Modeling and Multivariate Methods - SAS

662 Statistical Details Appendix A
The Factor Models
Table A.7 Two-Way Model with Interaction
A B A1 A2 B1 A1B1 A2B1
A1 B1 1 0 1 1 0
A2 B1 0 1 1 0 1
A3 B1 –1 –1 1 –1 –1
A1 B2 1 0 –1 –1 0
A2 B2 0 1 –1 0 –1
A3 B2 –1 –1 –1 1 1 Suppose this interaction is
missing.
The expected values for each cell are:
Table A.8 Expected Values
B1
B2
A1
A2
A3
μ + α 1
+ β 1
+ αβ 11
μ + α 1
– β 1
– αβ 11
μ + α 2
+ β 1
+ αβ 21
μ + α 2
– β 1
– αβ 21
μ – α 1
– α 2
+ β 1
– αβ 11
– αβ 21
μ – α 1
– α 2
– β 1
+ αβ 11
+ αβ 21
Obviously, any cell with data has an expectation that is estimable. The cell that is missing has an expectation
that is nonestimable. In fact, its expectation is precisely that linear combination of the design columns that
is in the singularity report
μ– α 1
– α 2
– β 1
+ αβ 11
+ αβ 21
Suppose that you want to construct a test that compares the least squares means of B1 and B2. In this
example, the average of the rows in the above table give these least squares means.
LSM(B1) = ( 1 ⁄ 3) ( μ+ α 1
+ β 1
+ αβ 11
+
μ+ α 2
+ β 1
+ αβ 21
+
μ– α 1
– α 2
+ β 1
– αβ 11
– αβ 21
)
= μ + β 1
LSM(B2) = ( 1 ⁄ 3) ( μ+ α 1
+ – β 1
– αβ 11
+
μ+ α 2
+ – β 1
– αβ 21
+
μ– α 1
– α 2
– β 1
+ αβ 11
+ αβ 21
)
= μ – β 1