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Cost-effectiveness analysis
using Markov modeling
Rahul Ganguly Ph.D.
November 25th, 2006
BITS, Pilani
1
Learning objective
• What is Markov modeling and why do we
need it?
• What are some of the important concepts
around Markov modeling?
• How do we apply Markov modeling to
answer research questions?
2
Types of modeling techniques
• Simple decision tree
– Deterministic
• Markov model
– Timing of event and recursive
• Monte-carlo simulation
– Stochastic
3
Limitations of simple decision tree
FATAL
DEAD
BLEED
ANTICOAGULANT
NON FATAL
POST BLEED
FATAL
DEAD
EMBOLUS
NON FATAL
NO EVENT
POST EMB
WELL
4
Limitations of simple decision tree
RECURRING EVENTS
TIMING OF EVENT
UTILITY
FATAL
DEAD
BLEED
BLEED
NON FATAL
EMBOLUS
NO EVENT
ANTICOAGULANT
FATAL
DEAD
EMBOLUS
NON FATAL
NO EVENT
POST EMB
WELL
5
Markov model
• Markov states
– Well
– Disabled (Non fatal Bleed, Embolus)
– Death
• Markov cycle
– During each cycle the patient may transition from one
state to another
– Cycle length is a clinically meaningful time interval
• Time spent in each state
– Cumulative cost / cumulative utility = CU ratio
6
Example
WELL
DISABLED
Utility
Cycles
DEAD
Well
1
3
Disabled Dead TOTAL
0.7
0
1
Quality adjusted
life expectancy
3
0.7
0
Cost/cycle/state
50000
100000
0
Cost
150000
100000
0
3.7
250000
Expected utility = ts X us
S = 1 to n
7
State transition probability
P1
P4
WELL
P9
P2
DISABLED
P3
P8
P5
DEAD
P6
MARKOV CHAIN
(CONSTANT PROBABILITY)
P matrix
TO
Well Disabled Dead
Well
0.6
0.2
0.2
FROM Disabled 0
0.6
0.4
Dead
0
0
1
P7
8
Carrom example
• Each piece is a “markov
state”
• Each strike is like a
“markov cycle”
• Each piece has
probability of moving to
another place
• Consider the net as an
“absorbing state”
– Entire cohort is ultimately
absorbed into this state
e.g. death
9
Markov states
STROKE
WELL
DEAD
DISABLED
DEAD
TEMPORARY STATE
POST
MI1
POST
MI
POST
MI2
POST
MI3
TUNNEL STATES
10
Markov cohort simulation
WELL
10 patients
DISABLED
DEATH
N1 cycles
WELL
5 patients
DISABLED
3 patients
DEATH
2 patients
N2 cycles
WELL
0 patients
DISABLED
0 patients
DEATH
10 patients
11
Markov cohort simulation
TOTAL
Utility
1
Utility
0.7
Utility
0
Cycle
Well
Disabled
Dead
Start
1
2
23
24
10000.0
6000.0
3600.0
0.1
0.0
15000
0.0
2000.0
2400.0
0.6
0.4
12499
0.0
2000.0
4000.0
9999.3
9999.6
Well
FROM Disabled
Dead
Well
0.6
0
0
Cycle Cumulative
sum
utility
7400.
5280.
0.5
0.3
7400.0
12680.0
23749.2
23749
TO
Disabled Dead
0.2
0.2
0.6
0.4
0
1
What do the numbers mean?
12
Markov cohort simulation
TOTAL
AVERAGE =
TOTAL/10000
Utility
1
Utility
0.7
Utility
0
Cycle
Well
Disabled
Dead
Start
1
2
23
24
10000.0
6000.0
3600.0
0.1
0.0
15000
0.0
2000.0
2400.0
0.6
0.4
12499
0.0
2000.0
4000.0
9999.3
9999.6
1.50
1.25
Cycle
sum
Cumulative
utility
7400.0
5280.0
0.5
0.3
7400.0
12680.0
23749.2
23749
2.37
13
Monte Carlo Simulation
WELL
AJAY
VIJAY
DISABLED
DEATH
N1 cycles
WELL
VIJAY
DISABLED
AJAY
DEATH
2 patients
N2 cycles
WELL
0 patients
DISABLED
0 patients
DEATH
AJAY
VIJAY
Random number generation
Can compute variance and Standard Deviation
14
Using Markov modeling
• Freedberg KA et al “The cost-effectiveness
of preventing AIDS-Related Opportunistic
infections” JAMA January 14, 1998; 279:
130-136
• Background:
– HIV results in various opportunistic infections
• Pneumonia (PCP)
• Mycobacterium
• Fungal infections
– Drug costs to treat vary ($60 to $15000)
15
Step 1: Research question
• What is the clinical impact, cost, and costeffectiveness of strategies for preventing
opportunistic infections in patients with
advanced HIV disease?
• Perspective: Societal
• How will we use the results?
– Decide which strategy is most beneficial
16
Step 2: Markov model
CD4 COUNT
Chronic
CD4 count
OI history
Acute
CD4 count
OI history
0.300 x 109/l
0.201 x 109/l
0.101 x 109/l
Opportunistic
Infections (OI)
PCP (Pneumonia)
Toxoplasmosis
MAC (Bacterial)
Fungal
CMV (VIRAL)
Death
0.051 x 109/l
Cycle length = 1 month
Cohort simulation = 1 million patients
0.00 x 109/l
Used C/C++ programming
Model can be built on Microsoft excel
Other software - Treeage
17
Step 3: Model parameters
• Drug efficacy
– % reduction in the incidence of opportunistic infection
• Transition probabilities
– From published literature and websites
– Remember to convert “rates” to “probabilities”
• Cost
– Existing data from surveys and clinical trials
– Cost to charge ratio
– Conversion to most recent rupees (accounting for inflation)
• Utilities
– From rating scales – have to convert to utilities
18
Rates to probabilities
r
r/100
p = 1- exp(-rt) p = 1 - exp(-r/12)
Yearly
Yearly incidence
Yearly
Monthly probability
incidence rate
rate/100
Probability
2
0.02
0.019799294
0.001665106
5
0.05
0.048765644
0.004157568
20
0.2
0.181252269
0.016526847
40
0.4
0.329652153
0.032780557
60
0.6
0.451154221
0.048765644
80
0.8
0.550633764
0.064486548
90
0.9
0.593392399
0.072249299
100
1
0.632082414
0.079947636
Beck JR, Paucker SG “The markov process in medical prognosis”
Medical Decision Making, 1983; 3: 419-458
19
Step 4: Report base case
Infection
Drug
No prophylaxis None
TrimethoprimPneumonia
sulfamethoxazole
Bacterial
Azithromycin
Clarithromycin
Rifabutin
Fungal
Fluconazole
Viral
Ganciclovir
Cost
Quality adjusted Incremental
life expectancy
CE ratio
$40,228
39.08
-
$44,786
42.56
$15,717
$40,749
$41,164
$41,068
$41,426
$46,009
39.24
39.26
39.21
39.22
39.3
$39,075
$62,400
$77,538
$102,686
$315,327
Research question
What is an acceptable incremental quality adjusted life year value
For India? (describe how will you estimate it)
20
Step 5: Sensitivity analysis
• “…when we doubled the incidence of each opportunistic
infection, prophylaxis became more cost-effective”
Policy implication
May be treatment should be targeted at more vulnerable
patients only
• “…to achieve a cost-effectiveness threshold of $50,000
per QALY saved, however, the cost of fluconazole would
have to be reduced to approx $100 per month”
Policy implication
Can the government negotiate a better price for the drug?
21
Are there any options you would
never consider?
Infection
No prophylaxis
TMP-SMX
TMP-SMX, Azithro
TMP-SMX, Fluconazole
TMP-SMX,Azithro, Fluconazole
TMP-SMX, Ganciclovir
TMP-SMX,Azithro, Ganciclovir
TMP-SMX,Fluconazole Ganciclovir
TMP-SMX, Azithro, Fluconazole,
Ganciclovir
$40,228
$44,786
$45,944
$47,046
$48,596
$54,628
$56,812
$58,082
Quality
adjusted life
expectancy
39.08
42.56
43.04
43.01
43.60
43.20
43.83
43.80
$61,119
44.62
Cost
22
Step 6: Conclusion
• “Pneumonia prophylaxis should be made
available to all patients”
• “Next priority should be MAC (Bacterial
infection) prophylaxis, where azithromycin
is most cost-effective”
• “Only when patients have access to those
medications is it reasonable, from CE
perspective, to consider fluconazole and
perhaps oral ganciclovir”
23
Markov modeling in India
• Agarwal R, Ghoshal UC, Naik SR “Assessment of costeffectiveness of universal hepatitis B immunization in
low-income country with intermediary endemicity using
markov model” Journal of hepatology 38 (2003) 215-222
Research question
Strategies to decrease Tuberculosis in Rural India?
?
24