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Survival models: representing uncertainty aboutthelengthoflife 245
Figure11.28 compares this two- part exponential survival model to the observation- based survival model for local
recurrence of breast cancer.
There are differences between the two survival curves shown in Figure11.28, but those are not large. Figure11.29
uses certainty equivalents to measure the importance of these differences. Recall that a similar curve (Figure11.12) was
used earlier in this chapter to compare the agreement between the simple exponential survival model and the
observation- based survival model. In that earlier comparison, the differences were very large– roughly half a year. On
the other hand, Figure11.29 shows that the deviations for the two-
part exponential model are reduced to less than a
single month for a wide range of risk attitudes.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1. 0
75 80 85 90 95 10
0
P[Alive at age x]
Age in years, x
Observation-based
survival model
Parametric two-part
survival model
Figure11.28 Parametric two- part survival model for 75- year- old woman with local recurrence of breast cancer. The corresponding observation-
based survival model is shown in grey.
0.0
1. 0
2.0
3.0
4.0
5.0
6.0
7. 0
0.0 0.1 0.2
0.3
Certainty equivalent (years)
Risk parameter (/year)
Calculated with the
observation-based
survival model
Calculated with the
parametric two-part
survival model
0.16/years
4.13 years
4.03 years
Figure11.29 Comparison of the certainty equivalents for experiencing the observation- based survival model shown in Figure11.26 and the
parametric two- part survival model shown in Figure11.28. The outcome utilities were calculated assuming an exponential utility model.
The outcome utilities were converted to the patient’s certainty equivalents. The utility model parameter is shown on the horizontal axis.
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246 Medical decision making
Summary
• Three key values characterize a survival model:
Survival probability:
time
Lifetime probability:
Dieattime1
Hazard rate:
H
tP tt
St St
St
Lt
St
DiebeforeAliveattime1
1
• If n
i
is the number of patients tracked at the end of the i
th
month and d
i
is the number of patients found to have
died during the i
th
month, then S(k), the value of the Kaplan–Meier survival model for month k, is given by
kk
11 1
11 22
// /
The ratio d
i
/n
i
estimates the hazard rate faced by patients during the i
th
month.
• With a constant hazard rate survival model the probability that death will occur during the next time interval,
given that the patient has survived until the start of the interval, is constant.
•
An observation- based survival model uses an age- adjusted actuarial survival model to extend the estimated
survival probabilities beyond the follow- up period for a study of survival in a population that matches the char-
acteristics of the patient.
• The exponential survival model is expressed:
t
Aliveattime
where the parameter λ is the probability of death during the next unit of time, as measured by t. The hazard
rate for the exponential survival model is the constant λ. Life expectancy with the exponential survival
modelis 1/λ.
• An actuarial survival model uses the annual mortality rates derived from the life tables published by the United
States. SSA as estimates for the hazard rates for members of the general population.
•
The parametric two- part survival model uses the following expression to determine the survival probabilities for a
patient with a life- threatening condition:
t
Alive in years
1
where
ρ is the probability the patient dies because of the condition
λ is the patient’s mortality rate if they die because of the condition
A(t) is the corresponding age- specific actuarial survival model
•
The following expression can be used to calculate the outcome utility when the uncertainty about the length of
life can be expressed as a parametric two- part survival model:
tcome UtilityExpected utility with actuarial
1
model
where
γ measures the patient’s risk attitudes in an exponential utility model
ρ is the probability the patient will die because of the condition
λ is the patient’s life expectancy if they die because of the condition
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Survival models: representing uncertainty aboutthelengthoflife 247
Epilogue
This chapter began with the two roles time plays in medical outcomes. The discussion in this chapter focused on one
of these roles– time as a measure of how long the patient will live. The survival models that were described quantify
the uncertainty a patient faces about the length of their life. The structure of these survival models can be adjusted to
match the patient’s age and gender as well as other factors affecting survival.
The other role for time in a medical outcome is representing when events will occur in the patient’s life. Those events
can affect the quality of the patient’s life as new symptoms and disabilities appear. Events also can affect the patient’s
survival by marking the onset of life-
threatening disease processes. The next chapter will show a representation of an
outcome that captures how these events unfold.
Bibliography
Beck, J.R., Kassirer, J.P., and Pauker, S.G. (1982) A convenient approximation of life expectancy (The “DEALE”). The American Journal of
Medicine, 73, 883–8.
Cox, D.R. (1972) Regression models and life- tables. Journal of the Royal Statistical Society Series B (Methodological), 34(2), 187–220.
Holli, K and Isola, J. (1997) Effect of age on the survival of breast cancer patients. European Journal of Cancer, 33(3), 425–8.
Kaplan, E.L. and Meier, P. (1958) Nonparametric estimation form incomplete observations. Journal of the American Statistical Association,
53(282), 457–81.
Lu, C.H., Lee, S.H., Liu, K.H. etal. (2017) Older age impacts on survival outcome in patients receiving curative surgery for solid cancer.
AsianJournal of Surgery, 41, 333–40.
Stokes, M.E., Thompson, D., Montoya, E.L. et al. (2008) Ten- year survival and cost following breast cancer recurrence: estimates from
SEER- Medicare data. Value in Health, 11(2), 213–20.
https://t.me/medicina_free
Medical Decision Making, Third Edition. Harold C. Sox, Michael C. Higgins, Douglas K. Owens, and Gillian Sanders Schmidler.
© 2024 John Wiley & Sons Ltd. Published 2024 by John Wiley & Sons Ltd.
248
CHAPTER12
Markov models
12.1 Introduction
The survival models described in the previous chapter quantify much of the uncertainty faced by a patient following
a decision. The exact number of remaining years of life is never certain for any patient. In many cases, relatively simple
mathematical expressions can represent that uncertainty.
However, the survival models that have been described are themselves static structures since they do not change as
time passes. In contrast, the effects of an illness or an injury on a patient’s survival often can change over time. Illnesses
and injuries can progress or heal, which can alter the mortality rates faced by the patient. This means that static models,
like the two- part survival models, sometimes are unable to fully represent the dynamics of how medical conditions can
evolve to change their effects on a patient’s survival.
With medical conditions that greatly shorten a patient’s remaining lifetime, the distortions caused by using a
static survival model usually are insignificant. That is why the two-
part survival model worked well for examples
involving breast cancer in the previous chapter. Recall that patients dying from a breast cancer recurrence usually
do so in 5 years or less. The mortality rates faced by those unfortunate patients can change during their few remain-
ing years. But the size of those changes will be small when compared to the high mortality rates causing those
shortened lifetimes.
On the other hand, with chronic diseases and behaviors that cause injury over long periods, the dynamics of how a
medical condition affects survival can be more pronounced. Consider smoking as an example. The increase in the
annual mortality rate faced by a first- time smoker is negligible. However, after a few decades that same behavior often
leads to illnesses that have very high annual mortality rates. Using a single average value to represent the mortality
rates associated with smoking greatly overestimates the risks faced by the first-
time smoker and greatly underesti-
mates the risk faced when one of those life- threatening conditions has developed.
Moreover, the length of life implied by a survival model is not the only important property of the corresponding
outcomes. Outcomes often include future changes in the symptoms and disabilities experienced by patients. The tim-
ings of those changes, and their durations, also affect how patients view the outcomes they face following a decision
because those symptoms and disabilities affect the quality of the patient’s life. As we saw in Chapter10, the quality of
life can be an important consideration in a clinical decision.
This chapter describes mathematical models that capture the dynamic nature of the uncertainty in outcomes, both
with regard to the length of life as well as the quality of life. These are called Markov models. The Russian mathematician
Andrey Andreyevich Markov first described the models that bear his name in the late nineteenth century. A full
description of Markovian analysis requires more mathematics than is appropriate for this book. Instead, this chapter
will follow a narrow path through this complex topic, touching only on the features of Markov models needed to
represent medical outcomes in the analysis of a decision.
12.1 Introduction 248
12.2
Markov model basics 249
12.3
Determining transition probabilities 259
12.4
Markov model analysis– anoverview 269
Epilogue 277
Bibliography 277
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Markov models 249
Definition: transition probability
Suppose that a Markov model includes health states A and B. The transition probability from health state A to
health state B is the probability that the patient is in B at the end of a time interval if the patient had been in health
state A at the beginning of the time interval.
The discussion in this chapter further focuses on how to develop a Markov model. This focused discussion begins
with the key concepts used in a Markov model, including the assumptions made about applying those concepts to
represent a patient’s possible outcomes. The second section provides examples of how to determine the probabilities
required by a Markov model. A short third section will conclude the chapter by outlining how Markov models can be
analyzed.
12.2 Markov model basics
12.2.1 Health states andtransition probabilities
Markov models use two fundamental concepts to represent the uncertainty in a medical outcome. Health states are the
first of these concepts. The unfolding of a medical outcome typically involves a sequence of stages during which the
patient experiences various symptoms, disabilities, and risks. For example, the reappearance of a cancer often starts
close to the site of the original diagnosis. As the returning cancer spreads the patient will experience changing symp-
toms and disabilities as different organs and tissue are affected. A Markov model of this progression uses health states
corresponding to the stages marking the spread of the disease.
Inevitably the definition of the health states in a Markov model requires the approximation of a continuous pro-
cess by a set of discrete stages. With the example of cancer recurrence, those stages typically are defined by a set of
landmarks encountered during the progression of the disease. The analyst adjusts the definition of the stages
according to the goals of the analysis. The only requirements are that (1) all of the possibilities are represented, and
(2) the patient’s condition fits into exactly one of the stages. In other words, recalling the terminology used in the
discussion of decision trees, the set of health states used in a Markov model must be collectively exhaustive and
mutually disjoint.
We used a concept similar to health states when we discussed the utility for the quality of life in Chapter10. A quality
state represents a possible combination of symptoms and disabilities experienced by a patient. Typically those symp-
toms and disability reduce the quality of the patient’s life. The quality adjustments made to outcome utilities were
based on the tradeoffs the patient would accept to avoid a less desirable quality state. Similarly, symptoms and disa-
bilities experienced by the patient at various stages in the progression of their medical condition define a health state.
However, the definition of a health state also includes changes in how the patient’s condition might progress to other
health states. This brings us to the second key concept in Markov models.
Transition probabilities are the second fundamental concept used in Markov models. The dynamic nature of an out-
come corresponds to changes between the health states. Transition probabilities quantify the uncertainty about when
those changes happen. More specifically, a transition probability quantifies the likelihood that a change takes place
during a time interval of a given length, such as a month or a year.
For example, suppose that a Markov model based on a 1-
year time interval includes health states A and B. That
Markov model would include a transition probability quantifying the likelihood that at the end of a year, the patient is
in health state B if they began the year in health state A.
Definition: health state
A health state is a collection of symptoms and levels of disability a patient will experience during the unfolding
of an outcome. The health states used in a Markov model are collectively exhaustive and mutually disjoint. This
means that at any point in the unfolding of an outcome, the patient’s condition must be represented by exactly
one of the health states in the Markov model.
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250 Medical decision making
Note that the use of a common time interval length is central to expressing a transition probability. All transition
probabilities used in a Markov model are expressed relative to the same length time interval. In effect, using a common
time interval length partitions the continuum of time into fixed- length time intervals in much the same way as health
states partition the progression of the patient’s condition into mutually exclusive stages.
12.2.2 Markov model diagrams andnotation
Figure12.1 shows a diagram for the simplest of all possible Markov models. This model consists of two health states:
(1) Alive and (2) Dead. The diagram uses circles to represent the two health states, each containing the corresponding
name and an index number, enclosed in paratheses. The possible transitions between health states are depicted by arcs
and labelled by the corresponding transition probabilities. For example, the transition probability p
12
, and the corre-
sponding arc, denotes a transition from the Alive health state to the Dead health state. Note that the first subscript “1”
for the transition probability p
12
indicates the starting health state (Alive). The second subscript “2” indicates the
destination health state (Dead).
The Markov model diagram in Figure12.1 also contains two arcs, each ending in the health state where they began.
These are called the survival transitions because they represent the continuation of the patient in the same health state.
For example, the survival transition with the probability p
11
represents the possibility that the patient will remain alive
at the end of the time interval if they were alive at the beginning of the time interval. The transition probability p
11
is
called the survival probability for the Alive health state.
Applying the survival probability terminology to the transition probability labelled p
22
in Figure12.1 may seem
peculiar because this transition represents the likelihood that a patient who has died will remain dead. However, step-
ping away from the grim reality of the Dead health state, notice that the underlying mathematical concept still applies.
The transition probability p
22
measures the likelihood of remaining in the corresponding health state during a time
interval. For the Dead health state, that survival probability is one.
The Dead health state in Figure12.1 is called a trapping state because it has no exiting transitions. Once the patient
enters this health state, they never move to another health state. The Markov models we will consider always have at
least one trapping state. When modeling survival, that trapping state will correspond to the end of life.
Markov models also can be built with less grim trapping states. Consider the analysis of a decision that affects the
length of recovery for a patient with a non- fatal medical condition, meaning the probability of death is not an issue.
The trapping state in this model might be the Recovered health state. Assuming relapse is not an issue, entering the
Recovered health state would mark the end of an outcome.
Dead
(2)
Alive
(1)
p
11
p
22
p
12
Figure12.1 Two- state Markov model of survival.
Definition: survival probability
The survival probability for health state A is the probability that the patient is in A at the end of a time interval if the
patient was in health state A at the beginning of the time interval.
Definition: trapping state
A health state is a trapping state if there are no possible transitions to other health states in the Markov model.
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Markov models 251
Returning to health states in general, recall that health states defining a Markov model must be collectively exhaus-
tive and mutually exclusive. This means that at any moment, the patient’s condition must match exactly one of the
health states in the model. It follows that the probabilities for the transitions originating from any health state must
sum to one because the patient must either stay in that health state or change to one of the other health states. Referring
to the two-
state Markov model in Figure12.1, notice that this means:
ppp
11 12 22
11 and
In general, for a Markov model with n possible heath states, the requirement that transition probabilities from any
health state must add up to one can be expressed as follows:
jj jj jn
:
12
1
Figure12.2 shows a slightly more complicated Markov model, involving three possible health states. The outcomes
represented by this model are for a patient who has been successfully treated for breast cancer but faces the possibility
of a future recurrence. The three health states in this second Markov model are: (1) Disease-
free, (2) Breast cancer
recurrence, and (3) Dead.
The two health states Disease- free and Breast cancer recurrence both have possible transitions to the third health
state Dead. In other words, the patient can die either while they are free of disease or after the cancer has recurred.
Presumably, death would be more likely during a time interval once there has been a recurrence. This change in the
probability of dying would be represented by the differences in the transition probabilities p
13
and p
23
. That is, p
23
would
be greater than p
13
.
The difference between the probability of dying for the Disease- free and Breast cancer recurrence health states
illustrates how Markov models represent the changes in the uncertainties involved with an outcome. Referring to the
discussion of dynamic processes in the introduction, Markov models provide a dynamic representation of what hap-
pens to a patient during an outcome.
The health states Disease-
free and Breast cancer recurrence in Figure12.2 also illustrate how Markov models can
represent the quality- of- life differences possible with an outcome. Consider the two possible outcomes for the Markov
model in Figure12.2, both lasting exactly 5 years.
Disease
free
(1)
Dead
(3)
Breast
cancer
recurrence
(2)
p
11
p
12
p
22
p
33
p
23
p
13
Figure12.2 Three- state Markov model of survival with possible breast cancer recurrence.
Transition probability requirement
Let n equal the total number of health states in a Markov model. Then:
jj jj jn
:
12
1
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252 Medical decision making
The first outcome is spent entirely in the Disease- free health state. Assume a time interval of 1month is used to
express the transition probabilities. This first outcome, lasting 5 years, would consist of 60 time intervals, all spent in
the Disease- free health state. The outcome ends after 60 time intervals because the patient died from some cause other
than the recurrence of the cancer.
Because the second outcome also lasts 5 years, it also would consist of 60 time intervals. However, suppose with
this second outcome the patient is in the Disease-
free health state only for the first 12months. At the end of a year,
the patient finds themselves in the Breast cancer recurrence health state. They stay in this less desirable health
state for the remaining 48 time intervals. At the end of 5 years, the patient succumbs to the recurrent breast
cancer.
Both of these two outcomes mean the same length of life for the patient. However, one is spent free of symptoms
while the other is mostly spent living with the pain and anxiety of breast cancer recurrence. The relative likelihood of
these two outcomes depends on the value for the transition probability p
12
. If p
12
is small, the first more desirable
outcome is more likely. As the value for p
12
increases, the likelihood of the second outcome grows.
12.2.3 Markov independence
So far, no mention has been made of the assumption that is defining property of Markov models. This key
assumption– called Markov Independence– is that transitions from any health state are independent of the health states
that were encountered prior to reaching the health state.
The Markov model in Figure12.3 illustrates the implications of Markov independence. Note that recurrence of
breast cancer is now represented by two health states: Local recurrence and Distant recurrence. This refinement of
the Markov model in Figure12.2 represents how the risk of death changes with progression of the cancer. The dis-
comfort and anxiety will increase for most patients as their health state changes from the Local recurrence health
state to the Distant recurrence health state. Therefore, these two health states also represent how quality of life can
change for thepatient.
Disease
free
(1)
Dead
(4)
Local
recurrence
(2)
Distant
recurrence
(3)
p
11
p
12
p
13
p
14
p
23
p
22
p
24
p
34
p
44
p
33
Figure12.3 Four- state Markov model of breast cancer, using separate health states for different stages of recurrence.
Markov independence
Markov independence means that the transition probabilities from any health state are independent of how the
health state is reached.
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Markov models 253
However, notice in Figure12.3 that there are two possible paths from the Disease- free health state to the Distant
recurrence health state. These two paths are highlighted in Figure12.4. With Path 1, the patient’s health state proceeds
first to the Local recurrence health state before reaching the Distant recurrence health state. Therefore, the change
from the Disease-
free health state to the Distant recurrence health state requires at least 2 time intervals. With Path 2,
the change from the Disease-
free health state to the Distant recurrence health state requires only 1 time interval.
There are at least two medical explanations for the second path. One explanation is that Path 2 represents a scenario
in which the physician treating the original breast cancer treatment overlooked a distant recurrence that was present
when breast cancer was first diagnosed. This would mean the untreated distant recurrence has had more time to affect
the organ or tissue where it resides. Another explanation is that the distant recurrence did first appear after the original
treatment but is the result of a transition from a local recurrence to a distant recurrence, all within a single time interval.
In other words, this is a very fast-
growing cancer.
Either explanation suggests that the breast cancer recurrence indicated by Path 2 is more aggressive than the breast
cancer recurrence indicated by Path 1. Therefore, arriving at the Distant recurrence health state by way of Path 2
should mean a greater value for mortality rate p
34
. In other words, the path to the Distant recurrence health state
changes the probabilities for a subsequent transition. This contradicts the Markov independence assumption, which
requires that the transition probabilities for a health state are independent of how the health state is reached.
Figure12.5 shows how to modify the Markov model for breast cancer recurrence to avoid contradicting the Markov
independence assumption. Notice in Figure12.5 that the possibility of a distant recurrence is represented by two
health states: Slow distant recurrence and Fast distant recurrence. The values for the transition probabilities p
35
and
p
45
represent the difference in the risk of death for the two paths to a distant recurrence. Path 1in Figure12.4 would
be represented in Figure12.5 by a transition from Disease- free to Local recurrence, followed by a second transition
from Local recurrence to Slow distant recurrence. On the other hand, Path 2in Figure12.4 would be represented
inFigure12.5 by a single transition from Disease- free to Fast distant recurrence. Differences between the transition
probabilities p
35
and p
45
would represent the difference between the mortality rates for slow and fast distant
recurrences.
In short, Markov independence can be interpreted as assuming history does not matter, which often is not the case
with medical outcomes. The approach demonstrated in Figure12.5 addresses the problem with assuming history
doesnot matter by including the relevant history in the definition of the health states. If the speed with which distant
recurrence appears matters, then the health states are modified to incorporate this relevant feature of how a cancer
recurrence progresses.
Disease
free
(1)
Dead
(4)
Local
recurrence
(2)
Distant
recurrence
(3)
Path 1
Path 2
p
11
p
12
p
22
p
24
p
23
p
14
p
34
p
33
p
44
p
13
Figure12.4 Copy of the Markov model in Figure12.3 showing two alternate paths to the Distant recurrence health state.
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254 Medical decision making
12.2.4 Stationarity assumption
Markovian analysis is often based on a second, closely related assumption, called the stationarity assumption. In the
context of a Markov model, stationarity means that the values for the transition probabilities do not change with time.
We will see that assuming stationarity often is problematic, particularly when the typical length of life is long. A long
length of life means the patient’s underlying health has a chance to change as it always does for all humans.
For example, the Markov model shown in Figure12.6 is a copy of the four- state model first shown in Figure12.3.
Consider the transition probability p
34
, which measures the mortality rate for patients with a distant recurrence of
breast cancer. These patients have a life expectancy of a little over 12months, which is not enough time for significant
changes due to aging. Therefore, it would be reasonable to assume that p
34
does not change much during the time this
probability represents the patient’s mortality rate.
Figure12.7 provides empirical support for the assertion that the mortality rate is constant for the Distant recurrence
health state. The curves in this figure are for women dying after diagnosis of a distant recurrence of breast cancer
(Stokes etal.,2008). The grey curve in Figure12.7 is the Kaplan–Meier survival curve fitted to observations from the
same dataset discussed in the previous chapter. The life expectancy for those patients was 12.6months. The black curve
is an exponential survival model fitted to the observations represented by the Kaplan–Meier survival curve. This
exponential survival model has the same life expectancy as the observed survival curve.
Recall that an exponential survival model, like a stationary Markov model, assumes that the risk of death is constant.
Therefore, the close agreement between the two survival curves supports the assumption of stationarity, at least for the
transition probability from the Distant recurrence health state to the Dead health state.
However, returning to the Markov model in Figure12.6, consider the transition probability p
14
. This probability is the
risk of death if the patient appears to be free of breast cancer. As depicted in the Markov model, this patient can experi-
ence a cancer recurrence in the future. However, until that happens, a patient in the Disease- free health state faces the
same risk of death as any similarly aged person.
As we saw in the discussion of the actuarial survival model in Chapter11, the survival model for individuals with-
out a known fatal disease differs greatly from an exponential survival model with its constant hazard rate. Therefore,
Stationarity
A Markov model has stationarity if the values for the transition probabilities are constant.
Disease
free
(1)
p
11
p
14
p
44
p
45
p
25
p
35
p
55
p
15
p
33
p
12
p
22
p
23
Dead
(5)
Local
recurrence
(2)
Fast
distant
recurrence
(4)
Slow
distant
recurrence
(3)
Figure12.5 Five- state Markov model of breast cancer recurrence representing different paths to distant recurrence.
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