Добавил:
Sekretar
kiopkiopkiop18@yandex.ru
t.me/Prokururor I Вовсе не секретарь, но почту проверяю
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_2946_Библиотеки_им_академика_М_И_Перельмана
.pdf
Survival models: representing uncertainty aboutthelengthoflife 225
Continuing this recursive calculation until the end of Month 10 generates the survival probabilities shown in
Figure 11.5. These survival probabilities constitute the Kaplan–Meier survival model for the observations in
Table 11.1. Figure 11.5 also shows the optimistic and pessimistic survival models for the hypothetical study.
Noticethat optimistic and pessimistic survival models provide an upper and lower bound for the Kaplan–Meier
survival model.
The general process used to generate a Kaplan–Meier survival model can be summarized as follows. Let n
i
denote
the number of patients tracked at the end of the i
th
month and d
i
denote the number of patients found to have died dur-
ing the i
th
month. The ratio d
i
/n
i
estimates the hazard rate faced by patients during the i
th
month and S(k), the value of
the Kaplan–Meier survival model for month k, is given by
kk
11 1
11 22
// /
Clearly, hazard rates play a central role in the Kaplan–Meier survival model. Note that the curve for the Kaplan–
Meier survival model is shown in Figure11.5 as a series of horizontal line segments. Each line segment depicts the
assumption that the hazard rate is constant between observations. Perhaps this is a dubious assumption but it is tradi-
tionally made in depictions of Kaplan–Meier survival curves.
Also, note that the hazard rate estimate is 0.0510 for the first month and 0.0435 for the second month. Do these
changes represent actual changes in the medical condition that determines the hazards faced by the patient or are these
changes simply the chance variations in the data? Given the lack of a clear trend in the hazard rates shown in Table11.1,
it might be reasonable to assume the latter explanation and average the hazard rates for the first 2months. Extending
this reasoning across the entire study implies the average hazard rate of 0.0448 for patients in the study.
This suggests what will be called the constant hazard rate survival model. Let ρ denote the average hazard rate observed
in a follow-
up study. The constant hazard rate survival model denoted by S
*
(t) is defined as follows:
**
1
0.0
0.2
0.4
0.6
0.8
1.0
012345
Months of follow-up, x
678
910
P [Alive at x months]
Optimistic
Pessimistic
Kaplan–Meier
Figure11.5 Optimistic, Pessimistic, and Kaplan–Meier survival curves for data from the hypothetical study of 100 patients shown in Table11.1.
Definition: Kaplan–Meier survival model
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.
https://t.me/medicina_free
226 Medical decision making
Based on the observations in Table 11.1, Figure 11.6 compares the Kaplan–Meier survival model to the constant
hazard rate survival model with ρ equal to the average value of 0.0448 for the study. Note that the alignment between
the two models is very good.
11.3 Medical example– survival after breast cancer recurrence
We now will move the discussion to representing a patient’s survival after discovery of a breast cancer recurrence. We
will focus on survival after discovery of a recurrence that is distant from the site of the original tumor. The data used
was obtained from the National Cancer Institute’s Surveillance Epidemiology and End Results (SEER) registry (Stokes
etal.,2008). Our analysis will focus on women who were over 65 years of age at the time of diagnosis. The average age
of the women in this subgroup was 76 years, which included 622women.
Figure11.7 shows the Kaplan–Meier survival curve derived from this dataset. We will consider this curve as the
basis for determining the survival model for a woman who was 75 years old at the time of diagnosis of a distant recur-
rence of breast cancer. Notice in Figure11.7 that the probability this 75- year- old woman will survive until the age of 77
is less than 25%. The survival curve in Figure11.7 only shows the uncertainty for the survival probabilities during the
first 10 years following diagnosis because that was the length of follow- up period reported in the study.
Not all of the deaths represented in Figure11.7 were caused by breast cancer. These were older women who faced
numerous other risks that could have caused their deaths. We might call those deaths from other causes the expected
deaths in the study population.
Therefore, the survival curve shown in Figure11.7 can be thought of as a combination of survival curves for two
different groups. One group consists of women dying from breast cancer. The other group consists of women dying
from other causes. What we will see is that the breast cancer deaths mostly occur during the first years of the follow- up
period. Deaths from other causes occurred through the follow- up period but dominate the deaths in the later part of
the follow- up period.
Later in this chapter, we will see how to estimate the number of deaths from other causes– the expected deaths–
based on what is called the actuarial survival model. For now, think of the actuarial model as representing the survival
0.0
0.2
0.4
0.6
0.8
1.0
012345678
910
P [Alive at x months]
Months of follow-up, x
Kaplan–Meier
Constant Hazard Rate
Figure11.6 Constant hazard rate model based on average hazard rate observed in data.
Definition: Constant hazard rate survival model
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.
https://t.me/medicina_free
Survival models: representing uncertainty aboutthelengthoflife 227
of a woman in the general population. In other words, for our patient, the actuarial survival model would provide her
survival probabilities if she no longer had the elevated risk of dying from breast cancer she faced back when she was
diagnosed with a recurrence at age 75.
Figure11.8 shows the overlap between the observed survival probabilities in Figure11.7 and the corresponding
actuarial survival model for a 75-
year- old woman. This second depiction of the survival curve has been restricted to
ages starting at 80 and ending at 85. Notice the close overlap between the two curves for this age range.
The number of patients represented by the Kaplan–Meier survival curve shown in Figure11.8 is small. Only about
10% of the original 622 patients survived beyond the first 5 years of follow-
up. However, the overlap shown in
Figure11.8 suggests that the uncertainty about the patient’s remaining lifetimes can be represented by the actuarial
survival model after the age of 80. From a medical perspective, this makes sense. 5-
year post- treatment survival
typically is taken as evidence that a cancer has been fully managed. In other words, after reaching 80 years of age,
thepatient’s survival model approximates that of any 80- year- old woman.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
75 76 77 78 79 80 81 82 83 84 85
P [Alive at age x]
Age in years, x
Portion of survival curve
expanded in Figure 11.8
Figure11.7 Kaplan–Meier survival curve observed in 622women with distant recurrences of breast cancer. Data is from a study of women treated
for metastatic cancer. Adapted from Stokes etal. (2008).
0.0
0.1
0.2
80 81 82 83 84
85
P [ Alive at age x]
Age in years, x
Kaplan–Meier survival
curve from Figure 11.8
Survival curve based
on expected deaths
Figure11.8 Comparison of the expected deaths for women between the ages of 80 and 85 and the Kaplan–Meier survival curve shown in Figure11.7.
https://t.me/medicina_free
228 Medical decision making
This alignment of survival models after age 80 leads to the composite survival model shown in Figure11.9.
Thedetails in how the survival model in Figure11.9 is assembled will be discussed later in this chapter. However, the
central idea is that the patient’s survival for the first 10 years following diagnosis is what was observed in the
622patients. After 10 years, the patient’s survival matches that of any comparably aged woman. In the following
discussion, this will be called the observation- based survival model.
In principle, an observation- based survival model, like the one shown in Figure11.9, achieves our goal of a credible
representation of the uncertainty about a patient’s length of life. Indeed, McNeil and her colleagues used this approach
to generate the survival models used in their studies of lung cancer and laryngeal cancer treatment. However, for com-
putational purposes, reducing the observation-
based survival model to a simpler parametric form would be useful.
How this is done starts with a discussion of the exponential survival model, which is the topic of the next section.
11.4 Exponential survival model
As said in earlier chapters, the exponential survival model provides a convenient representation of the uncertainty about
how long a patient will live. Like all parametric models, the validity of the exponential survival model rests on an assump-
tion. In this case, that assumption is that the hazard rate is constant. The exponential survival model assumes the
probability of death during a given period of time, such as a year, remains constant for the remainder of the patient’s life.
The mathematical expression for the exponential survival model is:
t
Aliveattime
where e is that constant known in mathematics as Euler’s number and λ is the probability of death, or hazard rate, dur-
ing the unit of time used to express t. For example, if t is expressed in years then λ is the probability of death during the
next year. In other words, λ is the hazard rate for the patient facing the uncertainty representing by the exponential
survival model. The exponential survival model is a constant hazard rate survival model because the parameter λ is
itself a constant.
Deaths observed during
follow up of 622 women
Age-specific
expected deaths
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
P[Alive at x years]
Years after diagnosis of recurrence, x
Figure11.9 Survival curve for 75- year- old woman with distant recurrence of breast cancer. Based on a 10- year follow- up of 622women and the
age- specific expected deaths. We call this an observation- based survival model.
Definition: Observation- based survival model
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
characteristics of the patient.
https://t.me/medicina_free
Survival models: representing uncertainty aboutthelengthoflife 229
One of the convenient features of the exponential survival model is that life expectancy is the reciprocal of the model
parameter (1/λ). This feature attracted considerable interest and became known by the acronym DEALE (Declining
Exponential Approximation of Life Expectancy) when the usefulness of the exponential survival model was recog-
nized by the medical decision analysis community (Beck etal.,1982).
This section will discuss how the exponential survival model can be adjusted to represent the possible outcomes
faced by a patient. However, first some technical details about the exponential survival model will be discussed.
11.4.1 Lifetime probabilities withthe exponential survival model
Figure11.10 shows the lifetime probabilities for an exponential survival model that has a life expectancy of 10 years.
Recalling the earlier discussion of time granularity, the lifetime probabilities shown in this figure are based on a time
granularity of 1 year. That discussion of granularity left unanswered the important question about how to determine
a patient’s expected utility given that probability has been determine for a range of lifetimes rather than a specific
number of years.
For example, consider a patient whose length of life is represented by the exponential survival model in Figure11.10.
According to this survival model, the probability is 0.0577 that the patient’s length of life is between 5 and 6 years.
What is the corresponding utility if the patient’s lifetime falls in this time interval?
Chapter8 described utility models for length of life; however, those models apply to specific lifetimes, not a range of
lifetimes. What specific length of life should be chosen to represent the range of lifetimes from 5 to 6 years? Selecting
5years would undervalue most of the possible lifetimes in this range. Conversely, selecting 6 years would overvalue the
possible lifetimes. So, now the question becomes “what is the corresponding utility if the patient’s lifetime is 5.5years?”.
What happens with an exponential survival model when the patient’s preferences also can be represented by an
exponential utility model? In Chapter9, we saw that with an exponential utility model, the patient’s preferences for
different lifetimes can be calculated by the expression
tility for life of yearst
e
t
1
Definition: Exponential survival model
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 model is 1/λ.
5 years
0.0577
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
01234567891011121314151617181
920
P [Lifetime between t and t +1 years]
Length of life, t (years)
Figure11.10 Lifetime probabilities for exponential survival model when life expectancy is 10 years and time granularity is 1 year.
https://t.me/medicina_free
230 Medical decision making
where γ is the parameter for the exponential utility model and t is the length of life measured in years. One feature of
this utility model is that the parameter γ quantifies the patient’s risk aversion. The stronger the patient’s risk aversion,
the larger the value of γ.
Normally, in order to calculate the expected utility, we would determine the expected utility by determining the
utility for each possible length of life, multiplying that utility by the corresponding probability for that length of life,
and summing the results. This would be a complicated calculation. However, the discussion in Chapter9 also noted
that if the patient’s survival with an outcome can be represented by an exponential survival model with a life expec-
tancy of 1/λ, then
xpected utility
1
This means that with the exponential utility model, the expected utility for an outcome can be determined without
that complicated calculation. Instead, the expected utility can be calculated directly from the parameters for the
survival model and the utility model.
Therefore, approximating the observation-
based survival curve shown in Figure 11.9 by an exponential survival
model would be convenient. This approximation would greatly simplify how we calculate the expected utility for a
patient faced with that survival model. Instead of adding up a lot of term, the expected utility could be calculated using
the simple formula shown earlier.
11.4.2 Fitting anexponential survival model toobservations– first attempt
We have just seen that assuming an exponential survival model represents the uncertainty for the length of life simpli-
fies how we incorporate risk attitudes into a decision analysis. However, is this assumption justified?
We will answer this important question in two stages. First, we will show that directly fitting an exponential
survival model to observations, like those shown in Figure11.9, can lead to misleading results. In other words, the
assumption is not strictly true. Observed survival probabilities often depart significantly from the probabilities
implied by an exponential survival model. Nevertheless, we will start with this direct approach to fitting an exponen-
tial survival model in order to demonstrate how this is done. We also will establish a meaningful metric for quantify-
ing the mismatch.
The second stage will examine how to combine the exponential and actuarial models to compose a survival model
that matches the observation- based survival model. This more complicated model is based on the earlier observation
that patients with a medical condition, like a breast cancer recurrence, actually face two possible causes of death. The
first of these is the medical condition itself. Distant recurrence of breast cancer often is a fatal condition for the patient.
We will see that survival for patients with distant recurrences often does align with an exponential survival model.
Thesecond cause of death are all the other reasons why people die. Ideally, the threat from these other causes of death
should be represented by the age-
adjusted actuarial model as was demonstrated in Figure11.9.
There are several ways to directly fit an exponential survival model to an observation-
based survival model. Perhaps
the simplest approach is to match the life expectancies for the exponential survival model and the observation- based
survival model.
Computing the life expectancy for the observation- based survival model in Figure 11.9 is time consuming but
straightforward. First, the lifetime probabilities are calculated from the survival probabilities. The lifetime probabilities
are multiplied by the corresponding lifetimes and the products are summed to calculate the life expectancy implied by
the survival model. For the survival model in Figure11.9, the result is a calculated life expectancy of 27.2months or
2.27 years.
Computing the life expectancy for an exponential survival model is far easier. Recall that for this survival
model, the life expectancy is one over the hazard rate (λ). Therefore, the exponential survival model fitted to the
observation- based survival model for distant recurrences of breast cancer would have a hazard rate of 1 over
2.72yearsor 0.4407/year.
Figure11.11 compares the resulting exponential survival curve with the original observation- based survival model.
There are similarities between the shape of the two survival curves but also some big differences. How significant are
those differences?
Recall how we answered this type of question in Chapter9 when we compared the survival curves for two lung
cancer treatments (see Subsection9.3.3). Those two treatments resulted in different survival curves for the patient.
Autility function representing the patient’s risk attitudes provided the outcome utilities for the possible lengths of
https://t.me/medicina_free
Survival models: representing uncertainty aboutthelengthoflife 231
life after treatment. Combining those outcome utilities with the corresponding survival probabilities determined an
expected utility for each of the two treatments. The differences between the resulting expected utilities meant one
treatment was preferred to the other for ranges of risk attitudes.
However, the significance of the difference between the two calculated expected utilities was difficult to decide. We
simply knew there was a difference between two abstract numbers determined by complicated calculations. Therefore,
we translated the expected utilities into certainty equivalents. Recall that a certainty equivalent for a gamble is a guar-
anteed outcome that has the same utility as the expected utility for the gamble. When we compared the certainty
equivalents for the two treatments we found that for a patient with a given level of risk aversion (γ=0.15/year),
undergoing surgery was equivalent to a guaranteed 3.26 years whereas undergoing radiotherapy was equivalent to a
guaranteed 2.97 years (see Figure9.9). In other words, switching from surgery to radiotherapy would be equivalent to
adding roughly 3months to a 3- year lifetime for this patient.
Figure11.12 shows the results of using the same approach to compare the two survival models in Figure11.11. As
usual, an exponential utility model is assumed to represent the patient’s risk attitudes. We adjusted that utility model
to represent different risk attitudes by changing the value for a term in the model we call the risk parameter (γ). A risk
parameter value of zero implies risk indifference. Increasing the value of the risk parameter increases the level of risk
aversion represented by the model.
According to Figure11.12, a patient with a risk parameter value of 0.15/year would consider facing the uncertainty
of the observation-
based survival model shown in Figure11.9 to be equivalent to a guaranteed lifetime of 1.93 years.
On the other hand, that same patient would consider living a guaranteed 1.46 years to be equivalent to facing the
uncertainty represented by the exponential survival model shown in Figure11.11. This means that for this level of risk
aversion, there is roughly a half- year difference between the exponential survival model and the observation- based
survival model.
Notice in Figure11.12 that the difference between the two survival models decreases as the level of risk aversion
decreases. When the patient is risk indifferent– in other words, they have a risk parameter value of 0– the two survival
models imply the same certainty equivalents. This result is to be expected. As we discussed in Chapter8, risk indiffer-
ence means the two survival models are equivalent to their respective life equivalents. By design, the two survival
models have the same life expectancies. Therefore, the two survival models would be equivalent to a patient who is
risk indifferent.
As noted at the beginning of this subsection, there is more than one way to directly fit an exponential survival model
to an observation- based survival model. However, the results are roughly the same for these alternative approaches.
The essential conclusion is that a simple exponential survival model does not always provide a good approximation
for the lifetime uncertainties faced by patients.
Observation-based
survival model
Exponential survival model implying same life
expectancy as the observation-based model
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
P [Alive at x years]
Years after diagnosis of recurrence, x
Figure11.11 Exponential survival model (black curve) fitted to the observation- based survival model for women with distant recurrence shown in
Figure11.10 (grey curve) by matching life expectancies.
https://t.me/medicina_free
232 Medical decision making
11.5 Actuarial survival models
The previous section described the exponential survival model, which is one of the two survival models often used to
represent the uncertainty faced by a patient in an outcome. As we have just seen, by itself, the exponential survival
model does not always align well with survival observations. This section describes a second survival model that can
be combined with the exponential survival model to more closely matched survival observations. This second model
is called the actuarial survival model.
The actuarial survival models used in this book are derived from the life tables published by the US Social Security
Administration (SSA). Each year these life tables report the number of deaths occurring in various age groups. The SSA
uses the life tables to determine the current age- specific annual mortality rates for the US population. We will use these
annual mortality rates as the hazard rates for a member of the general population.
Figure11.13 shows the annual mortality rates reported by the SSA in 2017. Notice that the vertical dimension in this
graph shows the logarithm of the mortality rates for various age groups. This is what data scientists call a semiloga-
rithmic graph. Using a semilogarithmic graph allows the dramatic changes that occur in hazard rates over a lifetime to
be displayed on a single page. Starting at approximately one chance in 10 000 for young children, the chance of dying
within a year climbs one thousand times to over one chance in 10 for centurions. Clearly, the survival model represent-
ing these hazard rates departs significantly from the exponential survival model, with its assumption of a constant
hazard rate.
What we mean by “general population” is that the mortality rates shown in Figure11.13 combine the risks from all
of the possible causes of death. When considering the survival of a patient with a potentially fatal disease, we recog-
nize the patient faces a risk of dying from that fatal disease. The patient also faces the risk from all of the other possible
causes of death. We will assume the mortality rates, observed in the entire population, quantify the risk of death from
those other causes.
Definition: Actuarial survival model
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.
Calculated using the observation-
based survival model for distant
breast cancer recurrence
Calculated using an exponential
survival model fitted to
observation-based survival model
0.15/years
1.93 years
1.46 years
0.0
1.0
2.0
3.0
0.0 0.1 0.2
0.3
Certainty equivalent (years)
Risk parameter (/year)
Figure11.12 Comparison of the certainty equivalents for experiencing the observation- based survival model shown in Figure11.10 and the
corresponding exponential survival model shown in Figure11.11.
https://t.me/medicina_free
Survival models: representing uncertainty aboutthelengthoflife 233
11.5.1 Age- andgender- specific actuarial survival models
Another perspective on the actuarial survival model is that it represents the survival of a patient based solely on their
age. Of course, age is seldom the only fact known about a patient. Earlier in this chapter we determined the survival
model for breast cancer patients. Almost all breast cancer patients are women. Therefore, the actuarial model we used
was limited to members of the general population who are women.
Figure11.14 compares the annual mortality rates for men and women. Notice the striking differences between the
annual mortality rates for young men and women. While the annual mortality rates are low for 20- year- olds, the prob-
ability of death during the next year for young men in this age group (0.0011/year) is almost 3 times as high as it is for
young women of a similar age (0.0004/year). This difference between men and women shrinks as we age, but it
continues throughout our lifetimes.
We have been discussing a 75- year- old patient with a distant recurrence of breast cancer. We can use the curve for
women in Figure11.14 as the actuarial survival curve for this patient. Back in the discussion of Kaplan–Meier
survival curves we saw how this was done. The process starts with the patient alive at age 75. Therefore, the survival
0.00001
0.00010
0.00100
0.01000
0.10000
1. 00000
0102030405060708090100 11
0120
Annual mortality rate
Age in years
Figure11.13 Age- specific annual mortality rates reported by the Social Security Administration for 2017.
0.00001
0.00010
0.00100
0.01000
0.10000
1. 00000
0102030405060708090100 11
0120
Annual mortality rate
Age in years
Both
Men
Women
75 years
0.0249
0.0011
0.0004
20 years
Figure11.14 Age- specific annual mortality rates for men and women reported by the Social Security Administration for 2017.
https://t.me/medicina_free
234 Medical decision making
probability at age 75 is 1. From the annual mortality rates (a.k.a., hazard rates) in Figure11.14 we know the probability
is 0.0249 that this woman will die from all causes before reaching 76 years of age. Therefore,
PP[] ]Aliveat[DiebeforeAliveat75 17675
1 0000 100249 0 9751...
Figure11.15 shows the results of repeating this calculation to the end of life for a 75- year- old woman. The result is
the age-
specific actuarial survival curve used back in Figure 11.9 to extend the survival model for our 75- year- old
patient.
Figure11.16 repeats the analysis we have just seen for men and women over a range of ages.
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
0105
P[Alive at age t]
Age in years, t
Figure11.15 Actuarial survival model for 75- year- old woman based on life tables reported by the Social Security Administration for 2017.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1. 0
01020
Initial age = 80 years
70 years
60 years
50 years
40 years
30 years
30 40 50 60
70
P[Alive at t years]
Years in the future, t
Men
Women
Figure11.16 Age- specific actuarial survival models for men and women based on life tables reported by the Social Security Administration for 2017.
Adapted from Social Security Administration for 2017.
https://t.me/medicina_free
Соседние файлы в папке Библиотека им академика М.И. Перельмана