Model criticism, comparison and selection in dynamic ... · Model criticism, comparison and selection in dynamic transmission models for HIV: Bayesian evidence synthesis Anne Presanis

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


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


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


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


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


Modelling HIV prevalence and incidence Incidence model

Incidence from prevalence

Susceptible

Infected Undiagnosed

Diagnosed

Susceptible


Diagnosed

Susceptible


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


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


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



π

δ

c

c ∈ {e, s, u, d}

data

SOPHID

Proportion of diagnosed

infection attributable to

each group

π(1− δ) ρπδ∑g ρπδ



SOPHID

Proportion of diagnosed

infection attributable to

each group


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


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


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)





























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


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


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?


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


0.0 0.1 0.2 0.30

10

20

30

40

50

Pr(T | Undiagnosed contact)D

ensi

ty

Model 3


0.0 0.1 0.2 0.30

10

20

30

40

50

Pr(T | Undiagnosed contact)

Den

sity

Model 4


Pr(T | diagnosed contact)

0

10

20

30

40

50D

ensi

tyPr(T | diagnosed contact)

0

10

20

30

40

50

Den

sity




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)



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



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



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


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.


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


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


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


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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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



Transmission probabilities - densities

0.0 0.2 0.4 0.6 0.8 1.0

0

20

40

60

80


Den

sity

Model 5

0.0 0.2 0.4 0.6 0.8 1.0

0

20

40

60

80


Den

sity

Model 6

0.0 0.2 0.4 0.6 0.8 1.0

0

20

40

60

80


Den

sity

Model 5

0.0 0.2 0.4 0.6 0.8 1.0

0

20

40

60

80


Den

sity

Model 6



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)


Documents

Model criticism, comparison and selection in dynamic ... · Model criticism, comparison and selection in dynamic transmission models for HIV: Bayesian evidence synthesis Anne Presanis