Test for interaction
We parameterised the model with interaction as a GLM based on the no-interaction model with (nX - 1)(nZ - 1) extra terms involving indicator variables.
This parameterisation makes it easy to test for whether X and Z interact — we simply test whether the (nX - 1)(nZ - 1) interaction parameters are zero.
Exercise and smoking
The scatterplot below shows the best-fitting (least squares) model with interaction for the data.
The analysis of variance table provides a test for whether there is interaction. The interaction row in the table tests whether the (3 - 1)(3 - 1) = 4 interaction parameters are zero.
We conclude that there is moderately strong evidence of an interaction between the effects of smoking and exercise type on time to maximum O2.
If we conclude that there is interaction, the main effects of exercise and smoking should neither be interpreted nor removed from the model, so they are greyed out in the table. (If you click the checkbox to remove the interaction, then the main effects can be examined.)
Smoking and exercise are orthogonal due to the design of the experiment — there were 3 individuals for each smoking-exercise combination. As a result, the order of adding the main effects does not affect their sequential sums of squares and there is only a single meaningful anova table (since the interaction must be added after the two main effects).
For observational data in which X and Z are not orthogonal, there would be two possible anova tables, corresponding to the two orders of adding the two main effects for X and Z. For such data, a table of Type 3 sums of squares may be preferred. However the test for interaction would be the same for the two approaches.