10(Literature_var). I obtained this but the problem is that if I compare Log10(Literature_var) with the raw values of the index I get the opposite (independently from the correlation). So I am not sure if when I am affirming that my Index_normalized is a good predictor I am mistaken because I should compare the raw values instead. $\endgroup$
    – M.S.
    Commented Jan 30 at 14:36
  • $\begingroup$ Still, you should know whether they are at least correlated. $\endgroup$
    – CaroZ
    Commented Jan 30 at 14:39
  • $\begingroup$ Do you mean that I could use the strength of the correlation between Log10(Literature_var) and Index_normalized as a point to statistically validate my Index_normalized? Thank you for your input $\endgroup$
    – M.S.
    Commented Jan 30 at 14:47

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