Modeling and Multivariate Methods - SAS

664 Statistical Details Appendix A
The Factor Models
It turns out that the design columns for missing cells for any interaction will always knock out degrees of
freedom for the main effect (for nominal factors). Thus, there is a direct relation between the
nonestimability of least squares means and the loss of degrees of freedom for testing the effect corresponding
to these least squares means.
How does this compare with what GLM does? GLM and JMP do the same test when there are no missing
cells. That is, they effectively test that the least squares means are equal. But when GLM encounters
singularities, it focuses out these cells in different ways, depending on whether they are Type III or Type IV.
For Type IV, it looks for estimable combinations that it can find. These might not be unique, and if you
reorder the levels, you might get a different result. For Type III, it does some orthogonalization of the
estimable functions to obtain a unique test. But the test might not be very interpretable in terms of the cell
means.
The JMP approach has several points in its favor, although at first it might seem distressing that you might
lose more degrees of freedom than with GLM:
1. The tests are philosophically linked to LSMs.
2. The tests are easy computationally, using reduction sum of squares for reparameterized models.
3. The tests agree with Hocking’s “Effective Hypothesis Tests”.
4. The tests are whole marginal tests, meaning they always go completely across other effects in interactions.
The last point needs some elaboration: Consider a graph of the expected values of the cell means in the
previous example with a missing cell for A3B2.
B1
A3B2?
B2
A3B2?
A1 A2 A3
The graph shows expected cell means with a missing cell. The means of the A1 and A2 cells are profiled
across the B levels. The JMP approach says you can’t test the B main effect with a missing A3B2 cell, because
the mean of the missing cell could be anything, as allowed by the interaction term. If the mean of the
missing cell were the higher value shown, the B effect would likely test significant. If it were the lower, it
would likely test nonsignificant. The point is that you don’t know. That is what the least squares means are
saying when they are declared nonestimable. That is what the hypotheses for the effects should be saying
too—that you don’t know.
If you want to test hypotheses involving margins for subsets of cells, then that is what GLM Type IV does.
In JMP you would have to construct these tests yourself by partitioning the effects with a lot of calculations
or by using contrasts.