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
Maximum Likelihood
The GLIMMIX procedure forms the log likelihoods of generalized linear models as
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where 
is the log likelihood contribution of the 
th observation with weight 
and 
is the value of the frequency variable. For the determination of and , see the WEIGHT and FREQ statements. The individual log likelihood contributions for the various distributions are as follows.
- Beta
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. See Ferrari and Cribari-Neto (2004). - Binary
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. - Binomial
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where
and 
are the events and trials in the events/trials syntax, and 
. 
. - Exponential
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. - Gamma
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. - Geometric
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. - Inverse Gaussian
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. - "Lognormal"
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.
If you specify DIST=LOGNORMAL with response variable Y, the GLIMMIX procedure assumes that
. Note that the preceding density is not the density of 
. - Multinomial
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. - Negative Binomial
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.
For a given
, the negative binomial distribution is a member of the exponential family. The parameter is related to the scale of the data, because it is part of the variance function. However, it cannot be factored from the variance, as is the case with the 
parameter in many other distributions. The parameter is designated as "Scale" in the "Parameter Estimates" table of the GLIMMIX procedure. - Normal (Gaussian)
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. - Poisson
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. - Shifted T
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.
Define the parameter vector for the generalized linear model as 
, if , and as 
otherwise. 
denotes the fixed-effects parameters in the linear predictor. For the negative binomial distribution, the relevant parameter vector is 
. The gradient and Hessian of the negative log likelihood are then
 |
The GLIMMIX procedure computes the gradient vector and Hessian matrix analytically, unless your programming statements involve functions whose derivatives are determined by finite differences. If the procedure is in scoring mode, 
is replaced by its expected value. PROC GLIMMIX is in scoring mode when the number 
of SCORING= iterations has not been exceeded and the optimization technique uses second derivatives, or when the Hessian is computed at convergence and the EXPHESSIAN option is in effect. Note that the objective function is the negative log likelihood when the GLIMMIX procedure fits a GLM model. The procedure performs a minimization problem in this case.
In models for independent data with known distribution, parameter estimates are obtained by the method of maximum likelihood. No parameters are profiled from the optimization. The default optimization technique for GLMs is the Newton-Raphson algorithm, except for Gaussian models with identity link, which do not require iterative model fitting. In the case of a Gaussian model, the scale parameter is estimated by restricted maximum likelihood, because this estimate is unbiased. The results from the GLIMMIX procedure agree with those from the GLM and REG procedure for such models. You can obtain the maximum likelihood estimate of the scale parameter with the NOREML option in the PROC GLIMMIX statement. To change the optimization algorithm, use the TECHNIQUE= option in the NLOPTIONS statement.
Standard errors of the parameter estimates are obtained from the inverse of the (observed or expected) second derivative matrix .