Reference for standardizing non-normal variables for comparison

Question

I need a reference to justify standardizing non-normal data. I will first explain what I am doing.

I had a survey with ~30 questions. The answers to the questions were numeric; the range of possible answers to these questions all had a minimum of 0 but the maximum was not all the same (for example, the range for some questions was 0-30, for others it was 0-4). I then divided the questions into two separate categories to create 2 scales. The questions were not in equal groupings; one scale has ~20 questions, the other has ~10. For each individual respondent I summed the answers to each question to create a score for each scale. Obviously the maximum possible score was much greater for the scale with ~20 questions than that of the scale with ~10 questions. The distribution of the scores for each scale has a heavy right skew.

I am using these scale in logistic regression. Each scale is being used in a separate model. The scale is the predictor variable of interest and is the only continuous variable, I have several control variables that are all either binary or categorical "dummy" variables and are the same in each model. Both models have the same dependent variable, in other words, both models are being used to predict the odds of the same event occurring.

I have chosen to standardize these scales in the usual way ((x-mean)/sd) so that when I draw conclusions about the models, I can say something along the lines of "for a 1 unit change in scale A, holding all other variables constant, there was a such and such change in odds of event Y, whereas for a 1 unit change in scale B, again holding all other variables constant, there was a such and such change in odds of event Y."

I have read conflicting reports on whether it is acceptable (by acceptable I mean that when I submit my paper for publication, the reviewers will accept my methods as legitimate) to standardize non-normally distributed variables. What I have been scouring the search engines, databases, and online texts for is a solid reference (could be a book or a published paper) that supports my decision of standardizing a non-normal variable. I have not had much luck and would very appreciative if anyone has suggestions for articles or texts that would help me achieve my goal.

Answer 1

There is one practical difficulty in standardizing your data in terms of having others be able to reproduce your results. If others used your same questions but had a different distribution of responses (different means/SDs) then their "scales" after standardization would be different from yours, even with the same questions. There is no need to standardize predictors for this analysis, and based on that difficulty I would argue against standardization.

As you describe it, it seems that you have tried to develop something like a Likert scale based on the sums of responses to multiple questions designed to assess the same underlying phenomenon. See this page for an introduction to ways to use Likert scales; they aren't always best treated as (quasi)-continuous variables in the way that you propose. You should look carefully at the literature on Likert scales to see if your approach, with different individual questions having different response values, really makes sense. For logistic regression, the most important issue is whether the log odds of the response variable is linearly related to the predictor variables. That has to be your main priority, and you might need some other transformations of your predictor variables to meet that goal.

Finally, your question suggests that you are trying to compare scale A and scale B. That raises a set of additional issues beyond the one you note about the different ranges of scale A and scale B. If comparing scales A and B is the main goal of your study, you might consider asking a new question about how best to compare different rating scales in this type of model. If you do ask such a question, please provide a bit more background about what scales A and B are designed to assess and what you are trying to accomplish with the comparison.