Markov Disease State

Modeling

Training in Clinical ResearchDCEA Lecture 6

UCSF Department of Epidemiologyand Biostatistics

March 8, 2007

James G. Kahn, MD, MPHjgkahn@ucsf.edu

Objectives:

• To understand the definition and uses of a Markov simulation.

• To understand steps in conducting a Markov simulation.

Background

• Many diseases progress through stages or states.e.g., physiologic abnormality, mild then moderate clinical disease, complications, end-stage.

• Ongoing risk of shifting between stages, over months or years.

• We’ve portrayed clinical outcomes with short-term or lifetime probabilities.

• Sometimes easier to represent diseases in stages and movement between stages.

• “Markov disease state simulation.”

Outline

1. What is a Markov simulation?

2. When should I do a Markov simulation?

3. Steps in a Markov simulation

1. What is a Markov simulation?

Portray progression of a disease over time

• Divide the disease into discrete “states”• Specify initial distribution, risks of

progression per unit time• Assign utilities and costs to each state/unit

time and transition• Conduct simulation with defined end-point

2. When should I do a Markov simulation?

• Probabilities/utilities change over time. e.g., risk of stroke (or disutility) increases with

age.• Data availability.

Data on risk of disease progression/intervention effectiveness more readily available for short time periods

• Face validity.Conceptualized by readers as having discrete,

progressive states. May tip the balance.• Multiple opportunities for intervention.

Portray effects of interventions occurring at multiple stages in disease progression.

Cumulative effectiveness ≠ point effectiveness.

Aneurysm CEA conducted with a Markov, for two reasons:

1. In older population all-cause mortality competes with the risk of SAH, and

increases as the cohort ages.

2. SAH risk data are available for short time periods only, easily translated to annual risk and not as easily to lifetime risk.

Course of HIV, impact of increased early HAART

Conducted with Markov for 3 reasons:

1. Data on HIV progression and treatment effectiveness focus on disease state transitions.

2. Face validity: clinicians, epidemiologists, others think of HIV disease in stages: infection, worsening CD4 and viral load, pre-AIDS disease, AIDS, and death.

3. HAART can be used in different stages of disease – in fact, the effect of HAART timing is the issue being assessed.

Diseases for which Markov adds little:

Key chance nodes occur in a relatively short time frame (a few years), i.e., quick resolution or stabilization of clinical condition

e.g., acute curable infections; management of acute cardiovascular events (MIs, strokes); cancers (if prognosis is captured with a few tree branches); and immunizations for non-epidemic childhood infections (e.g., hemophilus influenza).

3. Steps in a Markov - Overview

A. Portray disease states and transitions, structure simulation

B. Obtain data for the transition probabilities

C. Implement the model -- building, calibrating, quality control

Naimark: e.g., formulas to calculate cycle-specific utilities and costs (including discounting), reference tables

A. Disease states, transitions, structure

Portray disease states – principles

• Include all important states of the disease: --often stages of severity, e.g., renal disease in diabetes: severity of renal compromise (normal, micro-, macroalbuminuria, end-stage renal disease). --sometimes recurrent events e.g., recurrence/remission.

• Also often health states induced by therapy (e.g., side-effects).

Defining disease states- practical issues

Precisely what states?

• discrete shifts mark boundarieschanged health status, e.g., hypertension to

strokearbitrary/convention, e.g., micro/macro

albuminuria• working definitions in the field• data exist on progression• balance simplicity and completeness• interventions being studied• usually need absorbing state (e.g., death)

Aneurysm: long-term outcomes calculated by modeling movement among four states:

– Healthy

– Mild disability (due to surgery or SAH)

– Moderate-severe disability (ditto)

– Death

Renal disease in diabetes:

– Healthy

– Microalbuminuria

– Macroalbuminuria

– End-stage renal disease

– Death

Portraying transitions

Possible transitions between states

• Single “forward” transitions (i.e., from state 1 to 2, 2 to 3, 3 to death) always.

• Forward jumps (e.g., from 1 to 3, see HIV example) often.

• Rarely: backward transitions (e.g., from 3 to 2) … more realistic to add state (e.g., 3 in remission); state achieved via a sicker state ≠ state achieved via a healthier state.

• Death, in almost all Markov simulations, due to the disease or other causes.

Risk of progression

• Risk per unit time (i.e., per Markov cycle) in “source” state

e.g., “For individuals with microalbuminuria, there is 5% annual risk of progressing to macroalbuminuria.”

• Time-period risk can evolve

e.g., annual risk of mortality increases as individuals age.

Effectiveness of interventions

Usually represented as reduction in the risk of progression

e.g., “ACE-inhibitors decrease the risk of progressing from micro- to

macroalbuminuria by 70%.”

Disease state outcomes

Each disease state assigned utility (and cost) per cycle.

– Utility: If annual cycles, might be the portion of a QALY gained by being in that state for that year.

– Costs: direct, total, etc.

• Keep track of utilities and costs accumulated in each state in each cycle cumulative totals available at end.

Simulation structure

• Track movement between states over time.• Portray individual or group (e.g., 1000)• Cycle duration short -- real patient would not

have two state transitions in a cycle• Very quick on new computers!• End with specified duration, or when per cycle

utilities below threshold (e.g., 0.001 QALY). • Consequences of intervention strategies

captured by comparing similar tree structures with different input values

Graphic techniques

• Simple flow diagram effective for basic Markov states and transitions; limited transition probabilities possible without clutter.

Normal

Microalbuminuria

Macroalbuminuria

End-stage renal disease

De

ath

Multi-cycle bubble diagram

(Fig 1 Naimark)

• Clear and more information -- evolving state distributions and cumulative outcomes.

• Not often used in published Markov analyses, probably because unwieldy with more than 3-4 states.

“Markov subtree” (Naimark Fig 2ff)

• Markov with infinity symbol (∞) instead of chance node; states with branches; transitions with boxes at the end of each sub-branch.

• If complex, as in GCA example, multiple subtrees needed.

• Excellent at documenting structure, but requires understanding trees and has no natural way to report transition probabilities.

Transition matrix

• Efficiently summarizes states and transitions

• Corresponds to structure used to analyze Markov (pre fast computers)

Annual transition probability from source to target state.

Target states

Source states Normal Micro albuminuria

Macro albumin

uria

End-stage renal

disease

Death

Normal .96 .03 -- -- .01

Micro albuminuria

-- .90 .08 -- .02

Macro albuminuria

-- -- .94 .01 .03

End-stage renal disease

-- -- -- .85 .15

Multi-column table

• Allows more information (e.g., effectiveness) with some loss in organizational efficiency.

HIV disease:States match CDC definitions + common clinical distinctions.

Source State Target State Transition Probability per

Quarter

Reduction in Transition Rate with HAART

0. Uninfected 1    

1. Asymptomatic, CD4>500 2    

  3    

  4    

  5    

  6    

2. Asymptomatic, CD4<500 3    

  4    

  5    

  6    

3. Symptomatic, pre-AIDS 4    

  5    

  6    

4. AIDS, 1993 definition 5    

  6    

5. AIDS, 1987 definition 6    

6. Death (absorbing)    

B. Data for transition probabilities

• Precise extraction and adaptation of published (or custom) data.

• Plus usual data for CEA.

For the aneurysm Markov:

• annual probability of aneurysm rupture (SAH) from a prospective cohort, assumed constant over time

• one-time risks of death and disability from surgery / SAH from various studies.

• annual age-adjusted risk of death all causes from life tables, studies of individuals with disabilities.

For renal disease in diabetes:

• probability of progression and death from natural history studies

• effectiveness (reduction in progression) from trials

When data go missing

If detailed state-to-state transitions unavailable …

• data on larger jumps establish “benchmarks”

• benchmarks used to assign values to intervening transition probabilities (“calibration”)

HIV analysis benchmarks

• Data abstraction used all available natural history studies.

• Also legitimate to rely on a single definitive study.

• Each individual study review documented source and target states, study characteristics, ARV therapy in use.

STUDY % AT 2 YRS

% AT 5 YRS

% AT10 YRS

ARV THERAPY

Mellors, 1997n = 1,604 MSMs,

prospective cohort (MACS)

Source: 870 asympt. (P)Target: 1987 AIDS

p. 953: Authors report mixed ARV use, but not results by ARV use because baseline HIV RNA values (predictive of outcome) did not differ by ARV status.

AIDS Cum 5% 24% 55%

Died

Multiple studies summarized:

AIDS

N's 2 year 5 year 10 year

1b ==> AIDS 87 defn Little Mixed Most Little Mixed Most Little Mixed Most

I studies 1506 2352 0.070 0.030 0.228 0.107

Alcabes 1992 343 0.056 0.230

Hoover 1992 331 0.040 0.230

Mariotto 1992 420 0.220

Multicohort Ana. Proj Wrkshp 1994 1,744 0.080

Pehrson, 1997 188 0.030 0.110

Rezza 1989 205 0.210

Seaman, 1997 627 0.100 0.233

P studies 870 1637 0.050 0.240 0.120

Mellors, 1997 870 0.050 0.240

Volberding '95 1,637 0.120

P + I 1506 3222 1637 0.070 0.046 0.228 0.143 0.120

HAART effectiveness (versus no or fewer ARVs): in AIDS all clinical studies, laboriously reduced to table --

vs. dual vs. mono vs. none

ADE Death ADE Death ADE Death

Source state = AIDS (1993 or 1987 definition)

Cook 98 indinavir .53 -- --

Curriers 98 indinavir .5 -- --

Hammer 97 indinavir [.4?] .6 -- --

Lalezari 96 saq, dual n/a .70 --

Cameron 96 rit, dual n/a n/a .54 .43 -- --

McNaghten 98 mixed~ .4 .6 .85

Moore 98 mixed* -- .75 .75

Murphy 98 mixed** .78 .91 .60 .93

Palella 98 mixed .33 .67 .82

Sherer 98 mixed† >.6 >.6

Wong 98 mixed^ .5 +

Thiessard 98 mixed .73 >.9 .93

Values in model (median) .5 .6 .54 .75 [.6] .85

C. Implement the model

• Build the functional model

• Calibrate if relevant

• Maintain quality control

• Run the simulations

Build the functional model

• Standard protocols in decision analysis software

• Spreadsheet: custom programmed in a set of tables

• Successful model-building: – careful planning of states and transitions– programming from simple to complex, initially only

a few transitions– check results repeatedly to confirm that they make

sense

Calibration • If reliable transition probabilities and

no real-world benchmarks of disease progression, no further adjustment.

• If empirical benchmarks available, especially if more trustworthy than transition data -- calibration process: – goal = transition probabilities that produce

results consistent with real-world data.– time-consuming … proceed backwards

Documentation of part of the calibration process for HIV

model. Upright = benchmarks from cohort studies

Italics = from calibrated model2 year 5 year 10 year

Start at state 5 (AIDS 87 definition) a. Number alive with AIDS 27 10 1

Number in 5 26 11 1 b. Number dead 73 90 99

Number in 6 74 89 99 Transition risk AIDS 87 to death, per 3 months

= 0.128

Adjustments to calibrated transition risks to reflect changing non-HAART ARV use and lower death rates for AIDS model inputs:

Source State Target State

Transition Probability per

Quarter, no HAART (1)

Reduction in Transition Rate with

HAART (2)

0. Uninfected (3) 1 0.00004  

1. Asymptomatic, CD4 >500 2 0.02 0.91

  3 0.01 0.91

  4* 0.002 0.91

  5 0.0005 0.91

  6 0.0018 0.84

2. Asymptomatic, CD4 <500 3 0.04 0.9

  4* 0.005 0.9

  5 0.003 0.9

  6 0.0027 0.82

3. Symptomatic, pre-AIDS 4* 0.06 0.87

  5 0.02 0.87

  6 0.003 0.79

4*. AIDS, 1993 but not 1987 definition, incident (4) 4 0.12 0.54

4. AIDS, 1993 but not 1987 definition 5 0.03 0.54

  6 0.012 0.7

5. AIDS, 1987 definition 6 0.047 0.67

6. Death n/a n/a n/a

1. Transition probabilities without HAART assume current mix of non-HAART ARVs

2. HAART effects assume current mix of non-HAART antiretroviral use.

3. Stage 0 transition yields HIV incidence in U.S. of 40,000 per year

4. Stage 4* adjusts for the use of prevalent cohorts to calibrate progression from stage 4.

Quality control/debugging

• Markov models complex, rarely “transparent”

• Essential to monitor the accuracy of model outputs.

Quality control:

• Range checks: results plausible?

E.g., only 6.2 QALYs per person when mean survival = 12 years?

Quality control:

• 1-way SA: extreme values produce expected effects?

– E.g, zero effectiveness generate zero gain in QALYs? 100% effectiveness freeze disease progression? Each unit change in effectiveness (e.g., from 10% to 20% and from 80% to 90%) generate equal magnitude changes in outcome? Use narrowest outcome (e.g., QALYs expected rather than $/QALY).

Quality control:

• Markov trace: Shows distribution by state for each cycle. Shows evolution of disease progression, can reveal if odd patterns.

Run simulations

• Last step, culmination of process.

• Some models calculate all desired outcomes (e.g., QALYs and costs for all arms, net differences, and CE ratios), others require repeating analysis with different inputs.

Reporting of results

• Similar to that for any CEA – expected values for each arm and the net differences between arms

Year 1 Year 2 Year 3 Year 4 Year 5 Total

New AIDS diagnoses

Current insurance mix 49,745 44,535 40,451 37,294 34,893 206,918

Medicaid expansion 46,313 41,599 37,912 35,070 32,916 193,810

Difference -3,432 -2,935 -2,539 -2,225 -1,978 -13,108

Deaths

Current insurance mix 18,209 18,872 19,659 20,479 21,280 98,497

Medicaid expansion 17,763 18,390 19,134 19,909 20,665 95,862

Difference -446 -481 -524 -570 -615 -2,635

Life years

Current insurance mix 758,215 779,768 800,595 820,611 839,806 3,998,995

Medicaid expansion 758,380 780,390 801,715 822,272 842,053 4,004,811

Difference 165 623 1,120 1,661 2,248 5,816

Summary of Markov modeling

• Some diseases/problems more clearly/definitively modeled with explicit representation of disease states

• Markov simulations more complex than simple trees, but maybe only 50% more

• Biggest challenges: credible cumulative disease progression, quality control/debugging

Summary of DCEA course

• A perspective and set of tools to make difficult medical decision-making more explicit

• Imperfect art – e.g., problem definition, utility measurement, choice of SA

• Role in your professional life: (informed) skepticism about DCEA? Critical consumer? DCEA analyst?