Using slices to display the model
A quadratic model for two factors corresponds to a surface in 3 dimensions. Extending the model to three factors requires one extra dimension — it corresponds to a surface in 4 dimensions. This cannot be directly displayed.
It is difficult to visualise a response surface for 3 factors and almost impossible for 4 or more.
However it is possible to represent such a model with a series of slices of this surface at different values of one of the factors. An example will make this clearer.
Wastewater treatment in a refinery
The experiment that was actually conducted was intended to be a central composite design, but the star points were erroneously set at ±1.414 instead of ±1.682. (The star points should be ±1.414 if there are only two factors, but more extreme star points are needed to make the design rotatable if there are three or more factors.)
The table below shows the design points and response measurements from the experiment.
Run | Aluminium sulphate |
Polyelectrolyte | pH | Turbidity reduction, Y |
|
---|---|---|---|---|---|
Factorial points |
1 2 3 4 5 6 7 8 |
91 91 91 91 19 19 19 19 |
25 25 5 5 25 25 5 5 |
8.5 5.5 8.5 5.5 8.5 5.5 8.5 5.5 |
95.5 68.9 11.6 56.8 69.7 55.8 25.6 17.6 |
Star points |
9 10 11 12 13 14 |
106 4 55 55 55 55 |
15 15 29 1 15 15 |
7 7 7 7 9 5 |
56.8 52.6 41.5 17.6 58.8 35.7 |
Centre points |
15 16 17 18 |
55 55 55 55 |
15 15 15 15 |
7 7 7 7 |
30.9 28.7 20.2 25.9 |
The resulting response surface has the following equation.
Y = 269.5 - 0.352 Al - 3.14 Po - 66.8 pH + 0.00998 Al2
+ 0.0040 Po2 + 4.58 pH2
+ 0.0048 Al.Po - 0.0938 Al.pH + 0.647 Po.pH
For any fixed value of one factor, this surface takes the form of a 2-factor response surface. For example, when pH takes the level pH = 0, the equation is
Y = 269.5 - 0.352 Al - 3.14 Po + 0.00998 Al2
+ 0.0040 Po2
+ 0.0048 Al.Po
This equation shows how the mean turbidity reduction, Y, depends on Aluminium sulphate and Polyelectrolyte when pH is 0, and is a response surface that could be displayed graphically in 3 dimensions.
Similar slices at other values of pH also correspond to quadratic response surfaces showing how Y depends on Aluminium sulphate and Polyelectrolyte conditional on these values of pH.
The diagram above shows how turbidity reduction, Y, is modelled to depend on Aluminium sulphate and Polyelectrolyte for different values of pH. Drag the slider to adjust the value of pH.
Click the x-z rotation button and again adjust the value of pH.
If pH is low, the greatest turbidity reduction is when Aluminium sulphate is high, but when pH is high, the greatest turbidity reduction is when Polyelectrolyte is high.
The least turbidity reduction arises when pH is around 7, Aluminium sulphate is around 50 and Polyelectrolyte is low.
Select Selected contours from the pop-up menu and again use the slider to adjust the value of Bicarbonate.
The response surface is often represented on paper by these contour diagrams for different slices of the response surface.