Implications of the Non-Full-Rank Parameterization

For models with fixed effects involving classification variables, there are more design columns in constructed than there are degrees of freedom for the effect. Thus, there are linear dependencies among the columns of . In this event, all of the parameters are not estimable; there is an infinite number of solutions to the mixed model equations. The GLIMMIX procedure uses a generalized inverse (a -inverse, Pringle and Rayner 1971), to obtain values for the estimates (Searle 1971). The solution values are not displayed unless you specify the SOLUTION option in the MODEL statement. The solution has the characteristic that estimates are 0 whenever the design column for that parameter is a linear combination of previous columns. With this parameterization, hypothesis tests are constructed to test linear functions of the parameters that are estimable.

Some procedures (such as the CATMOD and LOGISTIC procedures) reparameterize models to full rank by using restrictions on the parameters. PROC GLM, PROC MIXED, and PROC GLIMMIX do not reparameterize, making the hypotheses that are commonly tested more understandable. See Goodnight (1978a) for additional reasons for not reparameterizing.