1. Generalized Linear Models
(know this)
Generalized Linear Models
An alternative to data transformations
Principle is to make the model fit the data, rather than changing the data to fit the model
Models include link functions that allow heterogeneous variances and nonlinearity
Analysis and estimation are based on maximum likelihood methods
Becoming more widely used - recommended by the experts
Need some understanding of the underlying theory to implement properly
Notes adapted from ASA GLMM Workshop, Long Beach, CA, 2010

2. Generalized Linear Models
ANOVA/Regression model is fit to a non-normal data set
Three elements:
Random component – a probability distribution for Yi from the exponential family of distributions (this is known)
Systematic component – represent the linear predictors (X variables) in the model
Link function – links the random and systematic elements
Form is mean + trt effect
No error term

3. Log of Distribution = “Log-Likelihood”
Binary responses (0 or 1)
Probability of success follows a binomial distribution
Logistic regression – response variables are binary (e.g. alive or dead); predictors can be categorical or continuous
MLE
Make multiple observations on Y and N
Apply likelihood formula for a range of P values
Determine which value of P has the greatest likelihood, given the data set
“canonical parameter” Takes the form Y * function of P

4. Use an inverse function to convert means to the original scale
Example – logit link
µ can only vary from 0 to 1
 can take on any value
Logistic regression – response variables are binary (e.g. alive or dead); predictors can be categorical or continuous
Use an inverse function to convert means to the original scale

5. Some Common Distributions & Link(s)
Log linear models – good for analysis of contingency tables; do not distinguish response and predictor variables – based on a Poisson distribution

6. RBD Mixed Model Analyses with SAS
(know this)
RBD Mixed Model Analyses with SAS
Distribution
Treatments Fixed
Blocks Fixed
Blocks Random
Normal
(continuous)
(PROC GLM)
Linear Model (LM)
(PROC MIXED)
Linear Mixed Model
(LMM)
Non-normal
(categories
or counts)
(PROC GENMOD)
Generalized Linear
Model (GLM)
(PROC GLIMMIX)
Mixed Model
(GLMM)
Mixed Models - contain both random and fixed effects
Note that PROC GLM will only handle LM!
PROC GLIMMIX can handle all of the situations above

7. Linear Models for an RBD in SAS
(know this)
Linear Models for an RBD in SAS
Treatments fixed, Blocks fixed
PROC GLM (normal) or PROC GENMOD (non-normal)
all effects appear in model statement
Model Response = Block Treatment;
Treatments fixed, Blocks random
PROC MIXED (normal) or PROC GLIMMIX (non-normal)
Only fixed effects appear in model statement
Model Response = Treatment;
Random Block;

8. GLIMMIX basic syntax for an RBD
proc glimmix;
class treatment block;
model response = treatment / link=log s dist=poisson;
random block;
lsmeans treatment/ilink diff;
fixed effects go in the model statement
random effects go in the random statement
default means and standard errors from lsmeans statement are on a log scale
ilink option gives back-transformed means on original scale and estimates standard errors on original scale
diff option requests significant tests between all possible pairs of treatments in the trial,

9. Estimation in LMM, GLM, and GLMM
(know this)
Estimation in LMM, GLM, and GLMM
Does not use Least Squares estimation
Does not calculate Sums of Squares or Mean Squares
Estimates are by Maximum Likelihood
Output includes
Source of variation
degrees of freedom
F tests and p-values
Treatment means and standard errors
Comparisons of means and standard errors