The GLIMMIX Procedure
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Overview
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Getting Started
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SyntaxPROC GLIMMIX StatementBY StatementCLASS StatementCODE StatementCONTRAST StatementCOVTEST StatementEFFECT StatementESTIMATE StatementFREQ StatementID StatementLSMEANS StatementLSMESTIMATE StatementMODEL StatementNLOPTIONS StatementOUTPUT StatementPARMS StatementRANDOM StatementSLICE StatementSTORE StatementWEIGHT StatementProgramming StatementsUser-Defined Link or Variance Function
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DetailsGeneralized Linear Models TheoryGeneralized Linear Mixed Models TheoryGLM Mode or GLMM ModeStatistical Inference for Covariance ParametersDegrees of Freedom MethodsEmpirical Covariance ("Sandwich") EstimatorsExploring and Comparing Covariance MatricesProcessing by SubjectsRadial Smoothing Based on Mixed ModelsOdds and Odds Ratio EstimationParameterization of Generalized Linear Mixed ModelsResponse-Level Ordering and ReferencingComparing the GLIMMIX and MIXED ProceduresSingly or Doubly Iterative FittingDefault Estimation TechniquesDefault OutputNotes on Output StatisticsODS Table NamesODS Graphics
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ExamplesBinomial Counts in Randomized BlocksMating Experiment with Crossed Random EffectsSmoothing Disease Rates; Standardized Mortality RatiosQuasi-likelihood Estimation for Proportions with Unknown DistributionJoint Modeling of Binary and Count DataRadial Smoothing of Repeated Measures DataIsotonic Contrasts for Ordered AlternativesAdjusted Covariance Matrices of Fixed EffectsTesting Equality of Covariance and Correlation MatricesMultiple Trends Correspond to Multiple Extrema in Profile LikelihoodsMaximum Likelihood in Proportional Odds Model with Random EffectsFitting a Marginal (GEE-Type) ModelResponse Surface Comparisons with Multiplicity AdjustmentsGeneralized Poisson Mixed Model for Overdispersed Count DataComparing Multiple B-SplinesDiallel Experiment with Multimember Random EffectsLinear Inference Based on Summary DataWeighted Multilevel Model for Survey DataQuadrature Method for Multilevel Model
- References
Satterthwaite Degrees of Freedom Approximation
The DDFM= SATTERTHWAITE option in the MODEL statement requests that denominator degrees of freedom in t tests and F tests be computed according to a general Satterthwaite approximation.
The general Satterthwaite approximation computed in PROC GLIMMIX for the test
![\[ H \colon \bL \left[ \begin{array}{c} \widehat{\bbeta } \\ \widehat{\bgamma } \end{array} \right] = \mb{0} \]](images/statug_glimmix0760.png)
is based on the F statistic
![\[ F = \frac{\left[ \begin{array}{c} \widehat{\bbeta } \\ \widehat{\bgamma } \end{array} \right]' \bL '(\mb{LCL}')^{-1} \bL \left[ \begin{array}{c} \widehat{\bbeta } \\ \widehat{\bgamma } \end{array} \right] }{r} \]](images/statug_glimmix0761.png)
where 
and 
is the approximate variance matrix of ![$[\widehat{\bbeta }’,\widehat{\bgamma }’-\bgamma ’]’$](images/statug_glimmix0594.png)
. See the section Estimated Precision of Estimates and the section Aspects Common to Adaptive Quadrature and Laplace Approximation.
The approximation proceeds by first performing the spectral decomposition 
, where 
is an orthogonal matrix of eigenvectors and 
is a diagonal matrix of eigenvalues, both of dimension 
. Define 
to be the jth row of 
, and let
![\[ \nu _ j = \frac{2 (D_ j)^2}{ \mb{g}_ j' \mb{A} \mb{g}_ j} \]](images/statug_glimmix0768.png)
where 
is the jth diagonal element of and 
is the gradient of 
with respect to 
, evaluated at 
. The matrix 
is the asymptotic variance-covariance matrix of , which is obtained from the second derivative matrix of the likelihood equations. You can display this matrix with the ASYCOV
option in the PROC GLIMMIX
statement.
Finally, let
![\[ E = \sum _{j=1}^{r} \frac{\nu _ j}{\nu _ j - 2}I(\nu _ j > 2) \]](images/statug_glimmix0772.png)
where the indicator function eliminates terms for which 
. The degrees of freedom for F are then computed as
![\[ \nu = \frac{2 E}{E - \mr{rank}(\mb{L})} \]](images/statug_glimmix0774.png)
provided E > r; otherwise 
is set to 0.
In the one-dimensional case, when PROC GLIMMIX computes a t test, the Satterthwaite degrees of freedom for the t statistic
![\[ t = \frac{\mb{l}' \left[ \begin{array}{c} \widehat{\bbeta } \\ \widehat{\bgamma } \end{array} \right]}{\mb{l}'\mb{Cl}} \]](images/statug_glimmix0775.png)
are computed as
![\[ \nu = \frac{2 (\mb{l}' \mb{Cl})^2}{ \mb{g}' \mb{Ag}} \]](images/statug_glimmix0776.png)
where 
is the gradient of 
with respect to , evaluated at .
The calculation of Satterthwaite degrees of freedom requires extra memory to hold q matrices that are the size of the mixed model equations, where q is the number of covariance parameters. Extra computing time is also required to process these matrices. The implemented Satterthwaite method is intended to produce an accurate F approximation; however, the results can differ from those produced by PROC GLM. Also, the small-sample properties of this approximation have not been extensively investigated for the various models available with PROC GLIMMIX.