Factored stochastic tree modeling for medical decision making

Gordon HazenNorthwestern University

Rowland ChangNorthwestern University

James PellissierLoyola University/Merck Pharmaceuticals

2

• What is a stochastic tree?– Basic concepts………………………………..

– Stochastic tree transformation and rollback….

– Approximating human survival……………….

– Factoring out mortality………………………..

• Factoring stochastic trees………………….• Discounting / Risk aversion……………….• Influence diagrams for stochastic models…• Our THA model…………………………...• The StoTree modeling environment……….• Cost-effectiveness for THA………………..

Outline of talk

3

What is a stochastic tree?• A stochastic tree is

– A decision tree with stochastic nodes added

– A continuous-time Markov chain with chance and decision nodes added

– A multi-state DEALE model

– A continuous-time version of a Markov cycle tree

Stochastic Trees

Continuous-time MCs

Decision Trees Discrete-time MCs /

Markov cycle trees

DEALE

4

…What is a stochastic tree?

me

pb ms

me

1 - pe

pe

pb

ms

me

.1 - pb

ms

Well

Stroke

Small Stroke Post Small Stroke

Big Stroke

Post Big Stroke

Stroke

Big Stroke

Dead

Dead

DeadDead

Matchar & Pauker (1986): Transient ischemic attacks in a man with coronary artery disease

5

…What is a stochastic tree?

Roach et al. (1988): Prostate cancer in a man with asymptomatic HIV

m0+mc+ma

m0+mc+ma

rc

m0+ma

m0+mc

ra

ra

rc

m0

No Disease

Dead

Cancer

AIDS

AIDS & Cancer

AIDS & Cancer

Dead

Dead

Dead

Dead

6

Markovcycletree

Die from other Causes Deado

Develop AIDS Cancer + AIDSWell Cancer recurs a

c No AIDS Cancer

Survive other a

o Develop AIDS AIDS

No cancer a

c No AIDS Well a

Die from other Causes Deado

Cancer Die from recurrent prostate cancer Deadc

Survive other Develop AIDS AIDS o Survive cancer a

c No AIDS Well a

Die from other Causes Deado

AIDS Die from AIDS Deada

Survive other Cancer recurs Cancer + AIDS o Survive AIDS c

a No cancer AIDS c

Die from other causes Deado

Cancer + AIDS Die from recurrent prostate cancer Deadc

Survive other Die from AIDS Dead o Survive cancer a

c Survive AIDS Cancer + AIDS a

Dead Dead

7

Transforming stochastic trees

=

3

1

2

x y2

y1

y3

321

• Superposition / Decomposition

p3

p2

p1

x

y3

y1

y2

iip

8

…Transforming stochastic trees

=

y2

y3

y1

• Eliminating self-transitions

2

3

1

y2

y3

y1

y2

9

…Transforming stochastic trees

pbms(1-pe)

me+pbmspe

me+pbmspe

pb(1-pe)

1-pb

pb pems

me

Well

Stroke

Post Small Stroke

Dead

Dead

Dead

Post Big Stroke

Dead

Post Big Stroke

10

Stochastic tree rollback

yy y

y y

y

)y(L)x(v)y(Lp

1)x(v)x(L

=

3

1

2

x y2

y1

y3

p3

p2

p1

x

y3

y1

y2

v(x) = Quality rate at x

L(x) = Mean quality-adjusted duration beginning at x

Recursive formula:

11

Stochastic tree rollback

3.0779.481

ms = 0.05 /yr

pe = 0.38 7.894

pb = 0.6667

me = 0.065 /yr

m0 = 0.0111 /yrqPBS = 0.2qPSS = 0.8

pbms(1-pe)

pb(1-pe)

me+pbmspe

1-pb

pb pe

me

me+pbmspe

ms

Well

Stroke

Post Small Stroke

Dead

Dead

Dead

Post Big Stroke

Dead

Post Big Stroke

12

Approximating human survival

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40

t (years)

P(T

> t

)

Exponential P(T > t)

WM Actual P(T > t)

13

Coxian approximation to human mortality

60-year-old white female

Stg 1 0.29 0.01Stg 2 0.28 0.02Stg 3 0.3 0Stg 4 0.3 0Stg 5 0.3 0Stg 6 0.3 0Stg 7 0.18 0.118Stg 8 0.298

Stg 7

Stg 8 Stg 7

Stg 6

Stg 6

Stg 5

Stg 5

Stg 4

Stg 4

Stg 3

Stg 3

Stg 2

Stg 2

Stg 1

Stg 1

1 2 3 4

5 6 7 8

14

Coxian approximation to human mortality

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60

t (Additional Years)

P(T

> t

)White FemaleAge 60

Coxian

15

Factoring out mortality

1 - pe

pe

s

0

Well

Stroke

Post Stroke

16

…Factoring out mortality

0

1

pe

1 - pe

s

Well

Stroke

Post Stroke

Background mortality

Stroke morbidity

17

…Factoring out mortality

pe

1 - pe

s

Well

Stroke

Post Stroke

Background mortality

Stroke morbidity

1 2 3

18

Equivalent product tree

W ell1

W e ll2

W e ll3

S troke1

S troke2

S troke3

P o s t s troke1

P o s t s troke2

P o s t s troke3

1

1

2

2

3

3

s

s

s

1

1

2

2p e

p e

p e

1 - p e

1 - p e

1 - p e

19

Rollback with Coxian mortality

s

W ell

Stroke

Poststroke

pe

1 - pe

3 .6 4 02 .4 9 81 .2 4 9

2 .2 5 71 .5 4 90 .7 7 5

5 .2 9 75 .1 2 84 .9 4 4

20

Factored stochastic trees

Cancer

AIDS

Background mortality

mcrc

No Disease Cancer Dead

mcrc

No Disease AIDS Dead

m0

Alive Dead

21

…Factoring stochastic trees

Systemic embolism

Pulmonary embolism

Systemic hemorrhage

Tsevat et al. (1986): Warfarin for dilated cardiomyopathy

1 - SE Surv

SE Surv

SE Rate

1 - SE MorbP

SE MorbP

SE Surv

SE Rate

1 - SE Surv

No Embolism

Embolism

Survive

Long-Term Morbidity

Subsequent Embolism

No Embolism

LongTerm MorbidityDead

Dead

PE Surv

1 - PE Surv

PE Rate

No Embolism

Embolism

No Embolism

Dead

SH Rate

1 - SH Surv

SH SurvSH MorbP

1 - SH MorbP

SH Surv

1 - SH Surv

SH Rate

NoHemorrhage

Hemorrhage

SurviveLong-Term Morbidity

SubsequentHemorrhage

No Hemorrhage

Long-Term Morbidity

DeadDead

22

Discounting / Risk aversion

)h(ut)x(v)h(uds)x(vhxut

0

t

)h(uedse)x(vhxu t)x(at

0

s)x(at

ix H i

i i

i ii

)x(a

)]H(u[E)x(v)]G(u[EG =

• Utility function yielding quality-adjusted duration

• Utility function yielding discounted quality-adjusted duration

• Rollback with discounting

23

Modeling risk attitude

Dead

W ell Dead

c

p

1 - p

W ell Dead

~

Vaccine scenario: What chance p of immediate death would you take to reduce your ongoing mortality rate by a percentage c?

Undiscounted quality-adjusted duration forces: p = c

Discounted quality-adjusted duration allows p < c (risk aversion)

24

Continuous-risk utility assessment

25

Influence diagrams

THA vsConservativeManagement

Initial THAOutcome

ACRFunctional

Status

Initial THA OutcomeProbabilities

ConservativeManagement

THA

Death (0.0045)

Poor (0.06)

Fair (0.24)

Good (0.69)

III

II

III

IV

Dead

Initial THAOutcome

ACR Functional Status

Decision tree Influence diagram

26

Influence diagrams with stochastic nodes

InfectionFailureCount

Infection RevisionOutcome Probabilities

Prosthesis Status

InfectionFailureRate

0 1 2 3

pISucc

pIFail

pIMortrInfection

Functioning Prosthesis

Infection Failure

Surgery

Functioning Prosthesis

Infection Failure

Death

27

THA model

THA vsConservativeManagement

Initial THAOutcome

AsepticFailureCount

ACRFunctional

Status

RevisionCount

Identity ofLast

Surgery

InfectionFailureCount

AsepticFailureRate

InfectionFailureRate

Infection RevisionOutcome Probabilities

Natural ProgressionRate

OA Progression underConservative Management

Background Mortality

Initial THA OutcomeProbabilities

Quality ofLife

Prosthesis Statusafter THA

Aseptic RevisionOutcome Probabilities

28

ACR functional status

Class Description

I Complete ability to carry on all usual duties without handicap

II Adequate for normal activities despite handicap of discomfort orlimited motion in the hip

III Limited only to little or none of duties of usual occupation or self-care

IV Incapacitated, largely or wholly bedridden or confined towheelchair, little or no self-care

29

THA vs. Conservative Management

THA

Conserv Mgmt

30

ACR Functional Status / Initial THA Outcome

pSuccess = 0.6925pFair = 0.243

pFailure = 0.06pSurgMort = 0.0045

qI = 1qII = 0.8

qIII = 0.5qIV = 0.3

Discount Rate = 3%

pSuccess

pFair

pFailure(Trigger Aseptic

Revision)

pSurgMort

Surgery

ACR Class III

ACR Class II

ACR Class I

Death

ACR Class IV

31

Prosthesis Status After THA

Last Surgery rAseptic rInfectionInitial THA 0.01 0.002 /yr

A or I Revision 1 0.04 0.02 /yrA or I Revision 2 0.05 0.035 /yrA or I Revision 3 0.1 0.05 /yr

Current Aseptic Revisio pASucc pAFail pAMort pISucc pIFail pIMort

Rev 1 0.7015 0.2865 0.012 0.7692 0.2115 0.0193Rev 2 0.6363 0.3517 0.012Rev 3 0.8477 0.1403 0.012

pISucc

No Revision

pIMort

pIFailInfection Revision

rInfection

pAMort

pAFail

pASucc

Aseptic Revision

No Revision

rAseptic

Daily Living

Aseptic Failure

Daily Living

Daily Living

Aseptic Failure

Death

Infection Failure

Infection Failure

Death

Daily Living

Daily Living

32

Last Surgery

Initial THA

Aseptic Revision

Infection Revision

33

Conservative Management

rNatural = 3.297% /yr

rNaturalIII IV

34

The StoTree modeling environment

35

…StoTree modeling environment

36

...StoTree modeling environment

37

Rollback in the THA model

85-year-old white male4.09

3/11/99 4:54 Rollback4.089 279 parameters examined

48 product nodes examined

2.12

THA

Conserv Mgmt4.383

pSuccess = 0.6925pFair = 0.243

3.533 pFailure = 0.06pSurgMort = 0.0045

4.0894.089

qI = 1qII = 0.8

qIII = 0.5qIV = 0.3

Discount Rate = 3%

1.333

pSuccess

pFair

pFailure(Trigger Aseptic

Revision)

pSurgMort

Surgery

ACR Class III

ACR Class II

ACR Class I

Death

ACR Class IV

38

Cost-effectiveness for THA85-year-old white male

-5770.33/11/99 5:01 Rollback

-20582 279 parameters examined48 product nodes examined

-20582

THA

Conserv Mgmt

-2648.7 pSuccess = 0.6925pFair = 0.243

pFailure = 0.06-2648.7 pSurgMort = 0.0045

-5770.3

-20582 Costs per case No THAPrimary

THA RevisionHospital -$ 17,000$ 20,000$

Physician -$ 5,000$ 5,000$ Rehabilitation -$ 3,000$ 3,000$

Total Per Case -$ 25,000$ 28,000$

Costs per yearMedical Per Yr III IV 775$ 775$

-158902 Custodial Per Yr IV 35,000$ 35,000$

Discount Rate = 3%

pSuccess

pFair

pFailure(Trigger Aseptic

Revision)

pSurgMort

Surgery

ACR Class III

ACR Class II

ACR Class I

Death

ACR Class IV

39

THA Cost-Effectiveness ResultsWhite Male Age 85 White Female Age 60

THA Conserv THA Conserv

Mean Dctd Years in

ACR Class I 2.944 0 9.717 0

ACR Class II 1.384 0 5.078 0

ACR Class III 0.063 3.952 0.608 11.39

ACR Class IV 0.022 0.49 0.48 4.634

Mean Dctd Life Expectancy 4.413 4.442 15.883 16.024

Mean Dctd QALY 4.089 2.123 14.23 7.087

Mean Dctd Costs

Primary Surgery, Rehab $25,000 - $25,000 -

Revision Surgery, Rehab $4,929 - $12,284 -

Ongoing Medical $66 $3,442 $843 $12,421

Custodial $776 $17,140 $16,789 $162,175

Total $30,771 $20,582 $54,916 $174,596

Marginal cost $10,189 $(119,680)

Marginal effectiveness (QALY) 1.966 7.143

Marginal CE ratio $5,183 -