Multicollinearity when controlling for a variable

Question

I have a few questions about multicollinearity in my data: I'm looking at a certain type of lesion seen on MRI scans; for each patient I know the volume of those lesions and a metric that captures the pattern of their distribution. I'm interested in the effect of the pattern metric on a clinical score, above and beyond the effect of the volume. The pattern metric is highly correlated with the volume (r = 0.7) though, which is in the nature of things because the more area the lesions cover in the brain, the more they are part of one cohesive area as opposed to being separate blobs. This made me wonder a few things:

1. Is there an issue of multicollinearity in a regression like clinical score ~ pattern + volume? Mitigating such an issue seems to defeat the purpose of what I am trying to see.
2. The pattern metric has been corrected for volume, i.e. each person's metric has been divided by that person's lesion volume. Do I still need to take volume into account in my regression if it is in a way already incorporated?
3. If I want to do a correlation instead of a regression, is there a difference between a) a partial correlation of clinical score and pattern metric after accounting for volume, and b) a correlation of clinical score and residualized pattern metric (after regressing the pattern metric on volume)?
4. When I calculated the correlation of clinical score and residualized pattern metric, the correlation was higher than before residualizing, is that a normal observation? I noticed that the correlation of clinical score and volume is negative while the one of clinical score and pattern is positive, I was not sure if that was a sign that something is wrong given that the correlation of pattern and volume is positive and large. Maybe this is exactly why the correlation with clinical score is higher after residualizing? Is there an analytical way of digging deeper into this?

Thank you in advance!

Answer 1

By completely arbitrary convention, you don't have a 'problem' with multicollinearity until the ${\rm VIF}\ge10$. With just two $X$ variables, that would require the pairwise correlation to be $r\ge0.95$. Thus, $r=0.7$ would not be multicollinearity. In fact, your ${\rm VIF}$ should be about $2$. I don't really see a problem here. Remember that: $$ {\rm VIF} = \frac{1}{1-r^2} $$ Moreover, what multicollinearity does is basically just increase the uncertainty of your estimates. That is, the standard errors / confidence intervals are wider, and you have less power and a higher p-value. If your result is currently significant (I can't quite tell from your write up), then there is no issue; if it's not significant, more power (perhaps rerunning the study with a larger dataset or controlling for another source of variability in the clinical score) might yield a clearer result.

A couple other thoughts: a correlation is a regression model presented differently, so this is a red herring. Using a residualized pattern vs unresidualized pattern amounts to attempting to answer a different question (albeit questions that will look very similar to most people). You need to decide which question you want to answer. Another possibility is to model an interaction between pattern and volume and see how pattern matters as volume changes—again, a slightly different question.

Answer 2

This is more of an extended comment than an answer.

As @gung - Reinstate Monica noted (+1) this should be handled with regression. There won't be multicollinearity of any magnitude in what you propose, and the downsides of multicollinearity are often overstated even when it's present.

What you need to address, based on your understanding of the subject matter, is just what your pattern represents, particularly if it's been "corrected for volume," and how you expect the combination of pattern and volume to be associated with the clinical_score outcome.

I suspect that the "correction" of pattern was something like a simple division by volume. The regression you propose also assumes that: (a) the association of each of pattern and volume with outcome is strictly linear, and (b) the association of either with outcome doesn't depend on the value of the other. Neither assumption seems very likely.

It looks like pattern and volume are both continuous measures. In general, it's a good idea to model such predictors flexibly, as with regression splines. If the association of either with outcome might depend on the value of the other, then you should include an interaction between them in the model to allow for that. Frank Harrell's Regression Modeling Strategies is a useful resource on this type of problem, in particular Chapters 2 and 4.

If you omit thar interaction then your estimate of the association of pattern with outcome, "above and beyond" the association of volume with outcome, is likely to be mis-specified even if that concept is well defined. I suspect that you might be better off not correcting pattern for volume, but without more information about what the uncorrected pattern represents it's hard to know.