Model for interaction

When a numerical variable, X, is modelled linearly and Z is categorical, the no-interaction model specifies parallel regression lines for the response against X at each level of Z,


yijk =  µ + 
(explained by X)
β xi

 + 
(explained by Z)
γj

 + 
(unexplained)
εijk

Interaction gives the model the additional flexibility of different regression slopes at different levels of Z,


yijk =  µ + 
(explained by X)
β xi

 + 
(explained by Z)
γj

 + 
(interaction)
δj xi

 + 
(unexplained)
εijk

Degrees of freedom

If factor Z has b levels, the two models have the following degrees of freedom.

  Degrees of freedom
No-interaction model b + 1
Model with interaction 2b

The difference gives the extra degrees of freedom for adding the interaction term.

  Degrees of freedom
Extra for interaction b - 1

Abrasion of coated fabric

A method for testing materials for abrasion resistance consists of measuring the loss in weight or decrease in thickness of specimens that are rubbed against an abrasive for a fixed time. The diagram below shows the results of testing 12 specimens of coated fabric that contained three different proportions of two types of filler.

The no-interaction model constrains the regression lines for the two fillers to be parallel. Click Least squares to display the best model of this type.

Click the checkbox Interaction to add an interaction to the model. This adds an extra draggable arrow to the diagram — interaction adds one degree of freedom to the model. Observe that the two lines for the filler types can be separately positioned.

Click Least squares then click the y-z rotation button. Increasing the percentage of filler F2 causes much more abrasion loss than a similar increase in the percentage of filler F1.

Testing for interaction

Interaction can again be tested using analysis of variance. The sum of squares explained by the interaction is the reduction in the residual sum of squares that arises from adding the interaction term.

The anova test compares the sum of squares explained by the interaction with the residual sum of squares in the usual way.