Model for interaction
A linear model for two numerical factors, X and Z, can be written as:
yijk = µ + |
(explained by X) β xi |
+ |
(explained by Z) γ zj |
+ |
(unexplained) εijk |
The simplest way to model an interaction between the effect of these two factors is to add a term involving the product of X and Z,
yijk = µ + |
(explained by X) β xi |
+ |
(explained by Z) γ zj |
+ |
(interaction) δ xizj |
+ |
(unexplained) εijk |
To help understand this model, it can be rewritten in the form,
yijk = µ + (β + δzj) xi + γ zj + εijk
In this form, it can be seen that the 'slope' for X depends on Z — the effect of changing x is different depending on the value of z. Similarly, writing the model as:
yijk = µ + β xi + (γ + δxj) zj + εijk
it can be seen that the effect of changing z depends on the value of x.
Catalyst surface area
Researchers conducted an experiment to investigate how the mole contents of cobalt and the calcination temperature affected the surface area of an ison-cobalt hydroxide catalyst. Four cobalt levels and five temperatures were used in the experiment and one sample of catalyst was created and tested at each of the 20 combinations of a cobalt level and temperature. The table below shows the results of the experiment.
Temperature (°F) | |||||
---|---|---|---|---|---|
Cobalt (mole) | 200 | 300 | 400 | 500 | 600 |
0.6 | 90.6 | 82.7 | 58.7 | 43.2 | 25.0 |
1.0 | 127.1 | 112.3 | 19.6 | 17.8 | 9.1 |
2.6 | 53.1 | 52.0 | 43.4 | 42.4 | 31.6 |
3.0 | 40.9 | 37.9 | 27.5 | 27.3 | 19.0 |
Note that the cobalt levels are not evenly spaced. This does not affect the analysis.
The diagram below shows a no-interaction model with three red arrows that allow the three parameters to be adjusted. Click Least squares to see the best-fitting model.
Click the checkbox Interaction and observe that a fourth green arrow appears in the diagram corresponding to the interaction parameter δ. Drag the arrows to get a feel for the types of relationship that can be modelled.
Click Least squares to fit the model to the data. Click y-x to rotate the model. Observe that the effect of increasing temperature is much greater when the cobalt level is low than when it is high.
Analysis of variance
In the analysis of variance table above, drag the red arrow down to display the sums of squares explained by the main effects of the factors and their interaction. All three explained sums of squares have 1 degree of freedom since they each involve one parameter.
From the p-value associated with the interaction, we would conclude that there is very strong evidence that the cobalt content and temperature interact in their effects on the catalyst surface area.