Graphical display of data with three variables
The relationship between two numerical variables, Y and X, can be effectively displayed in a 2-dimensional scatterplot. 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 3-dimensional scatterplot can be produced for data sets with a response and two explanatory variables.
A reasonable impression of such a 3-dimensional scatterplot can be obtained on a computer screen by showing it rotating.
Notation
In this chapter, we call the response variable Y when we are describing methods in general (but we use more meaningful names like 'Body fat' when discussing particular data sets). Similarly, the two explanatory variables will be denoted by X and Z.
By convention, the response variable is drawn on the vertical axis of the scatterplot.
Body fat (observational data)
We saw on the previous page that abdomen circumference had the highest correlation with body fat of the 13 measurements that were made from each of the 252 men. 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.
Energy expenditure of bees (experimental data)
In an experiment, an entomologist recorded energy expenditure (joules/sec) for bees drinking water with different sucrose concentrations (%) and at different temperatures. Energy expenditure is the response measurement.
Note that the data are experimental so the values of the two explanatory variables have been fixed by the entomologist. There were 3 replications of the experiment — 3 measurements were made for each combination of levels of sucrose and temperature. Again rotate the diagram to help understand how energy expenditure is related to sucrose concentration and temperature.