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![Page 1: Today: Lab 9ab due after lecture: CEQ Monday: Quizz 11: review Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions.](https://reader035.fdocuments.us/reader035/viewer/2022062718/56649eb35503460f94bba242/html5/thumbnails/1.jpg)
Today:Lab 9ab dueafter lecture: CEQ
Monday:Quizz 11: review
Wednesday:Guest lecture – Multivariate Analysis
Friday:last lecture: review – Bring questions
DEC 8 – 9am
FINAL EXAMEN 2007
![Page 2: Today: Lab 9ab due after lecture: CEQ Monday: Quizz 11: review Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions.](https://reader035.fdocuments.us/reader035/viewer/2022062718/56649eb35503460f94bba242/html5/thumbnails/2.jpg)
Biology 4605 / 7220 Name ________________
Quiz #10a 19 November 2012
1. What are the 2 main differences between general linear models and generalized linear models?
2. A generalized linear model links a response variable to one or more explanatory variables Xi according to a link function.
![Page 3: Today: Lab 9ab due after lecture: CEQ Monday: Quizz 11: review Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions.](https://reader035.fdocuments.us/reader035/viewer/2022062718/56649eb35503460f94bba242/html5/thumbnails/3.jpg)
Biology 4605 / 7220 Name ________________
Quiz #10a 19 November 2012
1. What are the 2 main differences between general linear models and generalized linear models?
Most common answers:A. Non –normal εB. ANODEV instead of ANOVA tableC. Link function
2. A generalized linear model links a response variable to one or more explanatory variables Xi according to a link function.
conceptual
implementation
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GLM, GzLM, GAM
A few concepts and ideas
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GLM
Model based statistics – we define the response and the explanatory without worrying about the name of the test
![Page 6: Today: Lab 9ab due after lecture: CEQ Monday: Quizz 11: review Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions.](https://reader035.fdocuments.us/reader035/viewer/2022062718/56649eb35503460f94bba242/html5/thumbnails/6.jpg)
GLM
t-test
ANOVA
Simple Linear Regression
Multiple Linear Regression
ANCOVA
GENERAL LINEAR MODELS
ε ~ Normal R: lm()
![Page 7: Today: Lab 9ab due after lecture: CEQ Monday: Quizz 11: review Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions.](https://reader035.fdocuments.us/reader035/viewer/2022062718/56649eb35503460f94bba242/html5/thumbnails/7.jpg)
GLM
An example from Lab 9
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GLM
Do fumigants (treatments) decrease the number of wire worms?
#ww = β0 + βtreatment treatment + βrow row + βcolumn column
treatment fixed
row random
column random
N=25
![Page 9: Today: Lab 9ab due after lecture: CEQ Monday: Quizz 11: review Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions.](https://reader035.fdocuments.us/reader035/viewer/2022062718/56649eb35503460f94bba242/html5/thumbnails/9.jpg)
GLM
0 2 4 6 8 10 12
-4-2
02
4
worm.lm$fitted.values
wor
m.lm
$res
idua
ls
N=25
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GLM
N=25-2 -1 0 1 2
-2-1
01
2
Theoretical Quantiles
Sta
ndar
dize
d re
sidu
als
lm(nw ~ trt + row + col)
Normal Q-Q
4
243
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GLM
N=25worm.lm$residuals
Fre
qu
en
cy
-4 -2 0 2 4
02
46
8
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GLM
N=25-4 -2 0 2 4
-4-2
02
4
worm.lm$residuals[1:24]
wor
m.lm
$res
idua
ls[2
:25]
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GLM
p-value borderline
Normality assumption not met
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GLM
N=25
p-value borderline
Normality assumption not met
n<30
Given that we do not violate the homogeneity assumption, randomizing will likely not change our decision… or will it?
Let’s try prand = 0.0626 (50 000 randomizations)
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GLM
0 1 2 3 4
-21
0
Treatment
Num
ber
of w
ire w
orm
sParameters:
Means with 95% CI
Anything wrong with this analysis?
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GLMResponse variable?
Counts
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GzLMPoisson error
#ww = eμ + ε μ = β0 + βtreatment treatment + βrow row + βcolumn column
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GzLMPoisson error
#ww = eμ + ε μ = β0 + βtreatment treatment + βrow row + βcolumn column
ALL fits > 0
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GzLMPoisson error
0 1 2 3 4
-21
0
Normal error
Treatment
Num
ber
of w
ire w
orm
s
0 1 2 3 4
-21
0
Poisson error
Treatment
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GzLMPoisson error
0 1 2 3 4
-21
0
Normal error
Treatment
Num
ber
of w
ire w
orm
s
0 1 2 3 4
-21
0
Poisson error
Treatment
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t-test
ANOVA
Simple Linear Regression
Multiple Linear Regression
ANCOVA
PoissonBinomial
Negative BinomialGamma
Multinomial
GENERALIZED LINEAR MODELS
Inverse Gaussian
Exponential
GENERAL LINEAR MODELS
ε ~ Normal
Linear combination of parameters
R: lm()
R: glm()
GzLM
![Page 22: Today: Lab 9ab due after lecture: CEQ Monday: Quizz 11: review Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions.](https://reader035.fdocuments.us/reader035/viewer/2022062718/56649eb35503460f94bba242/html5/thumbnails/22.jpg)
GzLM#ww = eμ + ε μ = β0 + βtreatment treatment + βrow row + βcolumn column
Generalized linear models have 3 components:
Systematic
Random
Link function
![Page 23: Today: Lab 9ab due after lecture: CEQ Monday: Quizz 11: review Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions.](https://reader035.fdocuments.us/reader035/viewer/2022062718/56649eb35503460f94bba242/html5/thumbnails/23.jpg)
GzLM#ww = eμ + ε μ = β0 + βtreatment treatment + βrow row + βcolumn column
Generalized linear models have 3 components:
Systematic
linear predictor
Random
Link function
![Page 24: Today: Lab 9ab due after lecture: CEQ Monday: Quizz 11: review Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions.](https://reader035.fdocuments.us/reader035/viewer/2022062718/56649eb35503460f94bba242/html5/thumbnails/24.jpg)
GzLM#ww = eμ + ε μ = β0 + βtreatment treatment + βrow row + βcolumn column
Generalized linear models have 3 components:
Systematic
linear predictor
Random
probability distribution poisson error
Link function
![Page 25: Today: Lab 9ab due after lecture: CEQ Monday: Quizz 11: review Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions.](https://reader035.fdocuments.us/reader035/viewer/2022062718/56649eb35503460f94bba242/html5/thumbnails/25.jpg)
GzLM#ww = eμ + ε μ = β0 + βtreatment treatment + βrow row + βcolumn column
Generalized linear models have 3 components:
Systematic
linear predictor
Random
probability distribution poisson error
Link function
log
![Page 26: Today: Lab 9ab due after lecture: CEQ Monday: Quizz 11: review Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions.](https://reader035.fdocuments.us/reader035/viewer/2022062718/56649eb35503460f94bba242/html5/thumbnails/26.jpg)
GzLM
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GLM
An example from Lab 6
2 4 6 8 10 12
01
02
03
04
0
period
dist
ance
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GLM
Do movements of juvenile cod depend on time of day?
distance = β0 + βperiod period
period categorical
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GLM
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GLM
2 4 6 8 10 12
01
02
03
04
0
period
dist
ance
Anything wrong with this analysis?
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GAM
2 4 6 8 10 12
01
02
03
04
0
Time
Dis
tanc
e
![Page 32: Today: Lab 9ab due after lecture: CEQ Monday: Quizz 11: review Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions.](https://reader035.fdocuments.us/reader035/viewer/2022062718/56649eb35503460f94bba242/html5/thumbnails/32.jpg)
t-test
ANOVA
Simple Linear Regression
Multiple Linear Regression
ANCOVA
PoissonBinomial
Negative BinomialGamma
Multinomial
GENERALIZED LINEAR MODELS
Inverse Gaussian
Exponential
Non-linear effect of covariates
GENERALIZED ADDITIVE MODELS
GENERAL LINEAR MODELS
ε ~ Normal
Linear combination of parameters
R: lm()
R: glm()
R: gam()GAM
![Page 33: Today: Lab 9ab due after lecture: CEQ Monday: Quizz 11: review Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions.](https://reader035.fdocuments.us/reader035/viewer/2022062718/56649eb35503460f94bba242/html5/thumbnails/33.jpg)
GAM
Generalized case of generalized linear models where the systematic component is not necessarily linear
distance ~ s(period)
y ~ s(x1) + s(x2) + x3 + ….
s: smooth function
Spline functions are concerned with good approximation of functions over the whole of a region, and behave in a stable manner
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GAMSmoothing - concept
![Page 35: Today: Lab 9ab due after lecture: CEQ Monday: Quizz 11: review Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions.](https://reader035.fdocuments.us/reader035/viewer/2022062718/56649eb35503460f94bba242/html5/thumbnails/35.jpg)
Degree of smoothness- +
GAM
How much smoothing?
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GAM
![Page 37: Today: Lab 9ab due after lecture: CEQ Monday: Quizz 11: review Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions.](https://reader035.fdocuments.us/reader035/viewer/2022062718/56649eb35503460f94bba242/html5/thumbnails/37.jpg)
t-test
ANOVA
Simple Linear Regression
Multiple Linear Regression
ANCOVA
GENERAL LINEAR MODELS
ε ~ Normal R: lm()
![Page 38: Today: Lab 9ab due after lecture: CEQ Monday: Quizz 11: review Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions.](https://reader035.fdocuments.us/reader035/viewer/2022062718/56649eb35503460f94bba242/html5/thumbnails/38.jpg)
t-test
ANOVA
Simple Linear Regression
Multiple Linear Regression
ANCOVA
PoissonBinomial
Negative BinomialGamma
Multinomial
GENERALIZED LINEAR MODELS
Inverse Gaussian
Exponential
GENERAL LINEAR MODELS
ε ~ Normal
Linear combination of parameters
R: lm()
R: glm()
Non-normal ε
Link function
![Page 39: Today: Lab 9ab due after lecture: CEQ Monday: Quizz 11: review Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions.](https://reader035.fdocuments.us/reader035/viewer/2022062718/56649eb35503460f94bba242/html5/thumbnails/39.jpg)
t-test
ANOVA
Simple Linear Regression
Multiple Linear Regression
ANCOVA
PoissonBinomial
Negative BinomialGamma
Multinomial
GENERALIZED LINEAR MODELS
Inverse Gaussian
Exponential
Non-linear effect of covariates
GENERALIZED ADDITIVE MODELS
GENERAL LINEAR MODELS
ε ~ Normal
Linear combination of parameters
R: lm()
R: glm()
R: gam()
Linear predictor involves sums of smooth functions of covariates