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.

Strength of asphaltic concrete

A civil engineer conducted an experiment to evaluate how different compaction methods and types of aggregate affect the strength of asphaltic concrete. Two types of aggegate and four levels of compaction were used in the experiment and three specimens were tested at each combination of levels for the two factors.

The diagram below shows the tensile strength of each specimen (psi) and the best-fitting model with no interaction between Aggregate and Compaction.

Rotate to y-z. Since there is no interaction, the difference between the mean strengths of specimens made with Basalt and Silicious aggregates is modelled to be the same for all compaction methods.

The fitted lines are all parallel.

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