Adding terms or imposing constraints

As with our models involving constraints in the previous section, the linear and quadratic models are intermediate between the general model with a parameter for each factor level and the most basic model in which the factor does not affect the response at all.

The linear and quadratic models can be considered as constraints on the parameters of the most general model (ensuring that they must change linearly with xi) and also corresponds to adding a term (and parameter) to the most basic model.

Hierarchy of models

The diagram below shows how the linear and quadratic models fit between the simplest model where the factor has no influence on the response and the most general model that allows any values for the mean responses at the factor levels, however unsmooth. Click the buttons Show linear and Show quadratic to insert the linear and quadratic models in the hierarchy.

Factor does not affect Y
yij  =  µ   +   εij
add term βi
Constrain all βi  = 0
Factor affects Y
yij
 =  µ   +   βi  +   εij