### Singly or Doubly Iterative Fitting

Depending on the structure of your model, the GLIMMIX procedure determines the appropriate approach for estimating the parameters of the model. The elementary algorithms fall into three categories:

1. Noniterative algorithms A closed form solution exists for all model parameters. Standard linear models with homoscedastic, uncorrelated errors can be fit with noniterative algorithms.

2. Singly iterative algorithms A single optimization, consisting of one or more iterations, is performed to obtain solutions for the parameter estimates by numerical techniques. Linear mixed models for normal data can be fit with singly iterative algorithms. Laplace and quadrature estimation for generalized linear mixed models uses a singly iterative algorithm with a separate suboptimization to compute the random-effects solutions as modes of the log-posterior distribution.

3. Doubly iterative algorithms A model of simpler structure is derived from the target model. The parameters of the simpler model are estimated by noniterative or singly iterative methods. Based on these new estimates, the model of simpler structure is rederived and another estimation step follows. The process continues until changes in the parameter estimates are sufficiently small between two recomputations of the simpler model or until some other criterion is met. The rederivation of the model can often be cast as a change of the response to some pseudo-data along with an update of implicit model weights.

Obviously, noniterative algorithms are preferable to singly iterative ones, which in turn are preferable to doubly iterative algorithms. Two drawbacks of doubly iterative algorithms based on linearization are that likelihood-based measures apply to the pseudo-data, not the original data, and that at the outer level the progress of the algorithm is tied to monitoring the parameter estimates. The advantage of doubly iterative algorithms, however, is to offer—at convergence—the statistical inference tools that apply to the simpler models.

The output and log messages contain information about which algorithm is employed. For a noniterative algorithm, PROC GLIMMIX produces a message that no optimization was performed. Noniterative algorithms are employed automatically for normal data with identity link.

You can determine whether a singly or doubly iterative algorithm was used, based on the Iteration History table and the Convergence Status table (Figure 43.17).

Figure 43.17: Iteration History and Convergence Status in Singly Iterative Fit

The GLIMMIX Procedure

Iteration History
Iteration Restarts Evaluations Objective
Function
Change Max
0 0 4 83.039723731 . 13.63536
1 0 3 82.189661988 0.85006174 0.281308
2 0 3 82.189255211 0.00040678 0.000174
3 0 3 82.189255211 0.00000000 1.05E-10

 Convergence criterion (GCONV=1E-8) satisfied.

The Iteration History table contains the `Evaluations` column that shows how many function evaluations were performed in a particular iteration. The convergence status message informs you which convergence criterion was met when the estimation process concluded. In a singly iterative fit, the criterion is one that applies to the optimization. In other words, it is one of the criteria that can be controlled with the NLOPTIONS statement: see the ABSCONV=, ABSFCONV=, ABSGCONV=, ABSXCONV=, FCONV=, or GCONV= option.

In a doubly iterative fit, the Iteration History table does not contain an `Evaluations` column. Instead it displays the number of iterations within an optimization (`Subiterations` column in Figure 43.18).

Figure 43.18: Iteration History and Convergence Status in Doubly Iterative Fit

Iteration History
Iteration Restarts Subiterations Objective
Function
Change Max
0 0 5 79.688580269 0.11807224 7.851E-7
1 0 3 81.294622554 0.02558021 8.209E-7
2 0 2 81.438701534 0.00166079 4.061E-8
3 0 1 81.444083567 0.00006263 2.311E-8
4 0 1 81.444265216 0.00000421 0.000025
5 0 1 81.444277364 0.00000383 0.000023
6 0 1 81.444266322 0.00000348 0.000021
7 0 1 81.44427636 0.00000316 0.000019
8 0 1 81.444267235 0.00000287 0.000017
9 0 1 81.444275529 0.00000261 0.000016
10 0 1 81.44426799 0.00000237 0.000014
11 0 1 81.444274843 0.00000216 0.000013
12 0 1 81.444268614 0.00000196 0.000012
13 0 1 81.444274277 0.00000178 0.000011
14 0 1 81.444269129 0.00000162 9.772E-6
15 0 0 81.444273808 0.00000000 9.102E-6

 Convergence criterion (PCONV=1.11022E-8) satisfied.

The `Iteration` column then counts the number of optimizations. The Convergence Status table indicates that the estimation process concludes when a criterion is met that monitors the parameter estimates across optimization, namely the PCONV= or ABSPCONV= criterion.

You can control the optimization process with the GLIMMIX procedure through the NLOPTIONS statement. Its options affect the individual optimizations. In a doubly iterative scheme, these apply to all optimizations.

The default optimization techniques are TECHNIQUE=NONE for noniterative estimation, TECHNIQUE=NEWRAP for singly iterative methods in GLMs, TECHNIQUE=NRRIDG for pseudo-likelihood estimation with binary data, and TECHNIQUE=QUANEW for other mixed models.