2128
The Use of Expected Value as an Aid to
Decisions Regarding Anticoagulation in
Patients With Atrial Fibrillation
Stephen E. Nadeau, MD
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Background: The method described provides a rational means for determining whether to institute
chronic anticoagulation to prevent stroke in patients with chronic atrial fibrillation under a variety of
clinical circumstances.
Summary of Comment: The concept of expected value is used in conjunction with data from clinical
studies to define the net value of anticoagulation to the patient. A full year of anticoagulation is warranted
in patients with recent stroke or transient ischemic attack thought to be due to cardiogenic embolism who
feel that stroke is a very serious event with nearly as much disvalue as death. If stroke has a lesser degree
of negative value to the patient, or it is uncertain whether the stroke was in a large-vessel distribution, or
it is uncertain whether a large-vessel distribution stroke was due to cardiogenic embolism, 6 months or
less of anticoagulation may be warranted. Indefinite anticoagulation is justifiable in most patients with
chronic atrial fibrillation without a history of stroke or transient ischemic attack but may be contraindicated in certain patients at extremely low risk for embolism and in patients who place a low value on
stroke relative to death and who have a modest increase in risk for fatal hemorrhage.
Conclusions: The method described provides a means readily usable by clinicians to make anticoagulation decisions in patients with chronic atrial fibrillation that will address risk-benefit tradeoffs with
somewhat greater precision than current approaches. (Stroke. 1993;24:2128-2134.)
KEY WoRDs * anticoagulants * atrial fibrillation * embolism
T here are a number of circumstances involving
patients at risk for stroke due to cerebrovascular disease or cardiogenic embolism in which
the potential risks and benefits of chronic anticoagulation are sufficiently close as to render clinical decision
making quite difficult. In part this reflects the fact that
the risks and benefits of anticoagulation have not yet
been completely defined at the population level, an
issue that can only be addressed by further study.
However, in part it reflects uncertainties intrinsic to
individual patients. Dealing with these uncertainties in a
rational, quantitative fashion will be the focus of this
article. Specifically, I will discuss anticoagulation in
patients with stroke or transient ischemic attack and
chronic atrial fibrillation in whom the relation of the
rhythm abnormality to the ischemic event is uncertain
or in whom the nature of the ischemic event itself is
uncertain (ie, microvascular or large-vessel distribution). I will also discuss anticoagulation of patients with
atrial fibrillation in the absence of stroke or transient
ischemic attack. Although a quantitative approach is
used, many uncertainties remain, and the results should
Received May 28, 1993; final revision received August 20, 1993;
accepted August 20, 1993.
From the Geriatric Research, Education, and Clinical Center,
Veterans Administration Medical Center, and the Department of
Neurology, University of Florida College of Medicine, Gainesville,
Fla.
Correspondence to Stephen E. Nadeau, MD, GRECC (182).
VA Medical Center, Gainesville, FL 32608-1197.
be used as a guide to decision making and not applied in
a formulaic fashion.
The general approach to be used makes use of the
concept of expected value, which is equal to the product
of the actual value of the end point and the likelihood of
the end point. Thus, the expected value of a 99:1
prospect at the horse races that pays $100 to win is
(0.01)($100)=$1. The concept of expected value provides a way of arriving at the value of an uncertain end
point in a rational fashion. The actual value of the end
point in the context of this article is to be determined by
the patient (see "Appendix'). The likelihood of the end
point can be ascertained from existing data. Anticoagulation decisions in patients at risk for cardiogenic
embolic stroke have been discussed previously in a
decision analytic framework,' but the method used
failed to include an explicit assessment of the value
system of the patient, something repeatedly shown to be
very important2-4; it also involved cumbersome calculations requiring the use of a decision-making computer
program. The method used in the present article is
intended to be easily usable by clinicians with access
only to the data included herein.
Stroke or Transient Ischemic Attack
Probable Cardiogenic Embolism
In the patient with a recent stroke due to atrial
fibrillation, the 1-year likelihood of stroke recurrence is
approximately 20%.5-7 By the end of 1 year, the stroke
rate asymptotically approaches 5%/y. The expected
Nadeau Expected Value
disvalue for the first year of no treatment at all, assuming the stroke was definitely due to cardiogenic embolism, is 0.20 Vs, where Vs is the value of recurrent stroke
as judged by the patient. Data from four major multicenter prospective randomized studies have indicated
that stroke occurrence in chronic atrial fibrillation can
be reduced by approximately 65% with chronic warfarin
administration.8-1 There are few studies on the value of
chronic anticoagulation in preventing stroke recurrence,5,12 none of a quality that meets present-day
standards. However, these data suggest that until controlled studies are completed, it is reasonable to assume
that anticoagulation is just as effective in preventing
stroke recurrence as stroke occurrence. The expected
1-year positive (prophylactic) value of anticoagulation
in the patient under discussion would then be
(0.65)(0.20)Vs, or 0.13Vs.
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The major risks of anticoagulation are systemic and
intracranial hemorrhage. I will assume that because
stroke is such an important event, the only other event
that bears serious consideration in an anticoagulation
decision is death. That is, I will assume that discomfort
and transient reductions in quality of life related to
nonfatal systemic hemorrhages are, compared with
stroke, relatively unimportant and can reasonably be
ignored in our decision analysis.
Data on risk of fatal hemorrhage in chronically anticoagulated stroke patients are quite limited, but they
suggest the risk is quite high, approximately 4%/y.12'13
Two thirds of fatalities are due to intracranial hemorrhage.12'13 These figures are derived in good part from
prospective randomized studies of chronic anticoagulation of stroke patients conducted 30 or more years ago,
using anticoagulation regimens of varying intensities.
The patients were more likely to have inadequately
controlled blood pressure than today (a known risk factor
for intracerebral hemorrhage), but other reasons for
excessive risk of intracranial hemorrhage are not immediately apparent. Most data suggest that unlike systemic
hemorrhage, which is overwhelmingly associated with
excessive anticoagulation (when it is not due to a potentially hemorrhagic lesion, such as a peptic ulcer), intracranial hemorrhage bears relatively little relation to
degree of anticoagulation13-'5 (but see Reference 16).
Thus, the defining and relatively irreducible risk to be
reckoned with in chronic anticoagulation is that of intracranial hemorrhage. Assuming a fatal hemorrhage rate of
4%/y, the 1-year expected negative value of chronic
anticoagulation attributable to death in a stroke patient
is 0.04VD, where VD is the disvalue of death as judged by
the patient (Table 1). As many as one third of intracranial hemorrhages during anticoagulation are not fatal,14"5 and most of these nonfatal hemorrhages are
associated with lasting residua equivalent to those of
ischemic stroke. Including these residua increases the
expected negative value of anticoagulation by 8%
(0.33 x 0.25) to 27% (0.33 x 0.8), depending on whether
the patient attaches modest (0.25VD) or high (0.8VD)
negative value to stroke, respectively (see below). Because of the uncertainty surrounding the 4%/y fatal
hemorrhage figure for anticoagulated stroke patients,
calculations have also been made assuming a rate of 2%/y
(Table 2).
For the patient with stroke definitely due to cardiogenic embolism, the 1-year expected positive value of
2129
chronic anticoagulation, 0.13VS, is clearly considerably
higher than the 1-year expected negative value of
chronic anticoagulation, 0.043VD to 0.05VD, as long as
the patient feels that stroke is a very serious event with
a negative value close to that of death. However, these
values are close enough that an explicit assessment of
the patient's values is warranted. This can often be done
in a relatively informal manner, but in many instances a
formal standard gamble discussion will be necessary
(see "Appendix"). In my experience with standard
gamble discussions in this context, patients who place a
high value on intact intellectual function and physical
independence will often conclude that stroke has 80%
of the disvalue of death; at the other extreme, patients
may conclude that stroke has as little as 25% of the
disvalue of death. Assuming the 25% figure, the expected positive value (EV+) of anticoagulation would
be (0.13)(0.25)VD=0.033VD, and the expected negative
value (EV-) of anticoagulation would be 0.04VD+ (0.33)
(0.25)(0.04)VD=0.043VD (taking into account the stroke
morbidity attributable to the one third of intracranial
hemorrhages that are not fatal). A full year of anticoagulation would have a negative net value. Assuming
the 80% figure, the expected positive value of anticoagulation would be (0.13)(0.8)VD=0.104VD, the expected
negative value would be 0.O4VD+(0.33)(0.8)(0.04)VD=
O.OSVD, and the net value of a full year of anticoagulation would be positive. If explicit assessment of the
patient's values indicates that stroke has less than 35%
{[0.04+(0.35)(0.33)(0.04)]/0.13} of the disvalue of
death, then the net expected value of anticoagulation
for 1 year becomes negative. In this circumstance, it
would make sense to anticoagulate for only 6 months,
reducing the risk of fatal hemorrhage to approximately
2% (EV- =0.02VD) but covering approximately 75%5 of
the 1-year riskof embolic stroke [EV+ = (0.75)(0.2)(0.65)
(0.3)VD=0.029VD] (Table 1). This assumes of course
that the risk of fatal hemorrhage is evenly distributed
over time; limited data suggest that it is.13
Possible Cardiac or Artery-to-Artery
Thromboembolism
It is often difficult to determine in a stroke patient
with atrial fibrillation whether the embolus came from
the heart or from a site of atheromatous disease in the
neck or the aorta. Carotid duplex or angiographic
evidence of carotid thrombus in situ or, to a lesser
extent, ulceration; a stuttering, waxing and waning,
progressive course17 (but see Reference 18); and multiple events in a single vascular territory, particularly the
posterior circulation, favor artery-to-artery thromboembolism. Evidence of multiple events in different cerebral
arterial distributions; stroke in conjunction with hematuria (presumably reflecting renal embolism19); and
hemorrhagic infarction6 favor cardiogenic embolism.
In the final analysis, however, in an elderly patient
with a single cerebrovascular event and carotid stenosis
without in situ thrombus, it is generally not possible to
determine the origin of the cerebral embolus with
certainty. Instead, this risk must be calculated on the
basis of probabilities. In patients with chronic atrial
fibrillation, the annual risk of stroke attributable to the
atrial fibrillation is 3.5%.2O I will assume that the annual
stroke risk in the distribution of a particular carotid is
45% of this, which is the fraction of total cerebral blood
2130
Stroke Vol 24, No 12 December 1993
TABLE 1. Summary of Expected Values of Anticoagulation for Patients With Atrial Fibrillation
None
1-Year
1-Year
1-Year
1-Year
Probability
Expected Expected
of Fatal
Probability
Positive Negative
of Stroke Hemorrhage
Value
Value
Procedure
Vs
0.20
0.04
0.80VD 0.1 04VD 0.051 V[ Anticoagulate 1 y
0.25VD 0.030VD 0.043VD Anticoagulate 6 mot
0.20*
0.04
0.80VD 0.035VDt 0.051 VD Anticoagulate 6-12 mot
None
0.04
0.25VD 0.01 1VDt 0.043VD
0.80VD 0.052VDt 0.051 VD
Anticoagulate 3-6 mot
Anticoagulate 1 y
0.25VD 0.01 6VDt 0.043VD
0.80VD 0.01 7VDt 0.051 V)
Anticoagulate 3-6 mot
Anticoagulate 3-6 mot
History of
Stroke/TEA
Recent Stroke or TIA
Definite cardiogenic embolic
(P[stroke is cardiogenic] = 1)
Large-vessel distribution
ipsilateral to >80% stenosis
(P[stroke is cardiogenic]=.34)
Large-vessel vs lacunar
stroke; normal carotids
(P[stroke is cardiogenic]=.5)
Large-vessel vs lacunar
stroke ipsilateral to >80%
stenosis
(P[stroke is cardiogenic]=.17)
None
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None; reduced risk for
systemic embolism
>1 Year
Prior
None
0.20
None
0.20*
0.04
None
0.05
0 005
None
0.02
0.005
0.25VD 0.005VDt
0.80VD 0.026VD
0.25VD 0.008VD
0.80VD 0.010VD
0.25VD 0.003VD
0.043VD No anticoagulation
0.0063VD Indefinite anticoagulation
0.0054VD Indefinite anticoagulation
if no excess risk for
hemorrhage
0.0063VD Indefinite anticoagulation
if no excess risk for
hemorrhage
0.0054VD No anticoagulation
0.051 VD No anticoagulation
0.05
Present
0.04
None
0.80VD 0.026VD
TIA indicates transient ischemic attack; Vs, value to patient of stroke; VD, value to patient of death. 1-year expected positive
value=(1-year probability of stroke)(probability that stroke is cardiogenic)(efficacy of warfarin)Vs. 1-year expected negative value=(1year probability of fatal hemorrhage)VD+(probability of nonfatal intracranial hemorrhage)Vs. Assumed efficacy of warfarin=65%;
probability of nonfatal intracranial hemorrhage is assumed to be 0.33 that of fatal hemorrhages of all types.
*Assumes patients with >80% stenosis and atrial fibrillation are not at higher than usual risk for stroke.
tAssumes warfarin does not reduce risk of lacunar stroke or artery-to-artery thromboembolic stroke.
tAssumes risk of fatal hemorrhage is evenly distributed over time.
flow supplied by each carotid. In patients with asymptomatic carotid stenosis of greater than 80%, the annual
risk of stroke without preceding transient ischemic
attack may be as high as 4%.21-26 1 will assume that three
fourths of these strokes are artery-to-artery thromboembolic and ipsilateral to the stenosis.26 In a patient
with both conditions who then experiences a largeartery distribution cerebral infarction in the territory of
the stenotic vessel, the probability that the stroke was
due to cardiogenic embolism is then (3.5)(0.45)/
[(3.5)(0.45)+(0.75)(4)]=0.34; this of course assumes
that there are no other possible sources of embolism, a
degree of oversimplification that is probably tolerable in
the present context.27 In patients with lesser degrees of
ipsilateral carotid disease, the likelihood of cardiogenic
embolism will be higher; it presumably approaches
100% in the patient with normal vessels. With extreme
carotid stenosis, the obstruction might actually prevent
cardiogenic emboli from reaching the brain; no good
data are available, but this consideration leads to the
common-sense conclusion that a cardiac embolus is
probably not the cause of the stroke in these circumstances and that consideration might be given to carotid
endarterectomy.
Assuming a 34% likelihood of cardiogenic embolism,
the 1-year positive expected value of anticoagulation in
the patient with high-grade ipsilateral carotid stenosis
would be (0.34)(0.65)(0.20)VS, or 0.044VS. This assumes
that the 1-year stroke rate in such a patient is not
greater than 20% and that chronic anticoagulation is of
no value in preventing recurrent stroke due to arteryto-artery thromboembolism; neither hypothesis has
been adequately tested. In this case, even if the patient
valued stroke very negatively (eg, Vs=0.8VD), the expected positive value of anticoagulation (EV+=(0.8)
(0.044)VD=0.035VD) would be less than the expected
negative value of anticoagulation, 0.05VD, and a full
year of anticoagulation could not be justified.
Possible Lacunar Infarction
There is a general consensus that lacunar infarction is
almost always due to intrinsic microvascular disease and
is uncommonly embolic in origin.28 Thus, if a patient in
chronic atrial fibrillation experiences a lacunar stroke, it
is safe to say that the stroke was unlikely to be due to
cardiogenic embolism, and the conditions of the clinical
decision are those of the patient with chronic asymptomatic atrial fibrillation (see below). Unfortunately,
situations frequently arise in which it is not possible to
be sure whether the patient has had a lacunar or a
large-vessel event. This is because of the fact that
clinical discrimination between the two types of stroke
is frequently inadequate, particularly when there is a
mild neurological deficit, and the fact that computed
tomography detects only 40% of lacunes and 70% of
large-vessel infarcts29 and magnetic resonance imaging
detects about 80% of lacunes3031 but is probably not
much better than computed tomography at detecting
Nadeau Expected Value
2131
TABLE 2. Analysis of the Effect of Different Estimates of Warfarin Efficacy and Risk in Patients With Prior
Cerebrovascular Events
Probability
That Stroke
Is
Recent Stroke or TIA
Definite cardiogenic embolic
Cardiogenic
1
1-Year
Probability
of Stroke
0.20
1-Year
Probability
of Fatal
Hemorrhage
0.04
Efficacy of
Warfarin
0.65
0.33
0.02
0.65
0.33
Large-vessel distribution
ipsilateral to 280% stenosis
0.34
0.20
0.04
0.65
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0.33
0.02
0.65
0.33
Large-vessel vs lacunar stroke;
normal carotids
0.50
0.20
0.04
0.65
0.33
0.02
0.65
0.33
Chronic atrial fibrillation; old
stroke
0.05
0.02
0.65
vs
0.80VD
0.25VD
0.80VD
0.25VD
0.80VD
0.25VD
0.80VD
0.25VD
0.80VD
0.25VD
0.80VD
0.25VD
0.80VD
0.25VD
0.80VD
0.25VD
0.80VD
0.25VD
0.80VD
0.25VD
0.80VD
1-Year
Expected
Positive
Value
0.104VD
0.O3OVD
0.052VD
0.01 6VD
0.104VD
0.030VD
0.052VD
0.01 6VD
0.035VD
0.011 VD
0.01 8VD
1-Year
Expected
Negative
Value
0.051VD
0.043VD
0.051 VD
0.043VD
0.025VD
0.022VD
0.025VD
0.022VD
0.051 VD
0.035VD
0.011 VD
0.01 8VD
0.005VD
0.043VD
0.051 VD
0.043VD
0.025VD
0.022VD
0.025VD
0.022VD
0.052VD
0.051 VD
0.01 6VD
0.043VD
0.051 VD
0.005VD
0.026VD
0.25VD
0.008VD
0.052VD
0.01 6VD
0.80VD
0.25VD
0.026VD
0.043VD
0.025VD
0.022VD
0.025VD
0.008VD
0.022VD
0.80VD
0.026VD
0.025VD
0.25VD
0.OO8VD
0.025VD
TIA indicates transient ischemic attack; Vs, value to patient of stroke; VD, value to patient of death. 1-year expected positive
value=(1-year probability of stroke)(probability that stroke is cardiogenic)(efficacy of warfarin)Vs. 1-year expected negative value=(1year probability of fatal hemorrhage)VD+(probability of nonfatal intracranial hemorrhage)Vs. Assumed efficacy of warfarin=65%;
probability of nonfatal intracranial hemorrhage is assumed to be 0.33 that of fatal hemorrhages of all types.
large-vessel events.32 In such cases, one might estimate
the probability of a large-vessel event at 50%. The
expected positive value of anticoagulation in a patient in
these circumstances in whom there is no evidence of
atheromatous cerebrovascular disease would then be
(0.5)(0.65)(0.20)Vs=0.065Vs. If the patient attached a
high negative value to stroke (eg, VS=0.8VD), the expected positive value of anticoagulation would be
(0.065)(0.8)VD=0.052VD, the expected negative value
0.05VD, and a full year of anticoagulation would be
justifiable. However, for Vs even slightly less than this,
some curtailment of duration of anticoagulation would
be indicated. I am assuming that the risk of fatal
hemorrhage is the same in patients with lacunar infarction as it is in patients with large-vessel infarction; this
is untested. If there is severe cerebrovascular disease
with greater than 80% ipsilateral carotid stenosis, such
that there is a probability of only 34% that the stroke, if
large-vessel, is due to cardiogenic embolism, then the
expected positive value of anticoagulation becomes
(0.34)(0.065Vs) =0.022Vs, which will always be less than
half the expected negative value of anticoagulation,
0.0433VD to 0.O5VD, indicating that the duration of
anticoagulation should be less than 6 months in all such
patients.
The risks of chronic anticoagulation can often be
substantially mitigated by successful cardioversion,
which allows early discontinuation of anticoagulants but
requires long-term follow-up to detect expeditiously
recurrent atrial fibrillation.
Because of the uncertainty as to whether warfarin is
just as effective in preventing stroke recurrence as it is in
2132
Stroke Vol 24, No 12 December 1993
preventing first cardiogenic embolic stroke (65%), I
have included a sensitivity analysis (Table 2) in which
the 1-year expected positive values are calculated for
the first three conditions in Table 1 assuming warfarin
efficacies of 0.65 and 0.33.
No Recent Embolism
The total annual stroke risk in patients with chronic
atrial fibrillation of greater than 1-year duration and in
patients with atrial fibrillation and a history of stroke
more than 1 year prior is approximately 5%,8-11,20 and
warfarin may reduce this by 65%.8-11 Thus, the expected
positive value of warfarin is (0.65)(0.05)Vs=0.0325VS. If
the disvalue to the patient of stroke is 0.8VD, the
expected positive value of anticoagulation becomes
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(0.0325)(0.8)VD=0.026VD. The annual rate of fatal
hemorrhage in recent randomized studies of anticoagulation in patients with atrial fibrillation was 0.4%.8,9,1
In older studies of chronic anticoagulation of patients
with ischemic heart disease it was 0.6%.13 I will assume
a rate of 0.5% for this analysis. The expected negative
value of warfarin would then be 0.0063VD, taking into
account morbidity from nonfatal intracranial hemorrhage. The net expected value of anticoagulation for 1
year would then be (0.026-0.0063)VD=0.0197VD, providing a strong rationale for chronic anticoagulation. At
the other extreme, patients may consider stroke to have
only 25% of the disvalue of death, yielding an expected
positive value of anticoagulation of (0.25)(0.0325)VD=
0.008VD. This is only 1.48 times the expected negative
value of anticoagulation (0.008/0.0054), and if the risk of
fatal hemorrhage with anticoagulation were suspected
to be much higher than usual, the net expected value of
chronic anticoagulation would become negative. In the
patient with a history of stroke, however remote, the
annual risk of fatal hemorrhage, as noted above, may be
4%, thus contraindicating anticoagulation in all stroke
patients after the first year (but see sensitivity analysis,
Table 2). Further studies are necessary to refine our
ability to predict fatal intracranial hemorrhage in anticoagulated patients.
I have assumed an annual stroke risk of 5% in
patients with chronic atrial fibrillation without a history
of stroke or transient ischemic attack in the preceding
year. Recent studies by the Stroke Prevention in Atrial
Fibrillation Investigators have identified several clinical
and echocardiographic features that predict increased
risk of stroke in patients with atrial fibrillation: a history
of hypertension, previous thromboembolism, recent
congestive heart failure, global left ventricular dysfunction, and left atrial size greater than 2.5 cm/M2 by
M-mode echocardiography.33,34 Annual stroke risk was
1% in patients with none of these risk factors, 6% in
patients with 1 or 2 risk factors, and 18.6% in patients
with 3 or more risk factors. Framingham Study data also
suggest that the risk of stroke during the first year of
chronic atrial fibrillation is 13%.35 Unfortunately, there
are no data on the efficacy of warfarin in curbing these
heightened stroke risks. However, these data do serve
to identify a population of patients with chronic atrial
fibrillation who are at such low risk of stroke that
anticoagulation is not warranted. Other studies have
identified patients younger than age 60 with lone atrial
fibrillation (unassociated with clinical, electrocardiographic, or chest roentgenographic evidence of heart
disease) and, less certainly, patients with paroxysmal
atrial fibrillation as having risks of stroke too low to
warrant anticoagulation.67
Conclusion
Many uncertainties remain regarding risks of stroke
and hemorrhage in the circumstances discussed in this
article. Both the institution and the withholding of
anticoagulation may be associated with major morbidity
or death. Nevertheless, clinicians are obliged to make a
decision, and they strive to make a maximally rational
one. My own bedside standard gamble discussions with
patients suggest that the disvalue of stroke may vary
between approximately 0.25VD and 0.8V, representing
a threefold difference. Because of the closeness of
benefits and risks of anticoagulation under many of the
circumstances I have considered, this threefold difference is frequently sufficiently large to have a major
impact on initiation or duration of anticoagulation
(Table 1). The use of an expected value approach
further permits the explicit handling of multiple sources
of uncertainty in estimating likelihood of an outcome.
Because of the uncertainties in risk estimates, the
method I have discussed cannot be used in a doctrinaire
or formulaic fashion but rather should yield some rough
guidelines for treatment that will address risk-benefit
tradeoffs with somewhat greater precision than current
approaches. The advantages of the method and the
impact of explicit assessment of patient values are likely
to be felt even with somewhat different assessments of
the various risks involved. When the difference between
expected positive and negative values is small, the
balance may be tipped by issues such as patient concerns regarding taking medication or discomfort and
reduction in quality of life related to nonfatal systemic
hemorrhage.
Appendix
Explicit ascertainment of the patient's rating of the disvalue
of stroke relative to death can be done using a standard gamble
technique.36 The following is a model of the dialogue the
physician might have with the patient:
You have experienced a stroke. Based on the tests we have
conducted, we believe that stroke was caused by a fragment of
blood clot that formed in your heart, broke off, and traveled to
your brain. We know that under these circumstances, there is
about a 1 in 5 chance that another clot will travel to your brain
within the next year, possibly causing an even larger stroke. A
medication is available, warfarin, that will reduce this risk of
having another stroke, perhaps by as much as 2/3. However,
warfarin can cause severe bleeding, including bleeding into the
head, which can be fatal. Thus, we need to carefully weigh the
benefits of warfarin against the risks. There is no uniformly
correct decision in a situation like this. Rather, you and I need
to work together to arrive at the correct decision for you. I
know about the medical aspects of this decision. However, this
is a decision involving a chance of having a stroke and a chance
of dying. Therefore, I need to know how you feel about having
a stroke and dying. In particular, I need to know how bad it
would be for you to have another stroke. This is a hard
question for anyone to answer. To help you answer. I am going
to show you a make-believe situation and ask you how you
would respond. This situation will undoubtedly make you feel
uncomfortable, so just do your best. Please keep in mind that
it is make-believe, designed to help me evaluate your feelings.
Pretend you are at a crossroads in life (show patient Figure).
Imagine you have an illness that will certainly cause a stroke
Nadeau Expected Value
% CHANCE OF
PERFECT HEALTH
% CHANCE OF
DEATH THIS WEEK
CERTAIN
STROKE THIS WEEK
Decision tree for standard gamble assessment of patient's valuation of stroke.
2133
to be workable at the bedside. The standard gamble method
used here, which is well accepted in the decision analysis
literature, makes this possible.
Acknowledgment
I am very grateful to John Meuleman for his many helpful
suggestions for this manuscript.
References
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within the next few days (point to the bottom path). However,
there is a radical treatment that could absolutely prevent you
from having a stroke (point to upper fork of top path).
Unfortunately, this treatment carries with it a risk of death
(point to lower fork of top path). What is the minimum chance
of a cure that you would have to have to accept the radical
treatment? I'm going to put some numbers in here and I want
you to tell me when it seems like it's a toss-up -the two paths
seem about the same. (Write in 50 on both forks of upper
path). Here, if you accept the treatment, you have a 50-50
chance of a cure-good health without a stroke-but also a
50-50 chance of dying. Over here (point to lower path), you
will definitely have a stroke. Which path would you choose?
Remember, these numbers do not apply directly to your
situation; however, they will help me to advise you whether to
take warfarin, which does involve accepting some risk of dying
in order to prevent stroke. (Substitute in successively higher or
lower numbers in 10% to 20% increments until the patient
understands the problem and settles on a set of numbers. The
percentages on the two upper branches must always add up to
100. The likelihood of death at which the patient is ambivalent
about the radical treatment constitutes the best estimate of his
valuation of stroke relative to death.)
This approach to estimating the disvalue of stroke makes
some simplifying assumptions. First, it treats stroke as a
permanent state, as opposed to an event followed by some
degree of recovery and return to at least some aspects of
normal life. Second, it treats all strokes as equal, when in fact
the impact of minor and major stroke is obviously quite
different. Third, it leaves to the patient the task of estimating
the long-term personal ramifications of nonfatal events that
occur during the treatment (or nontreatment) period. Other
investigators have used Markov and related models to deal
with these problems in various contexts.1,3738 These models in
essence calculate expected values for each of the remaining
months of the patient's life, entering into the calculation the
probability that the patient will experience a change in state,
eg, from health to stroke, stroke to death, or health to death,
in any particular month. Arbitrary disvalues are attached to
the various states, eg, major stroke and minor stroke, and a
discount factor is used to account for the fact that events that
occur or states that exist at some time in the future have less
value, or disvalue, than when they develop immediately. The
net result is an estimate of the value of a particular strategy
measured in quality-adjusted life expectancy. The major
source of appeal of this approach is that the product, qualityadjusted life years, is easily comprehended by the physician
and easily used as a "common currency" of decision making. I
have not used this approach because obtaining the patient's
estimates of the relative disvalue of major stroke, minor stroke,
and expected states of recovery from these events, not just in
the next year but for all the remaining years of his or her life,
is a formidably complex and probably impossible task. Markov
model advocates tacitly admit this by simply doing away with
the whole sticky issue of assessing patient values, notwithstanding that many studies have demonstrated substantial
variability in patient preferences and have therefore emphasized the need to assess these preferences on an individual
basis.2-4 Even in the absence of explicit value assessment, these
other models are often extremely complex.1 In this article I
have sought above all to achieve an approach that both
explicitly incorporates patient values and is sufficiently simple
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The use of expected value as an aid to decisions regarding anticoagulation in patients with
atrial fibrillation.
S E Nadeau
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Stroke. 1993;24:2128-2134
doi: 10.1161/01.STR.24.12.2128
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