Models with one two-factor interaction

The main effects model on the previous page assumes that the effect of changing any factor is the same whatever the values of the other factors. In many practical experiment, the effects of the factors are not additive — there is interaction between their effects.

The simplest types of interaction are two-factor interactions. A two-factor interaction between B and C means that the effect of changing B can depend on the value of C but not on D.


yijkl  =  µ

 + 
(explained by B)
βi

 + 
(explained by C)
γj

 + 
(explained by D)
δk
 
 + 
(B-C interaction)
θij

 + 
(unexplained)
εijkl

In this model, there is no interaction between factor D and the other factors so the effect of changing D is still the same whatever the values of B and C.

Models with several two-factor interactions

This model can be extended by adding two-factor interactions between other explanatory variables. Models with a single interaction term can be understood relatively easily but:

It is slightly harder to interpret models with two or three interaction terms.

Three-factor interaction

The most complex model for three factors contains all three 2-factor interactions and also a 3-factor interaction term between all three factors,


yijkl  =  µ

 +     ...     +  
(B-C-D interaction)
θijk

 + 
(unexplained)
εijkl

This model is the most flexible possible model for three factors — it places no restrictions on the mean response for the different combinations of factor levels (treatments) so the least squares estimate for each treatment is the mean observed response at that treatment.

Hierarchical models

A model with a 2-factor interaction between factors B and C implies that the effect of B may depend on the value of C. It therefore implies that we are allowing B to have an effect on the mean response so a main effect for B should also be included in the model.

If a model has a B-C interaction term, it should also include main effect terms for B and C.

Generalising to also include the 3-factor interaction term,

If a model has any interaction term, all main effects for these factors and all lower-level interaction terms between them should also be included in the model.

Models of this form are called hierarchical models.

Water uptake by toads and frogs

The diagram below illustrates the flexibility of hierarchical models for the water uptake experiment.

The diagram initially shows a model with all main effects but no interactions. There are four red arrows corresponding to the four non-zero parameters of this model, but two are initially superimposed, so start by dragging one arrow for the wet-toad combination to separate them. Drag the four arrows to investigate the flexibility of this model.

Click the checkbox to allow an interaction between Animal and Moisture. With this interaction term, we allow the effect of changing the pre-experiment moisture to be different for frogs and toads. One additional draggable arrows appears, corresponding to this model's extra flexibility. Again drag the arrows to investigate.

Click the y-x rotation button. Since there is no interaction between Injection and the other variables, the pattern of interaction between Animal and Moisture is the same for animals getting no hormone injection (purple lines) and those getting the hormone (green lines).

Other interactions

Add other 2-factor interaction terms to the model and observe the increasing flexibility.

Note that the diagram only allows you to specify hierarchical models.

Finally add all 2-factor interactions and the 3-factor interaction. This model is the most flexible possible and allows the mean Water uptake for all 8 treatments to be independently adjusted.