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Model criticism, comparison and selection in dynamictransmission models for HIV: Bayesian evidence synthesis
Anne Presanisin collaboration with Daniela De Angelis
and the Health Protection Agency
MRC Biostatistics Unit, Cambridge
16 March 2011All models are wrong
Groningen
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 1 / 23
Introduction
Outline
1 IntroductionMotivation: HIVEvidence synthesis
2 Modelling HIV prevalence and incidencePrevalence modelIncidence modelA joint model for incidence and prevalence
3 Transmission modellingMixing patterns
4 ResultsPosterior distributions
5 Model criticism & comparisonDeviance summariesInfluence & Identifiability
6 Concluding comments
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 2 / 23
Introduction Motivation: HIV
Motivation
Human Immunodeficiency Virus
Estimates of HIV prevalence and incidence are essential forunderstanding and monitoring the epidemic, as well as for assessingthe impact of public health interventions.
Challenges
HIV has a long asymptomatic incubation period, so manyinfections undiagnosed
Surveillance systems available only for certain risk groups andpopulations
Surveillance and other survey/ad-hoc data subject to biases
Data sometimes tell us only indirectly about the quantities ofinterest
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 3 / 23
Introduction Evidence synthesis
Evidence synthesis - a long-established idea
Methods for combining evidence are not new:
The Bayesian paradigm
combining prior knowledge with new [Bayes (1763), Efron(2010)]
Meta-analysis
combining studies of same type
Confidence Profile Method [Eddy et al (1992)]
combining information of different types/study designs(medical-decision making literature)
Multi-parameter evidence synthesis [Spiegelhalter et al (2004),Ades & Sutton (2006)]
epidemiology
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 4 / 23
Introduction Evidence synthesis
Statistical formulation
Interest: estimation of θ = (θ1, θ2 . . . , θk) on the basis of acollection of data y = (y1, y2 . . . , yn)
Each yi provides information on
a single component of θ, ora function of one or more components, i.e. on a quantityψi = f (θ)
Thus inference is conducted on the basis of both direct andindirect information.
Maximum likelihood: L =∏n
i=1 Li (yi | θ)
Bayesian: p(θ | y) ∝ p(θ)× L
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 5 / 23
Modelling HIV prevalence and incidence Prevalence model
Prevalence model
π
ρ
δ
π(1− δ) ρπδΣgρπδ
UA surveysPrevalence ofundiagnosedinfection
NATSALProportionof men whoare MSM
SOPHIDProportion of diagnosedinfection attributable to
each group
Stratified by time, risk group and region
Proportion in risk group
HIV prevalence
Proportion
diagnosed
SOPHIDTotal numberof diagnosedinfections
NΣgρπδ
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 6 / 23
Modelling HIV prevalence and incidence Incidence model
Incidence from prevalence
Susceptible
Infected Undiagnosed
Diagnosed
Susceptible
Infected Undiagnosed
Diagnosed
Susceptible
Infected Undiagnosed
Diagnosed
Not at risk Not at risk Not at risk
tt1 t2 t3
e(t)
s(t)
u(t)
d(t)
ρ(t) = s(t) + u(t) + d(t)
π(t) = (u(t) + d(t))/ρ(t)
δ(t) = d(t)/(u(t) + d(t))
proportion in risk group
HIV prevalence
proportion diagnosed
e(t) = 1− ρ(t)
s(t) = (1− π(t))ρ(t)
u(t) = (1− δ(t))π(t)ρ(t)
not at risk
susceptible
infected undiagnosed
d(t) = δ(t)π(t)ρ(t)infected diagnosed
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 7 / 23
Modelling HIV prevalence and incidence Incidence model
A multi-state model
s(t) u(t) d(t)ψ
e(t)α(t) λ(t) κ(t)
Migration in/outwards
Exits due to age/death
Incidence Diagnosis rateNew MSMNew 15 year olds
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 8 / 23
Modelling HIV prevalence and incidence A joint model for incidence and prevalence
Combined incidence and prevalence model
times {t2 . . . tK}
θ
π
ρ
time t1
c
c ∈ {e, s, u, d}Initial state of system
δ
π(1− δ) ρπδ∑g ρπδ
ρ
UA surveysPrevalence ofundiagnosedinfection
NATSALProportionof men whoare MSM
π
δ
c
c ∈ {e, s, u, d}
data
SOPHID
Proportion of diagnosed
infection attributable to
each group
π(1− δ) ρπδ∑g ρπδ
UA surveysPrevalence ofundiagnosedinfection
NATSALProportionof men whoare MSM
SOPHID
Proportion of diagnosed
infection attributable to
each group
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 9 / 23
Modelling HIV prevalence and incidence A joint model for incidence and prevalence
Results
Prior distribution: λ(t), κ(t) ∼ Unif(0, 1)Posterior distribution:
2002 2003 2004 2005 2006 2007
0.000
0.005
0.010
0.015
0.020
0.025
Incidence rateIncidence rateIncidence rateIncidence rateIncidence rateIncidence rate
2002 2003 2004 2005 2006 2007
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Diagnosis rateDiagnosis rateDiagnosis rateDiagnosis rateDiagnosis rateDiagnosis rate
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 10 / 23
Transmission modelling
Parameterisation of λ(t)
Incidence λJ(t) is a function of prevalence, the contact structure and theprobability of transmission given a contact.
CsJL IL
DLDL
TsJL
LJ
SJsJ
·
Random mixing
λJ(t) = χJ(t) {τDδL(t)πL(t)gJL + τU(1− δL(t))πL(t)gJL}
Presanis et al, Biostatistics, in press
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 11 / 23
Transmission modelling Mixing patterns
More realistic mixing patterns
Preferential mixing: avoiding diagnosed partners, so thatprevalence of diagnosed infection in chosen partners is smaller thanin all MSM:
λ(t) = χ(t) {τDφδ(t)π(t) + τU(1− δ(t))π(t)}
Model 1 Random mixing: φ = 1
Model 2 Completely avoid diagnosed partners (“serosorters”):φ = 0, λ(t) = χ(t) {τU(1− δ(t))π(t)}
Model 3 No information on proportion who avoid diagnosedpartners: φ ∼ Unif(0, 1)
Model 4 Informative prior: φ ∼ Beta(10, 40)
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 12 / 23
Transmission modelling Mixing patterns
More realistic mixing patterns
Preferential mixing: avoiding diagnosed partners, so thatprevalence of diagnosed infection in chosen partners is smaller thanin all MSM:
λ(t) = χ(t) {τDφδ(t)π(t) + τU(1− δ(t))π(t)}
Model 1 Random mixing: φ = 1
Model 2 Completely avoid diagnosed partners (“serosorters”):φ = 0, λ(t) = χ(t) {τU(1− δ(t))π(t)}
Model 3 No information on proportion who avoid diagnosedpartners: φ ∼ Unif(0, 1)
Model 4 Informative prior: φ ∼ Beta(10, 40)
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 12 / 23
Transmission modelling Mixing patterns
More realistic mixing patterns
Preferential mixing: avoiding diagnosed partners, so thatprevalence of diagnosed infection in chosen partners is smaller thanin all MSM:
λ(t) = χ(t) {τDφδ(t)π(t) + τU(1− δ(t))π(t)}
Model 1 Random mixing: φ = 1
Model 2 Completely avoid diagnosed partners (“serosorters”):φ = 0, λ(t) = χ(t) {τU(1− δ(t))π(t)}
Model 3 No information on proportion who avoid diagnosedpartners: φ ∼ Unif(0, 1)
Model 4 Informative prior: φ ∼ Beta(10, 40)
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 12 / 23
Transmission modelling Mixing patterns
More realistic mixing patterns
Preferential mixing: avoiding diagnosed partners, so thatprevalence of diagnosed infection in chosen partners is smaller thanin all MSM:
λ(t) = χ(t) {τDφδ(t)π(t) + τU(1− δ(t))π(t)}
Model 1 Random mixing: φ = 1
Model 2 Completely avoid diagnosed partners (“serosorters”):φ = 0, λ(t) = χ(t) {τU(1− δ(t))π(t)}
Model 3 No information on proportion who avoid diagnosedpartners: φ ∼ Unif(0, 1)
Model 4 Informative prior: φ ∼ Beta(10, 40)
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 12 / 23
Results Posterior distributions
Incidence by diagnosis status of contact
Grey: λU(t), incidence due to undiagnosed contacts
Magenta: λD(t), incidence due to diagnosed contacts
2002 2004 2006
Model 1
2002 2004 2006
0.000
0.002
0.004
0.006
0.008
0.010
0.012 Model 2
Model 3
0.000
0.002
0.004
0.006
0.008
0.010
0.012Model 4
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 13 / 23
Results Posterior distributions
Transmission probabilities
Prior distribution: τU ∼ Unif(0, 0.3), τD ∼ Unif(0, τU)Posterior distribution:
1 2 3 4
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Model
Probability of transmission given an undiagnosed contactProbability of transmission given a diagnosed contact
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 14 / 23
Model criticism & comparison Deviance summaries
Deviance Information Criteria (DIC)
Model n D̄ D(θ̄) pD DIC
1 174 175.7 22.3 153.4 329.12 174 176.7 22.9 153.8 330.53 174 175.4 22.4 153.0 328.44 174 176.0 22.5 153.4 329.4
What influences the estimates of λU , λD?
λ(t) = χ(t) {τDφδ(t)π(t) + τU(1− δ(t))π(t)}Equal fit to data informing prevalences, transition rates, sodata don’t have strong influence on estimates of λU , λD?
Only prior information on τU , τD , φ and model structurehaving effect?
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 15 / 23
Model criticism & comparison Influence & Identifiability
Transmission probabilities
Red: Prior Black: Posterior
0.0 0.1 0.2 0.3Pr(T | Undiagnosed contact)
Model 1
Pr(T | undiagnosed contact)
0.0 0.1 0.2 0.30
10
20
30
40
50
Pr(T | Undiagnosed contact)
Den
sity
Model 2
Pr(T | undiagnosed contact)
0.0 0.1 0.2 0.30
10
20
30
40
50
Pr(T | Undiagnosed contact)D
ensi
ty
Model 3
Pr(T | undiagnosed contact)
0.0 0.1 0.2 0.30
10
20
30
40
50
Pr(T | Undiagnosed contact)
Den
sity
Model 4
Pr(T | undiagnosed contact)
Pr(T | diagnosed contact)
0
10
20
30
40
50D
ensi
tyPr(T | diagnosed contact)
0
10
20
30
40
50
Den
sity
Pr(T | diagnosed contact)
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 16 / 23
Model criticism & comparison Influence & Identifiability
Where is the information coming from?
Bounds are well identified, so τU , τD partially identified
τD =λ(t)− χ(t)τU(1− δ(t))π(t)
χ(t)φδ(t)π(t)
and
0 ≤ τD ≤ τU
⇒λ(t)
χ(t)(1− δ(t)(1− φ))π(t)≤ τU ≤
λ(t)
χ(t)(1− δ(t))π(t)
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 17 / 23
Model criticism & comparison Influence & Identifiability
τU with limits, e.g. in year 2004
1 2 3 4
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Model
Lower limitPr(Transmission | Undiagnosed contact)Upper limit
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 18 / 23
Model criticism & comparison Influence & Identifiability
Lower limit informed by φ
Red: Prior Black: Posterior
0.0 0.2 0.4 0.6 0.8 1.0
Model 3
0.0 0.2 0.4 0.6 0.8 1.0
0
2
4
6
8
Den
sity
Model 4
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 19 / 23
Model criticism & comparison Influence & Identifiability
Influence of model structure on incidence
λ(t) = λU(t) + λD(t)
= χ(t) {τDφδ(t)π(t) + τU(1− δ(t))π(t)}
2002 2004 2006
Model 1
2002 2004 2006
0.000
0.002
0.004
0.006
0.008
0.010
0.012 Model 2
Model 3
0.000
0.002
0.004
0.006
0.008
0.010
0.012Model 4
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 20 / 23
Concluding comments
Summary
DIC only take us so far in discriminating between models,when priors/model structure have more influence ondifferences in inferences than data?
Importance of understanding influence of data, priors andmodel structure on inference (O’Hagan, 2003)
Idea that models/model structure are “priors”/indirectevidence (Efron, Stat. Sci. 2010)
How to judge/discriminate between different modelassumptions in this context, other than by presentingsensitivity analyses?
Here we conclude that further data required on behaviourchange once diagnosed (φ) - important endpoint.
Partial identifiability (Gustafson, Greenland) - but still able toinfer λU , λD based on mechanistic model assumptions.
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 21 / 23
Concluding comments
Further work
Investigating sources of data for behaviour
Expansion of model to 3 levels of risk amongst MSM?
Understanding influence of each part of model (data, priors,structure)
e.g. loosening assumed model structure, by usingbeta-binomial and negative binomial likelihoods instead ofbinomial and Poisson
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 22 / 23
Concluding comments
Acknowledgements
Daniela De Angelis (MRC Biostatistics Unit & Health ProtectionAgency)
Tony Ades (Bristol)
Aicha Goubar (INVS, Paris)
David Spiegelhalter, Paul Birrell, Chris Jackson (MRC BiostatisticsUnit)
Graham Medley (Warwick)
HIV Department, Health Protection Agency
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 23 / 23
Sensitivity analyses Partial identifiability
An aside on partial identifiability - Model 3
5000 6000 7000 8000 9000 10000
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Iterations
Pr(
T |
undi
agno
sed
cont
act)
5000 6000 7000 8000 9000 10000
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Iterations
Pr(
T |
diag
nose
d co
ntac
t)
5000 6000 7000 8000 9000 100000
5
10
15
20
25
30
35Pr(T | undiagnosed contact)
last iteration in chain
shrin
k fa
ctor
median97.5%
5000 6000 7000 8000 9000 10000
1.0
1.5
2.0
2.5
3.0
Pr(T | diagnosed contact)
last iteration in chain
shrin
k fa
ctor
median97.5%
0 10 20 30 40
−1.0
−0.5
0.0
0.5
1.0Pr(T | undiagnosed contact)
Lag
Aut
ocor
rela
tion
0 10 20 30 40
−1.0
−0.5
0.0
0.5
1.0Pr(T | diagnosed contact)
Lag
Aut
ocor
rela
tion
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0.00 0.05 0.10 0.15
0.00
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Pr(T | undiagnosed contact)
Pr(
T |
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nose
d co
ntac
t)
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A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 24 / 23
Sensitivity analyses Effect of further or less information
Further or less information
Model 5: τU ∼ N(0.4, 0.182)T (0, 1) from meta-analysis of Baggaley(2006)
Model 6: τU ∼ N(0.05, 0.0052)T (0, 1)
Model n D̄ D(θ̄) pD DIC
5 174 175.40 22.18 153.24 328.646 174 175.67 22.16 153.46 329.13
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 25 / 23
Sensitivity analyses Effect of further or less information
Transmission probabilities - densities
0.0 0.2 0.4 0.6 0.8 1.0
0
20
40
60
80
Pr(T | undiagnosed contact)
Den
sity
Model 5
0.0 0.2 0.4 0.6 0.8 1.0
0
20
40
60
80
Pr(T | undiagnosed contact)
Den
sity
Model 6
0.0 0.2 0.4 0.6 0.8 1.0
0
20
40
60
80
Pr(T | diagnosed contact)
Den
sity
Model 5
0.0 0.2 0.4 0.6 0.8 1.0
0
20
40
60
80
Pr(T | diagnosed contact)
Den
sity
Model 6
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 26 / 23
Sensitivity analyses Effect of further or less information
Transmission probabilities - traces
0 1000 2000 3000 4000 5000
0.00
0.05
0.10
0.15
Iterations
Pr(
T |
undi
agno
sed
cont
act)
Model 5
0 1000 2000 3000 4000 5000
0.00
0.05
0.10
0.15
Iterations
Pr(
T |
undi
agno
sed
cont
act)
0 1000 2000 3000 4000 5000
0.00
0.05
0.10
0.15
Iterations
Pr(
T |
diag
nose
d co
ntac
t)
Model 5
0 1000 2000 3000 4000 5000
0.00
0.05
0.10
0.15
Iterations
Pr(
T |
diag
nose
d co
ntac
t)
A. M. Presanis (MRC BSU) Bayesian dynamic transmission models 16 March 2011 27 / 23