Trondheim glmm

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


Outline
1 Precursors
Examples
Deﬁnitions
ANOVA vs. (G)LMMs
2 GLMMs
Estimation
Inference
3 Results
Coral symbionts
Glycera
Arabidopsis
4 Conclusions

GLMMs


Examples

Coral protection by symbionts

Number of predation events
10

8 2
Number of blocks

2
2
6 2
1
1
4
0
2 0 0
1
0
none shrimp crabs both

Symbionts

GLMMs


Examples

Environmental stress: Glycera cell survival
0 0.03 0.1 0.32 0 0.03 0.1 0.32

Anoxia Anoxia Anoxia Anoxia Anoxia
Osm=12.8 Osm=22.4 Osm=32 Osm=41.6 Osm=51.2 1.0

133.3

66.6 0.8

33.3

0.6
0
Copper

Normoxia Normoxia Normoxia Normoxia Normoxia
Osm=12.8 Osm=22.4 Osm=32 Osm=41.6 Osm=51.2
0.4

133.3

66.6
0.2

33.3

0 0.0

0 0.03 0.1 0.32 0 0.03 0.1 0.32 0 0.03 0.1 0.32

H2S

GLMMs


Examples

Arabidopsis response to fertilization & clipping
panel: nutrient, color: genotype

nutrient : 1 nutrient : 8
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Log(1+fruit set)

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unclipped clipped unclipped clipped

GLMMs


Deﬁnitions

Generalized linear models (GLMs)

non-normal data: binary, binomial,
count (Poisson/negative binomial)
non-linearity: log/exponential, logit/logistic:
link function L
ﬂexibility via linear predictor: L(response) = a + bi + cx . . .
stable, robust, fast, easy to use

GLMMs


Definitions

Random vs. fixed effects

Fixed effects (FE) Interested in specific levels (
“treatments”)
Random effects (RE): 2
Interested in distribution (
“blocks”)
Experimental
Temporal, spatial
Genera, species, genotypes
Individuals (
“repeated measures” )
inference on population of blocks
(blocks randomly selected?)
(large number of blocks [> 5 − 7]?)

GLMMs


ANOVA vs. (G)LMMs

Mixed models: classical approach

traditional approach to
non-independence
nested, randomized block,
split-plot . . .
sum-of-squares
decomposition/ANOVA:
ﬁgure out treatment SSQ/df,
error SQ/df
3

GLMMs


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 spatial
blocks)
no spatial or temporal correlation within blocks

GLMMs


ANOVA vs. (G)LMMs

“Modern” mixed models

Data still normal(izable), linear, but
unbalanced/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

GLMMs


ANOVA vs. (G)LMMs

Shrinkage (Arabidopsis)

Arabidopsis block estimates
5
11 2 5
7 9 4 9 q
3 6 10 5 q q q
4 2 q q q q
6 q q q
3
9 9 4 q q
q q q
Mean(log) fruit set

4 q q
10
8 q
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2 q
0 3
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10
q q
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−3

−15 q q

0 5 10 15 20 25

Genotype

GLMMs


ANOVA vs. (G)LMMs

Shrinkage (sparrows)
q

q q
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0.80 q
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q island
0.78 q
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q Hestmannøy
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q Sleneset
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q Gjerøy
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0.76 q q Indre Kvarøy
Heterozygosity

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q Husøy
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q Selvær
0.74 q Ytre Kvarøy
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q q q Aldra
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q q Myken
0.72 q
q Lovund
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q Onøy
q Nesøy
0.70 q Lurøy
q Sundøy
q q
0.68 q

2.0 2.5 3.0 3.5 4.0 4.5
Log(harmonic mean pop size)

GLMMs


ANOVA vs. (G)LMMs

GLMMs

Data not normal(izable), nonlinear
Standard distributions (Poisson, binomial etc.)
Speciﬁc forms of nonlinearity (exponential, logistic etc.)
Conceptually v. similar to LMMs, but harder

GLMMs


ANOVA vs. (G)LMMs

Challenges

Small # RE levels (<5–6)
Big data (> 1000 observations)
Spatial/temporal correlation structure (in GLMMs)
Unusual distributions of data (in GLMMs)

GLMMs


Estimation

Penalized quasi-likelihood (PQL) 1

ﬂexible (e.g. handles spatial/temporal correlations)
least accurate: biased for small samples (low counts per block)
SAS PROC GLIMMIX, R MASS:glmmPQL

GLMMs


Estimation

Laplace and Gauss-Hermite quadrature

more accurate than PQL: speed/accuracy tradeoﬀ
lme4:glmer, glmmML, glmmADMB, R2ADMB (AD Model Builder,
gamlss.mx:gamlssNP, repeated

GLMMs


Estimation

Bayesian approaches

usually slow but ﬂexible
best conﬁdence intervals
must specify priors, assess convergence
specialized: glmmAK, MCMCglmm 6 , INLA
general: BUGS (glmmBUGS, R2WinBUGS, BRugs, WinBUGS,
OpenBUGS, R2jags, rjags, JAGS)

GLMMs


Estimation

Extensions

Overdispersion Variance > expected from statistical model
Quasi-likelihood MASS:glmmPQL;
overdispersed distributions (e.g. negative
binomial): glmmADMB, gamlss.mx:gamlssNP;
observation-level random effects (e.g.
lognormal-Poisson): lme4, MCMCglmm
Zero-inflation Overabundance of zeros in a discrete distribution
zero-inflated models: glmmADMB, MCMCglmm
hurdle models: MCMCglmm

GLMMs


Inference

Wald tests/CIs

Widely available (e.g. summary())
Assume data set is large/well-behaved
Always approximate, sometimes awful; bad for variance
estimates

GLMMs


Inference

Likelihood ratio tests

Compare models (easy)
Conﬁdence intervals — expensive and rarely available
(lme4a for LMMs)
Asymptotic assumption
LMMs: F tests; estimate “equivalent” denominator df?
approximations 8;13 : doBy:KRmodcomp
don’t really know what to do for GLMMs
OK if number obs number of parameters and
large # of blocks . . .

GLMMs


Inference

Information-theoretic approaches

Above issues apply, but less well understood 4;5;7;11 :
AIC is asymptotic too
For comparing models with diﬀerent REs,
or for AICc , what is p?
“Level of focus” issue: what are you trying to predict? 5;14;15
(cAIC)

GLMMs


Inference

Bootstrapping

1 fit null model to data
2 simulate “data” from null model
3 fit null and working model, compute likelihood difference
4 repeat to estimate null distribution

simulate/refit methods; bootMer in lme4a (LMMs only!),
doBy:PBModComp, or “by hand”:
> pboot <- function(m0, m1) {
s <- simulate(m0)
2 * (logLik(refit(m1, s)) - logLik(refit(m0, s)))
}
> replicate(1000, pboot(fm2, fm1))

GLMMs


Inference

Bayesian inference

CIs, prediction intervals etc. computationally “free” after
estimation
Post hoc MCMC sampling:
(glmmADMB, R2ADMB, lme4:MCMCsamp)

GLMMs


Inference

Bottom line

Large data: computation slow, inference easy
Bayesian computation slow, inference easy
Small data: computation fast
Problems with zero variance (blme), correlations = ±1
Bootstrapping for inference?

GLMMs


Coral symbionts

Coral symbionts: comparison of results

Regression estimates
−6 −4 −2 0 2

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Added symbiont q
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Crab vs. Shrimp q
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q q GLM (fixed)
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q GLM (pooled)
Symbiont q
q q PQL
q q Laplace
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q AGQ

GLMMs


Glycera

Glycera ﬁt comparisons

qq qq
Osm:Cu:H2S:Anoxia q
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Cu:H2S:Anoxia q q
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Osm:H2S:Anoxia q
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Osm:Cu:Anoxia q

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Osm:Cu:H2S q
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H2S:Anoxia
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Cu:Anoxia q
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Osm:Anoxia qq
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Cu:H2S q
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Osm:H2S qq
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Osm:Cu q
q MCMCglmm
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Anoxia q q glmer(OD:2)
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H2S q
q q glmer(OD)
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Cu q
q q glmmML
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Osm
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q glmer

−60 −40 −20 0 20 40 60

Effect on survival (logit)

GLMMs


Glycera

Glycera: parametric bootstrap results
Osm Cu

0.5

0.1
0.05

0.01
0.005
Inferred p value

variable
0.001
normal
H2S Anoxia
t7

0.5 t14

0.1
0.05

0.01
0.005

0.001

0.001 0.0050.01 0.05 0.1 0.5 0.001 0.0050.01 0.05 0.1 0.5

True p value

GLMMs


Arabidopsis

Arabidopsis results

Regression estimates
−1.0 0.0 1.0

statusTransplant q

statusPetri.Plate q

rack2 q

nutrient8:amdclipped q

amdclipped q

nutrient8 q

GLMMs


What about space and/or time?

if in blocks, no problem (crossed random eﬀects) 10
test residuals, try to fail to reject NH of no autocorrelation
if normal (LMM), corStruct in lme, spdep
otherwise . . . spatcounts, geoRglm, geoBUGS, . . . ???
big data 9

GLMMs


Primary tools

Special-purpose:
lme4: multiple/crossed REs, (profiling): fast
MCMCglmm: Bayesian, fairly flexible
glmmADMB: negative binomial, zero-inflated etc.
General-purpose:
AD Model Builder (and interfaces)
BUGS/JAGS (and interfaces)
INLA 12
Tools are getting better, but still not easy!
Info: http://glmm.wikidot.com
Slides: http://www.slideshare.net/bbolker

GLMMs


Acknowledgements

Funding: NSF, NSERC, NCEAS
Data: Josh Banta and Massimo Pigliucci (Arabidopsis);
Adrian Stier and Seabird McKeon (coral symbionts); Courtney
Kagan, Jocelynn Ortega, David Julian (Glycera);
Co-authors: Mollie Brooks, Connie Clark, Shane Geange, John
Poulsen, Hank Stevens, Jada White

GLMMs


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GLMMs


Extras

Spatial and temporal correlation (R-side eﬀects):
MASS:glmmPQL (sort of), GLMMarp, INLA;
WinBUGS, AD Model Builder
Additive models: amer, gamm4, mgcv, lmeSplines
Ordinal models: ordinal
Population genetics: pedigreemm, kinship
Survival: coxme, kinship, phmm

GLMMs

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