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Markov Chain Population Models in
Medical Decision Making
Gordon HazenMin Huang
Northwestern University
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Markov models (individual-level)in medical decision making
Intervention that reduces disease mortality rate
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Conventional outcome measure—QALYs for an individual (or a cohort)
x 1 = 12.2
x 0.75 = 3.96
16.16 QALY
12.20 yr
5.28 yr
x 1 = 12.2
x 0.75 = 13.21
25.41 QALY
12.20 yr
17.62 yr
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From individual to population
Motivation:
To study a whole population
1.Equilibrium distribution of a population
2. Equilibrium measure of effectiveness of an intervention
Individual-level models — 1. no equilibrium2. no births
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Augment model by allowing “births”
Intervention that reduces disease mortality rate
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Population model and its routing
λ0
#Well #Diseased
0
μ0 μ1
λ0
Well Diseased
0
μ0 μ1
Population model
Routing process
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Population no longer dies out—reaches new equilibrium after intervention
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Jjjq
kjJkjkjq
j
jk
,...,0 ,)1,(
,,...,0, ,),(
Time-homogeneous individual-level Markov models
}0),({ ttX
{0,1,2,..,J,-1}, where ‘-1’ representing ‘Death’ is an absorbing state
Individual Markov model
State space
Transition rates
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Population models
Population Markov model }0),({ ttn
State space:
Transition rates:
},...,1,0interger negativenon a is :),...,,{( 110
1 J, jnRnnnZ jJ
JJ
jj
jjj
jjkjk
nTnq
kjJkjnnTnq
nnTnq
),(
,,...,0, ,),(
),(
,1
1,
— Open Jackson processes Serfozo
Serfozo R. Introduction to Stochastic Networks. Springer 1999.
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Routing processes
}0},1,,...,1,0{)({ tJtr
{0,1,2,..,J,-1}, where ‘-1’ is a source/sink node
Individual-level model
State space:
Transition rates
,),1(
,,...,0, ,)1,(
,),(
j
j
jk
jq
kjJkjjq
kjq
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PropertiesIf }0),({ ttr is irreducible, then at equilibrium:
Jnnn ,...,, 10• are independent,
Jjj ,...,0,
• Conditional on total population size |n|, n ismultinomial )~,...,~,~|,(| 10 Jn
0
jj J
jj
equilibrium population means
equilibrium population proportions
j )jn ~Poisson(
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Equilibrium population means
J
jkk
kjkj
J
jkk
jkjj00
)( Jj ,...,0
.
is the unique collection of positive numbersJjj ,...,0,
that satisfy balance equations of routing process
1 Q
i.e.
Here Q is a submatrix of the rate matrix of the routing process, and also a submatrix of the rate matrix of the underlying individual model, corresponding to all nonabsorbing states, i.e., health states {0,1,…,J}.
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What measures of quality are possible at the population level?
Equilibrium population measures
Individual QALYs
: QALYs for an individual starting in state j
Measures of health
jG
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x = 10.4
41%
59%x = 12.3
22.7 QALY
25.41
20.83
Average Lifetime QALY
ALQ
J
jjjG
0
~
Mean QALY of randomly selected individual from equilibrium population
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Total Lifetime QALY
TLQ
Mean total QALYs of all individuals in
equilibrium population
x = 310
12.2
17.6x = 367.1
677 person-QALYs
25.41
20.83
J
jjjG
0
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Average QALYs per Year
AQ/yr One-year QALY of randomly selected individual from equilibrium population
41% x 1.00 = 0.41
59% x 0.75 = 0.44
0.85 QALY/yr
J
jjjv
0
~
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12.2 x 1.00 = 12.2
17.62 x 0.75 = 13.22
25.42 person-QALY/yr
Total QALYs per Year
TQ/yr
One-year QALY of all individuals in equilibrium population
J
jjjv
0
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Discounted Total QALYs
DTQ
Mean total discounted QALYs for this and all subsequent generations of population.
406.5041 x 1.00 = 406.5
176.1518 x 0.75 = 132.1
538.6 person-QALYs
406.5 yr
176.2 yr
406.5041 x 1.00 = 406.5
400.3449 x 0.75 = 300.3
706.8 person-QALYs
406.5 yr
400.3 yr
Discount rate = 3%
0 ,
1,00
0
)](),()([ tdnjivedtvtneE ji
J
jiji
rtJ
jjj
rt
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DTQr
1
TLQ
J
jj
0
TQ/yr
J
jj
0
TG
TT LG
/
TQ/yr
AQ/yr
Relationships between measures
if the population is in equilibrium from t=0.
ALQ
AQ/yr
TQ/yr
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The simple illustrative example— differences among measures
Intervention that reduces disease mortality rate
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Evaluating interventions using these measures:
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• Problem: average measures do not account for population size increase due to better survival.
• Caution in choosing population measures
Insight
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Example: tamoxifen use to prevent breast cancerCol
Col N.F., Orr R.K., Fortin J.M. Survival impact of tamoxifen use for breast cancer risk reduction: projections from a patient-specific Markov model, Med Decis Making 2002; 22: 386-393.
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Non-homogeneous individual-levelMarkov models
1. Human background survival
}0},,{)({ tdeadalivetB
Background mortality rate )(1
0100 )( tabba ebt
2. The other factor: a homogeneous Markov process
}0},1,,...,1,0{)({ 0 tJtX
(Gompertz)
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Population models
Mean density with respect to age a of the population in state j at time t:
),( tam j
Theorem:
),0(
)(),(),(),(
tm
aQtama
tam
t
tam
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equilibrium mean density with respect to age a of the population in state j,
equilibrium expected total population count in state j.
Notations:
)(aj
j
),()( aPa
)(* aP
Conclusions:
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: QALYs for an individual starting from age a0 in state j
)( 0aG j
Individual QALYs
Measures of health
Equilibrium population measures
ALQ TLQ
AQ/yr TQ/yr
TLQ
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Example: tamoxifen use to prevent breast cancerCol
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SummarySummary
• Population Markov models for medical decision making.
• Population measures of interventions
• Age-dependency.