Figure 4.

Geometric proof why residuals and dependent variable are positively correlated in ordinary-least-squares linear regression, arguing against an approach that derives BM as the residual of predicting age from brain-structural independent variables. The dependent variable Y, the model estimate Ŷ, and the residuals ɛ are linearly dependent and form a triangle in R^N. Since Ŷ and ɛ are orthogonal, the subtending angle θ between ɛ and Y has to be smaller than 90°, implying that the correlation between ɛ and Y is positive.