No-interaction model for two factors

The no-interaction model for the effect of two factors X and Z on a response can be written in the form



yijk  =  µ


 + 
differences
between X-levels

βi


 + 
differences
between Z-levels

γj


 + 


εijk

If either factor has no effect on the response, the model is simplified by omitting the factor's term from the model.

There is therefore a network of meaningful models,

Neither X nor Z affects Y
yijk  =  µ   +  εijk
 
Only X affects Y
yijk  =  µ   +   βi  +   εijk
  Only Z affects Y
yijk  =  µ   +  γj   +   εijk
 
Both X and Z affect Y
yijk  =  µ   +   βi   +   γj   +   εijk

Each time the flexibility (i.e. freedom) of the model is increased by adding extra parameters, it is possible to get the fitted values closer to the actual response values — i.e. the residuals can be reduced. When extra parameters are added to the model and estimated by least squares, the residual sum of squares therefore decreases.

Warping of copper plates

The diagram below allows the different models to be fitted to the warping data from the previous page.

Initially the model used has no explanatory factors, so its residual sum of squares is high when fitted by least squares. (The fitted values are the overall mean reponse from all observations.)

Click the checkboxes to add the two factors to the model. Observe that:

Whenever a factor is added to the model, the residual sum of squares decreases.