I want to include two predictors (total brain volume and corrected gray matter volume) into one regression model in order to predict the level of cognition (dependent variable). However, this corrected gray matter volume is calculated as:

gray matter volume / total brain volume = corrected gray matter volume

making the regression model look like this:

Total brain volume + (gray matter volume / total brain volume) = cognition

Is this statistically sounds or are there issues with total brain volume essentially featuring twice in one model?

  • $\begingroup$ How to resolve this depends primarily on how well the two volumes are measured. Are the measurements accurate enough to ignore any measurement error, or would measurement error potentially be a concern? $\endgroup$ – whuber Jul 14 '15 at 14:57
  • $\begingroup$ Measurement error is a concern in this case, how does that affect this problem? $\endgroup$ – user48054 Jul 16 '15 at 6:25

Your suggested model for cognition score implies that the influence of gray-matter volume on cognition depends on total brain volume. Another way to investigate this is to do a simple linear regression of cognition on total brain volume and gray-matter volume but include an interaction term between those two predictors. In R this could be coded as:

cognition ~ totalBrainVolume + grayMatterVolume + totalBrainVolume:grayMatterVolume


cognition ~ totalBrainVolume * grayMatterVolume

don't know the SPSS syntax. This approach has the advantage of documenting how much the correction of gray matter volume for total brain volume, implicit in your model, matters.

  • $\begingroup$ Thank you for this comment, I have done this analysis and it apears that the interaction term is marginally signifiant, How to proceed? $\endgroup$ – user48054 Jul 16 '15 at 6:27
  • $\begingroup$ You could just report what you found. Try exploring the nature of the interaction, with plots of cognition against the predictors and of the residuals after the regression. You might find that a log transform of the volumes works better and removes the interaction term; then you could simply report two different coefficients for log(grayVolume) and log(totalVolume). If there precedent in your field for the model as you presented it, you may have to report your results that way to get published. If you can, however, try to report as simple a model as possible. $\endgroup$ – EdM Jul 16 '15 at 15:20

I think you may have collinearity problem if the gray matter volum and total brain volum are proportional. If two volumes are not very proportional, then I think it might be OK. The followings are some simulations.

y<-rnorm(100,20,2) #cognition
x1<-rnorm(100,10,1) #total brain volume
x2<-rnorm(100,2,1)  #gray matter volume
x3<-x1/x2 #gray mattervolume/total brain volume
x4<-x1/2 #gray matter volume and total volum are proporitonal
fit1 <- lm(y ~ x1 + x3)
fit2 <- lm(y ~ x1 + x4)

enter image description here

  • $\begingroup$ Thank you for your swift response. If I understand correctly, would it be OK to do this analysis when there is no indication of multicolliniearity in PRESxRESID plots and/or in multicolliniearity diagnostics (tolerance, variance inflation factor). I'm using SPSS by the way $\endgroup$ – user48054 Jul 14 '15 at 14:11
  • $\begingroup$ I think when you do a multiple linear regression, you study a predictor variable at one time and hold all other predictor variables constant (just like how you evaluate the multiple integration). For your case, you can say, when you fix the brain volume, how your gray matter proportion affect the cognition. I.e you have a guy has brain volume of 10, and 50%(5 volume) gray matter, and another guy also has brain volume 10, but with 40% (4 volume) gray matter. Then your question is wether 50% guy is more smart (more cognitive ) than 40% guy. $\endgroup$ – Deep North Jul 15 '15 at 0:14
  • $\begingroup$ Also you may consider EdM's suggestions to use gray matter volume directly add an interaction term. $\endgroup$ – Deep North Jul 15 '15 at 0:14

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