5.5.9.9.5. Motivation: What is the Form of the Model?

Models for various values of k

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 k = 1 factor case:

  \( Y = f(X_{1}) + \epsilon = c + B_{1}X_{1} + \epsilon \)

• For the k = 2 factor case:

  \( \begin{array}{lcl} Y & = & f(X_{1},X_{2}) + \epsilon \\ & = & c + B_{1}X_{1} + B_{2}X_{2} + B_{12}X_{1}X_{2} + \epsilon \end{array} \)

• For the k = 3 factor case:

  \( \begin{array}{lcl} Y & = & f(X_{1},X_{2},X_{3}) + \epsilon \\ & = & c + B_{1}X_{1} + B_{2}X_{2} + B_{3}X_{3} + \\ & & B_{12}X_{1}X_{2} + B_{13}X_{1}X_{3} + B_{23}X_{2}X_{3} + \\ & & B_{123}X_{1}X_{2}X_{3} + \epsilon \end{array} \)

• and for the general k case:

  \( \begin{array}{lcl} Y & = & f(X_{1},X_{2}, \ldots , X_{k}) + \epsilon \\ & = & c + \mbox{linear combination of all main} \\ & & \mbox{effects and all interactions} + \epsilon \end{array} \)

Note that the model equations shown above include coefficients that represent the change in Y for a one-unit change in X_i. To obtain an effect estimate, which represents a two-unit change in X_i if the levels of X_i are +1 and -1, simply multiply the coefficient by two.

Ordered linear combination

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

\( \begin{array}{lcl} Y & = & f(X_{1},X_{2}) + \epsilon \\ & = & c + B_{1}X_{1} + B_{2}X_{2} + B_{12}X_{1}X_{2} + \epsilon \end{array} \)

be rewritten as

\( \begin{array}{lcl} Y & = & f(X_{1},X_{2}) + \epsilon \\ & = & c + B_{2}X_{2} + B_{12}X_{1}X_{2} + B_{1}X_{1} + \epsilon \end{array} \)