GLMs 40+ years on: A Personal Perspective

GLMs 40+ years on: A Personal Perspective

GLMs 40+ years on: A Personal Perspective
John Hinde
Statistics Group,
Science Foundation Ireland funded Bio-SI Project
School of Mathematics, Statistics and Applied Mathematics
National University of Ireland, Galway
[email protected]
Research Supported by SFI Award 07/MI/012
RBras 2013, Campina Grande
23 July 2013
John Hinde (NUIG)
23 July 2013
1 / 47
Summary
1
The 1972 Paper
2
Spreading the word
3
Random effects
4
Brazil and biometry
5
Extensions
6
Acknowledgements
John Hinde (NUIG)
23 July 2013
2 / 47
The Paper — 1972: 40+ Anniversary
published in Series A
15 pages long
many examples — over half the paper
“useful way of unifying . . . unrelated statistical procedures”
John Hinde (NUIG)
23 July 2013
3 / 47
glms — the authors
John Nelder: 1924 — 2010
Statistician at National Vegetable Research Station (NVRS), now
Horticultural Research International, Wellesbourne — 1949-68
theory of general balance — unifying framework for the wide range of
designs in agricultural experimentation
initial work on GenStat
Head of the Statistics Department at Rothamsted — 1968-1984
theory of generalized linear models, with the late Robert Wedderburn
Applied Statistics Algorithms in Applied Statistics, JRSSC
further development of GenStat, with NAG
development of GLIM, first released in 1974
visiting Professor at Imperial College — 1972-2009
GLIMPSE “expert system” based on GLIM
theory of hierarchical generalized linear models (HGLMs), with Youngjo
Lee
Robert Wedderburn: 1947 —1975
Died aged 28 of anaphylactic shock from an insect bite.
John Hinde (NUIG)
23 July 2013
4 / 47
John Nelder: 1924 — 2010
John Hinde (NUIG)
23 July 2013
5 / 47
glms — the background
analysis of non-normal data — variance stabilising transformation of
the response
√
Poisson count data: square-root transformation, y
√
Binomial proportions: arc-sin-square-root, sin−1 ( y )
Exponential times: log transformation, log(y )
Probit analysis: Finney (1952) maximum likelihood for tolerance
distribution in toxicology
Dyke & Patterson (1952): logit model for analysis of proportions in
factorial experiment
transformations to linearity
Box-Cox transformation (1964)
Inverse polynomials, Nelder (1966)
Nelder (1968): . . . one transformation leads to a linear model and
another to normal error.
John Hinde (NUIG)
23 July 2013
6 / 47
glms — the idea
John Hinde (NUIG)
23 July 2013
7 / 47
glm Paper: contents
Intro: background ( 2 pages)
random component: 1-parameter exponential family
linear predictor: η = β0 + β1 x1 + · · · βp xp
link function: g (µ) = η
Model fitting: ( 3 pages)
maximum likelihood estimation using Fisher Scoring
Iteratively (Re)-Weighted Least Squares
sufficient statistics — canonical links
Analysis of Deviance
minimal ↔ complete (saturated) models
Special distributions, examples ( 6 pages)
Models in Teaching Statistics ( 1 page)
John Hinde (NUIG)
23 July 2013
8 / 47
glm Paper: examples
Normal: observations normal on log-scale; additive effects on inverse
scale
Poisson: Fisher’s tuberculin-test data — Latin square of counts
Poisson: multinomial distributions for contingency tables
Binomial: Probit & Logit models
Gamma: estimation of variance components in incomplete block
design
John Hinde (NUIG)
23 July 2013
9 / 47
Dissemination of glms
Conferences — “That’s a glm!”
Nelder (1984) Models for Rates with Poisson Errors: In a recent
paper, Frome (1983) described the fitting of models with Poisson
errors and data in the form of rates . . . fitted simply by GLIM . . . or
the use of a program that handles iterative weighted least squares
Nelder (1991) Generalized Linear Models for Enzyme-Kinetic Data:
Ruppert, Cressie, and Carroll (1989) discuss various models for fitting
the Michaelis-Menten equations to data on enzyme kinetics. I find it
surprising that they do not include, among the models they consider,
generalized linear models (GLMs) with an inverse link
The data-transformation approach suffers from the disadvantage that
normality of errors and linearity of systematic effects are still being
sought simultaneously
John Hinde (NUIG)
23 July 2013
10 / 47
Generalized Linear Models — Monograph
2013 Karl Pearson Prize - isi-web.org
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2013 Karl Pearson Prize
The ISI’s Karl Pearson Prize was established in 2013 to recognize a contemporary a
research contribution that has had profound influence on statistical theory,
methodology, practice, or applications. The contribution can be a research article or a
book and must be published within the last three decades. The prize is sponsored by
Elsevier B.V.
The inaugural Karl Pearson Prize is awarded to Peter McCullagh
and John Nelder[1] for their monograph Generalized Linear
Models (1983).
This book has changed forever teaching, research and practice in statistics. It provides
a unified and self-contained treatment of linear models for analyzing continuous,
binary, count, categorical, survival, and other types of data, and illustrates the
methods on applications from different areas. The monograph is based on several
groundbreaking papers, including “Generalized linear models,” by Nelder and
Wedderburn, JRSS-A (1972), “Quasi-likelihood functions, generalized linear models,
and the Gauss-Newton method,” by Wedderburn, Biometrika (1974), and “Regression
models for ordinal data,” by P. McCullagh, JRSS-B (1980). The implementation of GLM
was greatly facilitated by the development of GLIM, the interactive statistical package,
by Baker and Nelder. In his review of the GLIM3 release and its manual in JASA 1979
(pp. 934-5), Peter McCullagh wrote that "It is surprising that such a powerful and
unifying tool should not have achieved greater popularity after six or more years of
existence.” The collaboration between McCullagh and Nelder has certainly remedied
this issue and has resulted in a superb treatment of the subject that is accessible to
researchers, graduate students, and practitioners.
The prize will be presented on August 27, 2013 at the ISI
World Statistics Congress in Hong Kong and will be followed by
the Karl Pearson Lecture by Peter McCullagh.
Karl Pearson Lecture: Statistical issues in modern scientific
research
Peter McCullagh
University of Chicago, USA
John Hinde (NUIG)
http://www.isi-web.org/WSC/726-2013kpp[19/07/2013 14:12:54]
23 July 2013
11 / 47
Centre for Applied Statistics, Lancaster
Group set up by Murray Aitkin
Focus of activities:
EM algorithm
Mixture models — normal, latent class
mixed models
glms
statistical computing
count data and overdispersion
...
Advanced training courses
John Hinde (NUIG)
23 July 2013
12 / 47
Statistical Modelling in GLIM (1989)
An applied how to text with integrated GLIM code.
normal models
regression
analysis of variance
binomial responses
multinomial and Poisson
count data
multiway tables
survival models
parametric
Cox PH — piecewise exponential
discrete time
John Hinde (NUIG)
23 July 2013
13 / 47
GLIM Conferences, IWSM, Statistical Modelling
GLIM conferences — really on glms
IWSM: International Workshop on Statistical Modelling
Eventually led to Statistical Modelling Society
John Hinde (NUIG)
23 July 2013
14 / 47
Statistical Modelling Journal
In 2000, founding of journal Statistical Modelling
availability of data and code with papers → reproducible research
Statistical Modelling: An International Journal
http://stat.uibk.ac.at/SMIJ/
STATISTICAL MODELLING
Aims and Scope
Editorial Board
AN
INTERNATIONAL
JOURNAL
from
For Authors
Archives
Modelling Society
Statistical Modelling: An International Journal publishes original and high-quality articles that
recognize statistical modelling as the general framework for the application of statistical ideas.
Submissions must reflect important developments, extensions, and applications in statistical
modelling. The journal also encourages submissions that describe scientifically interesting,
complex or novel statistical modelling aspects from a wide diversity of disciplines, and submissions
that embrace the diversity of applied statistical modelling.
Indexed by Science Citation Index Expanded, ISI Alerting Services, and CompuMath
Citation Index, beginning with volume 3 (2003).
John Hinde (NUIG)
23 July 2013
15 / 47
John Nelder & Statistical Computing
Anti back-box packages
User should be in control
Default output should be minimal
System should not allow stupid models — marginality
Model specification using Wilkinson & Rogers formulæ
All structures available to the user — input to other routines
system should be open — user extendible (GLIM, GenStat, S/R, . . . )
Requires user expertise/knowledge
Principles embodied in GLIM
— a system specifically for fitting glms.
John Hinde (NUIG)
23 July 2013
16 / 47
GLIM: Interactive package (A Fistful of $’s!!)
[i] ? $yvar days $error p $
[i] ? $fit A*S*C*L $
[o] scaled deviance =
[o]
1173.9 at cycle 4
residual df =
118
Or, in John’s preferred style . . .
[i] ? $y days $e p $
[i] ? $f A*S*C*L $
John Hinde (NUIG)
23 July 2013
17 / 47
Current Software
Genstat
glm’s, glmm’s, ASREML, hglm’s, . . .
SAS
Stata
R
glm, gam, glmm, rbugs, r-inla, . . .
gnm — nonlinear glm’s (Firth & Turner)
gamlss - Mikis Stasinpoulos
John Hinde (NUIG)
23 July 2013
18 / 47
Normal Models
y = βT x + single error term includes
individual observation/measurement error
experimental unit variability
unobserved covariates
for simplest data structures/designs use normal linear model
more complex situations
structure in experimental unit variability
repeated measures/longitudinal observations
...
John Hinde (NUIG)
23 July 2013
19 / 47
Normal Mixed Model
y = βT x + γ T z + z unobserved random effects
shared random effects
multi-level/variance components models
longitudinal observations
spatial structure
z normal
normal model with structured covariance matrix
standard mixed model analyses – ML, REML
widely available in standard software
John Hinde (NUIG)
23 July 2013
20 / 47
Generalized Linear Models
Models for counts, proportions, times, . . .
y ∼ F (µ)
g (µ) = η = β T x
distributional assumption relates to the observation/measurement
process
how does this model incorporate
experimental/individual unit variability?
unobserved covariates?
It doesn’t!
hence overdispersion, etc
John Hinde (NUIG)
23 July 2013
21 / 47
Random Effect Models
Include random effect(s) in the linear predictor
η = βT x + γ T z
single conjugate random effect at individual level – standard
overdispersion models
negative binomial for count data
beta-binomial for proportions
z normal −→ generalized linear mixed models
z unspecified −→ nonparametric maximum likelihood
John Hinde (NUIG)
23 July 2013
22 / 47
Prof Clarice Demétrio, ESALQ/USP Brazil
Chance encounter at British Region Conference in Sussex led to ongoing
collaboration on extensions to glms
overdispersion modelling
counts
proportions
multicategory responses
models for data with excess zeros
truncated distributions
zero inflated models
score tests and model selection
applications to entomology, plant science,
...
John Hinde (NUIG)
23 July 2013
23 / 47
Biological Control
Entomological Biological Control —
control pests without damage to the
environment.
Boosted by the development of
techniques for intensive breeding of
insects.
In Brazil, Diatraea saccharallis is the
main pest in sugar cane cultivation.
Loss in São Paulo State approx.
US$119 million/year.
Control using natural parasites.
John Hinde (NUIG)
23 July 2013
24 / 47
Assay Experiment
two species of a parasite (Trichogramma galloi), AA and DA, were
put to parasitise 128 eggs of the Anagasta kuehniella, an
economically convenient alternative host;
a variable number of female parasites (2, 4, 8, . . ., 128) each with 10
replicates
AA species adapted to the alternative host; DA species adapted to
the natural host.
Response variable — number of parasitised eggs.
Compare the species and determine the optimal number of females
needed to produce the maximum number of parasitised eggs.
AA species: nonlinear pattern for the proportion of parasitised eggs;
DA species: pattern is unclear, many zeros observations.
John Hinde (NUIG)
23 July 2013
25 / 47
Parasitisation response: Biotypes AA & DA
Biotype DA
1.1
1.0
1.0
0.9
0.9
0.8
0.8
Proportion of Parasitised eggs
Proportion of parasitised eggs
Biotype AA
1.1
0.7
0.6
0.5
0.4
0.3
0.7
0.6
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0.0
0.0
1
2
3
Ln(Number of Females)
John Hinde (NUIG)
4
1
2
3
4
Ln(number of females)
23 July 2013
26 / 47
Models
Binomial
Overdispersed binomial
Overdispersed truncated binomial — removes zeros
Bernoulli-binomial mixtures — constant zero inflation
Bernoulli-overdispersed binomial mixtures
overdispersion + non-constant zero-inflation
John Hinde (NUIG)
23 July 2013
27 / 47
Potato Larvae Bioassay
Samples of potatoes were each infected with 30 larvae
Different concentrations of an insecticide applied to potato samples
Control sample (no insecticide) with 9 potatoes
Experiment conducted at 18o C, and after 60 days the numbers of
dead larvae were counted
Standard binomial model with log conc allowing for natural
mortality
Random effect at individual potato level — overdispersion
Random effect at concentration level — measurement error
John Hinde (NUIG)
23 July 2013
28 / 47
Potato Larvae Bioassay: Fitted Models
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observed values
fitted values
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0.8
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ED50 = 4.57
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0.4
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0.6
0.6
ED50 = 4.73
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0.2
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0.4
0.8
observed values
fitted values
0.0
Proportion of mortality
●
0.2
●
Random effect
Proportion of mortality
1.0
Binomial
−2
0
2
4
Log−concentration (OB/mL)
John Hinde (NUIG)
6
−2
0
2
4
6
Log−concentration (OB/mL)
23 July 2013
29 / 47
Dataset: Biological Pest Control
Termite Heterotermes tenuis: an important pest of sugarcane in
Brazil, causing damage of up to 10 metric tonnes/ha/year.
Fungus Beauveria bassiana: a possible microbial control.
Experiment: on the pathogenicity and virulence of 142 different
isolates of Beauveria bassiana.
Completely randomized experiment: five replicates of each of the 142
isolates.
Solutions of the isolates applied to groups (clusters) of n = 30 termites
kept in plastic Petri-dishes.
Mortality in the groups was measured daily for eight days
Data: 710 ordered multinomial observations of length eight.
John Hinde (NUIG)
23 July 2013
30 / 47
Cumulative Mortality: sample of isolates
2
1003
4
6
8
2
1006
1024
4
6
8
1028
1.0
0.8
0.6
0.4
0.2
0.0
879
883
885
957
787
823
848
852
1.0
0.8
0.6
proportion
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
732
743
745
767
1.0
0.8
0.6
0.4
0.2
0.0
2
4
6
8
2
4
6
8
days
John Hinde (NUIG)
23 July 2013
31 / 47
0.6
0.4
0.0
0.2
proportion
0.8
1.0
Cumulative Mortality: spaghetti plot of all isolates
1
2
3
4
5
6
7
8
days
John Hinde (NUIG)
23 July 2013
32 / 47
Multinomial Model: Cumulative Proportions
Because of natural time ordering consider models for the cumulative
proportions (isolate i, replicate k)
Rik,d = proportion of insects dead by day d,
γik,d = E(Rik,d ) = probability an insect dies by day d,
Rik = (Rik,1 , Rik,2 , . . . , Rik,D )T =
1
LYik
n
E[Rik ] = Lπik = γik
Var[Rik ] =
John Hinde (NUIG)
1
T
L[diag{π ik } − π ik π T
ik ]L = V(γ ik )
n
23 July 2013
33 / 47
Multinomial Model(ctd)
Use a glm with link function:
g (γik ) = Xik βi
Logit link function −→ cumulative logistic model


j
P
πik,s 


 s=1
g (γikj ) = logit(γikj ) = log  D+1
 = ηikj
P


πik,s
s=j+1
alternative models: discrete survival models, other ordinal models
Linear predictor: isolate specific factors, time dependency, . . .
e.g. Isolate specific linear time effect, constant over replicates
ηikj = β1i + β2i tj ,
John Hinde (NUIG)
23 July 2013
34 / 47
Random Effect Models
Incorporate random effects in the linear predictor:
Add random effect for each experimental unit (groups of insects).
simple time shifts
time dependent covariates with random coefficients
Replicate level random effect — accounts for overdispersion
Model isolates as a random effect.
ηikj = µ + timej + ui + ik
Non-parametric maximum likelihood techniques give a finite
mass-point distribution {ωk ; zk } for the isolate effects ui .
Using a small number of components may identify effective isolates –
look at the posterior distribution of ui .
John Hinde (NUIG)
23 July 2013
35 / 47
Drosophila Melongaster Data
2
0
−2
−6
−4
Expression level
4
Raw Data Fruitfly
0
5
10
15
20
Time
Gene expression profiles of genes in the embryo phase of Drosophila
Melongaster (fruitfly) data.
John Hinde (NUIG)
23 July 2013
36 / 47
Time-course gene expression data
Gene expression over time can be thought of as arising from a smooth
underlying curve/function.
Expression values usually measured with error/noise.
For each gene use the model
yj = g (tj ) + εj , j = 1, . . . , n
obs j at time tj , g (t) = smooth expression profile, εj is measurement
error.
Want to remove noise and estimate smooth expression profiles g (t).
Do this using basis function expansions.
John Hinde (NUIG)
23 July 2013
37 / 47
P-splines as mixed model
Represent P-spline smoothing as a linear mixed effects model.
Mixed effects model has the form
y = Xβ + Zu + ε.
For simplicity assume ε ∼ N(0, σε2 I).
For smoothing must also assume u ∼ N(0, σu2 I).
Estimates of β, σε2 , σu2 and u determined using (RE)ML and BLUP.
John Hinde (NUIG)
23 July 2013
38 / 47
Modelling gene expression clusters
Use P-splines to model mean curve µc (t) of cluster c.
Assumed the expression profile of gene i in cluster c follows the shape
of the mean curve of that cluster, but with a gene-specific shift (bi )
from that mean.
John Hinde (NUIG)
23 July 2013
39 / 47
Modelling gene expression clusters
Write the expression level for gene i in cluster c at time j as
yij = µc (tij ) + bi + εij , j = 1, . . . , ni ,
2 ).
where bi ∼ N(0, σbc
bi allows for gene-specific shift from the mean.
Equivalent to saying the ith gene in cluster c is distributed as
yi ∼ N(µc , Vc ),
2 E
2
where Vc = σbc
ni ×ni + σεc Ini ×ni .
Eni ×ni is a matrix where all the entries are 1.
John Hinde (NUIG)
23 July 2013
40 / 47
Modelling gene expression profiles
Full mixed model for estimating mean curve and individual expression
profiles in cluster c
Yc = Xc,s βc,s + Zc,s uc,s +Zc,b bc + εc ,
{z
}
|
µc (t)
Yc , Xc,s , Zc,s as defined previously.

Z1,b
0
···
0
 0
Z2,b · · ·
0

Zc,b =  .
..
..
.
.
.
 .
.
.
.
0
0
· · · ZNc ,b
2
uc,s ∼ N(0, σuc
I),
John Hinde (NUIG)
2
bc ∼ N(0, σbc
I)





2
I)
εc ∼ N(0, σεc
23 July 2013
41 / 47
Mixtures of mixed effects models
In practice do not know cluster membership.
Assume yi comes from a mixture of C clusters/groups
yi ∼ π1 N(µ1 , V1 ) + π2 N(µ2 , V2 ) + . . . + πC N(µC , VC ).
π1 , π2 , . . . , πC are mixing proportions,
C
P
πc = 1.
c=1
Obtain (posterior) probability that gene i comes from cluster c.
Estimate for each gene.
Also need to estimate (µ1 , V1 ), . . . , (µC , VC ).
Use the EM algorithm.
John Hinde (NUIG)
23 July 2013
42 / 47
Results Drosophila Melongaster Data
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10
15
20
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Time
John Hinde (NUIG)
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20
Time
23 July 2013
43 / 47
Simple mixture and random effect extensions
Aitkin, M, Francis, B, Hinde, J (2008)
Statistical Modelling in GLIM4, Oxford.
Aitkin, M, Francis, B, Hinde, J, Darnell, R (2009)
Statistical Modelling in R, Oxford.
John Hinde (NUIG)
23 July 2013
44 / 47
Extensions
Quasi-likelihood, . . .
Generalized additive models, smoothers, . . .
Multivariate glms
Random effect models, GLMMs, . . .
Mean and dispersion models, double glms, GAMLLs, . . .
Hierarchical glms
Generalized nonlinear glms
...
John Hinde (NUIG)
23 July 2013
45 / 47
The Future
Where next?
Over to you . . .
John Hinde (NUIG)
23 July 2013
46 / 47
Acknowledgements
Norma Coffey
Clarice Demétrio
Jochen Einbeck
Silvia de Freitas
Emma Holian
Naratip Jansakul
Marie-José Martinez
Georgios Papageorgiou
Martin Ridout
Mariana Ragassi Urbano
Afrânio Vieira
John Hinde (NUIG)
23 July 2013
47 / 47