Modeling and Multivariate Methods - SAS

Appendix A Statistical Details 661
The Factor Models
If there are missing cells or other singularities, the JMP tests are different than GLM tests. There are several
ways to describe them:
• JMP tests are equivalent to testing that the least squares means are different, at least for main effects. If
the least squares means are nonestimable, then the test cannot include some comparisons and, therefore,
loses degrees of freedom. For interactions, JMP is testing that the least squares means differ by more
than just the marginal pattern described by the containing effects in the model.
• JMP tests an effect by comparing the SSE for the model with that effect with the SSE for the model
without that effect (at least if there are no nested terms, which complicate the logic slightly). JMP
parameterizes so that this method makes sense.
• JMP implements the effective hypothesis tests described by Hocking (1985, 80–89, 163–166), although
JMP uses structural rather than cell-means parameterization. Effective hypothesis tests start with the
hypothesis desired for the effect and include “as much as possible” of that test. Of course, if there are
containing effects with missing cells, then this test will have to drop part of the hypothesis because the
complete hypothesis would not be estimable. The effective hypothesis drops as little of the complete
hypothesis as possible.
• The differences among hypothesis tests in JMP and GLM (and other programs) that relate to the
presence of missing cells are not considered interesting tests anyway. If an interaction is significant, the
test for the contained main effects are not interesting. If the interaction is not significant, then it can
always be dropped from the model. Some tests are not even unique. If you relabel the levels in a missing
cell design, then the GLM Type IV tests can change.
The following section continues this topic in finer detail.
Singularities and Missing Cells in Nominal Effects
Consider the case of linear dependencies among the design columns. With JMP coding, this does not occur
unless there is insufficient data to fill out the combinations that need estimating, or unless there is some
kind of confounding or collinearity of the effects.
With linear dependencies, a least squares solution for the parameters might not be unique and some tests of
hypotheses cannot be tested. The strategy chosen for JMP is to set parameter estimates to zero in sequence
as their design columns are found to be linearly dependent on previous effects in the model. A special
column in the report shows what parameter estimates are zeroed and which parameter estimates are
estimable. A separate singularities report shows what the linear dependencies are.
In cases of singularities the hypotheses tested by JMP can differ from those selected by GLM. Generally,
JMP finds fewer degrees of freedom to test than GLM because it holds its tests to a higher standard of
marginality. In other words, JMP tests always correspond to tests across least squares means for that effect,
but GLM tests do not always have this property.
For example, consider a two-way model with interaction and one missing cell where A has three levels, B has
two levels, and the A3B2 cell is missing.