From Linear to Generalized Linear Mixed Models: A Case Study in Repeated Measures
Compared to traditional linear mixed models, generalized linear mixed models (GLMMs) can offer better correspondence between response variables and explanatory models, yielding more efficient estimates and tests in the analysis of data from designed experiments. Using proportion data from a designed experiment with repeated measures, results from several candidate GLMMs implemented with different distributions (binomial and beta), likelihood estimation methods, covariance structures, and bias correction methods, are compared.
1996: ~10% grass cover 2009: ~10-40% grass cover
Darren James, USDA-ARS Jornada Experimental Range, New Mexico State University Email: [email protected]
• Design: 2 x 3 factorial in RCB • Experimental units: 18 1-acre plots • Response: grass foliar cover sampled at 3 times:
• 1996 (pre-trt); 2002, 2009 (post-trt)
P-values from Type III tests of fixed effects for well-performing models
Model: 1. Normal 2. Normal,
transformed 3. Binomial
R-side 4. Beta R-side
5. Binomial G-side
6. Beta G-side
Likelihood Estimation Method: REML REML RSPL RSPL Laplace Laplace Covariance Structure: UN UN UN UN UN ARH(1)
Bias Correction Method (DDFM): KR KR KR KR MBN MBN block 0.0952 0.113 0.2254 0.1457 0.3373 0.3491 block*year 0.1179 0.1144 0.2913 0.1775 0.4541 0.4668 year <.0001 <.0001 <.0001 <.0001 <.0001 <.0001 shrub treatment 0.0075 0.0096 0.0092 0.0036 0.0478 0.0397 year*shrub treatment 0.6154 0.752 0.7444 0.669 0.8739 0.8524 grazing treatment <.0001 <.0001 <.0001 <.0001 <.0001 <.0001 year*grazing treatment <.0001 <.0001 <.0001 <.0001 <.0001 <.0001 shrub treatment * grazing treatment 0.7152 0.819 0.6488 0.556 0.8204 0.8286 year*shrub treatment * grazing treatment 0.4026 0.8139 0.9987 0.9976 0.9989 0.999
Constructing an appropriate GLMM is not trivial, especially the case of repeated measures. The numerical estimation procedures GLMMs utilize can easily produce intractable or nonsensical results that are difficult to diagnose and rectify. Many common adjustments and modeling decisions fundamentally change the model’s inference space and alter appropriate interpretations of model parameters. Modelers must also confront mean-variance dependency, important differences between conditional (“G-side”) and marginal (“R-side”) formulations of random effects, and how to implement bias correction.
The map above shows the experimental layout. Repeated measured models must account for high between-year variability in the response variable.
Full Likelihood Quasi-likelihood
Linearization Integral Approximation
Pseudo-Likelihood
Penalized Quasi-likelihood
MLE (MSPL) REML
(RMPL) MLE
(MMPL)
REML (RSPL)
Full Integral
MLE REML
Laplace with MLE
Gauss-Hermite
Quadrature with MLE
Monte Carlo/ Markov Chain Monte Carlo (available with the MCMC
procedure but not with GLMMIX)
Likelihood Estimation methods available in the SAS GLIMMIX procedure
Likelihood Specification
Likelihood Estimation