Open source GLMM tools: Concordia

3,980 views

Published on

Talk on open source tools (mostly R, some BUGS and ADMB). Slightly different from actual presentation.

Published in: Technology, Education
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
3,980
On SlideShare
0
From Embeds
0
Number of Embeds
9
Actions
Shares
0
Downloads
34
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Open source GLMM tools: Concordia

  1. 1. Precursors GLMMs Results Conclusions References Open-source tools for estimation and inference using generalized linear mixed models Ben Bolker McMaster University Departments of Mathematics & Statistics and Biology 3 July 2011Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  2. 2. Precursors GLMMs Results Conclusions ReferencesOutline 1 Precursors Definitions Examples Challenges 2 GLMMs Estimation Inference 3 Results Coral symbionts Glycera Arabidopsis 4 Conclusions ConclusionsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  3. 3. Precursors GLMMs Results Conclusions ReferencesDefinitionsOutline 1 Precursors Definitions Examples Challenges 2 GLMMs Estimation Inference 3 Results Coral symbionts Glycera Arabidopsis 4 Conclusions ConclusionsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  4. 4. Precursors GLMMs Results Conclusions ReferencesDefinitionsDefinitions Fixed effects (FE) Predictors where interest is in specific levels Random effects (RE) Predictors where interest is in distribution rather 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-family responses, normal REs (on linearized scale) Generalized linear mixed models GLMMs = LMMs + GLMsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  5. 5. Precursors GLMMs Results Conclusions ReferencesDefinitionsDefinitions Fixed effects (FE) Predictors where interest is in specific levels Random effects (RE) Predictors where interest is in distribution rather 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-family responses, normal REs (on linearized scale) Generalized linear mixed models GLMMs = LMMs + GLMsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  6. 6. Precursors GLMMs Results Conclusions ReferencesDefinitionsGLMMs Distributions from exponential family (Poisson, binomial, Gaussian, Gamma, NegBinom(k), . . . ) Means = linear functions of predictors on scale of link function (identity, log, logit, . . . )Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  7. 7. Precursors GLMMs Results Conclusions ReferencesDefinitionsGLMMs (cont.) Linear predictor: η = Xβ + Zu Random effects: u ∼ MVN(0, Σ) Response: Y ∼ D g −1 η, φ (φ often ≡ 1)Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  8. 8. Precursors GLMMs Results Conclusions ReferencesDefinitionsMarginal likelihood Likelihood (Prob(data|parameters)) — requires integrating over possible values of REs to get marginal likelihood e.g.: likelihood of i th obs. in block j is L(xij |θi , σw ) 2 2 likelihood of a particular block mean θj is L(θj |0, σb ) marginal likelihood is 2 2 L(xij |θj , σw )L(θj |0, σb ) dθj Balance (dispersion of RE around 0) with (dispersion of data conditional on RE)Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  9. 9. Precursors GLMMs Results Conclusions ReferencesDefinitionsMarginal likelihood Likelihood (Prob(data|parameters)) — requires integrating over possible values of REs to get marginal likelihood e.g.: likelihood of i th obs. in block j is L(xij |θi , σw ) 2 2 likelihood of a particular block mean θj is L(θj |0, σb ) marginal likelihood is 2 2 L(xij |θj , σw )L(θj |0, σb ) dθj Balance (dispersion of RE around 0) with (dispersion of data conditional on RE)Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  10. 10. Precursors GLMMs Results Conclusions ReferencesDefinitionsBayesian solution? Bayesians should not feel smug: they are stuck with the normalizing constant Prior(β, θ, Σ)L(xij |β, θ)L(θ|Σ) Posterior(β, θ, Σ) = (!!) (. . .)dβ dθ dΣ and similar issues with marginal posteriorsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  11. 11. Precursors GLMMs Results Conclusions ReferencesExamplesOutline 1 Precursors Definitions Examples Challenges 2 GLMMs Estimation Inference 3 Results Coral symbionts Glycera Arabidopsis 4 Conclusions ConclusionsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  12. 12. Precursors GLMMs Results Conclusions ReferencesExamplesCoral 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 SymbiontsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  13. 13. Precursors GLMMs Results Conclusions ReferencesExamplesEnvironmental 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 H2SBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  14. 14. Precursors GLMMs Results Conclusions ReferencesExamplesArabidopsis response to fertilization & clipping panel: nutrient, color: genotype nutrient : 1 nutrient : 8 q q q q q q q q q 5 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Log(1+fruit set) q q q q q 4 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 3 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 2 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 q q q q q q q q q 0 q q q q q q q q q q q q unclipped clipped unclipped clippedBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  15. 15. Precursors GLMMs Results Conclusions ReferencesChallengesOutline 1 Precursors Definitions Examples Challenges 2 GLMMs Estimation Inference 3 Results Coral symbionts Glycera Arabidopsis 4 Conclusions ConclusionsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  16. 16. Precursors GLMMs Results Conclusions ReferencesChallengesData 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 . . . )Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  17. 17. Precursors GLMMs Results Conclusions ReferencesChallengesData challenges: computation Large n (of course) Multiple REs (dimensionality) Crossed REsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  18. 18. Precursors GLMMs Results Conclusions ReferencesChallengesInference Any departures from classical LMMs Small N (<40) Small n Inference on components of Σ (boundary effects, df)Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  19. 19. Precursors GLMMs Results Conclusions ReferencesChallengesRE examples Coral symbionts: simple experimental blocks, RE affects intercept (overall probability of predation in block) Glycera: applied to cells from 10 individuals, RE again affects intercept (cell survival prob.) Arabidopsis: region (3 levels, treated as fixed) / population / genotype: affects intercept (overall fruit set) as well as treatment effects (nutrients, herbivory, interaction)Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  20. 20. Precursors GLMMs Results Conclusions ReferencesEstimationOutline 1 Precursors Definitions Examples Challenges 2 GLMMs Estimation Inference 3 Results Coral symbionts Glycera Arabidopsis 4 Conclusions ConclusionsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  21. 21. Precursors GLMMs Results Conclusions ReferencesEstimationPenalized quasi-likelihood (PQL) alternate steps of estimating GLM using known RE variances to calculate weights; estimate LMMs given GLM fit 2 flexible (e.g. spatial/temporal correlations) biased for small unit samples (e.g. counts < 5, binary or low-survival data) widely used: SAS PROC GLIMMIX, R MASS:glmmPQL marginal models: generalized estimating equations (geepack, geese)Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  22. 22. Precursors GLMMs Results Conclusions ReferencesEstimationPenalized quasi-likelihood (PQL) alternate steps of estimating GLM using known RE variances to calculate weights; estimate LMMs given GLM fit 2 flexible (e.g. spatial/temporal correlations) biased for small unit samples (e.g. counts < 5, binary or low-survival data) widely used: SAS PROC GLIMMIX, R MASS:glmmPQL marginal models: generalized estimating equations (geepack, geese)Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  23. 23. Precursors GLMMs Results Conclusions ReferencesEstimationPenalized quasi-likelihood (PQL) alternate steps of estimating GLM using known RE variances to calculate weights; estimate LMMs given GLM fit 2 flexible (e.g. spatial/temporal correlations) biased for small unit samples (e.g. counts < 5, binary or low-survival data) widely used: SAS PROC GLIMMIX, R MASS:glmmPQL marginal models: generalized estimating equations (geepack, geese)Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  24. 24. Precursors GLMMs Results Conclusions ReferencesEstimationPenalized quasi-likelihood (PQL) alternate steps of estimating GLM using known RE variances to calculate weights; estimate LMMs given GLM fit 2 flexible (e.g. spatial/temporal correlations) biased for small unit samples (e.g. counts < 5, binary or low-survival data) widely used: SAS PROC GLIMMIX, R MASS:glmmPQL marginal models: generalized estimating equations (geepack, geese)Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  25. 25. Precursors GLMMs Results Conclusions ReferencesEstimationLaplace approximation approximate marginal likelihood for given β, θ find conditional modes by penalized, iterated reweighted least squares; then use second-order Taylor expansion around the conditional modes more accurate than PQL reasonably fast and flexible lme4:glmer, glmmML, glmmADMB, R2ADMB (AD Model Builder)Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  26. 26. Precursors GLMMs Results Conclusions ReferencesEstimation(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, repeatedBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  27. 27. Precursors GLMMs Results Conclusions ReferencesEstimationVariations Hierarchical GLMS (hglm, HGLMMM) Monte Carlo methods: MCEM 1 , MCMLE (bernor) 18 , sequential MC (pomp), data cloning (dclone)Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  28. 28. Precursors GLMMs Results Conclusions ReferencesEstimationBayesian approaches Monte Carlo approaches: MCMC (Gibbs sampling, Metropolis-Hastings, etc.) slow but flexible makes marginal inference easy must specify priors, assess convergence specialized: glmmAK, MCMCglmm 9 , INLA general: BUGS (glmmBUGS, R2WinBUGS, BRugs, WinBUGS, OpenBUGS, R2jags, rjags, JAGS)Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  29. 29. Precursors GLMMs Results Conclusions ReferencesEstimationExtensions Overdispersion Variance > expected from statistical model Quasi-likelihood approaches: MASS:glmmPQL Extended distributions (e.g. negative binomial): glmmADMB, gamlss.mx:gamlssNP Observation-level random effects (e.g. lognormal-Poisson): lme4 Zero-inflation Overabundance of zeros in a discrete distribution zero-inflated models: glmmADMB, MCMCglmm hurdle models: MCMCglmmBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  30. 30. Precursors GLMMs Results Conclusions ReferencesInferenceOutline 1 Precursors Definitions Examples Challenges 2 GLMMs Estimation Inference 3 Results Coral symbionts Glycera Arabidopsis 4 Conclusions ConclusionsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  31. 31. Precursors GLMMs Results Conclusions ReferencesInferenceWald tests/CIs Easy (e.g. typical results of summary): assume quadratic surface, based on information matrix @ MLE always approximate, sometimes awful (Hauck-Donner effect) often bad for variance estimates available from most direct-maximization packagesBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  32. 32. Precursors GLMMs Results Conclusions ReferencesInferenceLikelihood ratio tests/profile confidence intervals Model comparison is relatively easy Profiling is expensive — and not (yet) available . . . (lme4a for LMMs) in GLM(M) case, numerator is only asymptotically χ2 anyway: Bartlett corrections 3;4 , higher-order asymptotics: cond [neither extended to GLMMs!] OK if N − n, N 40?Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  33. 33. Precursors GLMMs Results Conclusions ReferencesInferenceConditional 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) or GLMMs, ν2 poorly defined: Kenward-Roger, Satterthwaite approximations 12;16 unimplemented except in SAS (partially in Genstat)Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  34. 34. Precursors GLMMs Results Conclusions ReferencesInferenceTests/CIs of variances [boundary problems] LRT depends on null hypothesis being within the parameter’s feasible range 6;13 violated e.g. by H0 : σ 2 = 0 In simple cases null distribution is a mixture of χ2 distributions (e.g. 0.5χ2 + 0.5χ2 : emdbook:dchibarsq) 0 1 simulation-based testing: RLRsimBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  35. 35. Precursors GLMMs Results Conclusions ReferencesInferenceInformation-theoretic approaches Above issues apply, but less well understood: 7;8 AIC is asymptotic “corrected” AIC (AICc ) 10 derived for linear models, widely used but not tested elsewhere 14 For comparing models with different REs, or for AICc , what is p? conditional AIC: 8;19 (cAIC) (level of focus issue: see also Deviance Information Criterion (DIC, 17 )Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  36. 36. Precursors GLMMs Results Conclusions ReferencesInferenceBootstrapping 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))Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  37. 37. Precursors GLMMs Results Conclusions ReferencesInferenceBayesian inference Marginal highest posterior density intervals (or quantiles) Computationally “free” with results of stochastic Bayesian computation Easily extended to prediction intervals etc. etc. Post hoc Markov chain Monte Carlo sampling available for some packages (glmmADMB, R2ADMB, eventually lme4a)Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  38. 38. Precursors GLMMs Results Conclusions ReferencesInferenceBottom line Large data: computation slow (maximization methods fastest), 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 bootstrappingBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  39. 39. Precursors GLMMs Results Conclusions ReferencesCoral symbiontsCoral symbionts: comparison of results Regression estimates −6 −4 −2 0 2 q q q q q q Added symbiont q q q q q q q Crab vs. Shrimp q q q q GLM (fixed) q q q GLM (pooled) q q PQL q q Laplace Symbiont q q AGQBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  40. 40. Precursors GLMMs Results Conclusions ReferencesGlyceraGlycera fit comparisons qq qq Osm:Cu:H2S:Anoxia q q q Cu:H2S:Anoxia q q q qq q Osm:H2S:Anoxia q q q qq q Osm:Cu:Anoxia q q q qq Osm:Cu:H2S q qqq qq H2S:Anoxia q qq q Cu:Anoxia q q q Osm:Anoxia qq q q q q Cu:H2S q q q q Osm:H2S qq q q q q q Osm:Cu q q MCMCglmm qqq q Anoxia q q glmer(OD:2) q qq H2S q q q glmer(OD) qq q Cu q q q glmmML q Osm qq qq q glmer −60 −40 −20 0 20 40 60 Effect on survival (logit)Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  41. 41. Precursors GLMMs Results Conclusions ReferencesGlyceraGlycera: MCMCglmm fit Osm : Cu : H2S : Oxygen q Osm : Cu : Oxygen q Osm : H2S : Oxygen q Cu : H2S : Oxygen q 3−way Osm : Cu : H2S q Osm : Cu q H2S : Oxygen q Osm : H2S q 2−way Cu : Oxygen q Osm : Oxygen q Cu : H2S q Oxygen q Osm q main effects Cu q H2S q −20 −10 0 10 20 30 Effect on survivalBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  42. 42. Precursors GLMMs Results Conclusions ReferencesGlyceraParametric 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 valueBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  43. 43. Precursors GLMMs Results Conclusions ReferencesArabidopsisArabidopsis: AIC comparison of RE models nointeract q int(popu) q int(gen) X int(popu) q int(gen) X nut(popu) q int(gen) X clip(popu) q nut(gen) X int(popu) q nut(gen) X nut(popu) q nut(gen) X clip(popu) q clip(gen) X int(popu) q clip(gen) X nut(popu) q clip(gen) X clip(popu) q 0 2 4 6 ∆AICBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  44. 44. Precursors GLMMs Results Conclusions ReferencesArabidopsisArabidopsis: fits with and without nutrient(genotype) Regression estimates −1.0 −0.5 0.0 0.5 1.0 1.5 q nutrient8:amdclipped q q statusTransplant q q statusPetri.Plate q q rack2 q q amdclipped q q nutrient8 qBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  45. 45. Precursors GLMMs Results Conclusions ReferencesConclusionsPrimary 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) INLA 15Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  46. 46. Precursors GLMMs Results Conclusions ReferencesConclusionsOutlook Computation: faster algorithms, parallel computation Inference: mostly computational? Implementation: extensions (e.g. L1-penalized approaches 11 ), consistency (profile, simulate, predict) Benefits & costs of staying within the GLMM framework Benefits & costs of diversity More info: http://glmm.wikidot.comBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  47. 47. Precursors GLMMs Results Conclusions ReferencesConclusionsAcknowledgements Data: Josh Banta and Massimo Pigliucci (Arabidopsis); Adrian Stier and Sea McKeon (coral symbionts); Courtney Kagan, Jocelynn Ortega, David Julian (Glycera); Co-authors: Mollie Brooks, Connie Clark, Shane Geange, John Poulsen, Hank Stevens, Jada WhiteBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  48. 48. Precursors GLMMs Results Conclusions References [1] Booth JG & Hobert JP, 1999. Journal of the 3867274916. URL http://www.cuvillier.de/ Royal Statistical Society. Series B, 61(1):265–285. flycms/en/html/30/-UickI3zKPS,3cEY= doi:10.1111/1467-9868.00176. URL http:// /Buchdetails.html?SID=wVZnpL8f0fbc. links.jstor.org/sici?sici=1369-7412(1999) [8] Greven S & Kneib T, 2010. Biometrika, 61%3A1%3C265%3AMGLMML%3E2.0.CO%3B2-C. 97(4):773–789. URL http: [2] Breslow NE, 2004. In DY Lin & PJ Heagerty, //www.bepress.com/jhubiostat/paper202/. eds., Proceedings of the second Seattle [9] Hadfield JD, 2 2010. Journal of Statistical symposium in biostatistics: Analysis of correlated Software, 33(2):1–22. ISSN 1548-7660. URL data, pp. 1–22. Springer. ISBN 0387208623. http://www.jstatsoft.org/v33/i02. [3] Cordeiro GM & Ferrari SLP, Aug. 1998. Journal [10] HURVICH CM & TSAI C, Jun. 1989. Biometrika, of Statistical Planning and Inference, 76(2):297 –307. 71(1-2):261–269. ISSN 0378-3758. doi:10.1093/biomet/76.2.297. URL doi:10.1016/S0378-3758(98)00005-6. URL http://biomet.oxfordjournals.org/content/ http://www.sciencedirect.com/science/ 76/2/297.abstract. article/B6V0M-3V5CVRT-M/2/ [11] Jiang J, Aug. 2008. The Annals of Statistics, 190f68a684dd08c569a7836ff59568e4. 36(4):1669–1692. ISSN 0090-5364. [4] Cordeiro GM, Paula GA, & Botter DA, 1994. doi:10.1214/07-AOS517. URL http: International Statistical Review / Revue //projecteuclid.org/euclid.aos/1216237296. Internationale de Statistique, 62(2):257–274. [12] Kenward MG & Roger JH, 1997. Biometrics, ISSN 03067734. doi:10.2307/1403512. URL 53(3):983–997. http://www.jstor.org/stable/1403512. [13] Molenberghs G & Verbeke G, 2007. The [5] Gelman A, 2005. Annals of Statistics, 33(1):1–53. American Statistician, 61(1):22–27. doi:doi:10.1214/009053604000001048. doi:10.1198/000313007X171322. [6] Goldman N & Whelan S, 2000. Molecular Biology [14] Richards SA, 2005. Ecology, 86(10):2805–2814. and Evolution, 17(6):975–978. doi:10.1890/05-0074. [7] Greven S, 2008. Non-Standard Problems in [15] Rue H, Martino S, & Chopin N, 2009. Journal of Inference for Additive and Linear Mixed Models. the Royal Statistical Society, Series B, Cuvillier Verlag, G¨ttingen, Germany. ISBN o 71(2):319–392.Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  49. 49. Precursors GLMMs Results Conclusions ReferencesConclusions [16] Schaalje G, McBride J, & Fellingham G, 2002. Journal of Agricultural, Biological & Environmental Statistics, 7(14):512–524. URL http://www.ingentaconnect.com/content/ asa/jabes/2002/00000007/00000004/art00004. [17] Spiegelhalter DJ, Best N et al., 2002. Journal of the Royal Statistical Society B, 64:583–640. [18] Sung YJ, Jul. 2007. The Annals of Statistics, 35(3):990–1011. ISSN 0090-5364. doi:10.1214/009053606000001389. URL http: //projecteuclid.org/euclid.aos/1185303995. Mathematical Reviews number (MathSciNet): MR2341695; Zentralblatt MATH identifier: 1124.62009. [19] Vaida F & Blanchard S, Jun. 2005. Biometrika, 92(2):351–370. doi:10.1093/biomet/92.2.351. URL http://biomet.oxfordjournals.org/cgi/ content/abstract/92/2/351.Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  50. 50. Precursors GLMMs Results Conclusions ReferencesConclusionsExtras Spatial and temporal correlation (R-side effects): MASS:glmmPQL (sort of), GLMMarp, INLA; WinBUGS, AD Model Builder Additive models: amer, gamm4, mgcv Penalized methods 11Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs

×