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talk at UWO 7 April 2011 on open-source tools for generalized linear mixed models

talk at UWO 7 April 2011 on open-source tools for generalized linear mixed models

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open-source GLMM tools open-source GLMM tools Presentation Transcript

  • 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 7 April 2011Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesOutline 1 Precursors Examples Definitions 2 GLMMs Estimation Inference: tests Inference: confidence intervals 3 Results Glycera Arabidopsis 4 ConclusionsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesExamplesOutline 1 Precursors Examples Definitions 2 GLMMs Estimation Inference: tests Inference: confidence intervals 3 Results Glycera Arabidopsis 4 ConclusionsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • 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
  • 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
  • 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
  • Precursors GLMMs Results Conclusions ReferencesExamplesGlossary: data Fixed effects Predictors where interest is in specific levels Random effects (RE) predictors where interest is in distribution rather than levels (blocks) (Gelman, 2005) Crossed RE multiple REs where levels of one occur in more than one level of another (ex.: block × year: cf. nested) http://lme4.r-forge.r-project.org/book/, Pinheiro and Bates (2000)Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesExamplesData challenges Estimation Computation Inference Small # RE levels (<5–6) Large n Small N (< 40) Overdispersion Multiple REs Small n Crossed REs Crossed REs Spatial/temporal correlation Unusual distributions (Gamma, neg. binom . . . )Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesDefinitionsOutline 1 Precursors Examples Definitions 2 GLMMs Estimation Inference: tests Inference: confidence intervals 3 Results Glycera Arabidopsis 4 ConclusionsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesDefinitionsGeneralized linear models Distributions from exponential family (Poisson, binomial, Gaussian, Gamma, neg. binomial (known k) . . . ) Means = linear functions of predictors on scale of link function (identity, log, logit, . . . ) Y ∼ D(g −1 (Xβ), φ) φ often set to 1 (Poisson, binomial) except for quasilikelihood approachesBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesDefinitionsGeneralized linear mixed models Add random effects: Y ∼ D(g −1 (Xβ + Zu), φ) u ∼ MVN(0, Σ) Synonyms: multilevel, hierarchical modelsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • 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
  • 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
  • Precursors GLMMs Results Conclusions ReferencesDefinitionsShrinkage 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 q 2 q 0 q 3 10 q q q −3 −15 q q 0 5 10 15 20 25 GenotypeBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesDefinitionsRE 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
  • Precursors GLMMs Results Conclusions ReferencesEstimationOutline 1 Precursors Examples Definitions 2 GLMMs Estimation Inference: tests Inference: confidence intervals 3 Results Glycera Arabidopsis 4 ConclusionsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • 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 (Breslow, 2004) flexible (allows spatial/temporal correlations, crossed REs) biased for small unit samples (e.g. counts < 5, binary or low-survival data) widely used: SAS PROC GLIMMIX, R MASS:glmmPQL: in ≈ 90% of small-unit-sample casesBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • 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 (Breslow, 2004) flexible (allows spatial/temporal correlations, crossed REs) biased for small unit samples (e.g. counts < 5, binary or low-survival data) widely used: SAS PROC GLIMMIX, R MASS:glmmPQL: in ≈ 90% of small-unit-sample casesBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • 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 (Breslow, 2004) flexible (allows spatial/temporal correlations, crossed REs) biased for small unit samples (e.g. counts < 5, binary or low-survival data) widely used: SAS PROC GLIMMIX, R MASS:glmmPQL: in ≈ 90% of small-unit-sample casesBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • 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 (Breslow, 2004) flexible (allows spatial/temporal correlations, crossed REs) biased for small unit samples (e.g. counts < 5, binary or low-survival data) widely used: SAS PROC GLIMMIX, R MASS:glmmPQL: in ≈ 90% of small-unit-sample casesBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesEstimationLaplace approximation approximate marginal likelihood for given β, θ (RE parameters), 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
  • Precursors GLMMs Results Conclusions ReferencesEstimationGauss-Hermite quadrature (AGQ) 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, repeatedBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesEstimationBayesian approaches Bayesians have to do nasty integrals anyway (to normalize the posterior probability density) various flavours of stochastic Bayesian computation (Gibbs sampling, MCMC, etc.) generally slower but more flexible solves many problems of assessing confidence intervals must specify priors, assess convergence specialized: glmmAK, MCMCglmm (Hadfield, 2010), INLA general: glmmBUGS, R2WinBUGS, BRugs (WinBUGS/OpenBUGS), R2jags, rjags (JAGS)Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesEstimationOverdispersion (slight tangent) Variance greater than expected from statistical model Quasi-likelihood approaches: MASS:glmmPQL Extended distributions (e.g. negative binomial): glmmADMB Observation-level random effects (e.g. lognormal-Poisson): lme4Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesEstimationComparison of coral symbiont 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
  • Precursors GLMMs Results Conclusions ReferencesInference: testsOutline 1 Precursors Examples Definitions 2 GLMMs Estimation Inference: tests Inference: confidence intervals 3 Results Glycera Arabidopsis 4 ConclusionsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesInference: testsWald tests [non-quadratic likelihood surfaces] For OLS/linear models, likelihood surface is quadratic; only asymptotically true for GLM(M)s Wald tests (e.g. typical results of summary) assume quadratic, based on curvature (information matrix) always approximate, sometimes awful (Hauck-Donner effect) do model comparison (F , score or likelihood ratio tests [LRT]) instead But . . .Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesInference: testsConditional F tests [Uncertainty in scale parameters] Model comparison: in general −2 log L = D = deviancei /φ Classical linear models: ˆ deviance and φ are both χ2 distributed so D ∼ F (ν1 , ν2 ) Denominator degrees of freedom (df) (ν2 ) for complex (unbalanced, crossed, R-side effects) models? Approximations: Satterthwaite, Kenward-Roger (Kenward and Roger, 1997; Schaalje et al., 2002) Is D really ∼ F in these situations? Scale parameters usually not estimated in GLMMs (Gamma, quasi-likelihood cases only). But . . .Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesInference: testsLikelihood ratio tests [non-normality of likelihood] What about cases where φ is specified (e.g. ≡ 1)? in GLM(M) case, numerator is only asymptotically χ2 anyway Bartlett corrections (Cordeiro et al., 1994; Cordeiro and Ferrari, 1998), higher-order asymptotics: cond [neither extended to GLMMs!]Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesInference: testsTests of random effects [boundary problems] LRT depends on null hypothesis being within the parameter’s feasible range (Goldman and Whelan, 2000; Molenberghs and Verbeke, 2007) violated e.g. by H0 : σ 2 = 0 In simple cases null distribution is a mixture of χ2 (e.g. 0.5χ2 + 0.5χ2 (emdbook:dchibarsq) 0 1 ignoring this leads to conservative tests (e.g. true p-value = 1 2 · nominal p-value) simulation-based testing: RLRsimBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesInference: testsInformation-theoretic approaches Above issues apply, but less well understood (Greven, 2008; Greven and Kneib, 2010) AIC is asymptotic “corrected” AIC (AICc ) (HURVICH and TSAI, 1989) derived for linear models, widely used but not tested elsewhere (Richards, 2005) For comparing models with different REs, or for AICc , what is p? AICcmodavg, MuMInBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesInference: testsParametric bootstrapping fit null model to data simulate “data” from null model fit null and working model, compute likelihood difference repeat to estimate null distribution > pboot <- function(m0, m1) { s <- simulate(m0) L0 <- logLik(refit(m0, s)) L1 <- logLik(refit(m1, s)) 2 * (L1 - L0) } > replicate(1000, pboot(fm2, fm1))Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesInference: testsFinite-sample problems How far are we from “asymptopia”? How much data (number of samples, number of RE levels)? How many parameters (number of fixed-effect parameters, number of RE levels, number of RE parameters)? Hope (#data) − (#parameters) 1 but if not?Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesInference: testsLevels of focus how many parameters does a RE take? Somewhere between q and r (e.g., 1 and the number of levels for a variance) . . . shrinkage Conditional vs. marginal AIC Similar issues with Deviance Information Criterion (Spiegelhalter et al., 2002)Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesInference: confidence intervalsOutline 1 Precursors Examples Definitions 2 GLMMs Estimation Inference: tests Inference: confidence intervals 3 Results Glycera Arabidopsis 4 ConclusionsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesInference: confidence intervalsWald tests a sometimes-crude approximation computationally easy, especially for many-parameter models use Wald Z (assume “residual df” large)? Or t, guessing at the residual df? Available from most packagesBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesInference: confidence intervalsProfile confidence intervals Tedious to program Computationally challenging Inherits finite-size sample problems from LRT lme4a (in development/soon!)Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesInference: confidence intervalsBayesian posterior intervals Marginal quantile or highest posterior density intervals Computationally “free” with results of stochastic Bayesian computation Easily extended to confidence intervals on predictions, 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
  • Precursors GLMMs Results Conclusions ReferencesInference: confidence intervalsSummary Large data computation can be limiting asymptotics better Small data RE variances may be poorly estimated/ set to zero (informative priors can help) inference trickyBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesGlyceraOutline 1 Precursors Examples Definitions 2 GLMMs Estimation Inference: tests Inference: confidence intervals 3 Results Glycera Arabidopsis 4 ConclusionsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesGlycera 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 survivalBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesGlycera 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
  • Precursors GLMMs Results Conclusions ReferencesGlyceraParametric bootstrap results 0.02 0.04 0.06 0.08 H2S Anoxia 0.08 0.06 0.04 Inferred p value 0.02 Osm Cu 0.08 0.06 0.04 0.02 0.02 0.04 0.06 0.08 True p valueBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesArabidopsisOutline 1 Precursors Examples Definitions 2 GLMMs Estimation Inference: tests Inference: confidence intervals 3 Results Glycera Arabidopsis 4 ConclusionsBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesArabidopsisArabidopsis: AIC comparison of REs 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
  • 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
  • Precursors GLMMs Results Conclusions ReferencesPrimary tools lme4: multiple/crossed REs, (profiling): fast MCMCglmm: Bayesian, very flexible glmmADMB: negative binomial, zero-inflated etc. Most flexible: R2ADMB/AD Model Builder, R2WinBUGS/WinBUGS/R2jags/JAGS, INLABen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesLoose ends Overdispersion and zero-inflation: MCMCglmm, glmmADMB Spatial and temporal correlation (R-side effects): MASS:glmmPQL (sort of), GLMMarp, INLA; WinBUGS, AD Model Builder Additive models: amer, gamm4, mgcv Penalized methods (Jiang, 2008) (?) Hierarchical GLMs: hglm, HGLMMM Marginal models: geepack, geeBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesTo be done Many holes in knowledge (but what can be done?) Faster algorithms, more parallel computation Lots of implementation and clean-up Benefits & costs of staying within the GLMM framework Benefits & costs of diversity More info: glmm.wikidot.comBen Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs
  • Precursors GLMMs Results Conclusions ReferencesAcknowledgements 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
  • Precursors GLMMs Results Conclusions ReferencesReferences Breslow, N.E., 2004. In D.Y. Lin and P.J. Heagerty, editors, Proceedings of the second Seattle symposium in biostatistics: Analysis of correlated data, pages 1–22. Springer. ISBN 0387208623. Cordeiro, G.M. and Ferrari, S.L.P., 1998. Journal of Statistical Planning and Inference, 71(1-2):261–269. ISSN 0378-3758. doi:10.1016/S0378-3758(98)00005-6. Cordeiro, G.M., Paula, G.A., and Botter, D.A., 1994. International Statistical Review / Revue Internationale de Statistique, 62(2):257–274. ISSN 03067734. doi:10.2307/1403512. Gelman, A., 2005. Annals of Statistics, 33(1):1–53. doi:doi:10.1214/009053604000001048. Goldman, N. and Whelan, S., 2000. Molecular Biology and Evolution, 17(6):975–978. Greven, S., 2008. Non-Standard Problems in Inference for Additive and Linear Mixed Models. Cuvillier Verlag, G¨ttingen, Germany. ISBN 3867274916. o Greven, S. and Kneib, T., 2010. Biometrika, 97(4):773–789. Hadfield, J.D., 2010. Journal of Statistical Software, 33(2):1–22. ISSN 1548-7660. HURVICH, C.M. and TSAI, C., 1989. Biometrika, 76(2):297 –307. doi:10.1093/biomet/76.2.297. Jiang, J., 2008. The Annals of Statistics, 36(4):1669–1692. ISSN 0090-5364. doi:10.1214/07-AOS517. Kenward, M.G. and Roger, J.H., 1997. Biometrics, 53(3):983–997. Molenberghs, G. and Verbeke, G., 2007. The American Statistician, 61(1):22–27. doi:10.1198/000313007X171322. Pinheiro, J.C. and Bates, D.M., 2000. Mixed-effects models in S and S-PLUS. Springer, New York. ISBN 0-387-98957-9. Richards, S.A., 2005. Ecology, 86(10):2805–2814. doi:10.1890/05-0074. Schaalje, G., McBride, J., and Fellingham, G., 2002. Journal of Agricultural, Biological & Environmental Statistics, 7(14):512–524. Spiegelhalter, D.J., Best, N., et al., 2002. Journal of the Royal Statistical Society B, 64:583–640.Ben Bolker McMaster University Departments of Mathematics & Statistics and BiologyOpen-source GLMMs