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I have a problem that is giving me a headache. I have measured two angles that I believe to be related to one another, and they are (this is a data set where I have measured the angle from a datum to two features on a bone. There are 14 bones in the data set):

Angles to feature 1 (F1):

15.225

14.2318

9.4301

12.2947

14.8846

7.6533

9.0948

11.9725

4.2773

14.1819

8.841

17.1037

20.2373

13.4599

Angles to feature 2 (F2):

3.1227

9.4799

7.9047

13.4962

8.5454

24.2871

11.443

12.6693

21.5271

4.0733

5.0085

4.0101

5.4445

16.424

When I plot F1 vs F2 and do a linear correlation I get an r^2 = 0.47. Related, but not very strongly.

The thing is, I'm doing this because I have a bunch partial bone specimens and cannot define the datum, so in general I'm going to have the angle between feature 1 and feature 2, and am hoping to be able to get the position of the datum from this angle. So if I plot (F1-F2) vs F1 I get a much better correlation (r^2 = 0.74) and (F1-F2) vs F2 gets even better (r^2 = 0.9)!

What I don't understand is, how can a linear combination of the two be better than either one alone? I have added no information and the equations are not linearly independent. What am looking at in the plots with the difference?

I have attached plots of F1 vs F2, Diff vs F1 and Diff vs F2 with the regressions plotted.

Thanks for your help.

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# Regression of linear combination better than just regression

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