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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?
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