Model without interaction

The additive model for two factors, X and Z, is


yijk  =  µ

 + 
(explained by X)
βi

 + 
(explained by Z)
γj

 + 
(unexplained)
εijk

The β-parameters and γ-parameters are called the main effects for X and Z.

In this model, the effects of the two factors on the mean response can separately described by the two sets of β-parameters and γ-parameters. This can be done because:

Since the effects of the two factors can be separately described, it is said that the two factors do not interact in their effect on the mean response.

Music and Alzheimer's disease

Researchers conducted an experiment to examine the effect of various types of music on agitation levels of patients at different stages of Alzheimer's disease. Fifteen patients in the early stages and another fifteen in the middle stages of the disease were selected to participate in the study. Three forms of music were tested: Easy listening, Mozart, and piano interludes and these were randomly allocated to patients in each of the two groups. While listening to music, agitation levels were recorded for the patients with a high score indicating a higher level of agitation.

The diagram below shows the agitation levels of the patients and the best-fitting model with no interaction.

Rotate to y-z. The coloured lines show how mean agitation level differs for the three types of music, both for early (purple) and middle (green) stage Alzheimer's. Observe that the fitted lines are all parallel.

Similarly, rotate to y-x. Again the three fitted lines (corresponding to different types of music) are all parallel.