# How to compare the effect of age on brain volume across different brain regions when the regions are of different sizes?

I am working on analyzing volumetric information extracted from the brain. In particular we would like to compare the strength of effects across three different regions (say x, y, and z). However, the three regions grossly differ in size (say y is 10 times larger than x, and z is 20 times larger than x). We would like to know if these three regions decline differentially with advancing age.

The differences in scale seem to introduce significant regional interactions by themselves. If I take the data from x and simply generate a fake y and z (by scaling x by 10 and 20, call them $y'$ and $z'$) I get significant interactions even though the underlying relationships are identical. This is clear when you look at B weights if you compute separate models for x, $y'$, and $z'$.

This does not appear to be an issue commonly addressed in studies that examine brain volume. Currently we are using partial correlations and then Steiger's Z to compare the effects of age across regions.

• Are there suggestions for dealing with such a problem with a more elegant approach?
• How can we compare the effect of age on brain volume across different brain regions when the regions are of different sizes?
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Your second paragraph isn't clear to me at all. Differences in scale cannot (in any way that I see) generate interactions by themselves. An interaction means that the effect of one independent variable is different at different levels of another. <br> Further, multiplying data by a constant will change the parameter values for a regression (or other method) but will not change the meaning. Measuring in millimeters or inches yields the same meaning. – Peter Flom Feb 27 '13 at 0:28
Why not standardize the data first so that scaling is identical for all three measures? – ReliableResearch Feb 27 '13 at 15:06
It seems like what the user is calling 'interactions' are differences in effect size across regions x, y, and z because they are looking at the raw regression coefficient. – rpierce Feb 27 '13 at 16:08
... but color me confused because "repeated measures ANOVA" is in the title. – rpierce Feb 27 '13 at 16:16
Maybe this suggestion is very stupid, but what if you use dummies to represent each of the three regions? – Herman Haugland Jul 28 '13 at 10:43

Rather than looking at (and comparing) your raw regression coefficients, you can calculate standardized regression coefficients ($\beta$s) by Z scoring your IVs and your DVs. These estimates will be on the same scale. Unfortunately, comparing them quantitatively might not be straightforward. Last time I checked, how to appropriately calculate the standard error of $\beta$ was an issue of some debate. So, your partial correlation approach (correlation of x and DV after partial-ing out correlation between y and DV and Z and DV) seems quite reasonable to me.