I can see why you would have lots of questions about the two kinds of parameterizations. Justification for the GLM parameterization would take lengthy explanations, not suited to a discussion answer (at least I can't think of easy and short explanations -- maybe others can). This all goes back to the 1970s when Jim Goodnight (founder and still CEO of SAS) was deciding on the parameterization for PROC GLM (and others). Basically, they needed to come up with a general approach that would work for any factorial, including nesting, and so on. He and colleagues made it quite clear why the GLM (over-)parameterization was needed, in general, for factorials. There are some old papers and reports that I don't have available right now. It has to to with estimable functions, a concept that is often difficult to understand. For the over-parameterization, it looks like those zero parameters are not there, but they really are. SAS does not just find the last levels (or whatever you make the reference) and makes them 0. They are present in the model fitting. But the generalized inverse of the matrix in the model-fit step ends ups with 0s for those terms (the constraint used to get the inverse). But they are still there. This may seem to be the same thing, but it's not. One cannot define expected values of main effects (and maybe other effects) unless those zero parameters are there. Likewise type3 (or other) tests of hypotheses (especially for main effects) can only be interpreted in terms of expected values (means) with the GLM parameterization. If you are only interested in the highest-level interaction, you don't need to be concerned about this (other than in defining the contrast of interest). Or if you have only one factor. Also, if all the variables in your model are factors, then all of these parameterizations are giving you the same model fit (just with different coefficients). But some things cannot be done with the reference parameterization. By the way, you can control the reference level with the GLM parameterization (I show this below). I can get LSMEANS, etc., etc., with this parameterization. (Be careful: SAS rearranges the parameters. See in the Solution table that T1 goes last. This changes the order of things in the estimate statement). This looks very much like your output with reference parameterization, but it has the subtle differences I describe above. Also, not so subtle differences (the type3 tests are totally different for the main effects of G and T with reference and glm parameterization). Finally, only GLM parameterization is possible directly with PROC GLM, MIXED, and GLIMMIX, the flagship procedures in SAS for factorials. This is not an accident. This is the parameterization that works best, and makes the most sense, as a general framework for factorials. For your application, it doesn't matter. proc genmod data=test; class G (ref='3') T (ref='1') / param=glm; model y = G|T / dist=bin type3 ; lsmeans G*T ; estimate 'G1 vs rest at T1, pos.' G 1 -0.5 -0.5 G*T 0 0 0 1 0 0 0 -0.5 0 0 0 -0.5 ; lsmestimate G*T 'G1 vs rest at T1, nonpos.' [1,1 4] [-0.5,2 4] [-0.5,3 4] / ilink ; run;
... View more
