open source glmm tools: concordia

Precursors GLMMs Results Conclusions References

Open-source tools for estimation and inferenceusing generalized linear mixed models

Ben Bolker

McMaster UniversityDepartments of Mathematics & Statistics and Biology

3 July 2011

Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology

Open-source GLMMs


Outline

1 PrecursorsDefinitionsExamplesChallenges

2 GLMMsEstimationInference

3 ResultsCoral symbiontsGlyceraArabidopsis

4 ConclusionsConclusions


Open-source GLMMs


Definitions

Outline






Open-source GLMMs


Definitions

Definitions

Fixed effects (FE) Predictors where interest is in specific levels

Random effects (RE) Predictors where interest is in distributionrather than levels (blocks)5

Mixed models Statistical models with both FEs and REs

Linear mixed models Linear effects, normal responses, normal REs

Generalized linear models Linearizable effects, exponential-familyresponses, normal REs (on linearized scale)

Generalized linear mixed models GLMMs = LMMs + GLMs


Open-source GLMMs


Definitions

GLMMs

Distributions from exponential family(Poisson, binomial, Gaussian, Gamma, NegBinom(k), . . . )

Means = linear functions of predictorson scale of link function (identity, log, logit, . . . )


Open-source GLMMs


Definitions

GLMMs (cont.)

Linear predictor:η = Xβ + Zu

Random effects:u ∼ MVN(0,Σ)

Response:

Y ∼ D(g−1η, φ

)(φ often ≡ 1)


Open-source GLMMs


Definitions

Marginal likelihood

Likelihood (Prob(data|parameters)) — requires integrating overpossible values of REs to get marginal likelihood e.g.:

likelihood of i th obs. in block j is L(xij |θi , σ2w )

likelihood of a particular block mean θj is L(θj |0, σ2b)

marginal likelihood is∫L(xij |θj , σ2

w )L(θj |0, σ2b) dθj

Balance (dispersion of RE around 0) with (dispersion of dataconditional on RE)


Open-source GLMMs


Definitions

Bayesian solution?

Bayesians should not feel smug: they are stuck with thenormalizing constant

Posterior(β,θ,Σ) =Prior(β,θ,Σ)L(xij |β,θ)L(θ|Σ)∫∫∫

(. . .)dβ dθ dΣ(!!)

and similar issues with marginal posteriors


Open-source GLMMs


Examples

Outline






Open-source GLMMs


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


Open-source GLMMs


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


Open-source GLMMs


Examples

Arabidopsis response to fertilization & clipping

panel: nutrient, color: genotype

Log(

1+fr

uit s

et)

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5

unclipped clipped

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unclipped clipped

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: nutrient 8


Open-source GLMMs


Challenges

Outline






Open-source GLMMs


Challenges

Data challenges: estimation

Small # RE levels (<5–6) [modes at zero]

Crossed REs [unusual setup]

Spatial/temporal correlation structure (“R-side” effects)

Overdispersion

Unusual distributions (Gamma, negative binomial . . . )


Open-source GLMMs


Challenges

Data challenges: computation

Large n (of course)

Multiple REs (dimensionality)

Crossed REs


Open-source GLMMs


Challenges

Inference

Any departures from classical LMMs

Small N (<40)

Small n

Inference on components of Σ (boundary effects, df)


Open-source GLMMs


Challenges

RE examples

Coral symbionts: simple experimental blocks, RE affectsintercept (overall probability of predation in block)

Glycera: applied to cells from 10 individuals, RE again affectsintercept (cell survival prob.)

Arabidopsis: region (3 levels, treated as fixed) / population /genotype: affects intercept (overall fruit set) as well astreatment effects (nutrients, herbivory, interaction)


Open-source GLMMs


Estimation

Outline






Open-source GLMMs


Estimation

Penalized quasi-likelihood (PQL)

alternate steps of estimating GLM using known RE variancesto calculate weights; estimate LMMs given GLM fit2

flexible (e.g. spatial/temporal correlations)

biased for small unit samples (e.g. counts < 5, binary orlow-survival data)

widely used: SAS PROC GLIMMIX, R MASS:glmmPQL

marginal models: generalized estimating equations(geepack, geese)


Open-source GLMMs

http://finzi.psych.upenn.edu/R/library/MASS/html/glmmPQL.html

http://cran.r-project.org/web/packages/geepack

http://cran.r-project.org/web/packages/geese


Estimation

Laplace approximation

approximate marginal likelihood

for given β, θ find conditional modes by penalized, iteratedreweighted least squares; then use second-order Taylorexpansion around the conditional modes

more accurate than PQL

reasonably fast and flexible

lme4:glmer, glmmML, glmmADMB, R2ADMB (AD Model Builder)


Open-source GLMMs

http://finzi.psych.upenn.edu/R/library/lme4/html/glmer.html

http://cran.r-project.org/web/packages/glmmML

https://r-forge.r-project.org/projects/glmmADMB/

https://r-forge.r-project.org/projects/R2ADMB/

http://admb-project.org/


Estimation

(adaptive) Gauss-Hermite quadrature (AGHQ)

as above, but compute additional terms in the integral(typically 8, but often up to 20)

most accurate

slowest, hence not flexible (2–3 RE at most, maybe only 1)

lme4:glmer, glmmML, gamlss.mx:gamlssNP, repeated


Open-source GLMMs

http://finzi.psych.upenn.edu/R/library/lme4/html/glmer.html

http://cran.r-project.org/web/packages/glmmML

http://finzi.psych.upenn.edu/R/library/gamlss.mx/html/gamlssNP.html

http://www.commanster.eu/rcode.html


Estimation

Variations

Hierarchical GLMS (hglm, HGLMMM)

Monte Carlo methods: MCEM1, MCMLE (bernor)18,sequential MC (pomp), data cloning (dclone)


Open-source GLMMs

http://cran.r-project.org/web/packages/hglm

http://cran.r-project.org/web/packages/HGLMMM

http://cran.r-project.org/web/packages/bernor

http://cran.r-project.org/web/packages/pomp

http://cran.r-project.org/web/packages/dclone


Estimation

Bayesian approaches

Monte Carlo approaches: MCMC (Gibbs sampling,Metropolis-Hastings, etc.)

slow but flexible

makes marginal inference easy

must specify priors, assess convergence

specialized: glmmAK, MCMCglmm9, INLA

general: BUGS (glmmBUGS, R2WinBUGS, BRugs, WinBUGS,OpenBUGS, R2jags, rjags, JAGS)


Open-source GLMMs

http://cran.r-project.org/web/packages/glmmAK

http://cran.r-project.org/web/packages/MCMCglmm

http://www.r-inla.org/

http://cran.r-project.org/web/packages/glmmBUGS

http://cran.r-project.org/web/packages/R2WinBUGS

http://www.biostat.umn.edu/~brad/software/BRugs

http://www.openbugs.info/w/

http://cran.r-project.org/web/packages/R2jags

http://cran.r-project.org/web/packages/rjags

http://www-ice.iarc.fr/~martyn/software/jags/


Estimation

Extensions

Overdispersion Variance > expected from statistical model

Quasi-likelihood approaches: MASS:glmmPQL

Extended distributions (e.g. negative binomial):glmmADMB, gamlss.mx:gamlssNPObservation-level random effects (e.g.lognormal-Poisson): lme4

Zero-inflation Overabundance of zeros in a discrete distribution

zero-inflated models: glmmADMB, MCMCglmmhurdle models: MCMCglmm


Open-source GLMMs



http://finzi.psych.upenn.edu/R/library/gamlss.mx/html/gamlssNP.html

http://cran.r-project.org/web/packages/lme4

http://cran.r-project.org/web/packages/glmmADMB




Inference

Outline






Open-source GLMMs


Inference

Wald tests/CIs

Easy (e.g. typical results of summary): assume quadraticsurface, based on information matrix @ MLE

always approximate, sometimes awful (Hauck-Donner effect)

often bad for variance estimates

available from most direct-maximization packages


Open-source GLMMs


Inference

Likelihood ratio tests/profile confidence intervals

Model comparison is relatively easy

Profiling is expensive — and not (yet) available . . . (lme4a forLMMs)

in GLM(M) case, numerator is only asymptotically χ2 anyway:Bartlett corrections3;4, higher-order asymptotics: cond

[neither extended to GLMMs!]

OK if N − n, N � 40?


Open-source GLMMs

http://lme4.r-project.org/

http://cran.r-project.org/web/packages/cond


Inference

Conditional F tests

What if scale parameter (φ) is estimated(e.g. Gaussian, Gamma, quasi-likelihood) ?

In classical LMMs, −2 log L ∼ F (ν1, ν2)

For non-classical LMMs (unbalanced, crossed, R-side) orGLMMs, ν2 poorly defined:Kenward-Roger, Satterthwaite approximations12;16

unimplemented except in SAS (partially in Genstat)


Open-source GLMMs


Inference

Tests/CIs of variances [boundary problems]

LRT depends on null hypothesis being within the parameter’sfeasible range6;13

violated e.g. by H0 : σ2 = 0

In simple cases null distribution is a mixture of χ2

distributions (e.g. 0.5χ20 + 0.5χ2

1: emdbook:dchibarsq)

simulation-based testing: RLRsim


Open-source GLMMs

http://finzi.psych.upenn.edu/R/library/emdbook/html/dchibarsq.html

http://cran.r-project.org/web/packages/RLRsim


Inference

Information-theoretic approaches

Above issues apply, but less well understood:7;8

AIC is asymptotic

“corrected” AIC (AICc)10 derived for linear models, widelyused but not tested elsewhere14

For comparing models with different REs,or for AICc , what is p? conditional AIC:8;19 (cAIC) (level offocus issue: see also Deviance Information Criterion (DIC,17)


Open-source GLMMs

http://www.bepress.com/cgi/viewcontent.cgi?filename=1&article=1202&context=jhubiostat&type=additional


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

confidence intervals?

simulate/refit methods; bootMer in lme4a (LMMs only!)

> pboot <- function(m0, m1) {

s <- simulate(m0)

2 * (logLik(refit(m1, s)) - logLik(refit(m0, s)))

}

> replicate(1000, pboot(fm2, fm1))


Open-source GLMMs

https://r-forge.r-project.org/projects/lme4a/


Inference

Bayesian inference

Marginal highest posterior density intervals (or quantiles)

Computationally “free” with results of stochastic Bayesiancomputation

Easily extended to prediction intervals etc. etc.

Post hoc Markov chain Monte Carlo sampling available forsome packages (glmmADMB, R2ADMB, eventually lme4a)


Open-source GLMMs


https://r-forge.r-project.org/projects/R2ADMB/

http://lme4.r-project.org/


Inference

Bottom line

Large data: computation slow (maximization methodsfastest), inference easy (asymptotics)

Bayesian computation slow, inference easy (posterior samples)

Small data: computation fast

RE variances may be poorly estimated/set to zero (upcoming:penalty/prior term in blmer within arm)inference tricky, may need bootstrapping


Open-source GLMMs

http://cran.r-project.org/web/packages/arm


Coral symbionts

Coral symbionts: comparison of results

Regression estimates−6 −4 −2 0 2

Symbiont

Crab vs. Shrimp

Added symbiont

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GLM (fixed)GLM (pooled)PQLLaplaceAGQ


Open-source GLMMs


Glycera

Glycera fit comparisons

Effect on survival (logit)

−60 −40 −20 0 20 40 60

Osm

Cu

H2S

Anoxia

Osm:Cu

Osm:H2S

Cu:H2S

Osm:Anoxia

Cu:Anoxia

H2S:Anoxia

Osm:Cu:H2S

Osm:Cu:Anoxia

Osm:H2S:Anoxia

Cu:H2S:Anoxia

Osm:Cu:H2S:Anoxia

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MCMCglmmglmer(OD:2)glmer(OD)glmmMLglmer


Open-source GLMMs


Glycera

Glycera: MCMCglmm fit

Effect on survival

−20 −10 0 10 20 30

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H2SCu

Osm

Oxygen

Cu : H2S

Osm : Oxygen

Cu : Oxygen

Osm : H2S

H2S : OxygenOsm : Cu

Osm : Cu : H2S

Cu : H2S : Oxygen

Osm : H2S : Oxygen

Osm : Cu : Oxygen

Osm : Cu : H2S : Oxygen

main effects

2−way

3−way

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Open-source GLMMs


Glycera

Parametric bootstrap results

True p value

Infe

rred

p v

alue

0.001

0.005

0.01

0.05

0.1

0.5

0.001

0.005

0.01

0.05

0.1

0.5

Osm

H2S

0.001 0.0050.01 0.05 0.1 0.5

Cu

Anoxia

0.001 0.0050.01 0.05 0.1 0.5

variable

normal

t7

t14


Open-source GLMMs


Arabidopsis

Arabidopsis: AIC comparison of RE models

∆AIC

clip(gen) X clip(popu)clip(gen) X nut(popu)clip(gen) X int(popu)

nut(gen) X clip(popu)nut(gen) X nut(popu)nut(gen) X int(popu)int(gen) X clip(popu)int(gen) X nut(popu)int(gen) X int(popu)

int(popu)nointeract

0 2 4 6

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Open-source GLMMs


Arabidopsis

Arabidopsis: fits with and without nutrient(genotype)

Regression estimates−1.0 −0.5 0.0 0.5 1.0 1.5

nutrient8

amdclipped

rack2

statusPetri.Plate

statusTransplant

nutrient8:amdclipped

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Open-source GLMMs


Conclusions

Primary tools

lme4: multiple/crossed REs, (profiling): fast

MCMCglmm: Bayesian, very flexible

glmmADMB: negative binomial, zero-inflated etc.

Flexible tools:

AD Model Builder (and interfaces)BUGS/JAGS (and interfaces)INLA15


Open-source GLMMs

http://cran.r-project.org/web/packages/lme4


http://cran.r-project.org/web/packages/glmmADMB

http://admb-project.org/



Conclusions

Outlook

Computation: faster algorithms, parallel computation

Inference: mostly computational?

Implementation: extensions (e.g. L1-penalized approaches11),consistency (profile, simulate, predict)

Benefits & costs of staying within the GLMM framework

Benefits & costs of diversity

More info: http://glmm.wikidot.com


Open-source GLMMs

http://glmm.wikidot.com


Conclusions

Acknowledgements

Data: Josh Banta and Massimo Pigliucci (Arabidopsis);Adrian Stier and Sea McKeon (coral symbionts); CourtneyKagan, Jocelynn Ortega, David Julian (Glycera);

Co-authors: Mollie Brooks, Connie Clark, Shane Geange, JohnPoulsen, Hank Stevens, Jada White


Open-source GLMMs


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Open-source GLMMs

http://links.jstor.org/sici?sici=1369-7412(1999)61%3A1%3C265%3AMGLMML%3E2.0.CO%3B2-C



http://www.sciencedirect.com/science/article/B6V0M-3V5CVRT-M/2/190f68a684dd08c569a7836ff59568e4



http://www.jstor.org/stable/1403512

http://www.cuvillier.de/flycms/en/html/30/-UickI3zKPS,3cEY=/Buchdetails.html?SID=wVZnpL8f0fbc



http://www.bepress.com/jhubiostat/paper202/

http://www.bepress.com/jhubiostat/paper202/

http://www.jstatsoft.org/v33/i02

http://biomet.oxfordjournals.org/content/76/2/297.abstract

http://biomet.oxfordjournals.org/content/76/2/297.abstract

http://projecteuclid.org/euclid.aos/1216237296



Conclusions

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content/abstract/92/2/351.


Open-source GLMMs

http://www.ingentaconnect.com/content/asa/jabes/2002/00000007/00000004/art00004

http://www.ingentaconnect.com/content/asa/jabes/2002/00000007/00000004/art00004



http://biomet.oxfordjournals.org/cgi/content/abstract/92/2/351

http://biomet.oxfordjournals.org/cgi/content/abstract/92/2/351


Conclusions

Extras

Spatial and temporal correlation (R-side effects):MASS:glmmPQL (sort of), GLMMarp, INLA;WinBUGS, AD Model Builder

Additive models: amer, gamm4, mgcv

Penalized methods11


Open-source GLMMs


http://cran.r-project.org/web/packages/GLMMarp


http://cran.r-project.org/web/packages/amer

http://cran.r-project.org/web/packages/gamm4

http://cran.r-project.org/web/packages/mgcv

open source glmm tools: concordia

Technology

biologyopensource glmms

linear mixed models

res linear mixed models

glmms estimation inference

linear models linearizable

b marginal likelihood

linear predictor

linear functions of