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Outcome utilities– adjusting forthe quality oflife 215
We now know enough to determine the utilities for each of the outcomes in the decision tree for this problem. The
last step is to combine the utility model for length of life, adjusted by the appropriate quality parameter, with the
corresponding survival model.
For example, consider the outcome labelled “A” in Figure10.15. The patient experiences this outcome if there is no
treatment and there is no pulmonary embolism. The following parametric utility model determines the utility for a life
of length t years with this outcome:
e
t
Life of years without complications
1
0 1300
0 1300.
.
The probability the patient will live t years during this outcome is derived from the exponential survival model with
parameter 1/35 years, or 0.0286/year. That is
t
Alive in years without complications
1
0 0286.
Recall from the discussion earlier in this chapter that if the quality- adjusted utility for life can be represented by the
parametric model
i
e
i
t
Life ofyears inquality state
th
1
and the length of life is determined by the exponential survival model
t
Alive in years
1
Then the expected utility is given by the expression:
xpected utility
i
For the outcome labeled A in the decision tree, the risk parameter θ
i
is 1 since this is life without complications.
Therefore, using the other parameter values for this outcome, we have that
xpected utility fornocomplications
i
1
0 0286 0 1300
6
..
..3063
What we have determined is the utility for any of the outcomes in the decision tree that follows from (1) the absence
of a pulmonary embolism and no treatment, (2) unnecessary treatment without intracranial hemorrhaging when there
is no pulmonary embolism, or (3) successful treatment of a pulmonary embolism without intracranial hemorrhaging.
Now consider the outcome labelled “B” in Figure10.15. The patient experiences this outcome if they survive a mild
stroke caused by treatment. The following parametric model utility determines the utility for a life of length t years
during this outcome:
e
Life of years with mild stroke disabilities
0 8000
1
.
00 1300
0 1300
.
.
t
The corresponding survival model is
1
0 0333. t
It follows that
xpected utility for mild stroke disabilities
i
0 8000.
00 0333 0 1300
4 8980
..
.
Repeating the same process for the outcome labelled “C” in Figure10.15 yields
xpected utility for severe stroke disabilities
i
030. 000
0 0667 0 1300
1 5254
..
.
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216 Medical decision making
Finally, Table10.4 lists the utilities for the possible outcomes of the pulmonary embolism treatment decision. Notice
that this table also contains a column labelled “Rescaled utilities.” This column contains the outcome utilities rescaled
so that utility for life with no complications is 1.0. The values for the rescaled outcome utilities are interesting because
they are the utilities that should have been determined by the direct approach to utility assessment described earlier.
The outcome utilities shown in this table will be used in a later chapter to demonstrate how the threshold for starting
treatment balances the possible benefits and harms of treatment.
Table10.4 Utilities forthe possible outcomes ofthe pulmonary embolism treatment decision.
Outcome Utility derived in narrative Rescaled utility
No complications 6.3063 1.0000
Mild stroke disabilities 4.8980 0.7767
Severe stroke disabilities 1.5254 0.2419
Death 0.0000 0.0000
Summary
• A quality- lifetime tradeoff model represents the utility for an outcome that has a given length of life, and a quality-
reducing symptom, by determining an equivalent outcome with a shorter length of life, but without the
symptom. The utility for the original outcome is the utility for that equivalent shorter symptom- free lifetime.
• A quality- lifetime tradeoff model can be expressed using the following notation:
ti
i
th
Utility for lifetimeinquality state
ti
i
th
Equivalent lifetimeinqualitystate
Then
wt
th
ii
years in quality state
Typically, the function u
i
(t) is assumed to be the same for all quality states.
• Let w
i
(t) denote the quality- lifetime tradeoff function assessed for the i
th
quality state and let ϕ
i
equal the value
for the ratio w
i
(t)/t averaged over a suitable time period, such as the patient’s life expectancy. The quality- lifetime
parametric utility model uses the following approximation to determine the patient’s utility for a life lasting t years
on the i
th
quality state:
t
th
i
years in quality state
The term ϕ
i
is called the quality- lifetime tradeoff parameter.
• Assume that utility for length of symptom- free life can be expressed as an exponential utility with risk param-
eter γ. Also assume that the quality-
lifetime tradeoff function for the i
th
quality state can be expressed by the
quality parameter ϕ
i
. The quality- lifetime parametric utility model can then be expressed:
th t
i
1
• Suppose that (1) the patient’s preferences for length of life can be represented by an exponential utility model
with parameter γ, (2) the patient’s quality-
lifetime tradeoff for life in thei
th
quality state can be represented by the
parameter ϕ
i
, and (3) the uncertainty for the length of life in an outcome can be represented by an exponential
survival model with parameter λ. The patient’s expected utility for that outcome is then:
i
i
Outcome
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Outcome utilities– adjusting forthe quality oflife 217
Epilogue
Chapters8,9, and10 have established a method for quantifying the patient’s preferences for the possible outcomes of
a decision. This method starts with the theoretical foundation established by von Neumann and Morgenstern. From
that foundation, these chapters built a framework for assessing a patient’s utilities for possible outcomes based on risk
attitudes for length of life and preferences for quality of life. Simple parametric models were developed that approxi-
mate outcome utilities in a manner that lends itself to use in a clinical setting. Those approximations require assump-
tions about how patients view the uncertainty in their lives.
What remains to be done is the establishment of survival models that provide the outcome probabilities needed to
complete the analysis. These survival models represent the uncertainty for the different lengths of life possible with an
outcome. The resulting probabilities can be combined with the corresponding outcome utilities to compute the
expected utility for each alternative in a clinical decision problem. The nature of the approach guarantees that the
alternative with the greatest expected utility is the alternative that best matches the preferences of the patient.
The next chapter will start the discussion of survival models. That discussion will describe the exponential survival
model, which has already been mentioned, as well as what will be called an actuarial survival model. These two
models can be combined to produce a general representation of the uncertainty faced by patients in many decision
problems.
Bibliography
McNeil, B.J., Weichselbaum, R., and Pauker, S.G. (1978) Fallacy of the five- year survival in lung cancer. New England Journal of Medicine,
299,1397–401.
McNeil, B.J., Weichselbaum, R., and Pauker, S.G. (1981) Speech and survival. Tradeoffs between quality and quantity of life in laryngeal
cancer. New England Journal of Medicine, 304, 982–7.
Shavelle, R.M., Brooks, J.C., Strauss, D.J., and Turner- Stokes, L. (2019) Life expectancy after stroke based on age, sex, and Rankin grade of
disability: a synthesis. Journal of Stroke and Cerebrovascular Diseases, 28(12), 104450.
Wang, C.C. and O’Donald, A.R. (1955) Cancer of the larynx: five- year results with emphasis on radiotherapy. New England Journal of Medicine,
252, 743–7.
• The quality- lifetime tradeoff model implies that risk aversion decreases as the quality of life decreases. With the
quality-
lifetime parametric model, an individual’s risk aversion for life with a symptom or disability is γϕ
i
whereϕ
i
is the quality- lifetime tradeoff parameter and γ is the individual’s risk parameter.
•
A quality- survival tradeoff model represents the utility for an outcome that has a given length of life, and a quality-
reducing symptom or disability, by determining the minimum probability of survival the patient would accept
in order to avoid the symptom or disability. The utility for the original outcome is that minimum survival
probability multiplied by the utility for living the same length of life free of the symptom or disability.
•
The quality- survival parametric model represents the utility for an outcome in which the patient will live for
timet in the i
th
quality state as follows:
tility for life of lengthin thequality stateti e
th
i
1
tt
where γ is the patient’s risk parameter expressed in the time units of tand θ
i
is the quality- survival tradeoff
parameter for thei
th
quality state.
•
Suppose that (1) the patient’s preferences for length of life can be represented by an exponential utility model
with parameter γ, (2) the patient’s quality-
survival tradeoff for life in thei
th
quality state can be represented by
the parameter θ
i
, and (3) the uncertainty for the length of life in an outcome can be represented by an exponential
survival model with parameter λ. The patient’s expected utility for that outcome is then:
i
Outcome
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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.
218
CHAPTER11
Survival models: representing uncertainty
aboutthelengthoflife
11.1 Introduction
Time plays critical roles in most medical outcomes. Those roles include measuring how long life lasts and marking
when events occur. This chapter focuses on the first of these roles– time as a measure of a patient’s lifetime.
Uncertainty is part of what complicates time’s role as an outcome measure. The range of possible lifetimes almost
always is known only as a set of probabilities for how long the patient might live. Those probabilities are represented
by what we call a survival model.
Earlier chapters already used survival models. In Chapter8, we saw how a survival model provided the length of
life probabilities needed to compute the expected utility for an outcome. In general, survival models usually are needed
to represent risk attitudes in the analysis of a decision. The relationship between a survival model and the correspond-
ing length of life probabilities will be a central focus of this chapter.
The methods to be discussed typically combine two basic survival models. One of these is the exponential survival
model. We often used this parametric model in earlier chapters to represent the uncertainty of how long a patient’s
life will last. The mathematical expression representing the exponential survival model has computational advan-
tages when combined with an exponential utility model, as was discussed in Chapter 9. This chapter explores the
validity of assuming that mathematical expression. Particular attention will be paid to cases where the length of a life
is controlled by a disease or injury. We will see that this simple survival model is more than a computational con-
venience in thesecases.
However, the purpose of medicine can be seen as lessening the control those diseases and injuries have on a patient’s
life. An alternative to the exponential survival model will be needed when a threat to a patient’s life has been success-
fully addressed. This need will be addressed by a second basic survival model– the actuarial survival model.
Curing a disease or successfully treating an injury does not mean the patient will live forever. Instead, the survival
of these fortunate patients usually resembles the survival of similar members of the general population. The observed
survival probabilities in the general population constitute what we will call an actuarial survival model. This chapter
will show how to combine exponential survival models and actuarial survival models to establish a credible represen-
tation of long life will last for a patient.
Mention should be made about what this chapter will not cover. Survival model analysis plays a central role in the
important field of predictive modelling. Statistical regression techniques, founded on the seminal work of David Cox
11.1
Introduction 218
11.2
Survival model basics 219
11.3
Medical example– survival after breast cancer recurrence 226
11.4
Exponential survival model 228
11.5
Actuarial survival models 232
11.6
Two- part survival models 235
Epilogue 247
Bibliography 247
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Survival models: representing uncertainty aboutthelengthoflife 219
(Cox,1972), have discovered important relationships between how long patients will live and the diseases and treat-
ments they experience. This chapter will not attempt to summarize this important but complex field. Instead, the goals
of this chapter are mathematical expressions that provide a creditable approximation for the uncertainty faced by a
patient after a decision. Combining those approximations with the utility models described in the preceding three
chapters will determine the expected utilities for the options available to that patient. The option with the greatest
expected utility will be the option the patient should choose.
Returning to what will be covered in this chapter, the discussion starts with the common properties of all survival
models. This will lead to the Kaplan–Meyer survival model, which provides a general framework for estimating
survival models from survival observations. Next, this chapter will discuss the two basic survival models: the
exponential survival model and the actuarial survival model. The final section describes how to combine those two
basic survival models to establish a mathematical expression that quantifies the uncertainty of how long a patient
will live.
11.2 Survival model basics
A survival model quantifies the probability that the patient will remain alive as time passes. Time typically is measured
relative to where the period of interest begins, such as the start of a chosen treatment. For example, Figure11.1 shows
the survival models for radiotherapy and surgery used by McNeil and her colleagues in their study of patient prefer-
ences for lung cancer treatments (McNeil, 1978). For a patient whose lifetime uncertainties are represented by one of
these two survival models, time zero is when the treatment began.
Focusing for a moment on the survival model for radiotherapy, labelled S
Rad
(t) in Figure11.1, notice that
Rad
Probability alive at time with radiotherapy
where t is measured relative to the start of treatment. The maximum value for a survival model always occurs at time
zero and decreases as time advances. The curve for S
Rad
(t) equals 1.0 at time zero because there is no immediate risk of
death for patients undergoing radiotherapy.
On the other hand, consider the survival curve for patients undergoing surgery in Figure11.1. That survival curve is
labelled S
Surg
(t). Notice that S
Surg
(0) is less than 1.0in order to account for the perioperative risk with lung cancer sur-
gery. The analysis performed by McNeil and her colleagues assumed a 5% operative mortality rate. This operative risk
is represented by setting S
Surg
(0) equal to 0.95.
Of course, after time zero, the survival probabilities always decrease as time advances since the probability of being
alive at some time in the future is never more than the probability of being alive today.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0510 15 20
25
Probability survival ≤ t years
Time since treatment, t (years)
Radiotherapy
Surgery
S
Surg
(t)
S
Rad
(t)
Figure11.1 Survival probabilities for survival curves use in comparison of radiotherapy and surgical treatment for lung cancer. Adapted from
McNeil (1978).
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220 Medical decision making
The two curves in Figure11.1 only show the patient’s survival probabilities out to 25 years after the start of treat-
ment. With either treatment, the probability of still being alive at 25 years is approximately 10% for the 60-
year- old man
represented in the analysis discussed back in Chapter8. Being alive in 25 years means that the patient is no longer
likely to die from lung cancer. However, they will not live forever. This means that if the survival models shown in
Figure11.1 were extended far enough into the future, the curves ultimately would reach zero. How those extensions
are made will be covered later in this chapter.
11.2.1 Survival probabilities
Three groups of values are associated with a survival model. The first of these are the values of the survival model
itself. These are called the survival probabilities and will be denoted by S(t). That is
Death after time
For example, Figure11.1 shows the patient’s survival probabilities for surgery and radiotherapy. Survival probabili-
ties usually are determined from observations of how long patients live in clinical situations that match that of the
patient of interest.
11.2.2 Lifetime probabilities
The second important group of values associated with a survival model are the probabilities that the patient’s life will
last exactly a specific length of time, such as 5 years. These probabilities will be called the lifetime probabilities and will
be denoted by L(t). That is
Length of life
Lifetime probabilities are important because they have a key role in expected utility analysis. Recall that we need to
know the lifetime probabilities for an outcome to determine the expected utility for that outcome. For example, these
were the probabilities listed in Table8.1 when we calculated the expected utility for the lung cancer treatments. The
uncertainty about the length of the patient’s life with an outcome is expressed by the lifetime probabilities. The expected
utility for that outcome is the sum of the lifetime probabilities multiplied by the patient’s corresponding utilities for
those lifetimes. Recall that Chapters8 and9 showed how to determine those utilities for the patient’s lifetimes.
11.2.3 Lifetime probabilities andthe representation oftime
There is a subtlety in how time is represented that must be understood in order to work with lifetime probabilities.
Alifetime probability only is meaningful if time has a granularity, such as days, months, or years. That granularity
represents the inexactness in how time is represented. For example, when we say that the patient lived for 1 year, we
do not mean that the patient’s life lasted exactly 12months or 52weeks or 365days. Instead, saying that the patient’s
lifetime was 1 year means that the patient lived approximately 1 year. In other words, granularity means that time is
expressed in terms of time intervals. For example, using a 1-
year granularity, saying that life lasted 5 years means that
death occurred after 5 years but before 6 years.
With this convention, the probability that death occurs during an interval is easily computed from a survival model.
Suppose that a time interval runs between times t
A
and t
B
. Then
BAA
Death during time interval Dead at Aliveat
SSt
B
Giving time a granularity means that time can only have discrete values such as 0, 1, and 2 years. The uncertainty for
discrete values is expressed by what statisticians call a probability mass function. A probability mass function expresses
the probability that a variable has a particular value. Without time granularity, the uncertainty about the length of life
would be expressed by something called a probability density function. Working with probability density functions
requires mathematics that is beyond the scope of this book. On the other hand, working with probability mass
functions only requires simple arithmetic.
For example, how do we determine the probability that a lung cancer patient’s life will last exactly 5 years after
radiotherapy in an analysis based on years? Figure11.2 shows an expanded view of the survival probability curve for
radiotherapy. Note that the survival probability for 5 years is 0.2143 and the survival probability for 6 years is 0.1811.
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Survival models: representing uncertainty aboutthelengthoflife 221
Therefore, the probability that the patient’s life will last more than 5 years but less than 6 years is the difference
between 0.2143 and 0.1811, or 0.0332.
Figure11.3 shows the lifetime probabilities derived by repeating this calculation for the survival models for radio-
therapy and surgery shown in Figure11.1.
In general, if L(t) denotes the lifetime probability at time t and S(t) is the corresponding survival model, then L(t) can
be determined by the following relationship:
1
The reader may wonder why the value for L(t) was not computed using a time interval centered on t. Remember that
L(t) is the probability that death occurs during an interval rather than at a specific point in time. Whether we associate
that probability with the midpoint of the interval or one of the endpoints is arbitrary. The lifetime probability measures
the likelihood of the outcome that death falls within the interval, whose length is the granularity of the time repre-
sentation. On the other hand, we must designate a specific time point in order to pair that lifetime probability with its
corresponding utility. That pairing will be required to compute an expected utility. How to choose that specific point
in the interval will be discussed later in this chapter.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0510 15 20
25
Probability survival ≤ t years
Time since treatment, t (years)
S
Surg
(t)
S
Rad
(t)
L(5 years) = 0.0332
0.2143
0.1811
5 years
6 years
Figure11.2 Expanded view of Figure11.1 showing how to calculate the lifetime probability at 5 years for radiotherapy (L(5 years)).
P[Lifetime between t and t +1 years]
Time since treatment, t (years)
0.00
0510 15 20
25
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Radiotherapy
Surgery
L
Rad
(t)
L
Surg
(t)
Figure11.3 Lifetime probabilities when time granularity is 1 year, derived from the survival models for radiotherapy and surgical treatment
shownin Figure11.1.
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222 Medical decision making
11.2.4 Hazard rates
The third important value associated with a survival model is called the hazard rate and will be denoted by H(t). The
hazard rate quantifies how the risk of death varies as the patient reaches different lifetimes by aging. For example, if
the patient survives 5 years, what is the probability the patient will die before year 6? The hazard rate quantifies this
risk of death.
Therefore, time granularity chosen for the analysis also is critical to the hazard rate. The hazard rate at time t is the
probability of death before time t + 1 if the patient lives until time t. Sometimes the hazard rate also is called the
mortality rate.
The hazard rate can be expressed as a conditional probability:
H
tP
tt
DiebeforeAliveattime1
From the definition of conditional probability
tt
Pt t
DiebeforeAliveattime
Diebeforeand Aliveattime
1
1
Pt
Aliveattime
But numerator in the ratio on the right is the probability that death falls within a specific time period. In order for
death to fall within a time period, the patient must be alive at the beginning of the time period and dead at the end of
the time period. That is
11
The denominator in the ratio is
Therefore
H
t
St St
St
Lt
St
1
Figure 11.4 shows the hazard rates for radiography and surgery derived from the survival curves shown in
Figure11.1. In other words, the curves in Figure11.1 show how the probability that death will occur within a year will
change over time for the two treatments.
Notice in Figure11.4 that at time zero, the hazard rate is higher for surgery than it is for radiotherapy. The high haz-
ard rate for surgery at time zero includes the operative risk associated with surgical treatment for lung cancer as well
0.00
0.10
0.20
0.30
0.40
0
510152
025
P[Death before t+1| Alive at time t]
Time since treatment, t (years)
L
Rad
(t)
L
Surg
(t)
Radiotherapy
Surgery
Figure11.4 Hazard rate when time granularity is 1 year, derived from the survival models for radiotherapy and surgical treatment shown in Figure11.1.
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Survival models: representing uncertainty aboutthelengthoflife 223
as the other risks faced by the patient during the first year. Also notice that, after the third year, the hazard rate drops
rapidly for both treatment options. The longer the patient survives, the more likely it is that their cancer has been
cured. This means the risk of death from lung cancer drops, which decreases the overall risk of death faced by the
patient.
Figure11.4 shows that 20 years after treatment, the hazard rates for the two treatments are nearly identical. The
patient is virtually certain to be free of lung cancer 20 years after treatment if they are still alive. By that age, the risks
now faced by the patient are the same risks any 80-
year- old person faces. The slow increase in the hazard rate after
15years reflects the increasing risk of death that we all face as we age.
We will now see that hazard rates play a central role in determining a survival model from observations of patient
survival.
11.2.5 Estimating asurvival model fromobservations
Several approaches can be used to estimate a survival model from a series of observations. For example, Table11.1
shows a hypothetical study that started with 100 patients. In this study, at the end of each month, the patients were
contacted to see if they were still alive. There are three possibilities:
1. Patient found to be still alive
2. Patient found to have died
3. Patient lost to follow up
Note that 93 of the original 100 patients were located at the end of the first month and found to be alive. Five patients
were located and found to have died. Two patients were lost to follow up. Therefore, the second month began with the
93 patients who were still alive and reachable by the study.
At the end of the second month, the number of patients who still could be tracked had dropped to 92 patients.
Among the 92 patients found, 88were found to be alive and 4were found to have died. The remaining 1 patient could
no longer be reached.
By the end of the tenth month, the total number of patients being tracked had dropped to 52 patients. Fifty of
thosepatients were still alive at 10months, 2had died and 4more were lost to follow up. Therefore, by the end of the
10- month study, a total of 34 patients were found to have died and 16were lost to follow up during the study. Those
16lost patients complicate how we would estimate the survival curve implied by this hypothetical follow- up study.
One approach to determining the survival model is to simply divide the total number of patients (100) into 100minus
the reported deaths (34). For example, 5 deaths were reported by the end of Month 1. This suggests that the survival
probability at the end of the first month is 95 divided by 100, or 0.95. By the end of Month 10, the total number of
reported deaths is 34. Continuing the same line of reasoning, the survival probability at the end of the tenth month
is66 divided by 100, or 0.66.
The survival model generated by this approach might be called the optimistic survival model since it assumes that
patients lost to follow up are still alive. Sixteen patients were lost to follow up by the end of the tenth month. Not
counting those lost patients as dead means we are assuming they are still alive. Of course, some of the 16 patients lost
to follow up may have died and those patients were lost to follow up because their families had more pressing con-
cerns than answering queries from a research team. Therefore, the optimistic survival function probably overestimates
the probability of survival because some of the deaths may have been missed.
Key concepts: Survival model values
Survival probability:
time
Lifetime probability:
Dieattime1
Hazard rate:
H
tP tt
St St
St
Lt
St
DiebeforeAliveattime1
1
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224 Medical decision making
A pessimistic survival model assumes that all patients lost to follow have died. Under this assumption, the total num-
ber of deaths for the end of Month 10would be the 34 patients known to have died and the 16 patients lost to follow
up, for a total of 50 deaths. This would imply a survival probability of 0.50 for Month 10. But a patient lost to follow up
might have moved and not notified the research team. Therefore, the pessimistic survival model is likely to underesti-
mate the probability of survival since some of the patients assumed to have died may still be alive.
11.2.6 Kaplan–Meier survival model
A third alternative is called the Kaplan–Meier survival model, after the two statisticians who independently discovered
this approach (Kaplan and Meier,1958). Recall that a hazard rate is the probability of death during the next time inter-
val. A Kaplan–Meier survival model interprets survival data as estimates of the hazard rate faced by members of the
study population. In turn, those estimated hazard rates imply a survival model.
Consider the number of patients located during a given month. These patients are at risk of dying during that
month. For example, 100 patients were at risk of dying during the first month in the hypothetical study shown in
Table11.1. However, two of those patients were lost to follow up. The 98 patients whose outcome is known are used to
estimate the hazard rate during the first month. Five of those 98 patients died during Month 1. Therefore, the probabil-
ity of dying– the hazard rate– for those 98 patients can be estimated by dividing 5 by 98, or 0.0510.
The Kaplan–Meier survival model assumes that patients lost to follow up faced the same hazard rate as patients
who were found during follow up. Under this key assumption, the probability of surviving until the end of
Month1would be
.
Now consider Month 2. The research team found 92 of the patients at the end of Month 2. Four of those patients were
reported to have died during Month 2. It follows that the hazard rate for the second month is estimated to be 4 over 92,
or 0.0435. Once again the hazard rate measures the probability of dying during the second month implying
P Surviving to endof MonthMonth hazard rate11 2
0 9490 100435..
Similarly, 4 of the 88 patients located at the end of Month 3were alive for a hazard rate of 4 over 88, or 0.0455. So,
P Surviving to endof MonthMonth hazard rate21 3
0 9077 100455..
Table11.1 Hypothetical follow- up study of100 patients.
Month Found alive Found dead Lost to follow up Estimated hazard rate
1 93 5 2 0.0510
2 88 4 1 0.0435
3 84 4 0 0.0455
4 80 3 1 0.0361
5 74 4 2 0.0513
6 70 3 1 0.0411
7 64 4 2 0.0588
8 60 3 1 0.0476
9 56 2 2 0.0345
10 50 2 4 0.0385
Total: 34 Total: 16 Average: 0.0448
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