Transformations - Crop and Soil Science
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Transcript Transformations - Crop and Soil Science
(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!
NY
NY
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