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