The Healthy-years Equivalents: How to Measure Them Using the Standard Gamble
Approach ABRAHAM MEHREZ, PhD, AMIRAM
GAFNI, PhD
The healthy-years equivalent (HYE) is a measure of outcome of health care programs that combines two outcomes of interest: quality of life and quantity of life. Unlike QALYs (qualityadjusted life years) HYEs fully represent patients’ (or other individuals’) preferences, as a result of the way they are calculated from each individual’s utility function. The authors suggest an algorithm to measure the HYE of any given lifetime health profile. The algorithm is based on the classic standard gamble method to measure individuals’ preferences under uncertainty, and consists of two lottery questions. Algorithms for the general case (any given lifetime health profile) and a simpler case—the chronic health state case—are provided, as is a modification of the algorithm aimed at shortening the length of the interview when an individual is faced with many possible lifetime health profiles. In addition, two questions are addressed. The first is theoretical and deals with the existence of HYE: do all lifetime health profiles, which are preferred to death, have hypothetical equivalents that can be measured in healthy years? The second is empirical and deals with the reproducibility of the measures obtained by using the measurement technique suggested. This is needed because the technique employs a combination of lottery questions that had not previously been used together. The results of an experiment performed to test the reproducibility of the measures were satisfactory. (Med Decis Making 1991;11:140-146)
Decisions about medical treatments and health programs involve both technical and value judgments. An important one, for example, is evaluating the tradeoff between quality of life and length of life. In recent years the concept of utility has been introduced into medical decision making to help estimate the preferences that individuals attach to the consequences of various courses of action. The most commonly used measure of outcome in such cases is the gain in quality-adjusted life years (QALYs). However, QALYs, a healthstatus index for which, at best, utility-based measures are used to obtain the weights, only partially incor-
tion. These conditions rance and Feeny~:
are
summarized
nicely by Tor-
quality and quantity must be mutually utility independent (preference for gambles on the one attribute are independent of the other attribute), the trade-off of quantity must exhibit the constant proportional trade-off property (the proportion of remaining life that one would trade-off for a specific quality improvement is independent of the amount of remaining life), and the single-attribute utility function for additional healthy life-years must be linear with time (for a fixed quality level one’s utilities are directly proportional to longevity, a property also referred to as risk neutrality with respect to
The two attributes of
porate patients’ preferences. The fact that a utility-weighted index (QALY) is not utility function has been acknowledged by Pliskin et a1.1 and Weinstein et al.’ They show that only under restrictive conditions would a QALY be a utility func-
time)
(p.
569).
a
These are veiy strong assumptions. Loomes and McKenzie4 reviewed the literature to see whether the above-mentioned assumptions were supported by empirical evidence. They conclude that, based on available empirical evidence, none of these assumptions holds. They summarize the limitations of various alternative QALY measures in two principal contexts:
Received October 17, 1989, from the Department of Industrial Engineering and Management, The Faculty of Engineering Sciences, Ben-Gurion University of the Negev, Beer Sheva, Israel (AM), and the Centre for Health Economics and Policy Analysis, Department of Clinical Epidemiology and Biostatistics, Faculty of Health Sciences, McMaster University, Hamilton, Ontario, Canada. Revision accepted for publication September 26, 1990. Supported by a grant from the Merck Company Foundation. Address correspondence and reprint requests to Dr. Gafni: De-
situations in which choices have to be made between different possible forms of treatment for the same individual and situations in which choices must be made between alternative ways of allocating limited social resources among diverse health care activities, serving different groups of people to different degrees. Notice that the assumptions identified by Pliskin et al.’ and Weinstein et allwhich are needed to equate
partment of Clinical Epidemiology and Biostatistics, Health Sciences Centre, McMaster
University,
1200 Main Street
West, Hamilton, On-
tario, Canada L8N 3Z5. 140
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141
QALYs with utility, are necessary and sufficient only when we deal with the specific circumstance of a chronic health state, i.e., the case of an individual who will live in a given health state that is less then full health for the rest of his or her life. As can be inferred from Mehrez and Gafnï,5more assumptions are needed to equate QALYs to utility for the more realistic situation of the lifetime health profile, i.e., that of an individual who will be in more than one health state during his or her remaining life. An example of such an assumption is that the utility function of the individual over his or her lifetime health profile should be of an additive form. Ignoring a patient’s preferences in the process of decision making can result in choosing the least preferred service, at least from the patient’s point of view.6-9 Thus, it is important to develop measures of outcome that fully (rather than partially) reflect patients’ preferences. Mehrez and GafnP have suggested a new utility measure of outcome, the healthy years equivalent (HYE). Like QALYs, the HYE combines both outcomes (quality of life and quantity of life) and preserves the intuitively appealing meaning. Unlike QALYs, HYEs fully represent patients’ (or other individuals’) preferences, stemming from the way they are calculated from each individual’s utility function. In other words, unlike the situation with QALYs, assumptions are not needed to equate the HYE with util-
ity. The methods of measurement suggested by Mehrez and GafnP could raise questions of the feasibility of measuring HYEs in some cases because they call for the exposure of individuals to much longer and more complex interviews. The feasibility question stems from two sources: the cost of such study (financial feasibility) and individuals’ willingness to participate and complete the interview and the reliability of the answers (performance feasibility). As mentioned by Mehrez and Gafnis the performance feasibility aspect of an existing utility measurement technique (the standard gamble method) is very satisfactory; thus, it is of lesser concern. (See also a recent four-part review by Froberg and Kane.l°) The area of major concern is financial feasibility, i.e., it would be useful to simplify the algorithm to shorten the necessary interviewing time. We have devised a simple algorithm to measure the healthy-years equivalent (HYE) of a given lifetime health profile. The algorithm which is based on the standard gamble method-the classic technique to measure individuals’ preferences under uncertainty-consists of two lottery questions. We describe the algorithm for the case of a chronic health state. Then we describe the algorithm for the general situation (any given lifetime health profile). A modification of the algorithm for the circumstance in which an individual is faced with many possible lifetime health profiles is also presented. The purpose of the modification is to reduce
the number of questions required and thus shorten the length of the interview. We then deal with the question of the existence of HYE, and describe the results of an experiment to test the reproducibility of the measures obtained by using the instrument (algorithm) suggested, because it employs a combination of lottery questions that had not previously been used
together.
Measuring HYEs: The Case of a Chronic Health State The simplest situation is that in which an individual has a particular number of remaining life years in a given constant health status followed by death. This application of the HYE is used, for example, by Pliskin et al.1 for patients with coronary artery disease and by Churchill et a1.11-13 for patients with end-stage renal disease. For the convenience of the reader, we repeat the definition of HYE’: Let Q and T denote two attributes of the outcome of concern (Q health state of the individual, T life years). Let Q represent the state of full health and Q death. T is defined in the following interval [0, T], where T is the maximal possible survival. Let U(Q,T) be a von Neumann-Morgenstern (vNM) utility function that describes the utility of being in a given health state Q, starting now, for a period of T years, followed by death, as viewed now by an individual. Assume that the individual is currently in health state Q and has T more years to live. Let H be years in full health and H* be the healthy-years equivalent of (Q,T). H* is defined as follows: =
Find H* such that
=
U(Q,H*) = U(Q,T)
(1)
The solution to equation 1 will hypothetical combination of H* years in full health (Q) that is equivalent, in terms of the individual utility, to living T years in health state Q. The algorithm suggested to measure H* is based on the standard gamble technique. This technique is the classic method of measuring cardinal preferences. In brief, the standard gamble method is a paired comparison in which the individual must choose between two alternatives. A detailed description of the different methods used is provided by Farquhar.14 For the application of the standard gamble technique to health care problems in providing weights for calculation of
yield a
QALYs
see
Torrance.ls
an individual who is currently in health and has T more years to live, e.g., Q Q being in a dialysis unit, T having ten more years to live. In this example ten years is assumed to be the maximum possible survival; thus, T T. However, in other T be T. Our task is to smaller than examples might find H* such that U(Q,H*) The U(Q,T). following twoH*. is to measure stage procedure suggested
Assume
state
=
=
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=
=
142
MEASURING THE UTILITY VALUE OF
U(Q,T)
The
following two-stage procedure
measure
Lottery questions are asked to determine the indifference level of P between:
is
suggested
to
H*.
MEASURING
U(QT)
Lottery questions are asked to determine the indifference level of P between:
Probability P is varied until the individual is indifferent between the two alternatives. Denote P* as the probability at the indifference point. At this point, assuming 1 and U(f~,0) = 0, the utility value of being U(Q,T) in the health state under consideration (Q) for T years is equal to P*. In other words, U(Q,T) P*. =
=
where
Q.~. = 4i) 1; -Qi
maximal survival. Probability P is varied until the individual is indifferent between the two alternatives. Denote P* as the probability at the indifferent point. At this point, assuming U(Qi) 1 and U(Qf) = 0, it implies that U(QT) P*. =
fgl, ’ ’ ’ gf]; T
=
possible
=
MEASURING
H*
=
Lottery questions are asked to determine the indifference level of H (i.e., H*) between:
MEASURING
H*
Lottery questions to determine the indifference level of H (i.e., H*) between: The number of years in full health (H) is varied until the individual is indifferent between the two alternatives. Denote H* as the number of healthy years at the indifference point. H* is the number of HYEs that is equivalent to living T years in health state Q.
Measuring HYEs: The General Case
is
The algorithm suggested can be extended to deal with the more general case described by Mehrez and Gafni.’ For the convenience of the reader, we provide the definition for HYE for the general case. A lifetime health profile of an individual can be described as a vector Q = [ qil where q, is the i th element of the vector. Let qi be the health state of the individual at the ith period (measured in years). For the sake of simplicity of presentation and without loss of generality, we assume that all periods are equal in terms of their time intervals. Denote q as perfect health during that period and g as death. Assume an individual who has T more years to live and faces the following lifetime health profile: QT = f ql, ’ ’ ’, qT]. Let U(QT) represent a vNM utility function over the individual’s lifetime health profile, which describes the utility as viewed now by the individual (i.e., the present utility). Let H be years in full health and H* be the healthy-years equivalent of QT. H* is defined as follows: Find H* such that
The number of period in full health (H) is varied until the individual is indifferent between the two alternatives. Denote H* as the number of healthy years at the indifference point. H* is the number of HYEs that
equivalent
where:
Q~.
Measuring HYEs: The Case of Many Possible Lifetime Health PmMes we describe a modification of the for the individual who is faced with many procedure health lifetime profiles (e.g., the case of a compossible The of the suggested moddecision tree). purpose plex ification is to reduce the number of questions needed in the interview and thus shorten the length of the interview. For simplicity and without loss of generality, we illustrate the modified procedure for the case of chronic health states. As a result we revert to the notations and definitions from the previous section dealing with the chronic health state case.
In this section
Assume an individual who is faced with many possible scenarios of lifetime health profiles. Denote (Q,/T,) as the ith scenario. Denote H*i as the healthy-years equivalent of (Q,,T~ ). The following three-stage proce-
dure is
U(QH*) = U(QT)
to
suggested
to measure
Hi.
ASSESS THE INDIVIDUAL’S UTILITY FOR LENGTH OF LIFE IN GOOD HEALTH
q QH* - [q;J such that q; = g otherwise
for = 1, ’ ’ ’ H*
i
In this
(2) in
step
step, lottery questions of the type described
1 of
previous
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sections
are
utilized to
plot
the
143
for length of healthy life. Depending on the maximal possible survival period (T), one can decide how many points to measure. Assume for the sake of simplicity that we decide to divide this period into four equal intervals. Because we define 0 and U(Q,T) 1 we have to measure only U(Q,O) the utility values for the following points in our example : (Q, 0.25 T), (Q, 0.5 T), (Q, 0.75 T). Measuring the utility values of these points is done by asking lottery questions as follows [For example for (Q, 0.5 T)]:
years in full health (Q) that is equivalent, in terms of the individual’s utility, to living T years in health state Q? The answer is-not necessarily. Depending on the individual’s specific utility function, it is easy to see that H* does not always exist (see the appendix for an
Find P such that the individual is indifferent between:
lottery question
individual’s
utility
curve
=
=
example). Consequently, can we determine, using a simple procedure, the set of all potential lifetime health profiles for which H* does not exist? For the two-dimensional case (chronic health state), one lottery question can help determine whether this set is not nil. The is:
Find the indifference level of P between
[(1 as the probability at the indifference point. P*. Then U(Q, 0.5 T) After measuring the utility values of the other survival periods we can plot the individual’s utility curve for length of healthy life.
f[p,(Q,T)I;
P),(g,0)]~ vs (Q,0)
Denote P*
=
MEASURING
U(O,,Ti)
Lottery questions are asked ference level of P between:
Denote
P*i as the probability U(Q,T¡) = P*,.
to
determine the indif-
at the
indifference
point.
H*¡
Having measured U(C~T,) for all i and having plotted the individual’s utility curve for length of healthy life, we can now calculate H*, for each i. By definition, H*, is such that U(Q,H*,) = U(Q,,T,). Since we know the value ofU(Q,/r,) from step 2, we can read (or calculate using simple extrapolation methods) the value of H*, directly from the individual’s utility curve for length of healthy life. The procedure described in this section is shorter (in terms of the number of questions presented during the interview) than the procedure described previously and thus affects the length of the interview. However, it is important to mention that it is less accurate. How much less accurate this modified procedure is compared with the original is yet to be resolved.
sure
event
(Q,0) and the bad seen
to
Reproducibility of Measures Obtained fpom the Instrument Three attributes are essential for any instrument to be useful as a measure of outcome: validity, reproducibility, and responsiveness. These have already been established for existing utility measures (see Mehrez and Gafnis for a summary of the literature). However, because the algorithm to measure HYE uses a combination of lottery questions, each of which had been used separately in the past in different studies, but which had not yet been used together, we performed an experiment to test the reproducibility of the new measure.
The Existence of HYE Does H*
outcome of the be meaningful comlottery (g,O) might of the outcome with the lottery (0,T). Thus, good pared individuals might have difficulty in defining the indifference level of P. The above-mentioned procedure is suggested to save time and shorten the interviews. However, if it causes more harm than good (interviewees have difficulty understanding the question) it is not recommended for use. If H* does not exist for some potential lifetime health profile, this condition is likely to be detected during the interviews using the procedures described in previous sections. tween the
not be
Notice that
CALCULATING
If such P (denoted as P*) exists, it is easy to show that H* does not exist for all potential lifetime health profiles (Q,,T,) for which 0 < U(Q,,T,) < P*. In practice, this might not be a simple question to explain to interviewees, as has already been noted,s because it involves distinguishing between two possible scenarios (g,0) and (Q,0). Also, the trade-off be-
exist? Can
always QH’ (defined above) be found for all possible lifetime health profiles (QT) such that U(QT) = U(QH.)? Returning to the example of a chronic health state, the question becomes: for all potential lifetime health profiles preferred to death (Q,T), can we always find a hypothetical combination of H*
A measure is reproducible if it yields the same results when repeated in stable subjects. Reproducibility is best measured by repeated administration of the instrument to subjects whose status has not changed. An experiment was conducted with 32 graduate students (ages 35-45 years) in an advanced course in decision sciences in the Department of Industrial En-
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144
Table 1 . Utility
Scores (P*) and Healthy-years Equivalent Values (H*) Assigned by 32 Subjects to Being on Dialysis for Ten Years Followed by Death, in Two Interviews Four Weeks Apart
values of P* (and the same for H*) for the two interviews. More precisely, the values (i.e., P* and H*) measured at the second round of interviews were used as the dependent variable. This analysis showed that the Pearson correlations for P* and H* were 0.9553 and 0.7758, respectively. The t-tests for the above-mentioned simple linear regressions were statistically significant (a 0.05). We than tested the null hypotheses that the variances of P* and H* for the two interviews were equal, using the F test for normal distribution. The results are presented in table 2. For both P* and H* the null hypothesis was not rejected (significance level 0.05). Stemming from these results, we tested the hypothesis that the difference between the means of P* and H* for the two interviews was equal to zero. This was examined by using a t-test under the following assumptions : 1) the observations are statistically independent ; 2) for each observation the repeated measurements might be statistically dependent; 3) the samples are taken from two normally distributed populations ; 4) there are unknown equal variances. The mean and variances are computed for each difference between the two repeated measurements. The results are presented in table 3. For both P* and H* the null hypothesis was not rejected (significance level 0.05). Thus, the results of the two interviews seem to be similar. In summary, the statistical analysis demonstrated in two different ways (by Pearson correlation coefficients and t-test) that at the individual and population levels the results of the interviews are similar up to the level of random error. An alternative approach could =
gineering and Management at Ben Gurion University of the Negev, Beer Sheva, Israel. The purpose of the experiment was to measure individuals’ preferences
(expressed in HYEs) with respect to being in the following health-state scenario: being on hospital dialysis for the next ten years followed by death. The description of the health scenario was taken from Sackett and Torrance.16 The scenario is a brief description of the physical, social, and emotional characteristics and
limitations characteristic of this health state. This scenario was used by Sackett and Torrance to measure the general public’s preferences with regard to this health state. The instrument, which is based on the algorithm described earlier for the case of a chronic health state, was administered twice to the same subjects. The interviews were separated by a four-week period. The interviews were conducted by two skilled interviewers who had had a two-hour training session prior to the interviews, in which they had been given the instrument. The instrument components are the health state description, the interview schedule, response recording forms, visual aids, and an interviewer manual. Such highly standardized instrument components are needed to conduct measurements in a systematic manner and reduce variability and bias. For the first round of interviews, the students were divided into two groups. An interviewer was assigned to each group. For the second round of interviews, the interviewers switched groups. This was done to blind the interviewers to the subjects’ previous responses. The average interview times were 7 minutes (range: 5-12 min) for the first round of interviews and 5 min (range: 3-9 min) for the second round of interviews. The means and standard deviations of the utility scores (P*) and the healthy-years equivalent values (H*) for the two interviews are presented in table 1. To examine the reproducibility of the measure suggested, we first analyzed the distribution of responses. The distributions of the values of P* and H* fit a normal distribution (according to a chi-square goodness-of-fit test). Second, the Pearson correlation was computed via a simple linear regression analysis between the
Table 2 o
Test Results of Equality of Variances of Normal Distributions for the Two Interviews
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145
Table 3 o
Test Results of Differences between Means for the Two Interviews
experiment were more sophisticated then the average patient. Thus, to learn more about participation and interview time, we would like to repeat the experiment with a more representative population.
might be tempting to use the time trade-off (TTO) technique to determine the hypothetical number of years in full health (HYE) that is equivalent to an individual’s possible lifetime health profile. In other words, time trade-off judgments might yield the needed equivalences that are currently determined in our algorithm by using two different types of lottery questions. The advantage of the time trade-off technique is that it would reduce the number of judgment questions needed and eliminate the demand on patients’ understanding of probabilities. As shown by Mehrez and Gafnp7 the TTO technique, although offered as a substitute to the standard gamble (SG) technique, was not related in a general way to any existing behavioral theory. Mehrez and Gafnil7 suggest such a link. They show that in the context of value function theory, the TTO technique makes posIt
be to
use a
analysis
cross-sectional, time-series, multivariate
under the normal distribution
assumption,
which, based on statistical theory, should provide similar results.
Diseussion We have described
a
relatively simple algorithm
to
the healthy-years equivalent of a given lifetime health profile. The algorithm is based on the standard gamble method. Lottery questions, which are widely used in health care to measure individuals’ preferences, were found to be valid and responsive. 10,15 The algorithm consists of two lottery questions that had not previously been asked together, so we performed an experiment to examine reproducibility. The results of this experiment were very satisfactory. The advantage of the HYE measure compared with the QALY measure is that it stems directly from the individual utility function and thus fully reflects the individual’s preferences. Mehrez and Gafnis raised the questions of financial and performance feasibility of the measure. The suggested algorithm offers answers to these questions. First, it consists of two lottery questions, compared with the one lottery question needed to calculate weights for the QALYs measure. This could increase the interviewing time, which might not be a burden where we compare only a few lifetime health profiles. We describe a modification of the algorithm for use when the analysis involves a comparison of many lifetime health profiles. Second, the measure was found to be responsive. Third, participation and the rate of completion of the interview were good, and the average interview time was reasonable. We acknowledge that the students used as subjects in our measure
sible identification of different points on an individual’s indifference curve in his or her evaluation space. Because the TTO technique stems from value function theory (i.e., measurement of preferences under certainty) while the SG technique stems from utility theory (i.e., measurement of preferences under uncertainty), the results generated by the two methods should not always be identical .17 This is supported by empirical evidence. Because medical interventions occur only in a world of uncertainty,’ the SG is the appro-
priate technique. We make a theoretical claim with regard to the existence of H* in this report. We show that it is not inevitable that for all possible lifetime health profiles one can always find a hypothetical combination of years in perfect health that is equivalent in terms of
the individual utility to living in those lifetime health profiles. This means that there might be lifetime health profiles that are better than immediate death but worse (from the individual’s perspective) than living the shortest possible period of time in full health. Whether such a situation exists is an empirical question that should be further explored. We have provided a lottery question that can help determine whether a given lifetime health profile falls into this category or not. It is interesting that the QALY measure assumes that this category does not exist: each lifetime health profile that is preferred to death has an &dquo;equivalent&dquo; in healthy years.
Finally, it is important to ask whether the use of HYEs instead of QALYs can affect the result of the analysis. Hypothetical examples’ illustrate that in spite of individuals’ statements that A to B (which
are
they prefer
supported by
their
alternative
answers
to SG
questions), QALY calculations indicate that they prefer alternative B
to A.
Also, in the
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context of economic
146
evaluation, the choice of QALY
as a measure of outthe use of ionic vs nonstudy comparing ionic contrast media was shown to predetermine the outcome of the anlaysis due to limitations in its ability
come
to
in
reveal
11.
patients’ preferences.&dquo;
examples mentioned illustrate the importance of using an outcome measure that coincides with individuals’ preferences. Yet more empirical work on the of this outcome measure is needed. To the best of our knowledge, the algorithm suggested in this paper is the only one that stems directly from vNM individuals’ utility functions. Thus, for all individuals who follow vNM axioms, it provides a measure of outcome that fully reflects their preferences. The extra lottery question needed in this algorithm (beyond the one already needed to measure the weights for QALY calculations) is the cost of having a measure with a solid theoretical foundation rooted in the vNM theory of expected utility.
continuous Med Decis
ambulatory peritoneal dialysis to Making. 1984;4:389-500. Churchill DN, Morgan J, Torrance GW. Quality of life in endstage renal disease. Peritoneal Dialysis Bull. 1986;4:20-3. Churchill DN, Torrance GW, Taylor DW, et al. Measurement of quality of life in end-stage renal disease: the time trade-off approach. Clin Invest Med. 1987;10:14-20. Farquhar PH. Utility assessment methods. Manage Sci. 1984;
hospital dialysis.
12.
13.
The
use
Churchill DN, Lemon BC, Torrance GW. Cost-effectiveness anal-
ysis comparing
a
14.
30:1283-300.
Utility approach to measuring health related qualof life. J Chronic Dis. 1987;40:593-600. 16. Sackett DL, Torrance GW. The utility of different health states as perceived by the general public. J Chronic Dis. 1978; 31:69715. Torrance GW.
ity
704.
17. Mehrez A, Gafni A.
Evaluating health related quality of life: an indifference curve interpretation for the time trade-off technique. Soc Sci Med. 1990;31:1281-3. 18. Gafni A, Zylak CJ. Ionic vs non-ionic contrast media: a burden or bargain. Can Med Assoc J. 1990;143:475-8.
APPENDIX The authors are grateful for helpful comments and suggestions on earlier version of this paper from Stephen Birch, David Feeny, and two anonymous referees. All errors and wrongheaded interpretations remain the responsibility of the authors.
The formulation of Pliskin et al.1 multiattribute utility function:
References
where
an
Pliskin JS, Shepard DS, Weinstein MC. Utility functions for life years and health status. Operations Res. 1980;28:206-24. 2. Weinstein MC, Fineberg HC, Elstein AS, et al. Clinical decision 1.
analysis. Philadelphia: W.
B.
Saunders, 1980.
3. Torrance GW, Feeny D. Utilities and quality-adjusted life years. Int J Technol Assess Health Care. 1989;5:559-78.
G, McKenzie L. The use of QALYs in health care decision making. Soc Sci Med. 1989;28:299-308. 5. Mehrez A, Gafni A. Quality adjusted life years, utility theory, and 4. Loomes
Med Decis Making. 1989;9:142-9. McNeil BJ, Weichselbaum R, Pauker SG. Fallacy of five year survival in living cancer. N Engl J Med. 1978;229:1397-401. McNeil BJ, Weichselbaum R. Pauker SG. Speech and survival: tradeoffs between quality and quantitiy of life in laryngeal cancer. N Engl J Med. 1981;305:982-7. Ben-Zion U, Gafni A. Evaluation of public investment in health care: is the risk irrelevant? J Health Econ. 1983;2:161-5. Mehrez A, Gafni A. The optimal treatment strategy—a patient perspective. Manage Sci. 1987;33:1602-12. Froberg DG, Kane RL. Methodology for measuring health-state preferences. J Clin Epidemiol. 1989;4:345-54 (I. Measurement strategy); 459-71 (II. Scaling methods); 585-92 (III. Population and context effects); 675-85 (IV. Progress and research agenda).
healthy years equivalents.
6.
7.
8. 9. 10.
UilT) = utility function U~(Q) = utility function a,b =
implies
the
following
for remaining life years for health states
constants
We also employ the following assumptions and definitions : 1) Q is the chronic health state preferred to death death and Q where Q healthy; 2) T is defined in the following interval [0,T] where T is the maximal possible sur=
vival ; 3) UQ(Q) b
=
0.3. Thus
=
UQ(Q) equation 1
=
0;
1; Ui10) becomes
=
=
0; UT(T) = 1;
equation
a
=
0.4;
2:
Assume an individual who is faced with the following potential lifetime health profile (Q,T). Using the standard gamble approach we measure U(Q,T) and find it to be 0.30. We can now calculate H by solving equation 3:
It follows that because
Downloaded from mdm.sagepub.com at UNIV OF CALIFORNIA SANTA CRUZ on March 19, 2015
U, (H)
Approach ABRAHAM MEHREZ, PhD, AMIRAM
GAFNI, PhD
The healthy-years equivalent (HYE) is a measure of outcome of health care programs that combines two outcomes of interest: quality of life and quantity of life. Unlike QALYs (qualityadjusted life years) HYEs fully represent patients’ (or other individuals’) preferences, as a result of the way they are calculated from each individual’s utility function. The authors suggest an algorithm to measure the HYE of any given lifetime health profile. The algorithm is based on the classic standard gamble method to measure individuals’ preferences under uncertainty, and consists of two lottery questions. Algorithms for the general case (any given lifetime health profile) and a simpler case—the chronic health state case—are provided, as is a modification of the algorithm aimed at shortening the length of the interview when an individual is faced with many possible lifetime health profiles. In addition, two questions are addressed. The first is theoretical and deals with the existence of HYE: do all lifetime health profiles, which are preferred to death, have hypothetical equivalents that can be measured in healthy years? The second is empirical and deals with the reproducibility of the measures obtained by using the measurement technique suggested. This is needed because the technique employs a combination of lottery questions that had not previously been used together. The results of an experiment performed to test the reproducibility of the measures were satisfactory. (Med Decis Making 1991;11:140-146)
Decisions about medical treatments and health programs involve both technical and value judgments. An important one, for example, is evaluating the tradeoff between quality of life and length of life. In recent years the concept of utility has been introduced into medical decision making to help estimate the preferences that individuals attach to the consequences of various courses of action. The most commonly used measure of outcome in such cases is the gain in quality-adjusted life years (QALYs). However, QALYs, a healthstatus index for which, at best, utility-based measures are used to obtain the weights, only partially incor-
tion. These conditions rance and Feeny~:
are
summarized
nicely by Tor-
quality and quantity must be mutually utility independent (preference for gambles on the one attribute are independent of the other attribute), the trade-off of quantity must exhibit the constant proportional trade-off property (the proportion of remaining life that one would trade-off for a specific quality improvement is independent of the amount of remaining life), and the single-attribute utility function for additional healthy life-years must be linear with time (for a fixed quality level one’s utilities are directly proportional to longevity, a property also referred to as risk neutrality with respect to
The two attributes of
porate patients’ preferences. The fact that a utility-weighted index (QALY) is not utility function has been acknowledged by Pliskin et a1.1 and Weinstein et al.’ They show that only under restrictive conditions would a QALY be a utility func-
time)
(p.
569).
a
These are veiy strong assumptions. Loomes and McKenzie4 reviewed the literature to see whether the above-mentioned assumptions were supported by empirical evidence. They conclude that, based on available empirical evidence, none of these assumptions holds. They summarize the limitations of various alternative QALY measures in two principal contexts:
Received October 17, 1989, from the Department of Industrial Engineering and Management, The Faculty of Engineering Sciences, Ben-Gurion University of the Negev, Beer Sheva, Israel (AM), and the Centre for Health Economics and Policy Analysis, Department of Clinical Epidemiology and Biostatistics, Faculty of Health Sciences, McMaster University, Hamilton, Ontario, Canada. Revision accepted for publication September 26, 1990. Supported by a grant from the Merck Company Foundation. Address correspondence and reprint requests to Dr. Gafni: De-
situations in which choices have to be made between different possible forms of treatment for the same individual and situations in which choices must be made between alternative ways of allocating limited social resources among diverse health care activities, serving different groups of people to different degrees. Notice that the assumptions identified by Pliskin et al.’ and Weinstein et allwhich are needed to equate
partment of Clinical Epidemiology and Biostatistics, Health Sciences Centre, McMaster
University,
1200 Main Street
West, Hamilton, On-
tario, Canada L8N 3Z5. 140
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141
QALYs with utility, are necessary and sufficient only when we deal with the specific circumstance of a chronic health state, i.e., the case of an individual who will live in a given health state that is less then full health for the rest of his or her life. As can be inferred from Mehrez and Gafnï,5more assumptions are needed to equate QALYs to utility for the more realistic situation of the lifetime health profile, i.e., that of an individual who will be in more than one health state during his or her remaining life. An example of such an assumption is that the utility function of the individual over his or her lifetime health profile should be of an additive form. Ignoring a patient’s preferences in the process of decision making can result in choosing the least preferred service, at least from the patient’s point of view.6-9 Thus, it is important to develop measures of outcome that fully (rather than partially) reflect patients’ preferences. Mehrez and GafnP have suggested a new utility measure of outcome, the healthy years equivalent (HYE). Like QALYs, the HYE combines both outcomes (quality of life and quantity of life) and preserves the intuitively appealing meaning. Unlike QALYs, HYEs fully represent patients’ (or other individuals’) preferences, stemming from the way they are calculated from each individual’s utility function. In other words, unlike the situation with QALYs, assumptions are not needed to equate the HYE with util-
ity. The methods of measurement suggested by Mehrez and GafnP could raise questions of the feasibility of measuring HYEs in some cases because they call for the exposure of individuals to much longer and more complex interviews. The feasibility question stems from two sources: the cost of such study (financial feasibility) and individuals’ willingness to participate and complete the interview and the reliability of the answers (performance feasibility). As mentioned by Mehrez and Gafnis the performance feasibility aspect of an existing utility measurement technique (the standard gamble method) is very satisfactory; thus, it is of lesser concern. (See also a recent four-part review by Froberg and Kane.l°) The area of major concern is financial feasibility, i.e., it would be useful to simplify the algorithm to shorten the necessary interviewing time. We have devised a simple algorithm to measure the healthy-years equivalent (HYE) of a given lifetime health profile. The algorithm which is based on the standard gamble method-the classic technique to measure individuals’ preferences under uncertainty-consists of two lottery questions. We describe the algorithm for the case of a chronic health state. Then we describe the algorithm for the general situation (any given lifetime health profile). A modification of the algorithm for the circumstance in which an individual is faced with many possible lifetime health profiles is also presented. The purpose of the modification is to reduce
the number of questions required and thus shorten the length of the interview. We then deal with the question of the existence of HYE, and describe the results of an experiment to test the reproducibility of the measures obtained by using the instrument (algorithm) suggested, because it employs a combination of lottery questions that had not previously been used
together.
Measuring HYEs: The Case of a Chronic Health State The simplest situation is that in which an individual has a particular number of remaining life years in a given constant health status followed by death. This application of the HYE is used, for example, by Pliskin et al.1 for patients with coronary artery disease and by Churchill et a1.11-13 for patients with end-stage renal disease. For the convenience of the reader, we repeat the definition of HYE’: Let Q and T denote two attributes of the outcome of concern (Q health state of the individual, T life years). Let Q represent the state of full health and Q death. T is defined in the following interval [0, T], where T is the maximal possible survival. Let U(Q,T) be a von Neumann-Morgenstern (vNM) utility function that describes the utility of being in a given health state Q, starting now, for a period of T years, followed by death, as viewed now by an individual. Assume that the individual is currently in health state Q and has T more years to live. Let H be years in full health and H* be the healthy-years equivalent of (Q,T). H* is defined as follows: =
Find H* such that
=
U(Q,H*) = U(Q,T)
(1)
The solution to equation 1 will hypothetical combination of H* years in full health (Q) that is equivalent, in terms of the individual utility, to living T years in health state Q. The algorithm suggested to measure H* is based on the standard gamble technique. This technique is the classic method of measuring cardinal preferences. In brief, the standard gamble method is a paired comparison in which the individual must choose between two alternatives. A detailed description of the different methods used is provided by Farquhar.14 For the application of the standard gamble technique to health care problems in providing weights for calculation of
yield a
QALYs
see
Torrance.ls
an individual who is currently in health and has T more years to live, e.g., Q Q being in a dialysis unit, T having ten more years to live. In this example ten years is assumed to be the maximum possible survival; thus, T T. However, in other T be T. Our task is to smaller than examples might find H* such that U(Q,H*) The U(Q,T). following twoH*. is to measure stage procedure suggested
Assume
state
=
=
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=
=
142
MEASURING THE UTILITY VALUE OF
U(Q,T)
The
following two-stage procedure
measure
Lottery questions are asked to determine the indifference level of P between:
is
suggested
to
H*.
MEASURING
U(QT)
Lottery questions are asked to determine the indifference level of P between:
Probability P is varied until the individual is indifferent between the two alternatives. Denote P* as the probability at the indifference point. At this point, assuming 1 and U(f~,0) = 0, the utility value of being U(Q,T) in the health state under consideration (Q) for T years is equal to P*. In other words, U(Q,T) P*. =
=
where
Q.~. = 4i) 1; -Qi
maximal survival. Probability P is varied until the individual is indifferent between the two alternatives. Denote P* as the probability at the indifferent point. At this point, assuming U(Qi) 1 and U(Qf) = 0, it implies that U(QT) P*. =
fgl, ’ ’ ’ gf]; T
=
possible
=
MEASURING
H*
=
Lottery questions are asked to determine the indifference level of H (i.e., H*) between:
MEASURING
H*
Lottery questions to determine the indifference level of H (i.e., H*) between: The number of years in full health (H) is varied until the individual is indifferent between the two alternatives. Denote H* as the number of healthy years at the indifference point. H* is the number of HYEs that is equivalent to living T years in health state Q.
Measuring HYEs: The General Case
is
The algorithm suggested can be extended to deal with the more general case described by Mehrez and Gafni.’ For the convenience of the reader, we provide the definition for HYE for the general case. A lifetime health profile of an individual can be described as a vector Q = [ qil where q, is the i th element of the vector. Let qi be the health state of the individual at the ith period (measured in years). For the sake of simplicity of presentation and without loss of generality, we assume that all periods are equal in terms of their time intervals. Denote q as perfect health during that period and g as death. Assume an individual who has T more years to live and faces the following lifetime health profile: QT = f ql, ’ ’ ’, qT]. Let U(QT) represent a vNM utility function over the individual’s lifetime health profile, which describes the utility as viewed now by the individual (i.e., the present utility). Let H be years in full health and H* be the healthy-years equivalent of QT. H* is defined as follows: Find H* such that
The number of period in full health (H) is varied until the individual is indifferent between the two alternatives. Denote H* as the number of healthy years at the indifference point. H* is the number of HYEs that
equivalent
where:
Q~.
Measuring HYEs: The Case of Many Possible Lifetime Health PmMes we describe a modification of the for the individual who is faced with many procedure health lifetime profiles (e.g., the case of a compossible The of the suggested moddecision tree). purpose plex ification is to reduce the number of questions needed in the interview and thus shorten the length of the interview. For simplicity and without loss of generality, we illustrate the modified procedure for the case of chronic health states. As a result we revert to the notations and definitions from the previous section dealing with the chronic health state case.
In this section
Assume an individual who is faced with many possible scenarios of lifetime health profiles. Denote (Q,/T,) as the ith scenario. Denote H*i as the healthy-years equivalent of (Q,,T~ ). The following three-stage proce-
dure is
U(QH*) = U(QT)
to
suggested
to measure
Hi.
ASSESS THE INDIVIDUAL’S UTILITY FOR LENGTH OF LIFE IN GOOD HEALTH
q QH* - [q;J such that q; = g otherwise
for = 1, ’ ’ ’ H*
i
In this
(2) in
step
step, lottery questions of the type described
1 of
previous
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sections
are
utilized to
plot
the
143
for length of healthy life. Depending on the maximal possible survival period (T), one can decide how many points to measure. Assume for the sake of simplicity that we decide to divide this period into four equal intervals. Because we define 0 and U(Q,T) 1 we have to measure only U(Q,O) the utility values for the following points in our example : (Q, 0.25 T), (Q, 0.5 T), (Q, 0.75 T). Measuring the utility values of these points is done by asking lottery questions as follows [For example for (Q, 0.5 T)]:
years in full health (Q) that is equivalent, in terms of the individual’s utility, to living T years in health state Q? The answer is-not necessarily. Depending on the individual’s specific utility function, it is easy to see that H* does not always exist (see the appendix for an
Find P such that the individual is indifferent between:
lottery question
individual’s
utility
curve
=
=
example). Consequently, can we determine, using a simple procedure, the set of all potential lifetime health profiles for which H* does not exist? For the two-dimensional case (chronic health state), one lottery question can help determine whether this set is not nil. The is:
Find the indifference level of P between
[(1 as the probability at the indifference point. P*. Then U(Q, 0.5 T) After measuring the utility values of the other survival periods we can plot the individual’s utility curve for length of healthy life.
f[p,(Q,T)I;
P),(g,0)]~ vs (Q,0)
Denote P*
=
MEASURING
U(O,,Ti)
Lottery questions are asked ference level of P between:
Denote
P*i as the probability U(Q,T¡) = P*,.
to
determine the indif-
at the
indifference
point.
H*¡
Having measured U(C~T,) for all i and having plotted the individual’s utility curve for length of healthy life, we can now calculate H*, for each i. By definition, H*, is such that U(Q,H*,) = U(Q,,T,). Since we know the value ofU(Q,/r,) from step 2, we can read (or calculate using simple extrapolation methods) the value of H*, directly from the individual’s utility curve for length of healthy life. The procedure described in this section is shorter (in terms of the number of questions presented during the interview) than the procedure described previously and thus affects the length of the interview. However, it is important to mention that it is less accurate. How much less accurate this modified procedure is compared with the original is yet to be resolved.
sure
event
(Q,0) and the bad seen
to
Reproducibility of Measures Obtained fpom the Instrument Three attributes are essential for any instrument to be useful as a measure of outcome: validity, reproducibility, and responsiveness. These have already been established for existing utility measures (see Mehrez and Gafnis for a summary of the literature). However, because the algorithm to measure HYE uses a combination of lottery questions, each of which had been used separately in the past in different studies, but which had not yet been used together, we performed an experiment to test the reproducibility of the new measure.
The Existence of HYE Does H*
outcome of the be meaningful comlottery (g,O) might of the outcome with the lottery (0,T). Thus, good pared individuals might have difficulty in defining the indifference level of P. The above-mentioned procedure is suggested to save time and shorten the interviews. However, if it causes more harm than good (interviewees have difficulty understanding the question) it is not recommended for use. If H* does not exist for some potential lifetime health profile, this condition is likely to be detected during the interviews using the procedures described in previous sections. tween the
not be
Notice that
CALCULATING
If such P (denoted as P*) exists, it is easy to show that H* does not exist for all potential lifetime health profiles (Q,,T,) for which 0 < U(Q,,T,) < P*. In practice, this might not be a simple question to explain to interviewees, as has already been noted,s because it involves distinguishing between two possible scenarios (g,0) and (Q,0). Also, the trade-off be-
exist? Can
always QH’ (defined above) be found for all possible lifetime health profiles (QT) such that U(QT) = U(QH.)? Returning to the example of a chronic health state, the question becomes: for all potential lifetime health profiles preferred to death (Q,T), can we always find a hypothetical combination of H*
A measure is reproducible if it yields the same results when repeated in stable subjects. Reproducibility is best measured by repeated administration of the instrument to subjects whose status has not changed. An experiment was conducted with 32 graduate students (ages 35-45 years) in an advanced course in decision sciences in the Department of Industrial En-
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144
Table 1 . Utility
Scores (P*) and Healthy-years Equivalent Values (H*) Assigned by 32 Subjects to Being on Dialysis for Ten Years Followed by Death, in Two Interviews Four Weeks Apart
values of P* (and the same for H*) for the two interviews. More precisely, the values (i.e., P* and H*) measured at the second round of interviews were used as the dependent variable. This analysis showed that the Pearson correlations for P* and H* were 0.9553 and 0.7758, respectively. The t-tests for the above-mentioned simple linear regressions were statistically significant (a 0.05). We than tested the null hypotheses that the variances of P* and H* for the two interviews were equal, using the F test for normal distribution. The results are presented in table 2. For both P* and H* the null hypothesis was not rejected (significance level 0.05). Stemming from these results, we tested the hypothesis that the difference between the means of P* and H* for the two interviews was equal to zero. This was examined by using a t-test under the following assumptions : 1) the observations are statistically independent ; 2) for each observation the repeated measurements might be statistically dependent; 3) the samples are taken from two normally distributed populations ; 4) there are unknown equal variances. The mean and variances are computed for each difference between the two repeated measurements. The results are presented in table 3. For both P* and H* the null hypothesis was not rejected (significance level 0.05). Thus, the results of the two interviews seem to be similar. In summary, the statistical analysis demonstrated in two different ways (by Pearson correlation coefficients and t-test) that at the individual and population levels the results of the interviews are similar up to the level of random error. An alternative approach could =
gineering and Management at Ben Gurion University of the Negev, Beer Sheva, Israel. The purpose of the experiment was to measure individuals’ preferences
(expressed in HYEs) with respect to being in the following health-state scenario: being on hospital dialysis for the next ten years followed by death. The description of the health scenario was taken from Sackett and Torrance.16 The scenario is a brief description of the physical, social, and emotional characteristics and
limitations characteristic of this health state. This scenario was used by Sackett and Torrance to measure the general public’s preferences with regard to this health state. The instrument, which is based on the algorithm described earlier for the case of a chronic health state, was administered twice to the same subjects. The interviews were separated by a four-week period. The interviews were conducted by two skilled interviewers who had had a two-hour training session prior to the interviews, in which they had been given the instrument. The instrument components are the health state description, the interview schedule, response recording forms, visual aids, and an interviewer manual. Such highly standardized instrument components are needed to conduct measurements in a systematic manner and reduce variability and bias. For the first round of interviews, the students were divided into two groups. An interviewer was assigned to each group. For the second round of interviews, the interviewers switched groups. This was done to blind the interviewers to the subjects’ previous responses. The average interview times were 7 minutes (range: 5-12 min) for the first round of interviews and 5 min (range: 3-9 min) for the second round of interviews. The means and standard deviations of the utility scores (P*) and the healthy-years equivalent values (H*) for the two interviews are presented in table 1. To examine the reproducibility of the measure suggested, we first analyzed the distribution of responses. The distributions of the values of P* and H* fit a normal distribution (according to a chi-square goodness-of-fit test). Second, the Pearson correlation was computed via a simple linear regression analysis between the
Table 2 o
Test Results of Equality of Variances of Normal Distributions for the Two Interviews
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145
Table 3 o
Test Results of Differences between Means for the Two Interviews
experiment were more sophisticated then the average patient. Thus, to learn more about participation and interview time, we would like to repeat the experiment with a more representative population.
might be tempting to use the time trade-off (TTO) technique to determine the hypothetical number of years in full health (HYE) that is equivalent to an individual’s possible lifetime health profile. In other words, time trade-off judgments might yield the needed equivalences that are currently determined in our algorithm by using two different types of lottery questions. The advantage of the time trade-off technique is that it would reduce the number of judgment questions needed and eliminate the demand on patients’ understanding of probabilities. As shown by Mehrez and Gafnp7 the TTO technique, although offered as a substitute to the standard gamble (SG) technique, was not related in a general way to any existing behavioral theory. Mehrez and Gafnil7 suggest such a link. They show that in the context of value function theory, the TTO technique makes posIt
be to
use a
analysis
cross-sectional, time-series, multivariate
under the normal distribution
assumption,
which, based on statistical theory, should provide similar results.
Diseussion We have described
a
relatively simple algorithm
to
the healthy-years equivalent of a given lifetime health profile. The algorithm is based on the standard gamble method. Lottery questions, which are widely used in health care to measure individuals’ preferences, were found to be valid and responsive. 10,15 The algorithm consists of two lottery questions that had not previously been asked together, so we performed an experiment to examine reproducibility. The results of this experiment were very satisfactory. The advantage of the HYE measure compared with the QALY measure is that it stems directly from the individual utility function and thus fully reflects the individual’s preferences. Mehrez and Gafnis raised the questions of financial and performance feasibility of the measure. The suggested algorithm offers answers to these questions. First, it consists of two lottery questions, compared with the one lottery question needed to calculate weights for the QALYs measure. This could increase the interviewing time, which might not be a burden where we compare only a few lifetime health profiles. We describe a modification of the algorithm for use when the analysis involves a comparison of many lifetime health profiles. Second, the measure was found to be responsive. Third, participation and the rate of completion of the interview were good, and the average interview time was reasonable. We acknowledge that the students used as subjects in our measure
sible identification of different points on an individual’s indifference curve in his or her evaluation space. Because the TTO technique stems from value function theory (i.e., measurement of preferences under certainty) while the SG technique stems from utility theory (i.e., measurement of preferences under uncertainty), the results generated by the two methods should not always be identical .17 This is supported by empirical evidence. Because medical interventions occur only in a world of uncertainty,’ the SG is the appro-
priate technique. We make a theoretical claim with regard to the existence of H* in this report. We show that it is not inevitable that for all possible lifetime health profiles one can always find a hypothetical combination of years in perfect health that is equivalent in terms of
the individual utility to living in those lifetime health profiles. This means that there might be lifetime health profiles that are better than immediate death but worse (from the individual’s perspective) than living the shortest possible period of time in full health. Whether such a situation exists is an empirical question that should be further explored. We have provided a lottery question that can help determine whether a given lifetime health profile falls into this category or not. It is interesting that the QALY measure assumes that this category does not exist: each lifetime health profile that is preferred to death has an &dquo;equivalent&dquo; in healthy years.
Finally, it is important to ask whether the use of HYEs instead of QALYs can affect the result of the analysis. Hypothetical examples’ illustrate that in spite of individuals’ statements that A to B (which
are
they prefer
supported by
their
alternative
answers
to SG
questions), QALY calculations indicate that they prefer alternative B
to A.
Also, in the
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context of economic
146
evaluation, the choice of QALY
as a measure of outthe use of ionic vs nonstudy comparing ionic contrast media was shown to predetermine the outcome of the anlaysis due to limitations in its ability
come
to
in
reveal
11.
patients’ preferences.&dquo;
examples mentioned illustrate the importance of using an outcome measure that coincides with individuals’ preferences. Yet more empirical work on the of this outcome measure is needed. To the best of our knowledge, the algorithm suggested in this paper is the only one that stems directly from vNM individuals’ utility functions. Thus, for all individuals who follow vNM axioms, it provides a measure of outcome that fully reflects their preferences. The extra lottery question needed in this algorithm (beyond the one already needed to measure the weights for QALY calculations) is the cost of having a measure with a solid theoretical foundation rooted in the vNM theory of expected utility.
continuous Med Decis
ambulatory peritoneal dialysis to Making. 1984;4:389-500. Churchill DN, Morgan J, Torrance GW. Quality of life in endstage renal disease. Peritoneal Dialysis Bull. 1986;4:20-3. Churchill DN, Torrance GW, Taylor DW, et al. Measurement of quality of life in end-stage renal disease: the time trade-off approach. Clin Invest Med. 1987;10:14-20. Farquhar PH. Utility assessment methods. Manage Sci. 1984;
hospital dialysis.
12.
13.
The
use
Churchill DN, Lemon BC, Torrance GW. Cost-effectiveness anal-
ysis comparing
a
14.
30:1283-300.
Utility approach to measuring health related qualof life. J Chronic Dis. 1987;40:593-600. 16. Sackett DL, Torrance GW. The utility of different health states as perceived by the general public. J Chronic Dis. 1978; 31:69715. Torrance GW.
ity
704.
17. Mehrez A, Gafni A.
Evaluating health related quality of life: an indifference curve interpretation for the time trade-off technique. Soc Sci Med. 1990;31:1281-3. 18. Gafni A, Zylak CJ. Ionic vs non-ionic contrast media: a burden or bargain. Can Med Assoc J. 1990;143:475-8.
APPENDIX The authors are grateful for helpful comments and suggestions on earlier version of this paper from Stephen Birch, David Feeny, and two anonymous referees. All errors and wrongheaded interpretations remain the responsibility of the authors.
The formulation of Pliskin et al.1 multiattribute utility function:
References
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an
Pliskin JS, Shepard DS, Weinstein MC. Utility functions for life years and health status. Operations Res. 1980;28:206-24. 2. Weinstein MC, Fineberg HC, Elstein AS, et al. Clinical decision 1.
analysis. Philadelphia: W.
B.
Saunders, 1980.
3. Torrance GW, Feeny D. Utilities and quality-adjusted life years. Int J Technol Assess Health Care. 1989;5:559-78.
G, McKenzie L. The use of QALYs in health care decision making. Soc Sci Med. 1989;28:299-308. 5. Mehrez A, Gafni A. Quality adjusted life years, utility theory, and 4. Loomes
Med Decis Making. 1989;9:142-9. McNeil BJ, Weichselbaum R, Pauker SG. Fallacy of five year survival in living cancer. N Engl J Med. 1978;229:1397-401. McNeil BJ, Weichselbaum R. Pauker SG. Speech and survival: tradeoffs between quality and quantitiy of life in laryngeal cancer. N Engl J Med. 1981;305:982-7. Ben-Zion U, Gafni A. Evaluation of public investment in health care: is the risk irrelevant? J Health Econ. 1983;2:161-5. Mehrez A, Gafni A. The optimal treatment strategy—a patient perspective. Manage Sci. 1987;33:1602-12. Froberg DG, Kane RL. Methodology for measuring health-state preferences. J Clin Epidemiol. 1989;4:345-54 (I. Measurement strategy); 459-71 (II. Scaling methods); 585-92 (III. Population and context effects); 675-85 (IV. Progress and research agenda).
healthy years equivalents.
6.
7.
8. 9. 10.
UilT) = utility function U~(Q) = utility function a,b =
implies
the
following
for remaining life years for health states
constants
We also employ the following assumptions and definitions : 1) Q is the chronic health state preferred to death death and Q where Q healthy; 2) T is defined in the following interval [0,T] where T is the maximal possible sur=
vival ; 3) UQ(Q) b
=
0.3. Thus
=
UQ(Q) equation 1
=
0;
1; Ui10) becomes
=
=
0; UT(T) = 1;
equation
a
=
0.4;
2:
Assume an individual who is faced with the following potential lifetime health profile (Q,T). Using the standard gamble approach we measure U(Q,T) and find it to be 0.30. We can now calculate H by solving equation 3:
It follows that because
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U, (H)
