Yes, I agree with you. That was the point I was trying to make (but probably not clearly). You should, and do, get different results with a weight versus no weight. I see no point in weighting the counts in the table. By the way, many statisticians strongly dislike the use of stepwise variable selection procedures (including backward selection). You can find many posts on this website about this. ... View more

I get different CTABLE results when I use WEIGHT versus when I do not use a weight. It is using the fitted logistic model to make the predictions, which depends on the weights. The procedure does not weight the actual observations, however, in the table; I don't think I would want to do that. ... View more

Your two versions of your original GLIMMIX code fit the same model, with same results. This is because you are using a normal distribution (the default), with an identity link function. In fact, you could fit the same model with MIXED (using the log as the response and a REPEATED statement) because of the normality. These two versions work equivalently when you have all times in all subjects. Also, as Steve indicates, with only two times, ARH(1) is not different from several other structures. The more complex questions would arise if you took the GLMM approach and model the response (not log response) as a member of the exponential family of distributions (with a log link). Then you would need to distinguish between Random day / subject = id(trt) type =ARH(1) residual; and Random day / subject = id(trt) type =ARH(1) ; You didn't ask about this, but now you would get different results for some non-normal distributions (e.g., Poisson). Here, the first one (with RESIDUAL option) would model overdispersion directly (R-side analysis) where the second version is modeling the linear predictor conditional on the random effect (G-side analysis). ... View more

Glad you are studying the problem in such detail. Notice that I started my previous post with "If you want a detailed explanation, you need to read some textbooks, or parts of textbook. I doubt if postings will go into the details." You misunderstood some of my comments, maybe because I couldn't write enough. Probably I was too casual in a few places.  I can only comment on a few of your statements here. GLM does a purely fixed-effects analysis, with post-model-fitting adjustments to get the random effects variances. Of course, you must use the right code (test option on random statement) to get the proper mean squares and F tests (the defaults clearly are wrong, as you point out). GLM is not capable of getting the right standard errors in many cases (with any options), especially when the SEs involve combinations of variances. This is why the estimate and contrast statements are totally wrong in GLM. GLM should be avoided when there are random effects. This was one of the points I was trying to make. When you use method=type3 in MIXED, you are getting, in part, the equivalence of the approach in GLM (assuming you used the right options in GLM), but it is not necessarily done in the same way. The expected mean square table, F statistics (based on the mean squares), and P values agree for the two procedures (once again, assuming you have the right code for GLM and method=type3 in MIXED). This is true for your example (by the way, you misspelled product in the interaction term in GLM). This is a reason why I said you were getting a fixed-type analysis, but clearly it is not an actual fixed effect analysis in MIXED. MIXED is much more sophisticated than GLM; MIXED is doing an actual mixed-effect analysis for all the method= options we have been discussing (when you have random statements). Sorry if I implied otherwise. Because MIXED knows how to use the random effects, you obviously can get a G-side covariance matrix. You are getting appropriate SEs, ESTIMATEs, and CONTRASTs with MIXED, but not necessarily the best choices with the method=type3 analysis (the latter may be open to some interpretation). Because MIXED is properly using random-effects terms as random effects, you get EBLUPs, and so on. Obviously, these do not exist when using GLM (one of many reasons not to use GLM). There is no point even trying to recover anything like an EBLUP in GLM. MIXED is a great procedure, and one you should be using (whether or not you use expected mean squares). The book I mentioned (SAS for Mixed Models, second edition) has several examples where method=type3 is utilized for interesting purposes. The limitation is that the method is appropriate for only a narrow class of mixed models. I would argue that likelihood-based methods are better for other reasons, but others could disagree. I did not bring up anything about narrow or broad inference. I think you can do either with the type3 method, with the right syntax. I don’t know if anyone has evaluated the effects of the model-fitting method on the accuracy of the results for broad and narrow inference. ... View more

If you want a detailed explanation, you need to read some textbooks, or parts of textbook. I doubt if postings will go into the details. The book SAS FOR MIXED MODELS, 2nd edition (2006) is a great reference. Here are a few things to consider. For PROC MIXED, the default is a likelihood approach (REML). This should tell you something. Here is another way to look at the problem. Method=type3 essentially duplicates the approach in PROC GLM, and the book I mentioned above makes it clear why one should not use GLM for models with random effects. The GLM approach (i.e., the method=type3 approach) actually fits the model with all terms as fixed effects, whether or not they are really random effects. Then, in a post-model-fitting step, expected mean squares are calculated and tests conducted based on the theoretical values of the mean squares. This approach (everything fixed to start with, then adjust later) can give appropriate results for a narrow set of circumstances (such as for balanced data, all effects independent, and so on); for this narrow set of circumstances, the method=type3 approach will duplicate the method=reml approach. Otherwise, the results will be different. For REML model fitting, the random effects are treated as random effects in the model-fitting process (as one would want, since they are random effects), and the tests of significance are derived based on contrasts of the expected values (there are no mean squares). The biggest effect of the model-fitting choice will show up in the estimated standard errors for estimated means and contrasts of means. For certain designs (e.g., split plots) , whether balanced or not, and for most (all?) unbalanced layouts, the standard errors will be incorrect when using method=type3, but correct with REML. So, I personally recommend likelihood-based methods for mixed models. But as I said above, there is a narrow set of circumstances where the results will be the same. The book I mentioned will also show you some situations (within this narrow set of circumstances), where valuable information can be obtained by using the type3 approach. ... View more

I really think you should stick with the REML estimation and testing, and not use method=type3 (based on expected mean squares). Contemporary mixed modeling has really moved away from sum of squares and mean squares. Although the latter can be fine for the simplest of situations, the expected mean square approach cannot be generalized for any of the interesting problems that are common. The likelihood-based analysis is the way to go. When I give workshops on mixed models, I tell the audience to forget everything about mean squares. ... View more