# Combining correlated predictors via binomial logistic regression

I'm new to the site and have tried to answer my question by reading old queries but it's a bit specific and I haven't been able to work it out - apologies for any unnecessary duplication if I've missed something.

I'm working on some analysis of a prospective clinical study where various predictive tests were performed with a view to predicting categorical outcomes of interest (birth before/after various gestations). Two of the predictive tests produce simple results, one produces complex spectral data. Previous analysts have taken the approach of summarising a portion of this data using logistic regression (i.e. values produced by measurements taken in a particular frequency range of interest). I think this is likely problematic due to the spectral measurements lack of independence (they are inevitably highly correlated). Am I correct in thinking this? The probabilities generated by the regression have then been used as input for ROC analysis to estimate accuracy of prediction by the measurements in combination and I'm concerned the original method to combine them was invalid. The references I've read have suggested that there are instances in which multicollinearity is less problematic but I'm not sure that applies here, and I'm wondering if I should look to use an alternative technique (?Principal component analysis/?partial least squares regression).

Apologies if this is an obvious question - i'm not a statistican/mathematician by background (clearly!)

• You might need to explain more about the spectral data--we are but lowly statisticians! This will tell us about your correlation structure, which will in turn provide your answer. Multi-level or mixed models are probably where this is headed, though. – user271536 Dec 10 '20 at 3:05
• Thank you for your reply! The spectral data measures the same tissue property at multiple different frequencies (on the same participant) (it’s more or less simultaneous in practice but I suppose could be viewed as an extremely short time series!) - so the frequency is the only varying factor if that makes sense? – Lou2016 Dec 10 '20 at 7:12
• If your response variable is by person, and frequencies $X_1$ $X_2$, $X_3$, etc. are covariates, you should be fine (unless you have clusters of people related to each other). But if your response is the measurements themselves, and so you have responses $y_1$, $y_2$, etc. coming from the same person, it's time to break out the generalized linear mixed models. In this case, a generalized linear mixed logistic regression. The question you want to ask yourself is: Do the rows of my dataset have relationships with each other? That's the type of independence we care about. – user271536 Dec 11 '20 at 0:07
• That is the more serious case. But you can have correlation, as you note, in the columns. This is easier to address, in my opinion. There are so many way. PCA, as you said, regularization methods are great too. But without a scientific knowledge of the subject matter, you are actually in the best position to say for sure. You can take a look at the data for yourself and explore multicollinearity. – user271536 Dec 11 '20 at 0:12
• Thanks again, that's really helpful. The response variable is indeed by person and no relationship between rows. I'm feeling slightly less alarmed! – Lou2016 Dec 11 '20 at 15:36