The GLIMMIX Procedure
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Overview
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Getting Started
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Syntax
PROC GLIMMIX Statement BY Statement CLASS Statement CONTRAST Statement COVTEST Statement EFFECT Statement ESTIMATE Statement FREQ Statement ID Statement LSMEANS Statement LSMESTIMATE Statement MODEL Statement NLOPTIONS Statement OUTPUT Statement PARMS Statement RANDOM Statement SLICE Statement STORE Statement WEIGHT Statement Programming Statements User-Defined Link or Variance Function
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Details
Generalized Linear Models Theory Generalized Linear Mixed Models Theory GLM Mode or GLMM Mode Statistical Inference for Covariance Parameters Satterthwaite Degrees of Freedom Approximation Empirical Covariance (Sandwich) Estimators Exploring and Comparing Covariance Matrices Processing by Subjects Radial Smoothing Based on Mixed Models Odds and Odds Ratio Estimation Parameterization of Generalized Linear Mixed Models Response-Level Ordering and Referencing Comparing the GLIMMIX and MIXED Procedures Singly or Doubly Iterative Fitting Default Estimation Techniques Default Output Notes on Output Statistics ODS Table Names ODS Graphics
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Examples
Binomial Counts in Randomized Blocks Mating Experiment with Crossed Random Effects Smoothing Disease Rates; Standardized Mortality Ratios Quasi-likelihood Estimation for Proportions with Unknown Distribution Joint Modeling of Binary and Count Data Radial Smoothing of Repeated Measures Data Isotonic Contrasts for Ordered Alternatives Adjusted Covariance Matrices of Fixed Effects Testing Equality of Covariance and Correlation Matrices Multiple Trends Correspond to Multiple Extrema in Profile Likelihoods Maximum Likelihood in Proportional Odds Model with Random Effects Fitting a Marginal (GEE-Type) Model Response Surface Comparisons with Multiplicity Adjustments Generalized Poisson Mixed Model for Overdispersed Count Data Comparing Multiple B-Splines Diallel Experiment with Multimember Random Effects Linear Inference Based on Summary Data
- References
| Exploring and Comparing Covariance Matrices |
If you use an empirical (sandwich) estimator with the EMPIRICAL= option in the PROC GLIMMIX statement, the procedure replaces the model-based estimator of the covariance of the fixed effects with the sandwich estimator. This affects aspects of inference, such as prediction standard errors, tests of fixed effects, estimates, contrasts, and so forth. Similarly, if you choose the DDFM=KENWARDROGER degrees-of-freedom method in the MODEL statement, PROC GLIMMIX adjusts the model-based covariance matrix of the fixed effects according to Kenward and Roger (1997) or according to Kackar and Harville (1984) and Harville and Jeske (1992).
In this situation, the COVB(DETAILS) option in the MODEL statement has two effects. The GLIMMIX procedure displays the (adjusted) covariance matrix of the fixed effects and the model-based covariance matrix (for ODS purposes, the name of the table with the model-based covariance matrix is "CovBModelBased"). The procedure also displays a table of statistics for the unadjusted and adjusted covariance matrix and for their comparison. For ODS purposes, the name of this table is "CovBDetails."
If the model-based covariance matrix is not replaced with an adjusted estimator, the COVB(DETAILS) option displays the model-based covariance matrix and provides diagnostic measures for it in the "CovBDetails" table.
The table generated by the COVB(DETAILS) option consists of several sections. See Example 40.8 for an application.
The trace and log determinant of covariance matrices are general scalar summaries that are sometimes used in direct comparisons, or in formulating other statistics, such as the difference of log determinants. The trace simply represents the sum of the variances of all fixed-effects parameters. If a matrix is indefinite, the determinant is reported instead of the log determinant.
The model-based and adjusted covariance matrices should have the same general makeup of eigenvalues. There should not be any negative eigenvalues, and they should have the same numbers of positive and zero eigenvalues. A reduction in rank due to the adjustment is troublesome for aspects of inference. Negative eigenvalues are listed in the table only if they occur, because a covariance matrix should be at least positive semi-definite. However, the GLIMMIX procedure examines the model-based and adjusted covariance matrix for negative eigenvalues. The condition numbers reported by PROC GLIMMIX for positive (semi-)definite matrices are computed as the ratio of the largest and smallest nonzero eigenvalue. A large condition number reflects poor conditioning of the matrix.
Matrix norms are extensions of the concept of vector norms to measure the "length" of a matrix. The Frobenius norm of an 
matrix 
is the direct equivalent of the Euclidean vector norm, the square root of the sum of the squared elements,
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The 
- and 1-norms of matrix are the maximum absolute row and column sums, respectively:
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These two norms are identical for symmetric matrices.
The "Comparison" section of the "CovBDetails" table provides several statistics that set the matrices in relationship. The concordance correlation reported by the GLIMMIX procedure is a standardized measure of the closeness of the model-based and adjusted covariance matrix. It is a slight modification of the covariance concordance correlation in Vonesh, Chinchilli, and Pu (1996) and Vonesh and Chinchilli (1997, Ch. 8.3). Denote as 
the 
model-based covariance matrix and as 
the adjusted matrix. Suppose that 
is the matrix obtained from the identity matrix of size 
by replacing diagonal elements corresponding to singular rows in with zeros. The lower triangular portion of 
is stored in vector 
and the lower triangular portion of 
is stored in vector 
. The matrix 
is constructed from an eigenanalysis of and is symmetric. The covariance concordance correlation is then
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This measure is 1 if = . If is orthogonal to , there is total disagreement between the model-based and the adjusted covariance matrix and 
is zero.
The discrepancy function reported by PROC GLIMMIX is computed as
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In diagnosing departures between an assumed covariance structure and 
—using an empirical estimator—Vonesh, Chinchilli, and Pu (1996) find that the concordance correlation is useful in detecting gross departures and propose 
to test the correctness of the assumed model, where 
denotes the number of subjects.