Linear, quadratic and categorical terms

We have already met two ways to model the effect of the i'th level of a numerical factor, X, on the response:

Linear term
This has the form β xi where xi is the numerical value of the factor and is only appropriate if the effect of the factor on the response is linear. This term has only one unknown parameter — the slope of the relationship.
Categorical term
This has the form βi and involves (a - 1) unknown parameters that allow any form of curvature in the relationship.

Intermediate between these is a quadratic relationship that imposes a degree of smoothness to the relationship but still allows some degree of curvature.

Quadratic term
This has the form β1 xi + β2 xi2.

Range of models for two numerical factors

If both factors in an experiment are numerical, we have a choice of linear, quadratic or categorical terms for each. Hypothesis test must be used to find the simplest model that is consistent with the data.

For example, the following model allows X to have a quadratic effect on the response and Z to have a linear effect.


yijk =  µ + 
(explained by X)
β1 xi + β2 xi2

 + 
(explained by Z)
γ zj

 + 
(unexplained)
εijk

We will not investigate the use of quadratic terms further in this section. The Response Surfaces chapter of the e-book describes quadratic models in much more detail.