Modeling and Multivariate Methods - SAS

660 Statistical Details Appendix A
The Factor Models
Table A.6 Nested Effects
B(A)
A1 A1 A2 A2 A3 A3
A B A13 A23 B13 B23 B13 B23 B13 B23
A1 B1 1 0 1 0 0 0 0 0
A1 B2 1 0 0 1 0 0 0 0
A1 B3 1 0 –1 –1 0 0 0 0
A2 B1 0 1 0 0 1 0 0 0
A2 B2 0 1 0 0 0 1 0 0
A2 B3 0 1 0 0 –1 –1 0 0
A3 B1 –1 –1 0 0 0 0 1 0
A3 B2 –1 –1 0 0 0 0 0 1
A3 B3 –1 –1 0 0 0 0 –1 –1
Least Squares Means across Nominal Factors
Least squares means are the predicted values corresponding to some combination of levels, after setting all
the other factors to some neutral value. The neutral value for direct continuous regressors is defined as the
sample mean. The neutral value for an effect with uninvolved nominal factors is defined as the average effect
taken over the levels (which happens to result in all zeroes in our coding). Ordinal factors use a different
neutral value in “Ordinal Least Squares Means” on page 670. The least squares means might not be
estimable, and if not, they are marked nonestimable. JMP’s least squares means agree with GLM’s
(Goodnight and Harvey 1978) in all cases except when a weight is used, where JMP uses a weighted mean
and GLM uses an unweighted mean for its neutral values.
Effective Hypothesis Tests
Generally, the hypothesis tests produced by JMP agree with the hypothesis tests of most other trusted
programs, such as SAS PROC GLM (Hypothesis types III and IV). The following two sections describe
where there are differences.
In the SAS GLM procedure, the hypothesis tests for Types III and IV are constructed by looking at the
general form of estimable functions and finding functions that involve only the effects of interest and effects
contained by the effects of interest (Goodnight 1978).
In JMP, the same tests are constructed, but because there is a different parameterization, an effect can be
tested (assuming full rank for now) by doing a joint test on all the parameters for that effect. The tests do
not involve containing interaction parameters because the coding has made them uninvolved with the tests
on their contained effects.