This could have been part of the "What does multiple linear regression look like?" note. However, I didn't want it to be seen as a footnote to the pretty pictures. This is the more important lesson.
A simple linear regression line is straight because we fit a straight line to the data! We could fit something other than a straight line if we want to. For example, instead of fitting
When homocysteine was regressed on CLC-folate and vitamin B12, why was the regression surface flat? The answer here, too, is because we fit a flat surface!
Let's take a closer look at the regession equation
If you draw the regression lines for various values of LCLC in the scatterplot of LHCY against LB12, you get a series of parallel lines, that is, you get the regression plane viewed by sighting down the LCLC axis.
The same argument applies to the regression surface for fixed LB12.
The first important lesson to be learned is that the shape of the
regression surfaces and the properties of the regression equation follow
from the model we choose to fit to the data. The second is that
we are responsible for the models we fit. We are obliged to
understand the interpretation and consequences of the models we fit. It
we don't believe a particular type of model will adequately describe a
dataset, we shouldn't be fitting that model! The responsibility is not
with the statistical software. It is with the analyst.