Modeling and Multivariate Methods - SAS

Appendix A Statistical Details 669
The Factor Models
Table A.13 Ordinal Interactions (Continued)
A*B
A2
A3
A B A2 A3 B2 B3 B2 B3 B2 B3
A3 B3 1 1 1 1 1 1 1 1
Note: When you test to see if there is no effect, there is not much difference between nominal and ordinal
factors for simple models. However, there are major differences when interactions are specified. We
recommend that you use nominal rather than ordinal factors for most models.
Hypothesis Tests for Ordinal Crossed Models
To see what the parameters mean, examine this table of the expected cell means in terms of the parameters,
where μ is the intercept, α 2 is the parameter for level A2, and so forth.
Table A.14 Expected Cell Means
B1 B2 B3
A1
A2
μ μ + αβ 2
+ αβ 12
μ+ β 2
+ β 3
μ+ α 2
μ + α 2
+ β 2
+ αβ 22
μ+ α 2
+ β 2
+ β 3
+ αβ 22
+ αβ 23
A3
μ+ α 2
+ α 3
μ + α 2
+ α 3
+ β 2
+ αβ 22
+
αβ 32
+ αβ 23
+ αβ 32
+ αβ 33
μ– α 2
+ α 3
+ β 2
+ β 3
+ αβ 22
+
αβ 23
+ β 32
+ αβ 33
Note that the main effect test for A is really testing the A levels holding B at the first level. Similarly, the
main effect test for B is testing across the top row for the various levels of B holding A at the first level. This
is the appropriate test for an experiment where the two factors are both doses of different treatments. The
main question is the efficacy of each treatment by itself, with fewer points devoted to looking for drug
interactions when doses of both drugs are applied. In some cases it may even be dangerous to apply large
doses of each drug.
Note that each cell’s expectation can be obtained by adding all the parameters associated with each cell that
is to the left and above it, inclusive of the current row and column. The expected value for the last cell is the
sum of all the parameters.
Though the hypothesis tests for effects contained by other effects differs with ordinal and nominal codings,
the test of effects not contained by other effects is the same. In the crossed design above, the test for the
interaction would be the same no matter whether A and B were fit nominally or ordinally.