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Fitting Linear Models > Stepwise Regression Models > Models with Crossed, Interaction, or Polynomial Terms
Publication date: 11/10/2021

Models with Crossed, Interaction, or Polynomial Terms

Some models, especially those associated with experimental designs, involve interaction terms. For continuous factors, these are products of the columns representing the effects. For nominal and ordinal factors, interactions are defined by model terms that involve products of terms representing the categorical levels.

When there are interaction terms, you often want to impose a restriction on the model selection process so that lower-order components of higher-order effects are included in the model. This is suggested by the principle of Effect Heredity. See Effect Heredity in the Design of Experiments Guide. For example, if a two-way interaction is included in a model, its component main effects (precedents) should be included as well.

Example of the Combine Rule

1. Select Help > Sample Data Library and open

2. Select Analyze > Fit Model.

3. Select Y and click Y.

4. In the Degree box, type 2.

5. Select F, Ct, A, T, and Cn and click Macros > Factorial to Degree.

6. For Personality, select Stepwise.

7. Click Run.

Figure 5.12 Initial Current Estimates Report Using Combine RuleĀ 

Initial Current Estimates Report Using Combine Rule

The model in Figure 5.12 contains all terms for up to two-factor interactions for the five continuous factors. The Combine, Restrict, and Whole Effects rules described in Rules enable you to control entry of interaction terms.

The Combine rule determines the entry of interaction terms based on two tests. See Combine. You can determine which of the two tests was used for the p-value based on the degrees of freedom, nDF. For example, the interaction term F*A has an nDF value of 3. This means that F*A is grouped with its precedent terms F and A, and is considered for entry based on the 3 degree of freedom joint F test. In contrast, the interaction term Ct*T has an nDF value of 1. This means that Ct*T is considered for entry based on the 1 degree of freedom F test that tests the significance of Ct*T after its precedent terms, Ct and T, are already included in the model. Click Step once to see that Ct is entered by itself.

8. Click Step again to see that Ct*T is entered, along with T (Ct is already in the model).

Figure 5.13 Current Estimates Report Using Combine Rule, One StepĀ 

Current Estimates Report Using Combine Rule, One Step

When there are significant interaction terms, several terms can enter at the same step. If the Step button is clicked twice, Ct*T is entered along with its two contained effects Ct and T. However, a step back is not symmetric because a crossed term can be removed without removing its two component terms. Notice that Ct and T now each have 2 degrees of freedom. This is because if Stepwise removes Ct or T, it must also remove Ct*T. If you change the Direction to Backward and click Step, Ct*T is removed and the degrees of freedom for Ct and T change to 1.

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