(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
Generalized Linear Models
ANOVA/Regression model is fit to a non-normal data set
Three elements:
1. Random component – a probability distribution for Yi from
the exponential family of distributions (this is known)
2. Systematic component – represent the linear predictors
(X variables) in the model
i    i
Form is mean + trt effect
No error term
3. Link function – links the random and systematic elements
i  g(i )
Log of Distribution = “Log-Likelihood”
 Binary responses (0 or 1)
 Probability of success follows a binomial distribution
Y
 N Y
N!
NY
NY

P 1  P 
  P 1  P 
Y! N  Y !
Y
 N  Y

N Y 
log   P 1  P  

 Y 
N
 P 
 Y log 
  N log(1  P )   log  
1 P 
Y
“canonical parameter” Takes the form Y * function of P
Example – logit link
  
link    log 

1  
µ can only vary from 0 to 1
 can take on any value
Use an inverse function to convert means to
the original scale

e


1 e
Some Common Distributions & Link(s)
Distribution
Variable
Type
Mean
Variance
Common
Link(s)
Normal
Continuous

2
Identity =
Binomial
Discrete
proportion
N(1  )
logit
probit
Poisson
Discrete
count


=log()

2
log(), 1/
Exponential Continuous


N
(know this)
RBD Mixed Model Analyses with SAS
Distribution
Treatments Fixed
Blocks Fixed
Treatments 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)
Generalized Linear
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
(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;
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,
(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