Outline
Introduction
Concepts
Stats
Examples
Software
Extensions
Discussion
Flexible modelling of the cumulative effects
of time-varying exposures
Applications in environmental, cancer and
pharmaco-epidemiology
Antonio Gasparrini
Department of Medical Statistics
London School of Hygiene and Tropical Medicine (LSHTM)
Centre for Statistical Methodology – LSHTM
28 November 2014
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
LSHTM
Outline
Introduction
Concepts
Stats
Examples
Software
Extensions
Discussion
Outline
1
Introduction
2
Conceptual model
3
Statistical model
4
Examples
5
Software
6
Extensions
7
Discussion
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
LSHTM
Outline
Introduction
Concepts
Stats
Examples
Software
Extensions
Discussion
Temporal aspects
The relationship between a risk factor and the associated health
effect always implies a temporal dependency: a common problem
in biomedical research
This issue encompasses study designs and statistical model:
Tobacco smoke and CVD risk
Occupational exposure and incidence of cancer
Drug intake and beneficial or side effects
Short-term temperature variation and mortality
A topic (somewhat) neglected in methodological research
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Outline
Introduction
Concepts
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Discussion
Previous research
Standard statistical approaches do not directly characterize this
temporal structure
Challenge: modelling (potentially complex) temporal patterns of
risk due to time-varying exposures
Models previously proposed in cancer epidemiology (Thomas
1988, Hauptmann 2000, Richardson 2009) and
pharmaco-epidemiology (Abrahamowicz 2012)
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Outline
Introduction
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Discussion
Limitations
Incomplete statistical development: e.g. no measures of
uncertainty
Poor software implementation: ad-hoc routines, computational
issues, convergence problems
Lack of a consistent conceptual and interpretational
framework
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Outline
Introduction
Concepts
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Extensions
Discussion
Distributed lag models
DLMs proposed by Almon (Econometrica 1965) in econometrics
for time series data, then applied in environmental epidemiology
by Schwartz (Epidemiology 2000).
Armstrong (Epidemiology 2006) extended them to distributed lag
non-linear models (DLNMs), applicable to non-linear
exposure-response associations
A far more developed statistical framework, but only applicable to
time series data
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Outline
Introduction
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Conceptual representation
Single exposure event
Forward perspective
●
Effect
●
●
0
●
t
t+1
…
t+2
●
●
…
t+L
Time
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Conceptual representation
Multiple exposure events
0
Effect
Backward perspective
t−L
…
…
t−2
t−1
t
Time
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Flexible modelling of the cumulative effects of time-varying exposures
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Assumptions
Under specific assumptions, these two perspectives can be merged
together:
assumption of identical effects
(fundamental) assumption of independency
These conditions underpin the conceptual framework for defining
and modelling DLNMs
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Outline
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Conceptual representation
New lag dimension
●
●
●
●
●
Effect
●
●
0
●
●
●
●
● ●
● ● ● ● ● ●
● ●
Forward
Backward
t0
0
t0 + L
te
lag
L
Time (Lags)
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Exposure-lag-response associations
The risk is represented by a function s(xt−` , . . . , xt−L ) defined in
terms of both intensity and timing of a series of past exposures,
expressed through:
an exposure-response function f (x) for exposure x
a lag-response function w (`) for lag `
Generating a bi-dimensional exposure-lag-response function
f ·w (x, `), whose integral provides:
Z
L
f ·w (xt−` , `) d` ≈
s(xt−` , . . . , xt−L ) =
`0
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
L
X
f ·w (xt−` , `)
`=`0
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Distributed lag models (DLMs)
Given a exposure history at time t for lags ` = `0 , . . . , L:
qxt = [xt−`0 , . . . , xt−` , . . . , xt−L ]T
and assuming a linear exposure-response, we can write:
T
s(qxt ; η) = qT
xt Cη = wxt η
where C is obtained from the lag vector ` = [`0 , . . . , `, . . . , L]T by
applying a specific basis transformation
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Flexible modelling of the cumulative effects of time-varying exposures
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Distributed lag non-linear models (DLNMs)
First the matrix Rxt is obtained applying a second basis
transformation to qxt
Then we define a tensor product:
T
Axt = (1T
v` ⊗ Rxt ) (C ⊗ 1vx )
which forms the crossbasis:
T
s(qxt ; η) = (1T
vx ·v` Axt )η = wxt η
The problem reduces to choosing a basis for each qxt and `,
defining exposure-response and lag-response functions,
respectively
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Alternative study designs
Time series
A
Case−control
A
B
C
tBj
B
●
tAj
●
Longitudinal
Cohort
A
C
A
tBj
B
●
tCk
tAj ●
C
tBj
●
tCj
tj
tAj ●
tCk
●
●
tk
tCj
tAk●
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
tAk●
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First example
Temperature and all-cause mortality
Research area where DLNMs were originally proposed
Time series data with daily death counts and temperature
measurements between 1st Jan 1993 and 31st Dec 2006 in London
(845,215 deaths in total)
In this setting, exposure histories are simply derived by ’lagging’
the temperature series
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Quasi-Poisson GLM
Analysis with a generalized linear model with quasi-Poisson family,
controlling for trends and day of the week
log(µt ) = α + sx (qxt ; η) +
P
X
sz (zt ; βz )
p=1
Here spline functions used to specify both f (x) and w (`)
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Flexible modelling of the cumulative effects of time-varying exposures
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Exposure-lag-response
1.10
1.05
RR
0
1.00
5
0.95
La
g
10
20
15
15
Tem
p
10
erat
ure 5
(C)
0
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
20
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Summaries
Exposure−lag−response
1.04
1.06
1.08
Lag−response at temperature 22C
1.02
RR
1.10
1.05
1.00
RR
1.00
0
0.98
5
0.95
20
La
g
10
15
15
Tem 10
pea
ture 5
(C
)
0
0
10
15
20
Lag
Overall cumulative exposure−response
1.4
0.8
0.98
1.0
1.00
1.2
RR
1.02
1.6
1.04
1.8
2.0
1.06
Exposure−response at lag 4
RR
5
20
−5
0
5
10
15
20
25
−5
Temperature (C)
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
0
5
10
15
20
25
Temperature (C)
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Second example
Radon exposure and lung cancer mortality
3,347 subjects working in the Colorado Plateau mines between
1950–1960, 258 lung cancer deaths
Yearly exposure history to radon (WLM) and tobacco smoke
(pack×100) reconstructed from 5-year age periods
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Proportional hazard model
Analysis with Cox proportional hazards model using age as time
axis, controlling for smoking and calendar year. For subject i:
log [h(it)] = log [h0 (t)] + sx (qxit ; ηx ) + sz (qzit ; ηz ) + γuit
Different functions used to specify f (x) and w (`): constant,
piecewise constant, quadratic B-spline
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Exposure-lag-response
Linear-by-constant
1.10
1.08
1.06
HR
1.04
1.02
1.00
0.98
250
200
5
10
150
W
15
LM
/ye 100
ar
20
25
30
50
35
0
g
La
)
ars
(ye
40
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Exposure-lag-response
Spline-by-constant
1.16
1.14
1.12
1.10
HR
1.08
1.06
1.04
1.02
1.00
0.98
250
200
5
10
150
W
15
LM
/ye 100
ar
20
25
30
50
35
0
g
La
)
ars
(ye
40
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Outline
Introduction
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Discussion
Exposure-lag-response
Linear-by-spline
1.10
1.08
1.06
HR
1.04
1.02
1.00
0.98
250
200
5
10
150
W
15
LM
/ye 100
ar
20
25
30
50
35
0
g
La
)
ars
(ye
40
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
LSHTM
Outline
Introduction
Concepts
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Discussion
Exposure-lag-response
Step-by-step
1.30
1.25
1.20
HR
1.15
1.10
1.05
1.00
250
200
5
10
150
W
15
LM
/ye 100
ar
20
25
30
50
35
0
g
La
)
ars
(ye
40
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Outline
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Discussion
Exposure-lag-response
Spline-by-spline
1.30
1.25
1.20
HR
1.15
1.10
1.05
1.00
250
200
5
10
150
W
15
LM
/ye 100
ar
20
25
30
50
35
0
g
La
)
ars
(ye
40
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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1.3
Lag-response curves from DLNMs
1.2
1.1
1.0
0.9
RR for 100 WLM/year
Spline−by−spline
Spline−by−piecewise
0
5
10
15
20
25
30
35
40
Lag (years)
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Flexible modelling of the cumulative effects of time-varying exposures
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1.3
Exposure-responses at different lags
Lag 15
Lag 25
1.1
0.9
1.0
RR at lag 15
1.2
Lag 5
0
50
100
150
200
250
WLM/year
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Flexible modelling of the cumulative effects of time-varying exposures
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Third example
MMR vaccine and ITP risk
Data from 35 children receiving the MMR (measles, mumps,
rubella) vaccine months and admitted to the hospital for idiopathic
trombocytopenic purpura (ITS) within 12-24 months of age.
Replicating and extending a previous analysis using the
self-controlled case series design (Whitaker 2006)
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Conditional Poisson regression
Analysis with conditional Poisson regression controlling for age.
For subject i at age a:
log(λiat ) = αi + sx (qxit ; ηx ) + f (ait ; γ)
Single exposure event modelled with a binary variable
Exposure-response assumed linear, lag-response modelled with
spline or piecewise constant functions
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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10
Spline
Piecewise constant
0
5
IRR
15
Lag-response
0
10
20
30
40
Lag (days)
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Flexible modelling of the cumulative effects of time-varying exposures
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Fourth example
Tobacco and lung cancer incidence
1,479 cases and 1,918 controls from three case-control studies
within the Synergy network
Yearly exposure history to tobacco smoke (cigarette/day)
reconstructed from questionnaires
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Logistic regression
Analysis with logistic regression controlling for sex
logit (µi ) = α + sx (qxi ; ηx ) + γui
Different functions used to specify f (x) and w (`): log, piecewise
constant, quadratic B-spline
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Exposure-lag-response
Log-by-spline
1.15
1.10
OR
1.05
1.00
80
60
0
r
a
ye
y/
da
g/
Ci
10
40
20
30
20
40
50
0
Lag
)
rs
(yea
60
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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1.00
1.05
1.10
Spline
Step
0.95
OR for 20 cig/day/year
1.15
Lag-response curves from DLNMs
0
10
20
30
40
50
60
Lag (years)
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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2.0
1.5
Relapse
10
20
50
40
30
20
10
0
Cessation
0.0
0
30
40
Cig/day/year
1.0
Quit
0.5
Cumulative OR
2.5
3.0
Dynamic prediction of risk
50
60
years
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Flexible modelling of the cumulative effects of time-varying exposures
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Fifth example
Trial on the effect of a drug
50 subjects followed for 4 weeks
Time-varying treatment randomly allocated in two of the four
weeks, each with a different dose selected at random
Outcome measured at the end of the 28 days
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Flexible modelling of the cumulative effects of time-varying exposures
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Linear regression
Analysis with linear regression controlling for sex
yi = α + sx (qxi ; ηx ) + γui + i
Exposure-response assumed linear
Lag-response modelled with spline or decay functions
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Exposure-lag-response
linear-by-spline
6
Effect
4
2
0
5
0
(d
ay
s)
10
15
60
Dos 40
e
20
g
80
La
100
20
0
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
25
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Lag-response
3
2
−1
0
1
Effect at dose 60
4
5
6
Spline function
0
5
10
15
20
25
Lag (days)
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Lag-response
3
2
−1
0
1
Effect at dose 60
4
5
6
Decay function
0
5
10
15
20
25
Lag (days)
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Software implementation
The framework is fully implemented in the R package dlnm,
available from the CRAN (Gasparrini JSS 2011)
The package contains a new vignette focusing on applications
beyond time series data
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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The R package dlnm
Example of code
library(dlnm)
cb <- crossbasis(Q,lag=c(2,40),
argvar=list(fun="bs",degree=2,knots=59.4,cen=0),
arglag=list(fun="bs",degree=2,knots=13.3,int=F))
model <- coxph(Surv(agest,ageexit,ind)~cb+smoke+caltime,data)
pred <- crosspred(cb,model,at=0:25*10)
plot(pred,"3d",xlab="WLM/year",ylab="Lag (years)",zlab="RR")
plot(pred,var=100,xlab="Lag (years)",ylab="RR")
plot(pred,lag=15,xlab="WLM/years",ylab="RR")
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Simulations
Linear−Constant
Plateau−Decay
1.5
1.5
1.10
1.08
1.4
1.3
1.3
HR
1.4
HR
HR
1.06
1.04
1.2
1.02
1.1
1.00
10
1.0
10
0
30
1.1
8
Ex 6
po
su 4
re
10
20
g
La
30
8
Ex 6
po
su 4
re
20
g
La
0 40
True
AIC avg
AIC samples
20
30
40
20
30
40
True
20
6
Exposure
8
10
30
40
AIC avg
AIC samples
True
HR
AIC avg
AIC samples
0.9
1.1
HR
1.5
AIC samples
1.1
4
10
Lag
0.9
2
0
Lag
AIC avg
AIC samples
HR
10
1.3
0.9 1.0 1.1 1.2 1.3
0
AIC avg
1.1
0
Lag
True
True
1.5
10
20
g
La
0.9
1.1
0
30
1.3
1.5
AIC samples
0
10
2
0 40
1.3
AIC avg
HR
True
2
0.9
HR
0.9 1.0 1.1 1.2 1.3
0 40
1.0
10
0
10
1.5
2
1.2
1.3
8
Ex 6
po
su 4
re
HR
Exponential−Peak
0
2
4
6
Exposure
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Flexible modelling of the cumulative effects of time-varying exposures
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10
0
2
4
6
8
10
Exposure
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Penalized DLNMs
Currently, the bi-dimensional exposure-lag-response function
f ·w (x, `) is specified using completely parametric methods
However, simple DLMs also proposed in a Bayesian (Welty 2008)
or penalized versions (Zanobetti 2000, Rushworth 2013, Obermeier
2015)
An obvious extension is to develop a semi-parametric version of
DLNMs through penalized splines
The development may be facilitated by ’embedding’ the R package
mgcv in dlnm, exploiting the existing GAM implementation
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Discussion
Interactions in DLNMs
Interactions in DLNMs would allow the exposure-lag-response
association varying depending on the value of other predictors (see
also Rushworth 2013)
This corresponds to relaxing the assumption of identical effects
This development extends the framework to a wide range of new
applications
However, it entails non-trivial methodological problems
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Discussion
Time-varying DLNMs
Japan
1.6
1985
Spain
3.0
2012
2010
2.5
RR
1.4
RR
1990
1.2
2.0
1.5
1.0
1.0
0.8
0.5
0
1
10
50
90 99
0
Summer temperature percentile
1
10
50
UK
1993
99
100
USA
1.4
2006
1.6
1.3
1.4
1.2
RR
RR
1.8
90
Summer temperature percentile
1.2
1.1
1.0
1.0
0.8
1985
2006
0.9
0
1
10
50
90
99
100
0
Summer temperature percentile
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
1
10
50
90
99 100
Summer temperature percentile
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Outline
Introduction
Concepts
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Examples
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Extensions
Discussion
Some advantages
DLNMs offer a flexible way to model exposure-lag-response
associations
Unified framework based on a general conceptual and statistical
definition, applicable in various study designs
Complete software implementation, models can be fitted with
standard regression routines
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Some limitations
The DLNM framework is only applicable to time-varying
(non-constant) exposures
It requires the availability of exposure histories (possibly
reconstructed)
Model selection procedures still under-developed
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
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Discussion
Main references
Gasparrini A. Modeling exposure-lag-response associations with
distributed lag non-linear models. Statistics in Medicine.
2014;33(5):881-899.
Gasparrini A & Armstrong B. The R package dlnm. http:
//cran.r-project.org/web/packages/dlnm/index.html
E-mail: [email protected]
Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
LSHTM
Outline
Introduction
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Other references (I)
Abrahamowicz et al (2006). Modeling cumulative dose and exposure duration provided insights regarding
the associations between benzodiazepines and injuries. Journal of Clinical Epidemiology, 59(4):393–403.
Abrahamowicz et al (2012), Comparison of alternative models for linking drug exposure with adverse
effects. Statistics in Medicine, 31:1014–1030.
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Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
LSHTM
Outline
Introduction
Concepts
Stats
Examples
Software
Extensions
Discussion
Other references (II)
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Gasparrini A
Flexible modelling of the cumulative effects of time-varying exposures
LSHTM

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