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: darren.james@ars.usda.gov

• 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