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Models for various values of k
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From the above discussion, we thus note and recommend a form
of the model that consists of an additive constant plus a linear
combination of main effects and interactions. What then is the
specific form for the linear combination?
The following are the full models for various values of k.
The selected final model will be a subset of the full model.
- For the trivial k = 1 factor case:
- For the k = 2 factor case:
- For the k = 3 factor case:
- and for the general k case:
Y = f(X1,
X2, ..., Xk) =
c + (1/2)*(linear combination of all main effects
and all interactions of all orders) +
Note that the above equations include a (1/2) term. Our reason
for using this term is discussed in some detail in
the next section. Other sources typically
do not use this convention.
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Ordered linear combination
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The listing above has the terms ordered with the main effects, then
the 2-factor interactions, then the 3-factor interactions, etc. In
practice, it is recommended that the terms be ordered by importance
(whether they be main effects or interactions). Aside from providing a
functional representation of the response, models should help
reinforce what is driving the response, which such a re-ordering does.
Thus for k = 2, if factor 2 is most important, the 2-factor
interaction is next in importance, and factor 1 is least important,
then it is recommended that the above ordering of
be rewritten as
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