Precursors GLMMs Results Conclusions References Generalized linear mixed models for ecologists: coping with non-normal, spatially and temporally correlated data Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology 30 August 2011 Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology GLMMs
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Precursors GLMMs Results Conclusions References
Generalized linear mixed models for ecologists:coping with non-normal, spatially and temporally
correlated data
Ben Bolker
McMaster UniversityDepartments of Mathematics & Statistics and Biology
30 August 2011
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
GLMMs
Precursors GLMMs Results Conclusions References
Outline
1 PrecursorsExamplesDefinitionsANOVA vs. (G)LMMs
2 GLMMsEstimationInference
3 ResultsCoral symbiontsGlyceraArabidopsis
4 Conclusions
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
GLMMs
Precursors GLMMs Results Conclusions References
Outline
1 PrecursorsExamplesDefinitionsANOVA vs. (G)LMMs
2 GLMMsEstimationInference
3 ResultsCoral symbiontsGlyceraArabidopsis
4 Conclusions
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
GLMMs
Precursors GLMMs Results Conclusions References
Examples
Coral protection by symbionts
none shrimp crabs both
Number of predation events
Symbionts
Num
ber
of b
lock
s
0
2
4
6
8
10
1
2
0
1
2
0
2
0
1
2
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
GLMMs
Precursors GLMMs Results Conclusions References
Examples
Environmental stress: Glycera cell survival
H2S
Cop
per
0
33.3
66.6
133.3
0 0.03 0.1 0.32
Osm=12.8Normoxia
Osm=22.4Normoxia
0 0.03 0.1 0.32
Osm=32Normoxia
Osm=41.6Normoxia
0 0.03 0.1 0.32
Osm=51.2Normoxia
Osm=12.8Anoxia
0 0.03 0.1 0.32
Osm=22.4Anoxia
Osm=32Anoxia
0 0.03 0.1 0.32
Osm=41.6Anoxia
0
33.3
66.6
133.3
Osm=51.2Anoxia
0.0
0.2
0.4
0.6
0.8
1.0
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
GLMMs
Precursors GLMMs Results Conclusions References
Examples
Arabidopsis response to fertilization & clipping
panel: nutrient, color: genotype
Log(
1+fr
uit s
et)
0
1
2
3
4
5
unclipped clipped
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: nutrient 8
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
non-linearity: log/exponential, logit/logistic:link function L
flexibility via linear predictor: L(response) = a + bi + cx . . .
stable, robust, fast, easy to use
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
GLMMs
Precursors GLMMs Results Conclusions References
Definitions
Random vs. fixed effects
Fixed effects (FE) Interested in specific levels (“treatments”)
Random effects (RE):2
Interested in distribution (“blocks”)ExperimentalTemporal, spatialGenera, species, genotypesIndividuals (“repeated measures”)inference on population of blocks(blocks randomly selected?)(large number of blocks [> 5 − 7]?)
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
GLMMs
Precursors GLMMs Results Conclusions References
Definitions
Random vs. fixed effects
Fixed effects (FE) Interested in specific levels (“treatments”)
Random effects (RE):2
Interested in distribution (“blocks”)ExperimentalTemporal, spatialGenera, species, genotypesIndividuals (“repeated measures”)inference on population of blocks(blocks randomly selected?)(large number of blocks [> 5 − 7]?)
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
GLMMs
Precursors GLMMs Results Conclusions References
Definitions
Random vs. fixed effects
Fixed effects (FE) Interested in specific levels (“treatments”)
Random effects (RE):2
Interested in distribution (“blocks”)ExperimentalTemporal, spatialGenera, species, genotypesIndividuals (“repeated measures”)inference on population of blocks(blocks randomly selected?)(large number of blocks [> 5 − 7]?)
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
GLMMs
Precursors GLMMs Results Conclusions References
Definitions
Random vs. fixed effects
Fixed effects (FE) Interested in specific levels (“treatments”)
Random effects (RE):2
Interested in distribution (“blocks”)ExperimentalTemporal, spatialGenera, species, genotypesIndividuals (“repeated measures”)inference on population of blocks(blocks randomly selected?)(large number of blocks [> 5 − 7]?)
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
GLMMs
Precursors GLMMs Results Conclusions References
ANOVA vs. (G)LMMs
Mixed models: classical approach
traditional approach tonon-independence
nested, randomized block,split-plot . . .
sum-of-squaresdecomposition/ANOVA:figure out treatment SSQ/df,error SQ/df
3
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
GLMMs
Precursors GLMMs Results Conclusions References
ANOVA vs. (G)LMMs
You can use an ANOVA if . . .
data are normal(or can be transformed)
responses are linear
design is (nearly) balanced
simple design (single or nested REs)(not crossed REs: e.g. year effects that apply across all spatialblocks)
no spatial or temporal correlation within blocks
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
GLMMs
Precursors GLMMs Results Conclusions References
ANOVA vs. (G)LMMs
“Modern” mixed models
Data still normal(izable), linear, butunbalanced/crossed/correlated
Balance(dispersion of observation around block mean)with(dispersion of block means around overall average)
Good for large, messy data. . . and when variation is interesting
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
GLMMs
Precursors GLMMs Results Conclusions References
ANOVA vs. (G)LMMs
Shrinkage (Arabidopsis)
● ●
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Arabidopsis block estimates
Genotype
Mea
n(lo
g) fr
uit s
et
0 5 10 15 20 25
−15
−3
0
3
●●
●
●●
●●
● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●
32
10
810
43 9 9 4 6
4 2 6 10 5 7 9 4 9 11 2 5 5
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
Data: Josh Banta and Massimo Pigliucci (Arabidopsis);Adrian Stier and Seabird McKeon (coral symbionts); CourtneyKagan, Jocelynn Ortega, David Julian (Glycera);
Co-authors: Mollie Brooks, Connie Clark, Shane Geange, JohnPoulsen, Hank Stevens, Jada White
Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology
GLMMs
Precursors GLMMs Results Conclusions References
[1] Breslow NE, 2004. In DY Lin & PJ Heagerty,eds., Proceedings of the second Seattlesymposium in biostatistics: Analysis of correlateddata, pp. 1–22. Springer. ISBN 0387208623.
[2] Gelman A, 2005. Annals of Statistics, 33(1):1–53.doi:doi:10.1214/009053604000001048.
[3] Gotelli NJ & Ellison AM, 2004. A Primer ofEcological Statistics. Sinauer, Sunderland, MA.
[4] Greven S, 2008. Non-Standard Problems inInference for Additive and Linear Mixed Models.Cuvillier Verlag, Gottingen, Germany. ISBN3867274916. URL http://www.cuvillier.de/
flycms/en/html/30/-UickI3zKPS,3cEY=
/Buchdetails.html?SID=wVZnpL8f0fbc.
[5] Greven S & Kneib T, 2010. Biometrika,97(4):773–789. URL http:
[8] Kenward MG & Roger JH, 1997. Biometrics,53(3):983–997.
[9] Latimer AM, Banerjee S et al., 2009. EcologyLetters, 12(2):144–154.
[10] Ozgul A, Oli MK et al., Apr. 2009. EcologicalApplications: A Publication of the EcologicalSociety of America, 19(3):786–798. ISSN1051-0761. URL http: