The GLIMMIX Procedure
-
Overview
-
Getting Started
-
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
-
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
-
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
| Generalized Linear Models Theory |
A generalized linear model consists of the following:
a linear predictor
a monotonic mapping between the mean of the data and the linear predictor
a response distribution in the exponential family of distributions
A density or mass function in this family can be written as
 |
for some functions 
and 
. The parameter 
is called the natural (canonical) parameter. The parameter 
is a scale parameter, and it is not present in all exponential family distributions. See Table 40.15 for a list of distributions for which 
. In the case where observations are weighted, the scale parameter is replaced with 
in the preceding density (or mass function), where 
is the weight associated with the observation 
.
The mean and variance of the data are related to the components of the density, 
, 
, where primes denote first and second derivatives. If you express as a function of 
, the relationship is known as the natural link or the canonical link function. In other words, modeling data with a canonical link assumes that 
; the effect contributions are additive on the canonical scale. The second derivative of , expressed as a function of , is the variance function of the generalized linear model, 
. Note that because of this relationship, the distribution determines the variance function and the canonical link function. You cannot, however, proceed in the opposite direction. If you provide a user-specified variance function, the GLIMMIX procedure assumes that only the first two moments of the response distribution are known. The full distribution of the data is then unknown and maximum likelihood estimation is not possible. Instead, the GLIMMIX procedure then estimates parameters by quasi-likelihood.