1. The document discusses generalized linear mixed models (GLMMs), which are statistical models that combine linear predictors, non-normal response distributions, link functions, and random effects. 2. It outlines some of the statistical, computational, and sociological challenges in using GLMMs, such as estimating models with large matrices and interpreting results accurately. 3. The conclusion emphasizes next steps like improving correlation structures and inference methods in GLMMs while addressing issues like proper interpretation and use by non-experts.

1. Denitions Statistics Computation Sociological Conclusions References
General-purpose tools
for generalized linear mixed models
Ben Bolker
McMaster University, Mathematics Statistics and Biology
13 September 2013
Ben Bolker
GLMMs

2. Denitions Statistics Computation Sociological Conclusions References
Outline
1 Denitions and context
2 Statistical challenges
3 Computational challenges
4 Sociological challenges
5 Conclusions
Ben Bolker
GLMMs

3. Denitions Statistics Computation Sociological Conclusions References
Outline
1 Denitions and context
2 Statistical challenges
3 Computational challenges
4 Sociological challenges
5 Conclusions
Ben Bolker
GLMMs

4. Denitions Statistics Computation Sociological Conclusions References
Generalized linear mixed models
GLMMs: 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 Statistics Computation Sociological Conclusions References
Generalized linear mixed models
GLMMs: 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 Statistics Computation Sociological Conclusions References
Generalized linear mixed models
GLMMs: 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 Statistics Computation Sociological Conclusions References
Generalized linear mixed models
GLMMs: 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 Statistics Computation Sociological Conclusions References
Technical denition
Yi
response
∼
conditional
distribution
Distr (g
−1(ηi )
inverse
link
function
, φ
scale
parameter
)
η
linear
predictor
= Xβ
xed
eects
+ Zb
random
eects
b
conditional
modes
∼ MVN(0, Σ(θ)
variance-
covariance
matrix
)
Ben Bolker
GLMMs

9. Denitions Statistics Computation Sociological Conclusions References
Outline
1 Denitions and context
2 Statistical challenges
3 Computational challenges
4 Sociological challenges
5 Conclusions
Ben Bolker
GLMMs

10. Denitions Statistics Computation Sociological Conclusions References
Estimation
Maximum likelihood estimation
L(Yi |θ, β)
likelihood
= · · · L(Yi |θ, β )
data|random eects
× L(β |Σ(θ))
random eects
dβ
deterministic: precision vs. computational cost:
penalized quasi-likelihood, Laplace approximation, adaptive
Gauss-Hermite quadrature (Breslow, 2004) . . .
Monte Carlo: frequentist and Bayesian (Booth and Hobert,
1999; Ponciano et al., 2009; Sung, 2007)
Ben Bolker
GLMMs

11. Denitions Statistics Computation Sociological Conclusions References
Estimation: example (McKeon et al., 2012)
Log−odds of predation
−6 −4 −2 0 2
Symbiont
Crab vs. Shrimp
Added symbiont
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
GLM (fixed)
GLM (pooled)
PQL
Laplace
AGQ
Ben Bolker
GLMMs

12. Denitions Statistics Computation Sociological Conclusions References
Inference
Big problem.
Inferential tools: either asymptotic
or taken from classical linear
models
boundary solutions (Stram and
Lee, 1994)
the great p-value/degrees of
freedom debate
small numbers of clusters
solutions: computational
and/or Bayesian
(parametric bootstrap, MCMC)
True p value
Inferredpvalue
0.02
0.04
0.06
0.08
0.02 0.06
Osm Cu
H2S
0.02 0.06
0.02
0.04
0.06
0.08
Anoxia
Ben Bolker
GLMMs

13. Denitions Statistics Computation Sociological Conclusions References
Outline
1 Denitions and context
2 Statistical challenges
3 Computational challenges
4 Sociological challenges
5 Conclusions
Ben Bolker
GLMMs

14. Denitions Statistics Computation Sociological Conclusions References
Sparse matrix algorithms
repeated decomposition of
large, matrices (especially Z)
ll-reducing permutation to
improve sparsity pattern
further improvements possible:
better matrix representation,
parallelization?
Ben Bolker
GLMMs

15. Denitions Statistics Computation Sociological Conclusions References
Bounded optimization
Parameterize
variance-covariance matrix Σ(θ)
(Pinheiro and Bates, 1996)
Positive denite or only
semi-denite?
Disadvantages of transforming
to unconstrain
(Disadvantages of boundary
solutions)
raw log
0
10
20
30
0 1 2 3 −3 −2 −1 0
deviance
Ben Bolker
GLMMs

16. Denitions Statistics Computation Sociological Conclusions References
Outline
1 Denitions and context
2 Statistical challenges
3 Computational challenges
4 Sociological challenges
5 Conclusions
Ben Bolker
GLMMs

17. Denitions Statistics Computation Sociological Conclusions References
Sociological issues
The curse of neophilia
Wide user base:
As usual when software for complicated statistical
inference procedures is broadly disseminated, there is
potential for abuse and misinterpretation.
(Breslow, 2004)
What if there is no good answer?
do no harm vs. better me than someone else
Diagnostics and warning messages
End users vs. downstream developers
Ben Bolker
GLMMs

18. Denitions Statistics Computation Sociological Conclusions References
Outline
1 Denitions and context
2 Statistical challenges
3 Computational challenges
4 Sociological challenges
5 Conclusions
Ben Bolker
GLMMs

19. Denitions Statistics Computation Sociological Conclusions References
Next steps
Alternative platforms/languages
Flexible correlation structures:
spatial, temporal, phylogenetic . . .
Improved MCMC methods?
Simulation tests of inferential tools (sigh)
Ben Bolker
GLMMs

20. Denitions Statistics Computation Sociological Conclusions References
Is it science?
Science is what we
understand well enough to
explain to a computer. Art
is everything else we do.
(Donald Knuth)
10
20
30
40
50
2006 2008 2010 2012
Date
articlespermonth
key
glmm
lme4
Ben Bolker
GLMMs

21. Denitions Statistics Computation Sociological Conclusions References
Acknowledgments
lme4: Doug Bates, Martin
Mächler, Steve Walker
Data: Adrian Stier (UBC/OSU),
Sea McKeon (Smithsonian),
David Julian (UF)
NSERC (Discovery)
SHARCnet
Ben Bolker
GLMMs

22. Denitions Statistics Computation Sociological Conclusions 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.
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.
Stram, D.O. and Lee, J.W., 1994. Biometrics, 50(4):11711177.
Sung, Y.J., 2007. The Annals of Statistics, 35(3):9901011. ISSN 0090-5364.
doi:10.1214/009053606000001389.
Ben Bolker
GLMMs