Downloaded 14 times
![Denitions
Estimation
Inference
Challenges open questions
Methods
Breslow (2004) on PQL
As usual when software for complicated statistical
inference procedures is broadly disseminated, there is
potential for abuse and misinterpretation. In spite of the
fact that PQL was initially advertised as a procedure for
approximate inference in GLMMs, and its tendency to
give seriously biased estimates of variance components
and a fortiori regression parameters with binary outcome
data was emphasized in multiple publications [5, 6, 24],
some statisticians seemed to ignore these warnings and to
think of PQL as synonymous with GLMM.
Ben Bolker
GLMMs
References](https://image.slidesharecdn.com/statssem2013-131112160056-phpapp01/85/Stats-sem-2013-27-320.jpg)
![Denitions
Estimation
Inference
Challenges open questions
References
Likelihood ratio tests
better, but still have to deal with two nite-size problems:
when scale parameter is free (Gamma, etc.), deviance is ∼ F
rather than ∼ χ2 , with poorly dened denominator df
in GLM(M) case, numerator is only asymptotically χ2 anyway
Bartlett corrections (Cordeiro and Ferrari, 1998; Cordeiro
et al., 1994), higher-order asymptotics: cond [neither extended
to GLMMs!]
Prole condence intervals: moderately dicult/fragile
Ben Bolker
GLMMs](https://image.slidesharecdn.com/statssem2013-131112160056-phpapp01/85/Stats-sem-2013-36-320.jpg)
Stats sem 2013
- 1. Denitions Estimation Inference Challenges openquestions Generalized linear mixed models: overview and open questions Ben Bolker McMaster University, Mathematics Statistics and Biology 12 November 2013 Ben Bolker GLMMs References
- 2. Denitions Estimation Inference Outline 1 Examples and denitions 2 Estimation Overview Methods 3 Inference 4 Challenges open questions Ben Bolker GLMMs Challenges open questions References
- 3. Denitions Estimation Inference Outline 1 Examples and denitions 2 Estimation Overview Methods 3 Inference 4 Challenges open questions Ben Bolker GLMMs Challenges open questions References
- 4. Denitions Estimation Inference Challenges openquestions References (Generalized) linear mixed models (G)LMMs: a statistical modeling framework incorporating: Linear combinations of categorical and continuous predictors, and interactions Response distributions in the exponential family (binomial, Poisson, and extensions) Any smooth, monotonic link function (e.g. logistic, exponential models) Flexible combinations of blocking factors (clustering; random eects) Applications in ecology, neurobiology, behaviour, epidemiology, real estate, . . . Ben Bolker GLMMs
- 5. Denitions Estimation Inference Challenges openquestions References (Generalized) linear mixed models (G)LMMs: a statistical modeling framework incorporating: Linear combinations of categorical and continuous predictors, and interactions Response distributions in the exponential family (binomial, Poisson, and extensions) Any smooth, monotonic link function (e.g. logistic, exponential models) Flexible combinations of blocking factors (clustering; random eects) Applications in ecology, neurobiology, behaviour, epidemiology, real estate, . . . Ben Bolker GLMMs
- 6. Denitions Estimation Inference Challenges openquestions References (Generalized) linear mixed models (G)LMMs: a statistical modeling framework incorporating: Linear combinations of categorical and continuous predictors, and interactions Response distributions in the exponential family (binomial, Poisson, and extensions) Any smooth, monotonic link function (e.g. logistic, exponential models) Flexible combinations of blocking factors (clustering; random eects) Applications in ecology, neurobiology, behaviour, epidemiology, real estate, . . . Ben Bolker GLMMs
- 7. Denitions Estimation Inference Challenges openquestions References (Generalized) linear mixed models (G)LMMs: a statistical modeling framework incorporating: Linear combinations of categorical and continuous predictors, and interactions Response distributions in the exponential family (binomial, Poisson, and extensions) Any smooth, monotonic link function (e.g. logistic, exponential models) Flexible combinations of blocking factors (clustering; random eects) Applications in ecology, neurobiology, behaviour, epidemiology, real estate, . . . Ben Bolker GLMMs
- 8. Denitions Estimation Inference Challenges openquestions References Examples ecology survival, predation, etc. (experimental plots) genomics presence/absence of polymorphisms, gene expression (individuals) educational assessment student scores (students × teachers) psychology/sensometrics decisions, responses to stimuli (individuals) epidemiology disease prevalence (postal codes, provinces, countries) Ben Bolker GLMMs
- 9. Denitions Estimation Inference Challenges openquestions References Examples ecology survival, predation, etc. (experimental plots) genomics presence/absence of polymorphisms, gene expression (individuals) educational assessment student scores (students × teachers) psychology/sensometrics decisions, responses to stimuli (individuals) epidemiology disease prevalence (postal codes, provinces, countries) Ben Bolker GLMMs
- 10. Denitions Estimation Inference Challenges openquestions References Examples ecology survival, predation, etc. (experimental plots) genomics presence/absence of polymorphisms, gene expression (individuals) educational assessment student scores (students × teachers) psychology/sensometrics decisions, responses to stimuli (individuals) epidemiology disease prevalence (postal codes, provinces, countries) Ben Bolker GLMMs
- 11. Denitions Estimation Inference Challenges openquestions References Examples ecology survival, predation, etc. (experimental plots) genomics presence/absence of polymorphisms, gene expression (individuals) educational assessment student scores (students × teachers) psychology/sensometrics decisions, responses to stimuli (individuals) epidemiology disease prevalence (postal codes, provinces, countries) Ben Bolker GLMMs
- 12. Denitions Estimation Inference Challenges openquestions References Examples ecology survival, predation, etc. (experimental plots) genomics presence/absence of polymorphisms, gene expression (individuals) educational assessment student scores (students × teachers) psychology/sensometrics decisions, responses to stimuli (individuals) epidemiology disease prevalence (postal codes, provinces, countries) Ben Bolker GLMMs
- 13. Denitions Estimation Inference Challenges openquestions Coral protection by symbionts Number of predation events Number of blocks 10 8 6 2 2 2 2 1 1 4 0 2 0 shrimp crabs 0 1 0 none Symbionts Ben Bolker GLMMs both References
- 14. Denitions Estimation Inference Environmental stress: 0 Anoxia Osm=12.8 0.03 Challenges open questions References Glycera cell survival 0.1 0.32 0 Anoxia Osm=22.4 Anoxia Osm=32 0.03 0.1 0.32 Anoxia Osm=41.6 Anoxia Osm=51.2 1.0 133.3 66.6 0.8 33.3 0.6 Copper 0 Normoxia Osm=12.8 Normoxia Osm=22.4 Normoxia Osm=32 Normoxia Osm=41.6 Normoxia 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 H2S Ben Bolker GLMMs 0.32 0 0.03 0.1 0.32
- 15. Denitions Estimation Inference Challenges openquestions Arabidopsis response to fertilization clipping panel: nutrient, color: genotype nutrient : 1 nutrient : 8 q q q q Log(1+fruit set) 3 2 1 0 Ben Bolker GLMMs q q q q q q q q q q q q q q q q q q q q q unclipped 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 q q q q 5 clipped unclipped clipped 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 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 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 References
- 16. Denitions Estimation Inference Challenges openquestions References Coral demography Before q Mortality probability 1.00 qqq q Experimental q q q q q q q qqqqqqqqq qq qq q qq q q qqq qq q q qqqq qq q q q qq qq qq q q q q qq q q q q q 0.75 Treatment q 0.25 q q qq q qq q qqq q qqq q q q qq q q q q q qq qq qqqq q q q q q qq q q 0.00 0 10 q q q q 20 30 q q q q qq q q q qq qq qq q q qq q q q q qq qqq qq qq q q q qqq qqqqq q q q q q q q q qq q 40 50 0 10 Previous size (cm) Ben Bolker GLMMs 20 q q q 30 qq q 40 50 Present q 0.50 Removed
- 17. Denitions Estimation Inference Challenges openquestions Technical denition conditional distribution Yi ∼ Distr response η linear predictor b conditional modes Ben Bolker GLMMs = Xβ xed eects (g −1 (η ), i φ ) scale inverse parameter link function + Zb random eects ∼ MVN(0, Σ(θ) ) variancecovariance matrix References
- 18. Denitions Estimation Inference Outline 1 Examples and denitions 2 Estimation Overview Methods 3 Inference 4 Challenges open questions Ben Bolker GLMMs Challenges open questions References
- 19. Denitions Estimation Inference Challenges openquestions References Overview Maximum likelihood estimation L(Y |θ, β) = i L(Y |β, b ) ··· i likelihood data|random eects × L(b |Σ(θ)) d b random eects Best t is a compromise between two components (consistency of data with β and random eects, consistency of random eect with RE distribution) Ben Bolker GLMMs
- 20. Denitions Estimation Inference Challenges openquestions Overview Integrated (marginal) likelihood L (x|b, β) Scaled probability 1.0 0.8 L prod 0.6 L (b |σ2) 0.4 0.2 0.0 −10 −5 0 5 conditional mode value (u ) Ben Bolker GLMMs 10 References
- 21. Denitions Estimation Inference Challenges openquestions Overview Shrinkage Mean(log) fruit set Arabidopsis block estimates 5 9 11 2 5 10 5 7 9 4 q q 4 2 6 q q q q q q 4 6 9 q q 3 9 q q q 4 q q q 10 q q 8 q q 3 2 10 q q q q q 3 0 −3 −15 q q 0 5 10 15 Genotype Ben Bolker GLMMs 20 25 References
- 22. Denitions Estimation Inference Challenges openquestions Methods Estimation methods deterministic: precision vs. computational cost: penalized quasi-likelihood, Laplace approximation, adaptive Gauss-Hermite quadrature (Breslow, 2004) . . . stochastic (Monte Carlo): frequentist and Bayesian (Booth and Hobert, 1999; Ponciano et al., 2009; Sung, 2007) Ben Bolker GLMMs References
- 23. Denitions Estimation Inference Challenges openquestions References Methods Penalized quasi-likelihood (PQL) alternate steps of estimating GLM using known RE variances to calculate weights; estimate LMMs given GLM t (Breslow, 2004) exible (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: 90% of small-unit-sample cases descendants: higher-order PQL, hierarchical GLM Ben Bolker GLMMs in ≈
- 24. Denitions Estimation Inference Challenges openquestions References Methods Penalized quasi-likelihood (PQL) alternate steps of estimating GLM using known RE variances to calculate weights; estimate LMMs given GLM t (Breslow, 2004) exible (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: 90% of small-unit-sample cases descendants: higher-order PQL, hierarchical GLM Ben Bolker GLMMs in ≈
- 25. Denitions Estimation Inference Challenges openquestions References Methods Penalized quasi-likelihood (PQL) alternate steps of estimating GLM using known RE variances to calculate weights; estimate LMMs given GLM t (Breslow, 2004) exible (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: 90% of small-unit-sample cases descendants: higher-order PQL, hierarchical GLM Ben Bolker GLMMs in ≈
- 26. Denitions Estimation Inference Challenges openquestions References Methods Penalized quasi-likelihood (PQL) alternate steps of estimating GLM using known RE variances to calculate weights; estimate LMMs given GLM t (Breslow, 2004) exible (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: 90% of small-unit-sample cases descendants: higher-order PQL, hierarchical GLM Ben Bolker GLMMs in ≈
- 27. Denitions Estimation Inference Challenges openquestions Methods Breslow (2004) on PQL As usual when software for complicated statistical inference procedures is broadly disseminated, there is potential for abuse and misinterpretation. In spite of the fact that PQL was initially advertised as a procedure for approximate inference in GLMMs, and its tendency to give seriously biased estimates of variance components and a fortiori regression parameters with binary outcome data was emphasized in multiple publications [5, 6, 24], some statisticians seemed to ignore these warnings and to think of PQL as synonymous with GLMM. Ben Bolker GLMMs References
- 28. Denitions Estimation Inference Challenges openquestions References Methods Laplace approximation for given β, θ (RE parameters), nd 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 exible lme4:glmer, glmmML, glmmADMB, R2ADMB Ben Bolker GLMMs (AD Model Builder)
- 29. Denitions Estimation Inference Challenges openquestions Methods Gauss-Hermite quadrature (GHQ) as above, but compute additional terms in the integral (typically 8, but often up to 20) most accurate slowest, hence not exible (23 RE at most, maybe only 1) lme4:glmer, glmmML, repeated Ben Bolker GLMMs References
- 30. Denitions Estimation Inference Challenges openquestions Methods Adaptive vs. non-adaptive GHQ Adaptive GHQ is more expensive at a given n , but makes up for it in accuracy Ben Bolker GLMMs References
- 31. Denitions Estimation Inference Challenges openquestions References Methods Stochastic approaches Mostly Bayesians (Bayesian computation handles high-dimensional integration) various avours: Gibbs sampling, MCMC, MCEM, etc. generally slower but more exible simplies many inferential problems must specify priors, assess convergence/error specialized: bernor glmmAK, MCMCglmm (Hadeld, 2010), INLA, glmmBUGS, R2WinBUGS, BRugs (WinBUGS/OpenBUGS), R2jags, rjags (JAGS) general: Ben Bolker GLMMs
- 32. Denitions Estimation Inference Challenges openquestions Methods Estimation: example (McKeon et al., 2012) Log−odds of predation −6 −4 −2 0 2 q q q q q Added symbiont q q q q q Crab vs. Shrimp q Symbiont Ben Bolker GLMMs q q q q GLM (fixed) GLM (pooled) PQL Laplace AGQ References
- 33. Denitions Estimation Inference Outline 1 Examples and denitions 2 Estimation Overview Methods 3 Inference 4 Challenges open questions Ben Bolker GLMMs Challenges open questions References
- 34. Denitions Estimation Inference Challenges openquestions Wald tests Wald tests (e.g. typical results of summary) based on information matrix assume quadratic log-likelihood surface exact for regular linear models; only asymptotically OK for GLM(M)s computationally cheap approximation is sometimes awful (Hauck-Donner eect) Ben Bolker GLMMs References
- 35. Denitions Estimation Inference Challenges openquestions 2D proles for coral predation −2 −4 −6 −8 tttboth −10 −12 0 1 2 3 −2 −6 −4 −2 −4 tttshrimp −6 −8 −10 0 1 2 3 15 0 −4 −2 0 −2 −4 tttcrabs −6 −8 0 1 2 3 −10 10 15 10 (Intercept) 5 0 0 1 2 3 2 4 6 8 101214 .sig01 0 −1 −2 −3 Scatter Plot Matrix Ben Bolker GLMMs References
- 36. Denitions Estimation Inference Challenges openquestions References Likelihood ratio tests better, but still have to deal with two nite-size problems: when scale parameter is free (Gamma, etc.), deviance is ∼ F rather than ∼ χ2 , with poorly dened denominator df in GLM(M) case, numerator is only asymptotically χ2 anyway Bartlett corrections (Cordeiro and Ferrari, 1998; Cordeiro et al., 1994), higher-order asymptotics: cond [neither extended to GLMMs!] Prole condence intervals: moderately dicult/fragile Ben Bolker GLMMs
- 37. Denitions Estimation Inference Challenges openquestions Parametric bootstrapping t null model to data simulate data from null model t null and working model, compute likelihood dierence repeat to estimate null distribution should be OK but ??? not well tested (assumes estimated parameters are suciently good) Ben Bolker GLMMs References
- 38. Denitions Estimation Inference Challenges openquestions Parametric bootstrap results 0.02 0.06 Inferred p value H2S Anoxia 0.08 0.06 0.04 0.02 Osm Cu 0.08 0.06 0.04 0.02 0.02 0.06 True p value Ben Bolker GLMMs References
- 39. Denitions Estimation Inference Challenges openquestions References Bayesian approaches Provided that we have a good sample from the posterior distribution (Markov chains have converged etc. etc.) we get most of the inferences we want for free by summarizing the marginal posteriors Model selection is still an open question: reversible-jump MCMC, deviance information criterion Ben Bolker GLMMs
- 40. Denitions Estimation Inference Outline 1 Examples and denitions 2 Estimation Overview Methods 3 Inference 4 Challenges open questions Ben Bolker GLMMs Challenges open questions References
- 41. Denitions Estimation Inference Challenges openquestions Next steps Dealing with complex random eects: regularization, model selection, penalized methods (lasso/fence) Flexible correlation structures: spatial, temporal, phylogenetic hybrid improved MCMC methods (mcmcsamp, Stan) Reliable Ben Bolker GLMMs assessment of out-of-sample performance References
- 42. Denitions Estimation Inference Challenges openquestions Glycera estimates q q q q q Osm:Cu:H2S:Anoxia q q Cu:H2S:Anoxia q q q q q Osm:Cu:Anoxia qq q q q Osm:Cu:H2S q qq q q H2S:Anoxia qq qq q q q q q q Cu:Anoxia Osm:Anoxia Cu:H2S q qq qq q q q Osm:H2S:Anoxia q q q qq q q qq q qq qq q q qq q q Osm:H2S Osm:Cu Anoxia qq qq q qq q q q H2S Cu q q q q q Osm −60 −40 −20 0 q q q q q MCMCglmm glmer(OD:2) glmer(OD) glmmML glmer 20 Effect on survival Ben Bolker GLMMs 40 60 References
- 43. Denitions Estimation Inference Challenges openquestions Acknowledgments lme4: Doug Bates, Martin Mächler, Steve Walker Data: Adrian Stier (UBC/OSU), NSERC (Discovery) Sea McKeon (Smithsonian), SHARCnet David Julian (UF), Jada-Simone White (Univ Hawai'i) Ben Bolker GLMMs References
- 44. Denitions Estimation Inference Challenges openquestions References Booth, J.G. and Hobert, J.P., 1999. Journal of the Royal Statistical Society. Series B, 61(1):265285. doi:10.1111/1467-9868.00176. 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 122. Springer. ISBN 0387208623. Cordeiro, G.M. and Ferrari, S.L.P., 1998. Journal of Statistical Planning and Inference, 71(1-2):261269. 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):257274. ISSN 03067734. doi:10.2307/1403512. Hadeld, J.D., 2010. Journal of Statistical Software, 33(2):122. ISSN 1548-7660. McKeon, C.S., Stier, A., et al., 2012. Oecologia, 169(4):10951103. ISSN 0029-8549. doi:10.1007/s00442-012-2275-2. Pinheiro, J.C. and Bates, D.M., 1996. Statistics and Computing, 6(3):289296. doi:10.1007/BF00140873. Ponciano, J.M., Taper, M.L., et al., 2009. Ecology, 90(2):356362. ISSN 0012-9658. Sung, Y.J., 2007. The Annals of Statistics, 35(3):9901011. ISSN 0090-5364. doi:10.1214/009053606000001389. Ben Bolker GLMMs