Lecture Notes – Statistics 892 – Mixed Models – Spring 2005 – page 109Estimation of the dispersion parameterThe overdispersion parameter is unknown and therefore must be estimated. There are twosuggested methods of estimating the overdispersion parameter.McCullagh (1983) gives the estimate as12ˆ ˆ( ) ' ( )Pearsons ˆy V yN p N pmm mcf-- -= =- -where N-p is the df for lack of fit and Vµ is the diagonal matrix of variance functions.McCullagh and Nelder (1989) suggest using the deviance DevianceˆN pf =-So for the two models that we have investigated so far, the estimates of the dispersion parameters using the two methods are:Model Using Pearsons Using Deviance1 1.01 .642 .79 .59PROC GENMOD fixes the scale parameter at 1.0, unless you specify that you want to use the dispersion parameter. If the dispersion parameter is significantly greater than one,indicating overdispersion (variance greater than the mean), then the scale parameter should be used to adjust the variance. Failing to account for the overdispersion can resultin inflated test statistics. However, when the dispersion parameter is less than one, then the test statistics become more conservative, which is not considered as much of a problem. PROC GLIMMIXSAS has available an experimental GLIMMIX procedure available for download. This GLIMMIX procedure replaces the GLIMMIX macro that was previously available. The GLIMMIX procedure is easier to use and has a manual available online. The manual is 220 pages long.To download the GLIMMIX procedure, go to the following web addresshttp://support.sas.com/rnd/app/da/glimmix.htmlClick on the download button. This will bring you to a page where you can download the procedure. Before you can download the software, SAS requires that you register.Lecture Notes – Statistics 892 – Mixed Models – Spring 2005 – page 110There is a button on the right to click to go through the registration process. If you are already a registered SAS user, you will need to enter your email address and password. Once you are logged into SAS, you can download the software. Click on the request download button. This will download an executable file for installing the procedure into your SAS 9.1 software. If you do not have SAS 9.1 on your computer, you can purchase a license for SAS 9.1 from the Statistics Department for $40, where you will check out a set of CD's for installing SAS 9.1.Model 3 - fixed effects of baseline and treatment, random centerThe model that we are fitting with Model 3 is the followinglog(µij/(1-µij))=a + xij+i +CjThis looks just like Model 2, but now we are considering centers as random rather than fixed. The assumption isCj ~N(0,\s\up 6(2 ))Model 3 can be fit with the following SAS programproc glimmix data=ldbp;class trt center;model cfb/one=cf1b trt/ddfm=satterth cl;lsmeans trt/diff pdiff;random center;run;In the model statement the CL option requests a t type CI be constructed for each fixed effect parameter. Below is selected output from the analysis. The GLIMMIX Procedure Model Information Data Set WORK.LDBP Response Variable (Events) cfb Response Variable (Trials) one Response Distribution Binomial Link Function Logit Variance Function Default Variance Matrix Not blocked Estimation Technique Residual PL Degrees of Freedom Method SatterthwaiteFirst we look at the model information to make sure that we have set up the model correctly. The variance function is the default variance function for a bernoulli (µ(1- µ)). PROC GLIMMIX does allow you to define a different variance function. The variance function is not blocked. In other words, we did not do any grouping of the observations. The class information is listed next.Lecture Notes – Statistics 892 – Mixed Models – Spring 2005 – page 111 Class Level Information Class Levels Values trt 3 A B C center 29 1 2 3 4 5 6 7 8 9 11 12 13 14 15 18 23 24 25 26 27 29 30 31 32 35 36 37 40 41 Number of Observations Read 283 Number of Observations Used 283 Number of Events 41 Number of Trials 283 Dimensions G-side Cov. Parameters 1 Columns in X 5 Columns in Z 29 Subjects (Blocks in V) 1 Max Obs per Subject 283 The dimension give information about the sizes of the X and Z design matrices. Note that X has one column for each fixed effect and Z has one column for each center effect. The next information presented is the optimization information. The default optimizationis the Quisi-Newton method. The iteration history displays information about the progress of the optimization process. After the initial optimization, the GLIMMIX procedure performed 6 updates before the convergence criterion was met. Optimization Information Optimization Technique Dual Quasi-Newton Parameters in Optimization 1 Lower Boundaries 1 Upper Boundaries 0 Fixed Effects Profiled Starting From DataLecture Notes – Statistics 892 – Mixed Models – Spring 2005 – page 112 Iteration History Objective MaxIteration Restarts Subiterations Function Change Gradient 0 0 1 1341.5683928 2.00000000 3.241048 1 0 3 1475.3792347 2.00000000 0.000068 2 0 3 1524.8573372 0.89553871 0.000114 3 0 3 1525.1622045 0.07024552 2.49E-6 4 0 2 1524.9247004 0.00481471 1.48E-8 5 0 1 1524.8876009 0.00004290 1.249E-9 6 0 0 1524.8872551 0.00000000 2.949E-7 The fit statistics are output next. The word "Pseudo" preceding the different statistics indicates that these are computed from a pseudo-likelihood. Just like with the previous models, the ratio of the Pearson Chi-Square over