Graphical display of data and models
The relationship between two numerical variables, Y and X, can be effectively displayed in a 2-dimensional scatterplot and the least squares line can be superimposed on it.
For each additional variable, an extra dimension would be needed to produce a corresponding display. This is clearly impossible for three or more explanatory variables, but a rotating 3-dimensional scatterplot can be drawn if there are only two explanatory variables, X and Z.
Linear models with three or more explanatory variables cannot be displayed graphically
In the remaining pages of this section, we will use 3-dimensional diagrams to illustrate least squares with two explanatory variables. The ideas extend algerbraically (though not graphically) to models with more explanatory variables.
Notation
We again use the generic name Y to denote the response when we are describing methods in general (though more meaningful names like 'Body fat' are used when discussing particular data sets). Similarly, we give the two explanatory variables the generic names X and Z.
Body fat
Of the 13 potential explanatory variables that could be used to predict body fat, abdomen circumference has the highest correlation with body fat. The diagram below initially shows a scatterplot of these two variables.
This display is actually 3-dimensional and the 3rd dimension (which initially points towards you and therefore cannot be seen) represents the heights of the men. Position the mouse in the centre of the display and drag towards the bottom right of the screen. This rotates the display and should give you a reasonable impression of the 3-dimensional nature of this scatterplot. The buttons under the display can also be used for rotation.
Click on crosses to see how the three measurements from each man relate to the position of its cross on the three axes.