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Cardinal Scales for Health Evaluation
Charles M. Harvey
Department of Decision and Information Sciences, University of Houston, Houston, Texas 77204,
[email protected]
Lars Peter Østerdal
Department of Economics, University of Copenhagen, DK-1353 Copenhagen K, Denmark,
[email protected]
P
olicy studies often evaluate health for an individual or for a population by using measurement scales that
are ordinal scales or expected-utility scales. This paper develops scales of a different type, commonly called
cardinal scales, that measure changes in health. Also, we argue that cardinal scales provide a meaningful and
useful means of evaluating health policies. Thus, we develop a means of using the perspective of early neoclassical welfare economics as an alternative to ordinalist and expected-utility perspectives.
Key words: health scales; population health; cardinal utility; neoclassical welfare economics; social welfare;
preference intensity
History: Received on October 2, 2009. Accepted on April 14, 2010, after 2 revisions. Published online in Articles
in Advance June 17, 2010.
1.
Introduction
called a difference relation, an intensity relation, or a
strength-of-preference relation, and a cardinal scale is
also defined as any scale that is unique up to a positive linear transformation, e.g., an expected-utility
function. Thus, it is not possible to adopt definitions
that agree with all those in the literature.
Each of the three types of binary relations may or
may not satisfy conditions that imply the existence
of a scale that represents the relation. For example, a
so-called lexicographic ordinal relation has no ordinal
scale. A disadvantage of the terms cardinal relation
and cardinal scale is that they do not indicate this
important distinction.
This paper presents models that contain cardinal
scales for the health of an individual and for the
health of members of a population. In the first part,
we argue that changes in individual health and in
population health can be meaningfully compared in a
policy study of population health, and thus cardinal
relations have an operational meaning in such a context. And as a stronger point, we argue that changes
in population health are often the relevant objects to
compare in such a study as a means to compare the
effectiveness of policy options.
In the second part of this paper, we develop cardinal scales for the health of an individual. Here we
A policy analyst who wishes to evaluate the effects
of a social policy on the health of an individual or
within a population has a variety of measurement
scales and supporting procedures to choose from.
Typically, the scale is an ordinal-utility scale or an
expected-utility scale.
Both types of scales represent binary relations that
compare health alternatives; an ordinal-utility scale
represents a relation on health outcomes such that
greater scale amounts correspond to better outcomes,
and an expected-utility scale represents a relation on
probability distributions of health such that greater
expected values of scale amounts correspond to better
distributions.
This paper develops models and procedures for a
different type of binary relation and scale; here, better
changes in health correspond to greater differences in
scale amounts. A relation that compares changes in
outcomes is commonly called a cardinal relation, and a
scale that represents a cardinal relation in this manner
is commonly called a cardinal scale.
But other terminology is used with these meanings,
and this terminology is used with other meanings. For
example, a cardinal relation (as defined here) is also
256
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consider health outcomes of two types: health states
and health–duration pairs. By a health state, we mean
the health of a person, measured by one or more
variables. And by a health–duration pair, we mean a
health state and, as additional information, the duration of the health state. We do not consider health
outcomes that are sequences of health states or that
are streams of health states over an interval of time.
We present models in which a cardinal relation on
a person’s health outcomes satisfies certain conditions
if and only if it is represented by a cardinal scale that
has a certain form. For example, we show that a cardinal relation on health–duration pairs satisfies certain conditions if and only if it has a cardinal scale
that is a product of a cumulative discounting function and a scale defined on health states. An open
question is whether parallel results can be obtained
for health outcomes that are sequences of health
states or streams of health states over a bounded or
unbounded interval of time.
The third part of this paper discusses the aggregation of cardinal scales for the members of a population into a cardinal scale for the population as a
whole. We adapt to the context of health a model of
Harvey (1999) in which a cardinal relation for population welfare is represented by a cardinal scale that is
a weighted sum of cardinal scales for the population
members. This model is parallel to the well-known
model of Harsanyi (1955) that aggregates expectedutility functions for the members of a population into
an expected-utility function for the population.
The third part of this paper also discusses conditions on social values that imply simplifications in a
cardinal health model, e.g., equal weights, and it discusses assessment procedures by which a policy analyst can construct cardinal scales for individual and
population health in such a model.
The fourth part of this paper discusses procedures
by which a model that has been constructed can be
used to evaluate a change in population health, for
example, a change that is predicted to result from a
policy option. In brief, a policy analyst can calculate a
change in population health that is indifferent to the
given change and that is much simpler to report. Also,
he can compare the effectiveness of policy options by
comparing these simple equivalent changes. A utility
scale that is not a cardinal scale cannot provide such
an evaluation or comparison of policy options.
This paper concludes with a hypothetical application that evaluates a proposed therapy for the reduction of hypertension, commonly known as high blood
pressure.
The interest throughout this paper is prescriptive.
The models and procedures are intended for studies
of population health that are concerned with valuing
as well as predicting the effects of alternative policies. As in the case of expected-utility, there surely are
many types of violations of the conditions on values
that are presented here. That important subject we do
not address.
Most of the analytic results in this paper, as well as
in Harvey (1999), are based on work by the Danish
mathematician J. L. W. V. Jensen (1905, 1906). Proofs
of results are in the appendix.
2.
Meaningful Interpretations of a
Cardinal Relation
This section and the next discuss cardinal relations
and cardinal scales, respectively. These discussions
are intended as a background, although a very incomplete one, for the models and procedures presented in
later sections.
Pareto (1896) and Fisher (1918) pointed out long
ago that the term “utility” had two different meanings: (i) an older, hedonic meaning in utilitarianism
and classical welfare theory as the degree of pleasure and pain (or welfare, well-being, etc.) that a person experiences as part of a specified consequence, and
(ii) a newer, choice-oriented meaning in consumer
theory as the degree to which a consequence satisfies a person’s preferences. Pareto gave an illustration
in which the two meanings differ: a child may experience better health by taking medicine even though
the child much prefers not to take the medicine.
This distinction in the meaning of utility is logically
independent of that between a cardinal relation and an
ordinal relation. Historically, however, the two distinctions have sometimes been confused. An insufficient
recognition of the distinction in the meaning of utility
made by Pareto (1896) and Fisher (1918) (and a disagreement as to whether cardinal comparisons were
needed in economics) led to a schism between early
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258
neoclassical welfare economists, who used hedonic
utility and cardinal relations, and so-called new welfare economists, who used choice-oriented utility and
ordinal relations. Cooter and Rappoport (1984, p. 507)
note that “[t]he term ‘ordinalist revolution’ refers to
the rejection of cardinal notions of utility and to the
general acceptance of the position that utility was not
comparable across individuals.” In recent years, there
has been a renewed interest in the hedonic meaning
of utility; see, e.g., Broome (1991a, b; 2004), Kahneman
et al. (1997), Kahneman (2000), Loewenstein and Ubel
(2008), and Dolan and Kahneman (2008).
In this paper, cardinal relations can have either a
hedonic or a choice-oriented interpretation. Below, we
argue that either interpretation is meaningful in a
study of population health—and thus one should be
able to use cardinal relations with either interpretation as a part of the study.
Explicitly or implicitly, a study of population health
assumes value judgments concerning health, often
called social values. Judgments that compare changes
in a person’s health can be made by the person himself or they can be made by someone else (an expert,
an agency, or an idealized person). Such an entity
often is called a social planner. Usually but not always,
the social planner will follow the hedonic evaluations
or the preferences of the affected individuals.
If a cardinal relation on changes in a person’s health
is regarded as comparing changes in what the person experiences, then the comparisons are possible in
principle, no matter how difficult they may be in practice. And if a cardinal relation on changes in health
for a population has a similar hedonic meaning, then
the comparisons of changes are possible in a similar
fashion.
But if a cardinal relation on changes in health is
regarded as the preferences of a person or a group, then
an important question is whether or not the comparisons are meaningful. Here, we argue that a cardinal
relation regarded as preferences can have meaning.
Other arguments for meaningfulness are presented in
von Winterfeldt and Edwards (1986, pp. 208–211).
The argument that cardinal relations as preferences
are not meaningful has two parts. The first part is to
observe that an individual facing a decision is at a
single initial position. Changes from that position can
Harvey and Østerdal: Cardinal Scales for Health Evaluation
Decision Analysis 7(3), pp. 256–281, © 2010 INFORMS
be identified with their final outcomes, and comparisons of changes are not needed. It suffices to observe
the behavior of the individual in choosing outcomes.
The second part of the argument is as follows.
A classic technique of economic analysis is to specify
a set of alternatives to be compared so that it contains
alternatives that are not possible but are similar in
structure to the possible alternatives. Then, a binary
relation can be extended, at least in principle, from the
set of possible alternatives to the larger set of alternatives. But outcomes and changes in outcomes do not
have a similar structure; thus, it is not meaningful to
extend a relation from a set of outcomes to a set of
both outcomes and changes.
We think that this argument is valid but that for
prescriptive studies of population health it does not
preclude the meaningfulness of cardinal relations. To
explain, we must first describe what we mean by
such a study. We mean a study whose purpose is to
examine policy options. Part of the study is empirical, describing and predicting the consequences of
the options. And part is normative, assigning values to the consequences. The values may be hedonic, e.g., percents of infections prevented, or they may
be choice oriented, e.g., preferences in the form of
trade-offs between categories of injury. In either case,
the values are prescriptive: when hedonic, they reflect
choices by the policy analyst as to which harms and
benefits to include, and when choice oriented, they
are preferences that the analyst has assigned by a process of assumption and assessment (which the analyst
should make explicit) rather than by a direct reporting of choice behavior.
The actions of the policy options may be interventions in a single population (including the option of
nonintervention), they may be a single intervention
in many populations (including the case of a population at different times), or they may involve multiple populations and multiple types of intervention.
The consequences of the interventions are the changes
in health that they are estimated to have effected or
are predicted to effect. Often the initial distribution of
health in a population will not be uniform, i.e., not
everyone has the same initial health. And often when
more than one population is involved (as in the hypothetical application in §10), the populations will not
have the same initial distribution of health.
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Consider the second part of the argument that cardinal relations as preferences lack meaning, namely,
the argument that a relation cannot be extended from
outcomes to changes in outcomes. For studies of population health, the argument is valid but not relevant;
the consequences to be compared are changes. There
is no question of extending a relation that compares
outcomes to one that compares changes, that is, of
extending an ordinal relation to a cardinal relation.
The first part of the argument, like the second, is
valid but not relevant. Even when there is a single population, most likely the individuals are not
at the same initial position; their changes in health
will entail different initial positions as well as different final positions. And when there is more than one
population, then most likely the initial distributions
of health in the populations are different; changes in
distributions of health will entail different initial distributions of health as well as different final distributions of health.
The preferences in a prescriptive study of population health are not descriptive; they are introduced
according to a process other than direct observation of choice behavior. The process constitutes their
meaning, and the above arguments fail to deny that
meaning.
Actually, we take a stronger position than that
argued for above. We believe that cardinal relations
as preferences are meaningful even when there is
only one population and one initial distribution of
health. We will argue for this position indirectly by
making an analogy with preferences between uncertain outcomes. Suppose that a decision analyst wishes
to construct a prescriptive model of preferences for
a situation in which the consequences are prospects
having two outcomes, the first being a specified status quo position and the second being an outcome
in a specified set. To make the situation less realistic
but a better analogy, assume that the two outcomes
are equally likely. Here, the prospects can be identified with their second outcomes, and comparisons of
prospects are not needed. Perhaps the decision analyst will judge that an ordinal-utility model that compares the second outcomes is appropriate, or perhaps
he or she will judge that an expected-utility model
that compares probability distributions is appropriate.
But we doubt that the analyst would like to be told
that an expected-utility model is meaningless.
The motivating question for this paper is not
whether cardinal relations as preferences are meaningful; rather, it is whether they are useful in prescriptive models on population health. We believe that the
answer will depend on the situation that a policy analyst faces and that it also will and should depend on
the proclivities of the analyst. The goal of this paper,
and the focus of the following sections, is to provide
a few tools for an analyst who chooses to use cardinal
relations.
3.
Cardinal Scales
Suppose that h denotes a health outcome of any
type for an individual or for a population, and that
H denotes a set of health outcomes h. Suppose also
that a change from a health outcome h to a health
outcome h is denoted by h → h . We will refer to comparisons of these changes as preferences even though
they may also have a hedonic interpretation. A statement that a change h → h is at least as preferred as a
change k → k will be denoted by h → h k → k .
Cardinal relations and cardinal scales will be
defined as follows. A set of statements, h → h k → k , where the health outcomes are in a set H , will
be called a cardinal relation and will be denoted by .
We will say that is defined on the set H . Other
types of comparisons, e.g., “is preferred to” and “is
indifferent to,” will be defined in terms of . And a
function wh defined on H will be called a cardinal
scale for provided that h → h k → k if and only
if wh − wh ≥ wk − wk for any health outcomes
in H .
A cardinal relation induces a relation, called an
ordinal relation, that compares outcomes themselves.
Such a relation ord can be defined by h ord h if and
only if h → h h → h. A cardinal scale wh for is
an ordinal scale for ord (because h ord h if and only
if wh − wh ≥ wh − wh if and only if wh ≥
wh), but not every ordinal scale for ord is a cardinal
scale for . An ordinal scale is ordinally unique; that
is, if wh is an ordinal scale, then a function w ∗ (h
is also if and only if there exists a strictly increasing
function f w such that w ∗ (h = f wh for h in H .
As in the case of expected utility, a cardinal relation must satisfy certain conditions in order to have a
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cardinal scale. The condition that corresponds to the
independence principle is often called dynamic consistency. It states that h → h k → k and h → h k → k imply h → h k → k for any health outcomes in the set H. Various models for the existence
of a cardinal scale have been constructed; see, e.g., Alt
(1936), Debreu (1960), Scott (1964), Pfanzagl (1968),
and Krantz et al. (1971). Except for the model of Scott
(1964), the conditions also imply that the cardinal
scale is cardinally unique; that is if wh is a cardinal
scale, then a function w ∗ h is also if and only if there
exist constants a > 0 and b such that w ∗ h = awh + b
for h in H .
The models of Alt (1936), Debreu (1960), and
Pfanzagl (1968) assume that the set H is topologically
connected in a given topology, and they establish that
a cardinal relation satisfies the conditions if and only
if it has a continuous cardinal scale, that is, a continuous function wh. Because H is connected, the range
of such a function is an interval.
It will simplify our models to assume that there
exist health outcomes that are not indifferent. This
will be the case if and only if any cardinal scale has a
nonpoint range.
In a policy study, the analyst should determine the
appropriateness of conditions that imply the existence
of a cardinal scale having the properties described
above. However, for the purpose of developing the
models in this paper, we will directly assume these
implications.
Definition 1. A cardinal or ordinal relation will be
called proper provided that it has a topologically continuous scale whose range is a nonpoint interval. Any
such scale will be called proper.
Lemma 1. If a cardinal relation is proper, then, (i) the
cardinal scales for are cardinally unique, (ii) any cardinal scale for is proper, and (iii) the relation ord is
proper.
If the health outcomes in a set H are described
by one or more variables whose domains are intervals, then the set H is (topologically) connected. And
if H is connected and satisfies conditions that
imply the existence of a continuous, nonpoint cardinal scale, then is proper. Here, the assumption
that the set H is connected can be weakened. If the
Decision Analysis 7(3), pp. 256–281, © 2010 INFORMS
outcomes in H are described by one or more variables whose domains are intervals and (possibly) also
by one or more variables whose domains are finite
and have sufficiently fine gradations, then H has a
more general property, called “preferential connectedness.” Harvey (2008) shows that the assumption of
connectedness can be weakened to that of preferential
connectedness.
4.
Health States
This section and the next present models of cardinal
scales that evaluate changes in health for an individual. In this section, a health outcome is a state of a
person’s health. For example, the health outcome may
be a morbidity or an injury, or it may be the condition
of the person’s health in general. A health outcome
defined in this manner will be called a health state.
In a policy study, the health states are described
by one or more health variables (often called ratings,
dimensions, health scales, etc.). A health variable may
have either an interval domain or a finite domain. Typically, the set of health states to be compared is defined
as the product set of the domains of the variables.
However, we will not require this condition here.
In the discussion below, the term “ordinal scale”
will mean any scale (including an expected-utility
scale) that compares health states and has not been
verified to be a cardinal scale. Suppose that a policy
analyst wishes to construct a cardinal scale for a set
of health states. The method that we develop is for
the analyst first to obtain an ordinal scale and then
to either verify that the ordinal scale is cardinal or to
construct a cardinal scale that is based on the ordinal
scale.
An advantage of this approach is that it enables the
analyst to separate two different issues of preferences.
The ordinal scale represents the issue of trade-offs
between the health variables, e.g., for a given decrease
in one variable, how much increase in another variable is needed to produce an indifferent health state.
The cardinal scale can then be defined on the single ordinal scale, rather than on the health variables. It represents the issue of preference intensity
between the values of the ordinal scale, e.g., for a
given increase and a given third value, how much
does the third value need to increase to produce a
change that is indifferent to the given increase.
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A second advantage of the approach is that a wide
variety of ordinal scales are available for the analyst
to use. Often, such a scale is called “health-related
quality of life” or “quality of life” (see, e.g., Gold et al.
1996, Dolan et al. 1996, Dolan 2000). For brevity, we
will use the term quality scale to refer to any ordinal
scale when the health outcomes are health states.
Quality scales have been developed both for morbidity and trauma. Examples of this type of scale are
the Functional Capacity Index, the Quality of Wellbeing Scale (Kaplan and Anderson 1996), the Health
Utilities Indices (Torrance et al. 1982, 1995, 1996;
Feeny et al. 1996; Furlong et al. 1998), the Health and
Activities Limitation Index, the SF-36 metric for health
status measurement, and the EuroQoL quality of life
scales (Kind 1996, Dolan 1997, Richardson et al. 2001).
Procedures to assess a quality scale have been
developed, either as part of a health study or as
a separate undertaking. Examples of these assessment procedures (commonly called scales) include
time trade-off scales, standard gamble scales, person
trade-off scales, and rating scales; see, e.g., Torrance
(1976, 1986) and Dolan et al. (1996).
Below we develop models of cardinal scales that
depend on a predetermined quality scale. The quality scale can be an established scale, a scale that is
constructed as part of a study, or a health variable
if health states are described by a single variable.
A health state will be denoted by s, a set of health
states will be denoted by S, and a quality scale will
be denoted by qs.
Theorem 1. Suppose that a cardinal relation for a
set S of health states has a proper quality scale qs. Then,
is proper if and only if it has a cardinal scale of the
form ws = f qs), where f q is a continuous, strictly
increasing function defined on the range of qs.
A function f q as described in Theorem 1 will be
called a conversion function. It converts an ordinal scale
q to a cardinal scale f q, i.e., a scale in which differences are meaningful. Differences in the scale q will
be meaningful if and only if q is a cardinal scale itself.
Theorem 1 provides a means of reducing the task
of constructing a cardinal scale ws that is defined
on multivariable health states to that of constructing
a conversion function f q that is defined on a single
variable qs. The conditions below imply (and are
implied by) the existence of conversion functions f q
that have special forms, and thus are relatively simple
to construct.
Definition 2. A health state ŝ will be called middling health (or middling) relative to two health
states s and s provided that the changes s → ŝ and
ŝ → s are indifferent.
Cardinal neutrality for a quality scale qs. For any
health states s ŝ, and s , if qŝ = 12 qs + 12 qs , then ŝ
is middling relative to s and s .
Cardinal constancy for a quality scale qs. If
a health state ŝ is middling relative to two health
states s and s , and t t̂, and t are health states with
qt = qs + q, qt̂ = qŝ + q, and qt = qs + q,
for some number q, then t̂ is middling relative to t
and t .
Theorem 2. Suppose that is a proper cardinal relation, and qs is a proper quality scale for a set S of health
states. Then,
(i) has cardinal neutrality for qs if and only if qs
is a cardinal scale for ;
(ii) has cardinal constancy for qs if and only if has a cardinal scale of the
 form

r > 0
erqs

(1)
ws = qs
r = 0


−erqs r < 0
for some amount of the parameter r. For a health state ŝ
that is middling relative to two health states s and s , its
health quality qŝ is greater than, equal to, or less than
the average health quality, 12 qs + 12 qs , when r is greater
than, equal to, or less than zero, respectively.
The hypothetical application in §10 concerning
hypertension illustrates the use of cardinal constancy.
To oversimplify somewhat, here the quality scale is
blood pressure and the cardinal scale is the risk of
heart disease.
The proof of part (i) is based on Jensen’s (1905, 1906)
functional equation, which in the present notation is
f 12 q + 12 q = 12 f q + 12 f q . The proof of part (ii) is
based on a similar functional relationship introduced
in Harvey (1990) for models of risk attitudes.
5.
Health–Duration Pairs
In this section, a health outcome consists of a single health state and its duration. A health outcome
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262
defined in this manner will be called a health–duration
pair. The duration of a health state can be defined as
the time from a specified event, e.g., an accident or
the onset of a disease, in which case different individuals in a population will have different initial times,
or it can be defined as the time from a common initial
time, e.g., a specified present time.
A health state will again be denoted by s, and its
duration in years will be denoted by t. Thus, a health–
duration pair will be denoted by (s t). A quality scale
will again be denoted by qs.
Most models of health–duration pairs are expectedutility models; see, e.g., Bleichrodt et al. (1997) for a
discussion of the advantages of such models. The simplest utility functions are of the form us t = tqs,
where the linear factor t represents risk neutrality
toward duration. In these models, the utility us t
of a health–duration pair (or the expected utility of a
probability distribution) is an amount that represents
an equivalent duration in a state of optimal health;
this amount is called “quality-adjusted life years”
(QALYs) or “healthy-years equivalents” (HYEs).
We regard the various QALY models due to Pliskin
et al. (1980) as the basic expected-utility models
for health–duration pairs. Alternative conditions
on preferences are presented in Bleichrodt et al.
(1997), Miyamoto et al. (1998), and Hazen (2004) for
expected-utility models, and in Doctor and Miyamoto
(2003) and Østerdal (2005) for deterministic models. QALY models for health–duration pairs are also
discussed in Loomes and McKenzie (1989), Broome
(1993), Johannesson et al. (1994), Gold et al. (1996),
and Dolan (2000).
Using an ordinal-utility approach, Mehrez and
Gafni (1989) developed QALY/HYE models. They
proposed a procedure in which a health–duration
pair is measured by assessing an indifferent health–
duration pair in which HYE years (i.e., the healthyyears equivalent) of optimal health are followed by
death. See also, e.g., Gafni et al. (1993), Loomes (1995),
and Johannesson (1995).
Describing health outcomes as health states or
as health–duration pairs is a simplification. It may
be appropriate because of data limitations, but it
entails assumptions (whether stated or not) regarding what occurs after the health state. In this section,
Harvey and Østerdal: Cardinal Scales for Health Evaluation
Decision Analysis 7(3), pp. 256–281, © 2010 INFORMS
we assume that a common health state occurs afterward. An alternative assumption would be to assume
that what occurs afterward is stochastically independent of the health state; in §9, we examine a related
assumption of stochastic independence for a population of individuals.
This section develops cardinal scales for a cardinal
relation concerning health–duration pairs. We introduce various conditions on and its associated ordinal relation, and show that they imply (and are implied by) the existence of cardinal scales for that
have various properties or special forms. Throughout,
we assume that is a proper cardinal relation defined
on a product set S × T , where S is a set of health states
as in the previous section and T is a set of durations.
Suppose that s0 denotes the health state that occurs
after the health state in a health–duration pair. The
state s0 will be called the standard state. It may be
specified as, e.g., death, optimal health, or a return
to previous health. We assume that s0 is in the set S
of health states, and we make no assumption as to
whether S contains health states that are better than
or worse than s0 . In particular, when s0 is a state
of death, the set S can contain health states that are
worse than death. For expected-utility models having
states worse than death, see Miyamoto et al. (1998).
Ethical arguments have been made for an “equal
value of life” principle that society should not compare fatalities with nondeath health states; see, e.g.,
Harris (1987) and Nord (2001). In the case that s0
is a state of death, this principle is not consistent
with the models presented here. For reasons why
social preferences among health outcomes should
depend on degrees of health and duration; see, e.g.,
Singer et al. (1995), Williams (1997), and Hasman and
Østerdal (2004).
Health–duration pairs of the form (s0 t describe
the same health outcome because the state s0 occurs
after the time t. Thus, pairs of the form (s0 t should
be modeled as indifferent.
Moreover, for a fixed duration t, the cardinal relation defines a cardinal relation S t for health states
by s1 → s2 S t s3 → s4 provided that (s1 t → s2 t s3 t → s4 t). When the relations S t are the same
for any duration t, they can be regarded as a single
cardinal relation. We will denote such a cardinal relation by S and say that induces the relation S .
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These two comments can be formalized as the following conditions on a cardinal relation :
(A) Any two health–duration pairs with the standard state s0 are (ordinally) indifferent.
(B) The relation restricted to health–duration
pairs with a fixed duration t is independent of t, and
the relation S induced by is a proper cardinal
relation.
Condition (B) is not appropriate when the set T
contains the duration t = 0. For it seems natural to
assume that any pairs (s 0) are indifferent. For positive durations, however, (B) seems appropriate, at
least for prescriptive uses in population health studies. In medical decision making, (B) can be violated by
the stated preferences of a patient; see, e.g., Miyamoto
et al. (1998).
Lemma 2. Suppose that is a proper cardinal relation
for a set S × T of health–duration pairs. Then, satisfies
conditions (A) and (B) if and only if there exists a cardinal
scale for of the form
ws t = atgs
(2)
where at is a positive, continuous function on the set T
of durations, and gs is a continuous function on the set
S of health states such that gs has an interval range and
gs0 = 0.
Next, we introduce three additional conditions on
a cardinal relation for health–duration pairs. These
conditions are concerned with the ordinal relation
that is associated with . They imply (and are
implied by) further properties of the functions at and
gs in (2).
Assume that T is the semi-infinite interval T =
0 . This choice avoids two issues: that of interpreting health outcomes of zero duration, and that of
assigning an upper bound on duration.
As before, imagine that a quality scale qs on the
set S of health states has been specified. We will
assume that qs is a proper ordinal scale for the cardinal relation S .
If a health state s is preferred to the standard
state s0 , then pairs (s t with greater durations t
should be preferred (because the standard state s0 will
occur after the state s. Similarly, if a state s is less preferred than s0 , then pairs (s t with lesser durations t
should be preferred.
A pair (s ) with a brief duration should be preferentially close to pairs with the standard state s0 in
the sense defined by condition (E) below. By contrast,
Bleichrodt et al. (1997) and Miyamoto et al. (1998)
define T = 0 ) and introduce a “zero-condition,”
which states that any two outcomes (s1 0) and (s2 0)
are indifferent. This condition and the condition that
ordinal preferences are continuous on the set S ×
0 ) imply condition (E) on the subset S × 0 ).
These ideas can be stated as conditions on the ordinal relation associated with a relation :
(C) qs is a proper ordinal scale for the cardinal
relation S defined by condition (B).
(D) For a health state s and two durations t1 < t2 ,
if s is preferred to s0 , then (s t2 is preferred to (s t1 ,
and if s is less preferred than s0 , then (s t2 is less
preferred than (s t1 ).
(E) For a duration t and two health states s1 and
s2 , if s1 and s2 are preferred to s0 , then there exists a
(small) duration such that (s2 ) is less preferred
than (s1 t), and if s0 is preferred to s1 and s2 , then
there exists a (small) duration such that (s2 ) is
preferred to (s1 t).
Theorem 3. Suppose that is a proper cardinal relation for a set S × T of health–duration pairs. Suppose also
that T = 0 ) and that qs is a function defined on the
set S. Then, and qs satisfy conditions (A)–(E) if and
only if there exists a proper cardinal scale for of the form
ws t = Dtf qs
(3)
where the following are true:
(i) f q is a continuous, strictly increasing function
defined on the range of the function qs, and f qs0 = 0.
Also, induces a proper cardinal relation S on changes
in health states, and f qs) is a proper cardinal scale
for S .
(ii) Dt is a continuous, strictly increasing function
defined on the union [0 ) of the interval T = 0 ) and
zero, and D0 = 0. Also, induces a proper cardinal relation T on changes in duration (as explained below), and
Dt is a proper cardinal scale for T .
Moreover, each of the functions Dt and f q is unique
up to a positive multiple.
We present part of the proof of (ii) here because
it provides a needed explanation. According to a
cardinal scale ws t) of the product form (3), if a
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264
health state s is preferred to the standard state s0 (and
thus f qs > 0), then greater durations in s are preferred, and if s is less preferred than s0 (and thus
f qs < 0), then lesser durations in s are preferred.
Because the scale ws t) is proper and thus has a nonpoint range, at least one of these cases occurs.
With this motivation, we will define a cardinal relation T s for a fixed health state s that is preferred
to s0 by t1 → t2 T s t3 → t4 , provided that (s t1 →
s t2 s t3 → s t4 ). And for a fixed state s that is
less preferred than s0 , we will define T s by: t1 →
t2 T s t3 → t4 provided that (s t3 → s t4 s t1 →
s t2 ). In each of these cases, the function Dt is a cardinal scale for T s . Therefore, the cardinal relations
T s , where s is not indifferent to s0 are the same. We
define T as this common cardinal relation.
As shown below, the special conditions defined in
§4 for health states can be applied in the context of
health–duration pairs as conditions on a relation S ,
and they imply the same special forms of a conversion function f q. Thus, it suffices in this section to
discuss conditions on a cardinal relation that imply
special forms of a function Dt. Moreover, a restriction for a function, f q or Dt, is independent of a
restriction on the other function.
Cardinal timing neutrality. For any health state s,
durations t1 < t2 , and shift t > 0, the changes (s t1 →
s t2 ) and (s t1 + t → s t2 + t) are indifferent.
Cardinal timing aversion. For any health state s,
durations t1 < t2 , and shift t > 0, if s is preferred to
s0 , then (s t1 → s t2 ) is preferred to (s t1 + t →
s t2 + t), and if s is less preferred than s0 , then
(s t1 → s t2 ) is less preferred than (s t1 + t →
s t2 + t).
Cardinal constant timing aversion (I). Preferences
are timing averse, and for any s1 s2 t1 < t2 t > 0,
and t > 0, if two changes, (s1 t1 → s1 t1 + t ) and
(s2 t2 → s2 t2 + t , with the same change t in duration are indifferent, then the changes, (s1 t1 + t →
s1 t1 + t + t) and (s2 t2 + t → s2 t2 + t + t), are
indifferent.
Cardinal constant timing aversion (II). Preferences
are timing averse, and for any s t1 < t2 t3 < t4 , and
t > 0, if two changes, (s t1 → s t2 ) and (s t3 →
s t4 ), with the same health state s are indifferent, then the changes, (s t1 + t → s t2 + t) and
(s t3 + t → s t4 + t), are indifferent.
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As the terminology indicates, these conditions on
cardinal preferences between health–duration pairs
are parallel to well-known conditions on ordinal preferences between temporal sequences of events, that
is, conditions on discounting. Specifically, the conditions are parallel to conditions often called zero discounting, positive discounting, and constant positive
discounting.
Theorem 4. Suppose that is a proper cardinal relation for a set S × T of health–duration pairs. Suppose also that T = 0 ) and that qs is a function
defined on S. Assume that and qs satisfy conditions
(A)–(E), and thus by Theorem 3, has a cardinal scale,
ws t = Dtf qs), as described there. Define q0 = qs0 ).
Then,
(i) has cardinal neutrality for qs if and only if f q
can be chosen as f q = q − q0 ;
(ii) has cardinal constancy for qs if and only if f q
can be chosen as

rq
rq0

r > 0

e − e
f q = q − q0
(4)
r = 0


erq0 − erq r < 0
for some amount of the parameter r;
(iii) has cardinal timing neutrality if and only if Dt
can be chosen as Dt = t;
(iv) has cardinal timing aversion if and only if Dt
is strictly concave;
(v) cardinal constant timing aversions (I) and (II) are
equivalent, and they are satisfied if and only if Dt can
be chosen as Dt = r −1 1 − e−rt ) for some amount of the
parameter r > 0.
Moreover, the implications (i) and (ii) for the function
f q are independent of the implications (iii)–(v) for the
function Dt.
A function Dt is an indefinite integral, Dt =
du du, if and only if Dt is absolutely continu0
ous on any bounded subinterval of [0 ). Then, dt
is called a discounting function, and Dt is called a
cumulative discounting function.
We will call any continuous, strictly increasing
function Dt a cumulative discounting function even
though not every such function is absolutely continuous. Any linear or strictly concave function Dt such
as those in Theorem 4 is absolutely continuous.
t
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In the case of cardinal timing neutrality, the discounting function dt can be chosen as dt =
D t = 1; in the case of cardinal constant timing aversion, dt can be chosen as dt = D t = e−rt r > 0;
and in the case of cardinal timing aversion, dt can
be chosen to be positive and strictly decreasing. In the
last case, dt may tend to infinity as t tends to zero,
and thus represent extreme discounting of the future
compared to the present.
6.
Cardinal Scales for
Population Health
Here we come to a division between medical decision
making, i.e., the choice of a medical treatment for an
identified individual, and health policy analysis, i.e.,
the evaluation of policy options concerning the health
of an identified population. We think that describing health as a health state or as a health–duration
pair is often appropriate for health policy analysis,
mainly because of the absence of extensive information on each individual in the population, but we
think that describing health as a sequence or a stream
of health states is appropriate for medical decision
making, mainly because of the availability of extensive information on the individual.
The objective of this paper is to develop models for health policy analysis. We conjecture, however, that the models in the previous sections could
serve as a partial basis for cardinal-utility models
for medical decision making. The medical decisionmaking models could be developed by combining
these models with discounting models for discreteand continuous-time outcomes in Harvey (1986; 1988;
1995; 1998a, b). The models in Harvey (1988) are
expected-utility models, and those in the other papers
are ordinal-utility models.
This section is concerned with the aggregation of
cardinal scale amounts for the individuals in a population into a cardinal scale amount for the population
as a whole. We present a cardinal-utility model that is
parallel to the expected-utility model due to Harsanyi
(1955). Except for one generalization (as explained
below), the model presented is a specialization for
the context of population health of the cardinal-utility
social welfare model in Harvey (1999).
An alternative cardinal-utility welfare model has
been developed by Dyer and Sarin (1979). However,
265
it is an “algebraic” model, and hence does not provide health scales that are continuous functions of the
health outcomes. For this reason, we use the “topological” model in Harvey (1999).
As in §2, the health outcomes can be of any type.
Suppose that the individuals in a population are
indexed by i = 1 n. A health outcome for an
ith individual will be denoted by hi , and the set of
such outcomes will be denoted by Hi . A distribution of health outcomes over the population will be
called a health distribution and will be denoted by h =
h1 hn ). We assume that the set of health distributions is the product set H = H1 × · · · × Hn .
The social welfare model in Harvey (1999) assumes
(as stated in terms of social health) that for each ith
individual there is a cardinal relation i that compares changes in (population) health distributions.
Because we wish to develop models for the presciptive purpose of examining social health policies, we
will assume that i reflects social values regarding health outcomes for the ith individual. Thus, i
depends only on the ith components hi in health distributions. Indeed, we define i as a cardinal relation
that compares changes hi → hi in health outcomes for
the ith individual. We will call i an individual cardinal
relation.
This simplification excludes models in which i
reflects the personal values of an individual who is
altruistic. Such preferences are included in the social
welfare model in Harvey (1999).
The social welfare model also assumes that there
is a cardinal relation for changes in health distributions. We will denote such a relation by P and call
it a population cardinal relation. A cardinal scale for an
individual cardinal relation i will be called an individual scale and will be denoted by wi hi , and a cardinal scale for a population cardinal relation P will be
called a population scale and will be denoted by wP (h).
The generalization of the social welfare model in
Harvey (1999) is to omit the assumption that each
set Hi i = 1 n, of health outcomes is topologically connected. Instead, we assume that the individual and population cardinal relations are proper. As
discussed in §3, this approach permits some of the
variables that describe the health outcomes to have
finite domains rather than interval domains provided
that each set Hi is “preferentially connected.”
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The key assumption, both in Harvey (1999) and
here, is the condition below that connects a population cardinal relation to the individual cardinal
relations. This condition is analogous to Pareto conditions in additive-utility models and in expected-utility
models.
Cardinal Pareto agreement. For any changes
h → h and k → k in health distributions, if there is
an ith individual such that hj → hj is indifferent to
kj → kj for all j = i, then preferences between h → h
and k → k according to the population cardinal
relation P agree with preferences between hi → hi
and ki → ki according to the individual cardinal
relation i .
In other words, when society does not need to make
trade-offs between changes in the health of different
persons, then preferences between changes in public health agree with preferences between changes in
health for the one person who matters. In this sense,
the Pareto condition is a requirement of individual
sovereignty.
Suppose that (hi = h; hk k = i) denotes a health distribution in which the ith individual has the health
outcome h and the other individuals have the outcomes indicated. Cardinal Pareto agreement implies
that preferences between changes (hi = h; hk k = i →
hi = h ; hk k = i) according to the relation P agree
with preferences between changes h → h according
to the relation i (and thus are independent of the
health outcomes hk k = i).
Theorem 5. A proper population cardinal relation P
has cardinal Pareto agreement with a set of proper individual cardinal relations i i = 1 n, if and only if
for any individual scales wi hi ) there exist weights ai > 0,
i = 1 n, such that the function
wP h = a1 w1 h1 + · · · + an wn hn (5)
is a population scale. The weights ai in (5) are unique up
to a common positive multiple.
The above model compares changes in population
health by a scale that is a linear function of scales that
compare changes in health for the members of the
population. Hence, it provides a foundation for the
utilitarian principle that cardinal utility for a society
is the sum (or a weighted sum) of cardinal utilities
for the members of the society. The weights ai in the
linear function (5) will be called interpersonal weights,
and the model will be called a linear health model.
The result that cardinal Pareto agreement implies a
linear function (5) may seem surprising. The explanation lies in the difference between ordinal and
cardinal uniqueness. Pareto agreement for ordinal
preferences implies that a population ordinal scale is
a function of individual ordinal scales that is strictly
increasing in each scale, but it does not imply any
form of the function. As noted above, Pareto agreement for cardinal preferences implies that for fixed
outcomes hk k = i, a population scale wP (h) restricted
to health distributions of the form (hi = h; hk k = i)
is a cardinal scale for the individual cardinal relation i . For an individual scale wi hi , it follows that
for these health distributions, wP h = ai wi hi + bi ,
for some ai > 0 and bi . Theorem 5 can be viewed
as an extension of this uniqueness property. Moreover, the social welfare model developed by Harsanyi
(1955) for expected utility can be viewed in the same
manner.
Next, we digress to discuss the special but common
case in which there are only two health outcomes of
interest. We show that in this case a policy analyst
can assume that the individual cardinal relations i
are the same even though it is apparent that they are
not; indeed, the analyst does not even need to assess
a common individual scale.
Suppose that an analyst wishes to use a linear
health model in a policy study. Most likely, he or
she can envision a wide range of health outcomes for
the individuals in the defined population. Assume,
however, that the health distributions to be examined
in the study contain only two health outcomes. The
health distributions in the study differ in that different subgroups of the population have one outcome or
the other. Our impression is that this situation is common. It may occur for various reasons; e.g., the available data provide needed health information only on
the two outcomes, or the analyst decides that only the
two outcomes are relevant to the study.
Corollary 1. In a linear cardinal model, assume that
there exist health outcomes h0 and h1 such that they are
in each set Hi , and h1 is preferred to h0 by each individual. Then, there exist unique individual scales wi hi ) such
that wi h0 = 0 and wi h1 = 1 for i = 1 n. (Hence,
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the individual scales wi hi are equal for h0 and are equal
for h1 .)
Suppose that wP h = a1 w1 h1 + · · · + an wn hn is
an associated population scale. For a health distribution h, define Sh = i = 1 n: hi = h1 }. Then, for
any health distribution whose only health outcomes
are h0 and h1 , the population scale wP h) reduces to
the form
ai (6)
wP h =
i∈Sh
In a model as described in Corollary 1, the only
assessment task for the analyst is to assess the interpersonal weights ai i = 1 n. Section 8 discusses a
procedure for determining the weights ai by assessing
so-called person trade-offs.
7.
Equal Individual Scales and Equal
Interpersonal Weights
This section discusses two simplifications of a linear
health model. First, we present a model in which conditions on preferences imply equal individual scales,
and then we present a model in which conditions on
preferences imply both equal scales and equal interpersonal weights. These special linear health models
will be much easier to implement than the general
model.
7.1. Equal Individual Scales
We expect that in most health policy studies the health
outcome sets Hi can be chosen as a common set H ,
and the individual cardinal relations i can be chosen as a common relation I . The set of possible outcomes for an individual can still vary significantly
from one person to another; for example, younger
people and older people may have quite different sets
of possible outcomes. What is assumed is that preferences between changes in outcomes (whether possible
or not) do not vary significantly from one person to
another. This condition does not mean that society is
indifferent between a change for one person and the
same change for another person.
Theorem 6. In a linear cardinal model, assume that the
health outcome sets Hi are equal and that the individual
cardinal relations i are equal. Then, for any individual
scale wI hi for the common relation i , there exist weights
ai > 0 i = 1 n, such that the function
wP h = a1 wI h1 + · · · + an wI hn (7)
is a population scale. The weights ai in (7) are unique up
to a common positive multiple.
For the purposes of comparing policy options and
of evaluating a single option, the gain in simplicity
of a population scale (7) over a population scale (5)
appears to outweigh its loss in generality; indeed, we
regard Theorem 6 as the basic model for purposes of
application. Sections 8 and 9 discuss reasons for this
opinion. The potential use of a population scale (7) to
evaluate a policy option as well as to compare policy
options is reflected in the terminology below.
Definition 3. A model as described in Theorem 6
will be called a population health evaluation model, and
a scale (7) in such a model will be called a population
evaluation scale.
7.2. Equal Interpersonal Weights
In this subsection, we ask when the interpersonal
weights in a population health evaluation model will
be equal. A common individual scale wI hi is cardinally unique, and for a given scale wI hi , the interpersonal weights are unique up to a positive multiple.
It follows that whenever there exist equal weights for
some population scale, then any weights for any population scale are equal. Hence, we can speak loosely
of “equal weights.”
The result below provides conditions on preferences that imply both equal individual cardinal relations and equal interpersonal weights. We will call a
model that satisfies these conditions an equal weights
model. In this model, the only assessment task is to
assess a common individual scale wI hi . For any such
scale, the corresponding function wP (h) is a population scale.
Theorem 7. Suppose that the health outcome sets Hi in
a linear cardinal model are equal, and that h0 is a fixed but
arbitrary health outcome. Then, the following conditions
on preferences are equivalent, and they are satisfied if and
only if the model is an equal weights model.
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(a) For any individuals i and j and any health outcome
h, the changes in health distributions
hi = h0! hk = h0 k = i
→ hi = h! hk = h0 k = i and
hj = h0! hk = h0 k = j
→ hj = h! hk = h0 k = j
(8)
are indifferent according to the population cardinal
relation.
(b) The individual cardinal relations i are equal, and
for any individuals i and j there exists a health outcome
h nonindifferent to h0 such that the changes (8) in health
distributions are indifferent according to the population
cardinal relation.
(c) For any individuals i and j and any health outcome
h, the health distributions
hi = h! hk = h0 k = i
and
hj = h! hk = h0 k = j
(9)
are indifferent according to the ordinal relation associated
with the population cardinal relation.
(d) The individual cardinal relations i are equal, and
for any individuals i and j there exists a health outcome h
nonindifferent to h0 such that the health distributions (9)
are indifferent according to the ordinal relation associated
with the population cardinal relation.
7.3.
Should Different Interpersonal Weights Be
Assigned to Different Age Groups?
What differences among the members of a population should imply unequal importance for the same
changes in health outcomes for different individuals?
One possibility is differences in age. Here we discuss
several reasons why different interpersonal weights
might be assigned to people of different ages.
First, age can be related to the social importance of
individuals. Murray (1994) and Murray and Acharya
(1997) argue that because working-aged adults make
greater economic contributions than do children or
seniors, their health has greater social importance.
The authors assign unequal weights to persons of different ages as part of a “disability-adjusted life years”
(DALY) scale for health outcome-streams. Often, in a
DALY scale the weight assigned to a person aged 25
is about twice the weight assigned to someone aged 6
or 67. See Anand and Hanson (1997) for a critical
review. In our opinion, it is not justified to infer
unequal social importance from unequal economic
roles. Moreover, U.S. health policies (e.g., health programs for children and Medicare) implicitly assign
greater weights to children and to seniors than to
working-aged adults.
Second, age can be related to concerns for equality. Greater age tends to imply greater lifetime health
and longevity. Williams (1997) argues for assigning weights that favor equality in people’s lifetime quality-adjusted life years. The weights resulting
from this “fair-innings argument” will be greater for
younger people than for older people.
Greater age also tends to imply lesser future health
and longevity. One can argue for assigning weights
that favor equality in people’s future quality-adjusted
life years. The weights resulting from this futureequality argument will be greater for older people
than for younger people.
Another criterion—one in the spirit of utilitarianism—is to assign weights that favor the sum of
improvements in health and longevity. This criterion
is meaningful for cardinal utility but not for ordinal or expected utility. First, assume that health outcomes are defined as health states—with the duration
of a health state and any ensuing states unknown. A
change in health states (e.g., from fatality to a state
of no health problem) is likely to produce a greater
improvement in a person’s health and longevity for
a younger person than for an older person. So when
health outcomes are defined as health states, the
resulting weights may be greater for younger people. But when health outcomes are defined as health–
duration pairs, a change in health outcomes entails
the same improvement in health and longevity for an
older person as for a younger person. So for health–
duration pairs, knowing the ages of the population
members does not provide a reason for assigning
unequal weights.
Typically, the comparison of policy options in an
equal weights model with health states will differ
from a comparison of the same options in an equal
weights model with health–duration pairs or health
outcome streams. In particular, fatalities will count the
same regardless of age in the model with health states,
whereas a fatality for an older person will count less
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than a fatality for a younger person in the model with
health–duration pairs or health outcome streams. In
such a context, one cannot assign equal weights
both in the model with health states and in the
model with health–duration pairs or health outcome
streams. Similar observations have been made for
expected-utility models; see, e.g., Bordley (1994) and
Hammitt (2002).
Below, we define two other types of assessments
that an analyst can use to determine an individual
scale or interpersonal weights. In the case of an individual scale, the analyst can first use assessments of
the above type to imply that the scale has a functional
form or belongs to a parametric family and then use
assessments of the types defined below to determine
the scale.
8.
8.2.
Assessment Procedures
In this section and the next, we describe two types
of procedures concerning a cardinal scale; those in
this section are for constructing a cardinal scale, and
those in the next section are for using a cardinal scale.
In both sections, we assume that social values satisfy the conditions for a population health evaluation
model.
Here, we describe statements of social values that
a policy analyst can use to determine an individual
scale and interpersonal weights (and hence a population scale). The statements might be, for example, interview responses by a policy maker, modeling
judgments by the analyst, or assumptions of social
values that the analyst uses to evaluate policy options
conditional on the assumptions. For brevity, we will
refer to any such statements as assessments, and we
will refer to their use to determine a scale or weights
as assessment procedures.
We limit the discussion to assessment procedures
that require both individual and population cardinal
relations and Pareto agreement between them. We do
not discuss assessment procedures such as time tradeoffs that have been developed for noncardinal preference relations.
8.1. Assessing Conditions on Cardinal Preferences
One method of assessment is to assess a condition on cardinal preferences (such as those defined
in previous sections) that implies a property of an
individual scale or that implies equal interpersonal
weights. Some of the conditions regarding an individual scale imply that the scale has a functional form,
e.g., wI s = f qs and wI s t = Dtf qs; some
determine a scale, e.g., wI s = qs and wI s t =
tqs; and some imply that the scale belongs to a parametric family, e.g., wI s = erqs , r > 0, and wI s t =
r −1 1 − e−rt qs, r > 0.
Assessing Person Trade-Offs for Changes
in Health Quality
The assessments defined here state social trade-offs
between a change in health outcomes and the number
of people who experience the change. Similar assessments, called person trade-offs, have been proposed in
contexts that do not involve cardinal utility; see, e.g.,
Richardson (1994), Nord (1995), Murray and Lopez
(1996), and Green (2001). Østerdal (2009) showed that
an ordinal additive social welfare model based on person trade-off assessments must be of a certain type
in order to satisfy normative conditions such as the
Pareto principle.
For our discussion, first suppose that the health
outcomes are health states and that the model is an
equal weights model. Then, Theorems 1 and 7 imply
that there is a population scale of the form wP s =
f qs1 + · · · + f qsn ), where s = s1 sn denotes
a health distribution.
Select a best health state s 1 and a worst health state
0
s , and normalize f q such that f qs 1 = 1 and
f qs 0 = 0. Because the conversion function f q is
now unique, it remains to find more of its values and
then to interpolate between these values.
To find a value of f q, an analyst can select a health
state s that is easy to visualize and assess a number k
in one of the following statements.
(a) Society is indifferent between an improvement
from s 0 to s for the entire population and an improvement from s 0 to s 1 for k members of the population.
(b) Society is indifferent between a worsening from
s 1 to s for the entire population and a worsening
from s 1 to s 0 for n − k members of the population.
In statement (a) or (b), society is making a tradeoff between a degree of improvement or worsening
and a number of population members that experience
the improvement or worsening. Statement (a) implies
that nf qs − f qs 0 " = kf qs 1 − f qs 0 ))],
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270
and statement (b) implies that nf qs 1 − f qs" =
n − kf qs 1 − f qs 0 ))]. Either equation implies
that the conversion function has the value f qs =
k/n. Thus, a value f qs of the conversion function can be interpreted as a proportion p = k/n of the
population.
The person trade-offs in (a) and (b) can also be
stated as comparisons of health distributions or as
comparisons of changes in health distributions. The
following statement is an example.
(c) Society is indifferent between a health distribution in which everyone in the population has the
health state s and a health distribution in which k
members have the health state s 1 and the remaining
n − k members have the health state s 0 .
Statements (a)–(c) are equivalent in the sense that
the assessed number k should be the same in each
one. As a question in behavioral economics, one
could investigate whether people do in fact assess
the same number k in these statements or exhibit
a systematic discrepancy similar to that between
people’s willingness-to-pay and willingness-to-accept
amounts.
When the interpersonal weights are unequal, one
can determine a population scale in two steps. First,
select a subpopulation in which the weights are equal
and use conditions on cardinal preferences and/or
assessments of person trade-offs to determine an individual scale, wI s = f qs). Then, order the population members in some fashion and use assessments
of person trade-offs analogous to (a) and (b) to determine a set of interpersonal weights.
To explain the second step for (a) or (b), suppose
that the phrase “for k members” in (a) is replaced
by “for the first k members,” and the phrase “for
n − k members” in (b) is replaced by “for the last
n − k members.” Suppose that wP s = a1 f qs1 +
· · · + an f qsn ) denotes a population scale with the
given ordering of the individuals and that the weights
are normalized such that a1 + · · · + an = c for a specified number c (e.g., c = 1 or c = n). Then, a modified assessment (a) or (b) implies that a1 + · · · + ak =
cf qs). Here, cf qs) is known. The analyst can
select a list of health states with different degrees of
health quality, and thus obtain a list of equations. The
weights can then be calculated by solving the simultaneous equations.
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Next, suppose that the health outcomes are health–
duration pairs, the weights are unequal, and social
values satisfy the conditions in Theorem 3. Then,
there is a population scale, wP s t = a1 Dt1 f qs1 + · · · + an Dtn f qsn ), where (s t = s1 t1 sn tn ) denotes a health distribution. The analyst can
select a duration t and consider health distributions in
which every health–duration pair has the duration t.
Then, the scale wP s t) reduces to a population scale
regarding health states, and the analyst can use the
assessment procedures described above to determine
a conversion function f q and interpersonal weights
ai . Finally, the analyst can use any of a wide variety of methods to determine a cumulative discounting
function Dt.
8.3.
Assessing Social Attitudes Toward Inequality
in Health Quality
In an assessment (a)–(c) as described above, an analyst selects a health state s and thus a health quality
q = qs, and assesses a number k and thus a proportion k/n. This procedure can be reversed; the analyst can select a number or a proportion and assess
a health state or a health quality. In such a reverse
person trade-off procedure, one assesses amounts, q =
f −1 k/n), and thereby finds a value, f q = k/n, of the
conversion function at each assessed amount q.
By using a reverse person trade-off procedure, an
analyst can select the proportion k/n = 12 . This proportion is relatively easy to visualize; half the population
has the best health state, s 1 , and the other half has the
worst health state, s 0 . The inverse amount q = f −1 12 is the health quality of a middling health state s relative to s 0 and s 1 ; that is, the changes, s 0 → s and
s 0 → s 1 , are indifferent according to the individual
relation.
A reverse person trade-off involves population
health and thus is an assessment of social values.
Below, we explain how it can be interpreted as an
assessment of social attitude toward inequality in
health quality. Suppose that q 1 = qs 1 ) and q 0 =
qs 0 ) are the best and worst amounts of health quality, respectively, and that there is a population scale,
wP s = f qs1 + · · · + f qsn ).
Suppose that s0 1 is a health distribution in which
half the population has the health quality q 1 , whereas
the other half has the health quality q 0 , and that savg
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is a health distribution in which the entire population
has the average health quality, q avg = 12 q 0 + 12 q 1 . The
average health quality in both s0 1 and savg is q avg .
But savg has no inequality in health quality, whereas
s0 1 has an extreme degree of inequality in health
quality. Is savg preferred to s0 1 ?
To examine this question, suppose that sindiff is a uniform distribution that is indifferent to s0 1 and that
q indiff is the common health quality in sindiff . Then, the
comparison between the distributions savg and s0 1
is the same as that between the health quality q avg
in savg and the health quality q indiff in sindiff . If q avg >
q indiff , then the distribution savg with no inequality in
health quality is preferred to the distribution s0 1
with extreme inequality in health quality. In this sense,
society is averse to inequality in health quality with
respect to q 0 and q 1 . In the case that q avg > q indiff for
any pair of health qualities q 0 < q 1 , we will say that
society is averse to inequality in health quality. The
reader can check that the population relation satisfies
this condition if and only if the conversion function
f q is strictly concave.
Suppose that an analyst assesses that social values
have cardinal constancy with respect to a quality scale
qs. Then, by Theorem 2, the conversion function f q
has a linear-exponential form (1). A single reverse person trade-off suffices to determine the parameter r in
(1), and the value of r determines society’s attitude
toward inequality in health quality for any q 0 < q 1 .
By an extension of the above terminology, society is
averse, neutral, or prone to inequality in health quality when r is negative, zero, or positive, respectively.
9.
Evaluation Procedures
Suppose that in a study of population health, an analyst has identified a set of policy options, has estimated the changes h → h in population health that
the options may produce, and has constructed a population health evaluation model that represents social
values relevant to the study. This section describes
procedures by which he or she can use a population
scale in the model to evaluate a policy option, that
is, to calculate and report information on that particular option. Most of these procedures lead directly to
methods for comparing two options: the analyst simply determines the difference or the percentage difference in their evaluations.
271
These procedures are meaningful only in a context
of cardinal relations. For a given policy study, an analyst can choose a procedure that he or she judges to
be appropriate for the situation.
In most policy studies, the issue is how to modify current policy (or whether to modify it). Typically, each policy option (including the current policy)
implies a probability distribution of health distributions. In this paper, however, we discuss only the case
in which it suffices to replace any probability distribution by a predicted health distribution. Thus, we
focus on the difficulty in recommending a policy that
is due to the complexity of the health distributions.
We assume that the analyst models each policy
option as a change from a initial health distribution
to the health distribution that is predicted to result
from the option. To provide a more concrete discussion, we will assume that the initial distribution is the
health distribution that is predicted to result from the
current policy. The reader should note, however, that
the evaluation procedures that are discussed do not
depend on this choice; in particular, the analyst could
choose a simple, hypothetical health distribution as
the initial distribution.
Assume that the population scale is a weighted
average, wP h = a1 wI h1 + · · · + an wI hn , where
a1 + · · · + an = 1. The predicted health distributions for
the current policy and for a policy option will be
denoted by hcur and by hopt , respectively. Thus, the
analyst wishes to evaluate each change hcur → hopt .
The health outcomes can be of any type.
9.1. Reporting a Change in Health Distributions
One type of procedure for reporting a change hcur →
hopt for a policy option is to calculate an indifferent change h → h with simpler health distributions
and report the calculated change. The change h → h
can be calculated by the formula wP h − wP h =
wP hopt − wP hcur .
A variety of procedures are possible. For example, the analyst could select the initial health distribution h to be a uniform distribution (i.e., a distribution
h = h h) in which everyone has the same health
outcome h), and for each policy option calculate a
final distribution h that is also a uniform distribution.
The advantage of these procedures is that they do
not require the population scale to have a simple interpretation, e.g., as a quality scale or as a QALY scale.
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The reported information is a change in health distributions and not a difference in population scale
amounts. The disadvantage of the procedures is that
they lack the usual features of measurement; for example, they do not measure a change hcur → hopt as an
amount of a single variable.
9.2. Reporting a Subpopulation
Suppose that the health outcomes in a policy study
range from a worst health outcome, denoted by h0 ,
to a best health outcome, denoted by h1 , and that a
policy option implies a change hcur → hopt that is an
improvement, or at least not a worsening, in population health.
First, assume that the population health evaluation
model is an equal weights model. For each k = 0
n, suppose that hk denotes a health distribution in which k individuals have the health outcome
h1 , and the other n − k individuals have the health
outcome h0 . In particular, h0 denotes the health
distribution in which everyone has the worst outcome h0 , and hn denotes the health distribution in
which everyone has the best outcome h1 .
In such a situation, one can calculate for any change
cur
h → hopt a change h0 → hk that is approximately
indifferent to hcur → hopt in the sense that the difference, wP hk − wP h0 ), is approximately equal to the
difference, wP hopt − wP hcur ). Then, one can report
either the number k or the proportion k/n as an evaluation of the change hcur → hopt .
If the population health evaluation model is not an
equal weights model, then the population members
must be ordered in some fashion, e.g., by increasing
or decreasing age. Suppose that hk denotes a health
distribution in which the first k individuals have the
health outcome h1 . Then, a number k or a proportion
k/n can be calculated as above.
If the individual scale wI hi is normalized such
that wI h0 = 0 and wI h1 = 1, then in the case of
equal weights, k is determined by the first formula
that follows, and in the more general case of (possibly) unequal weights, k is determined by the second
formula:
k = nwP hopt − wP hcur a1 + · · · + ak = wP hopt − wP hcur These evaluation procedures can be used when the
population scale lacks a simple interpretation because
they use a scale measured in units which differ from
those of the population scale.
9.3. Reporting an Average Change in Health
We first discuss procedures for the case in which the
health outcomes are health states, and then discuss
procedures for the case in which the health outcomes
are health–duration pairs. In each case, a weighted
average population scale wP h) will serve as an evaluation scale. The key is to identify circumstances in
which the scale wP h) has a simple interpretation.
9.3.1. Evaluation Scales for Health States. Assume
that an individual scale has the form wI s = f qs),
as described in §4. For a health distribution s =
s1 sn ), suppose that q̄ denotes its weighted average quality scale amount, that is, q̄ = a1 qs1 + · · · +
an qsn .
If preferences have cardinal neutrality for a quality scale qs, then by Theorem 2 one can choose the
individual scale as wI s = qs. Then, wP s = a1 qs1 + · · · + an qsn = q̄. Hence, the scale wP s) has a simple
interpretation, namely, as an average health quality,
and thus it can be used as an evaluation scale. To
evaluate a change scur → sopt , an analyst can use the
formula
wP sopt − wP scur = q̄ opt − q̄ cur opt
cur
(10)
Because the difference, q̄ − q̄ , in average health
quality from scur to sopt equals the average change in
health quality from scur to sopt , the analyst can report
the average change in health quality calculated by (10)
as an evaluation of the change scur → sopt .
If preferences do not have cardinal neutrality for
qs, then the conversion function f q is nonlinear.
In this case, the population scale amount of a health
distribution may be unequal to its average health
quality, and different health distributions with the
same average health quality may be nonindifferent.
Moreover, a change scur → sopt in health distribu
tions may be preferred to a change scur → sopt , even
though the average change in health quality is less for
scur → sopt than it is for scur → sopt . For this reason,
it can be misleading to evaluate a change scur → sopt
by its average change in health quality when preferences do not have cardinal neutrality for the health
quality scale.
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9.3.2. Evaluation Scales for Health–Duration Pairs.
Assume that an individual scale has the product form wI s t = Dtf qs), as described in §5.
A health distribution will be denoted by (s t =
s1 t1 sn tn ).
First, consider the use of duration as an evaluation
scale. As mentioned in §5, the QALYs of a health–
duration pair are defined as follows. First, specify a
state s 1 of good health. Then, for any health–duration
pair (s t), find a health–duration pair (s 1 &) such
that (s t) and (s 1 &) are indifferent, i.e., society is
indifferent between a person spending a duration t in
a health state s and spending a (presumably shorter)
duration & in the (presumably better) health state s 1 .
Here, it follows that Dtf qs = D&f qs 1 ).
Typically, the scale of durations & defined in this
manner is called a QALY scale. We will call a duration & the QALY duration of a health–duration pair
(s t), where the redundant term “duration” is added
to emphasize that a QALY scale is a time scale.
Define = &1 &n ) and s1 = s 1 s 1 ).
Pareto agreement implies that a health distribution
(s t) is indifferent to the health distribution (s1 )
with a uniform health state of s 1 because for each
individual the corresponding health–duration pairs
are indifferent.
For convenience, we will assume that the conversion function f q is normalized such that
f qs 1 = 1. It follows that & = D−1 Dtf qs).
If preferences are cardinal timing neutral, and thus
present and future outcomes have the same importance, then by Theorem 4 one can choose Dt = t.
Then, & = tf qs). Suppose that &¯ denotes the average QALY duration of a health distribution (s t), that
is, &¯ = a1 &1 + · · · + an &n . Then, wP s t = wP s1 , =
a1 &1 f qs 1 + · · · + an &n f qs 1 = &.
¯ Therefore, the
population scale can be interpreted as an average
QALY duration, and thus it can be used as an evaluation scale. To evaluate a change (s tcur → s topt , an
analyst can use the formula
wP s topt − wP s tcur = &¯ opt − &¯ cur (11)
Because the difference, &¯ opt − &¯ cur , in average QALY
duration from (s tcur to (s topt equals the average
change in QALY duration from (s tcur to (s topt , the
analyst can report the average change in QALY duration as an evaluation of the change (s tcur → s topt .
If preferences are not cardinal timing neutral, and
thus future outcomes are discounted, then the cumulative discounting function Dt is nonlinear. In this
case, the population scale amount of a health distribution may be unequal to its average QALY duration, and different health distributions with the same
average QALY duration may be nonindifferent. Moreover, a change, scur → sopt , in health distributions may
be preferred to a change, scur → sopt , even though
the average change in QALY duration is less for
scur → sopt than it is for scur → sopt . In a model that
discounts future outcomes, it can be misleading to
evaluate a change scur → sopt by its average change in
QALY duration.
What should be done in the case of nonzero discounting? Here, we propose a procedure that is dual
to the QALY procedure discussed above in that it uses
health qualities rather than durations as an evaluation scale. To define a scale based on health quality, first select a duration t 1 that is to play the role
of the health state s 1 . Perhaps, t 1 is an extremely
long duration (what might be called a Methuselahn
duration). Then, for any health–duration pair (s t),
find a health–duration pair (s t 1 such that (s t)
and (s t 1 ) are indifferent, i.e., society is indifferent
between a person spending a duration t in a health
state s and spending a (presumably longer) duration
t 1 in the (presumably worse) health state s . Suppose
that ' denotes the health quality of the health state s ,
that is, ' = qs ). We will call ' the years-adjusted life
quality of the health–duration pair (s t), and we will
abbreviate this phrase as YALQ. To emphasize that a
YALQ scale is a health quality scale, we will call a
health quality ' a YALQ quality.
Define t1 = t 1 t 1 ) and s = s1 sn ).
Pareto agreement implies that a health distribution
(s t) is indifferent to the health distribution (s t1 )
with a uniform duration of t 1 because for each individual the corresponding health–duration pairs are
indifferent.
As in the discussion of health states, suppose that
social preferences have cardinal neutrality for qs.
Then, by Theorem 4 one can choose the individual
scale as wI s t = Dtqs − qs0 ), where s0 is the
standard state defined in §5. We will assume that
the quality scale qs has been normalized such that
qs0 = 0, and thus wI s t = Dtqs. We also assume
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that the cumulative discounting function Dt has
been normalized such that Dt 1 = 1.
Suppose that '¯ denotes the average YALQ quality
of a health distribution (s t, that is, '¯ = a1 '1 + · · · +
an 'n . Then, wP s t = wP s t1 = a1 Dt 1 '1 + · · · +
an Dt 1 'n = '.
¯ Hence, wP s t) can be interpreted as
an average YALQ quality, and thus it can be used as
an evaluation scale. To evaluate a change (s tcur →
s topt , an analyst can use the formula
wP s topt − wP s tcur = '¯ opt − '¯ cur (12)
Because the difference, '¯ opt − '¯ cur , in average YALQ
quality from (s tcur to s topt equals the average
change in YALQ quality from (s tcur to s topt , the
analyst can report the average change in YALQ quality
calculated by (12) as an evaluation of s tcur → s topt .
10.
A Hypothetical Application for
Hypertension
This section illustrates how the models and procedures in this paper can be applied. The example we
present concerns the evaluation of a proposed therapy
for hypertension, known informally as high blood
pressure. The example is hypothetical, both in the
data from clinic trials and in the assessment of individual and population scales. It is also hypothetical
in that neither of us has expertise on hypertension.
We take most of our information from a summarizing report, Chobanian et al. (2003), published by the
American Heart Association.
Optimal blood pressure is defined as a systolic
blood pressure (SBP) of 115 mm Hg and a diastolic
blood pressure (DBP) of 75 mm Hg (or is defined
as SBP less than 120 mm Hg and DBP less than
80 mm Hg). Hypertension is defined as an SBP greater
than 140 mm Hg or a DBP greater than 90 mm Hg.
The World Health Organization estimates that suboptimal blood pressure is responsible for about twothirds of cerebrovascular disease and about one-half
of ischemic heart disease worldwide.
Blood pressure can be measured by two general
methods. The more accurate method, called ambulatory blood pressure monitoring (ABPM), is to have
a person wear a monitoring device during a 24-hour
period. The less accurate method, which we will call
office measuring (OM), is to measure the person’s
blood pressure in a medical office. As discussed in
Chobanian et al. (2003), there are significant and variable differences in the measurement results of the two
methods.
Stage I hypertension is defined in Chobanian et al.
(2003) as an SBP in the range of 140–160 mm Hg and
a DBP in the range of 90–100 mm Hg as measured
by the OM method. Because Stage I hypertension is
defined in terms of the OM method, individuals with
this diagnosis can have a wider range of actual SBP
and DBP amounts.
Suppose that as part of larger project a research
team wished to evaluate the effectiveness of a
proposed therapy for reducing blood pressure in
individuals with Stage I hypertension. The therapy
is a combination of a drug regimen and counseling
sessions to promote lifestyle changes. It was expected
to have tolerable but not insignificant side effects of
the drugs and disruption due to the lifestyle changes.
In part because the therapy includes lifestyle counseling in addition to drugs, it was anticipated to have
significantly different effects on blood pressure for
different populations defined by factors such as level
of education, age, and other health variables.
The goal of the subproject was to evaluate the ability
of the therapy to reduce the risks of cerebrovascular
disease and ischemic heart disease associated with
hypertension for prospective populations of individuals who are diagnosed with Stage I hypertension—and
if possible to compare the predicted risk reduction of
the therapy in different defined populations.
First, the researchers defined six categories of individuals, called Categories I–VI, based on information
other than blood pressure. Then, they conducted clinical trials of the therapy for persons with Stage I
hypertension. For each participant, extensive information was gathered, and the person was placed
in one of six sample populations, also called Categories I–VI, according to the information. Measurements of blood pressure were taken at the beginning
and at the end of a four month period (with different periods for different individuals). The more accurate ABPM method was used because the researchers
were interested in the ability of the therapy to reduce
an individual’s actual blood pressure. Each person’s
blood pressure measurements were aggregated into
a single SBP amount and a single DBP amount such
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that greater SBP and DBP amounts indicated greater
health risks according to previous studies (see below).
A health outcome was defined as a health state
measured by a person’s calculated SBP and DBP.
Thus, s = (SBP, DBP). The data for each sample population consisted of an initial distribution sint of health
states s int at the beginning of the trial periods and a
final distribution sfin of health states s fin at the end
of the trial periods. Hence, the data from the clinical trials consisted of the information in the changes
fin
int
sint
k → sk k = 1 6. The initial distributions sk
were appreciably different; in particular, they had different averages of SBP and of DBP.
The researchers next consulted studies that related
blood pressure and risks of cerebrovascular disease
and ischemic heart disease. The studies had considered different types of risk, i.e., different measures of
the risk of different categories of morbidity and fatality. One type of risk was the probability of a major
stroke during a person’s lifetime. For persons with
Stage I hypertension, the risk measures seemed to
have linear relationships with one another, i.e., greater
differences in one risk measure seemed to correspond
with greater differences in the other risk measures.
Because of this concurrence between types of risk,
the researchers were able to define an individual cardinal relation I on changes s → s in blood pressure
in terms of the consequent changes in risk without
specifying the type of risk. For the same reason, they
were able to define a population cardinal relation P
on changes s → s in health distributions in terms of
changes in the sum of the risks for the population
members without specifying the type of risk.
The researchers judged that these cardinal relations
satisfied the conditions for the existence of cardinal scales, e.g., the condition of dynamic consistency
mentioned in §3. They also judged that the cardinal relations satisfied the conditions for a population health evaluation model with equal interpersonal
weights.
The blood pressure variables and the risk scales
measured harms and disutilities rather than goods
and utilities. Recognizing this, the researchers translated the population health evaluation model into a
harm/disutility model. In this discussion, however,
we will translate as we proceed.
275
The second part of the project was to assess individual and population scales. Based on the risk studies, the researchers judged that for fixed SBP and
DBP amounts, the health states (SBP + 20, DBP)
and (SBP, DBP + 10) are associated with about the
same risk, and that intermediate health states, e.g.,
(SBP + 10, DBP + 5), have about the same risk. Based
on these judgments, they assessed a linear quality
scale, qs = −20−1 (SBP − 115 − 10−1 (DBP − 75).
Later, the researchers decided for reasons of convenience to redefine the quality scale as one-half this
amount, namely, as qs = 12 −20−1 (SBP − 115 − 10−1
(DBP − 75)].
The researchers next observed that an increase in
SBP or DBP from a higher level has a greater effect
on risk than the same increase from a lower level;
e.g., an increase from 140 to 155 mm Hg is worse
than an increase from 115 to 130 mm Hg. Hence, the
functional dependence of risk on blood pressure is
convex over the observed range. By translating from
harms to benefits, the conversion function f q, which
represents the dependence of an individual scale on
qs, is concave.
The researchers modeled f q by verifying the condition of cardinal constancy, at least as a rough
approximation. This condition can be stated by fixing, for example, the variable DBP and considering
different amounts of the variable SBP. Then, cardinal constancy states that if two changes SBP → SBP∗
and SBP∗ → SBP incur equal changes in risk, then for
any amount the changes SBP + → SBP∗ + and
SBP∗ + → SBP + incur equal changes in risk.
By Theorem 2, cardinal constancy implies that
there is a conversion function f q having the linearexponential form in (1). The concavity of f q then
implies that there is a conversion function of the form
f q = −erq , and hence an individual scale of the form
wI s = −erqs , for some parameter amount r < 0.
To assess the parameter r, the researchers used the
studies relating blood pressure to risks. Chobanian
et al. (2003, p. 1210) states that, “[f]or every 20 mm
Hg systolic or 10 mm Hg diastolic increase in BP
[blood pressure], there is a doubling of mortality from
both ischemic heart disease and stroke.” This statement does not specify what change in DBP accompanies a 20 mm Hg increase in SBP or vice versa.
Because individuals with a 20 mm Hg higher SBP also
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have a 10 mm Hg higher DBP much more often than
no higher DBP, we guess that the statement refers to
an increase of 20 mm Hg in SBP and an increase of
10 mm Hg in DBP. Such increases in SBP and DBP
correspond to a decrease of one unit in qs.
A risk measure is linearly related to the individual
scale; that is, risk = af q + b. Hence, the above discussion implies that af q − 1 + b = 2af q + b). To
solve this equation for q, differentiate with respect to
q to obtain f q − 1 = 2f q. Because f q = −erq , this
implies that r = −ln 2. Hence, by the definition of qs
wI SBP DBP
= − exp 12 ln 220−1 SBP − 115 + 10−1 DBP − 75" Because the researchers had constructed an equal
weights model, there was no need to assess interpersonal weights. Moreover, a population scale amount
wP s) did not depend on how the individuals in the
population were indexed.
The Category I–VI sample populations had different sizes, as would prospective populations of persons with Stage I hypertension. For that reason, the
researchers chose an averaging population scale, that
is, they chose the scale
wP s =
n
1
w SBPi DBPi n i=1 I
where n is the size of the population, (SBPi , DBPi ) is
the health state of the ith individual in the population,
and wI (SBPi , DBPi ) is as specified above.
The third part of the project was to use the population scale to evaluate (and if possible to compare) the
effectiveness of the therapy to reduce risk in prospective populations consisting of individuals with Stage I
hypertension who have the characteristics of Categories I–VI. Such a population was called a category k
population where k = 1 6. As described above,
fin
the clinical trials provided the data, sint
k → sk k =
1 6, from the six sample populations.
First, the researchers calculated evaluations of the
therapy. They considered each of the three types of
evaluation discussed in §9. Their conclusions were as
follows.
(1) The researchers specified the uniform distribution s0 with s0 = 150, 95) and for each k = 1 6 calfin
culated a uniform distribution sk such that sint
k → sk
is indifferent to s0 → sk . The calculated distributions
ranged from s1 = 137, 88.5) to s6 = 141, 90.5). The
researchers reported that for a category k population,
the reductions in risk due to the therapy would be
equivalent to the reductions in risk due to a uniform
reduction in blood pressure from s0 to sk .
(2) The researchers specified sl = 140, 90) and su =
160, 100) as extreme health states for Stage I hypertension, and for each k = 1 6 calculated a fraction
fin
pk such that the change sint
k → sk is indifferent to a
change from a uniform distribution with su to a distribution in which blood pressure is reduced from su
to sl for a fraction pk of the population and remains
at su for the remaining fraction of the population.
The calculated fractions ranged from p1 = 053 to p3 =
0.41. The researchers reported that for a category k
population, the reductions in risk due to the therapy
would be equivalent to the reductions in risk due to
an extreme reduction in blood pressure from su to sl
in a fraction pk of the population.
(3) The researchers observed that because the confin
version function f q is concave, a change sint
k → sk
would not be indifferent to the change from a uniform distribution with the average amounts of SBP
and DBP in sint
to a uniform distribution with the
k
average amounts of SBP and DBP in sfin
k . For this reason, they did not report the reductions in risk due to
the therapy in category k populations as equivalent
reductions in risk due to changes in average blood
pressure.
The researchers were also able to compare the
effectiveness of the therapy to reduce risk for pairs
of different category k populations. To do so, they
first calculated the differences, wk = wP sfin
k −
int
wP sk k = 1 6. The differences ranged from
w1 = 141 to w6 = 107.
The differences wk show that the therapy would
be most effective for Category I patients and least
effective for Category VI patients. The researchers
were also able to make quantitative comparisons. The
cardinal uniqueness of the population scale implies
that ratios wj /wk are independent of the particular
scale wP s). Moreover, the risk measures in the studies
of cerebrovascular disease and ischemic heart disease
are linearly related to the scales wP s) (because both
are linearly related to wI s). Thus, from the fact that
w1 was 32% greater than w6 , the researchers were
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able to estimate that the reductions in risks would be
about 32% greater for a Category I population than
for a Category VI population and that the risk reductions would be intermediate for the other types. They
reported the percents as concise comparisons of effectiveness for the six types of prospective populations
of persons with Stage I hypertension.
Acknowledgments
The authors thank James K. Hammitt for helpful comments on a memo that initiated this paper, and Alec Morton for helpful comments on this paper. They also thank
the reviewer and associate editor for their comments, especially for the suggestion that they include a hypothetical
application. Charles Harvey is retired from the University
of Houston. His address is 5902 NW Pinewood Place, Corvallis, Oregon 97330.
Appendix. Proofs of Results
Proof of Lemma 1. Suppose that wh is a proper cardinal scale and zh is a cardinal scale (proper or not).
Let W and Z denote the ranges of wh and zh, respectively. Because wh and zh are ordinal scales, we have
zh = f wh for some strictly increasing function f w.
The domain W of wh is a nonpoint interval, and the range
Z of zh and f w can any set of numbers.
= 12 w + 12 w . Then, w
is in W ,
For w w in W , define w
and there exist h h̄ h in H such that wh = w wh̄ =
− w = w − w
implies h → h̄ ∼ h̄ → h ,
wh = w . Thus, w
w
which implies zh̄ − zh = zh − zh̄, which implies
= 12 f w + 12 f w ). Jensen (1905, 1906) proved that any
f w
continuous function defined on an interval that satisfies the
above equation is linear (see, e.g., Aczél 1966, p. 43), and
essentially the same argument can be used to show that any
increasing function that satisfies the equation is linear. The
further arguments are straightforward. Proof of Theorem 1. Because qs is proper, it is continuous and has a nonpoint interval range Iq .
If is proper, then it has a cardinal scale ws with a
nonpoint interval range Iw . Both ws and qs are ordinal
scales, and thus, ws = f qs), for some strictly increasing function f q. Moreover, f q is continuous because its
domain Iq and range Iw are intervals.
Conversely, suppose that has a cardinal scale, ws =
f qs), where the function f q is continuous and strictly
increasing. Then, the function ws is continuous, and thus
its range is an interval. Moreover, its range is nonpoint
because the range of the quality scale qs is nonpoint
and f q is a strictly increasing function defined on that
range. Proof of Theorem 2. We will prove only the forward
implications as the converse implications are straightforward to verify. By Theorem 1, has a cardinal scale f qs,
where the function f q is continuous and strictly increasing
function on the nonpoint interval range Iq of qs.
To show part (i), assume that has cardinal neutrality
for qs. For any amounts q q in Iq , there exist health states
s s ŝ with qs = q qs = q , and qŝ = 12 q + 12 q . Then,
qŝ = 12 qs + 12 qŝ, and thus s → ŝ ∼ ŝ → s by the condition of cardinal neutrality. It follows that f qŝ − f qs =
f qs − f qŝ), and thus f qŝ = 12 f qs + 12 f qs ).
Therefore, the function f q is a solution of Jensen’s functional equation, f 12 q + 12 q = 12 f q + 12 f q . Because f q
is continuous and strictly increasing, it follows that f q =
aq + b for some constants a > 0 and b. And because a positive linear transformation of a cardinal scale is a cardinal
scale, qs is a cardinal scale for .
To show part (ii), assume that the relation has cardinal
constancy for qs. Then, for any amounts q q̂ q and q +
q q̂ + q, q + q in Iq , there are health states s ŝ, s and
t t̂ t such that qs = q, qŝ = q̂ qs = q and qt = q +q,
qt̂ = q̂ + q, qt = q + q.
Assume that f q̂ = 12 f q + 12 f q . Then, f qŝ −
f qs = f qs − f qŝ, and thus ŝ is middling relative to s and s . By the condition of cardinal constancy,
it follows that t̂ is middling relative to t and t . Then,
f qt̂ − f qt = f qt − f qt̂, and thus f q̂ + q =
1
f q + q + 12 f q + q.
2
In summary, the function f q is a solution of the
following functional relationship:
f q̂ = 12 f q + 12 f q implies
f q̂ + q = 12 f q + q + 12 f q + q
(13)
This functional relationship is studied in Harvey (1990,
Theorem B1). In the present notation, it is shown that a
continuous, strictly increasing function f q is a solution
of (13) if and only if it has a linear-exponential form: f q =
a exprqs + b r > 0; f q = aqs + b r = 0; or f q =
−a exprqs + b r < 0, for some a > 0 b, and r. And
because a positive linear transformation of a cardinal scale
is a cardinal scale, has a cardinal scale of a form in (1).
Unlike the case for Jensen’s equation, the assumption that
f q is continuous cannot be omitted. Proof of Lemma 2. Assume that ws t) is a proper cardinal scale for . Let ws t = ws t denote the function
of s obtained from ws t by fixing a duration t. Because S
is proper by condition (B), Lemma 1 implies that each function ws t is a proper cardinal scale for S and that the
scales ws t are related by positive linear transformations.
Choose a duration t ∗ and define w ∗ s = ws t ∗ ). Each
function ws t is a positive linear transformation of w ∗ s,
that is, ws t = atw ∗ s + bt for some amounts at > 0
and bt that can depend on the duration t. But condition (A) implies that ws0 t = ws0 t ∗ ) for any t. Hence,
w ∗ s0 = ws0 t ∗ = ws0 t = atw ∗ s0 + bt, and thus
bt = atw ∗ s0 + w ∗ s0 . Therefore, ws t = ws t =
atw ∗ s − w ∗ s0 + w ∗ s0 .
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278
Define zs t = ws t − w ∗ s0 . Then, zs t = atgs
where gs = w ∗ s − w ∗ s0 . Thus, zs t has the product
form (2). It is a proper cardinal scale for because it differs
from ws t by a constant. Hence, it is continuous, and thus
the functions at and gs are continuous. As noted above,
the function at is positive. And the function gs has an
interval range because w ∗ s is a proper cardinal scale for
the relation S . Moreover, gs0 = w ∗ s0 − w ∗ s0 = 0.
For the reverse implications, assume that a function
ws t = atgs is a proper cardinal scale for where
the functions at gs are as described. Then, ws0 t =
atgs0 = 0 for any t, and thus condition (A) is satisfied. It
remains to show that condition (B) is satisfied.
For any fixed duration t, the function, ws t = ws t =
atgs, is a cardinal scale for the cardinal relation S t
defined by restricting to health–duration pairs with the
duration t. Because the functions ws t = atgs differ
from one another by a positive multiple, gs is a cardinal
scale for each relation S t . Hence, the relations S t are the
same (and thus they can be denoted by S . By assumption,
the function gs is continuous and has an interval range. If
its range is a point, then gs = gs0 = 0 for any health state
s, and thus ws t = 0 for any health–duration pair (s t.
But this contradicts the assumption that ws t is proper
scale. Proof of Theorem 3. Assume that satisfies conditions (A)–(E). Then, it has a cardinal scale, ws t = atgs,
as described in Lemma 2. And as shown in the proof of
Lemma 2, gs is a proper cardinal scale for the cardinal relation S that is induced by . Hence, condition (C)
implies that gs = f qs for a continuous, strictly increasing function f q with a nonpoint interval range. Moreover,
f qs0 = gs0 = 0, and there exists a health state s with
f qs = 0.
Define a function Dt on the interval [0 ) by Dt =
at t in T = 0 ), and D0 = 0. By Lemma 2, Dt is positive and continuous on T . For any health state s that is not
indifferent to the standard state s0 Dt = f qs−1 ws t
is a cardinal scale for the cardinal relation T s that is
defined by . Thus, condition (D) implies that Dt is
strictly increasing on T = 0 ).
The function Dt is continuous and strictly increasing
on the larger interval [0 if and only if it tends to
zero as t tends to zero. Assume that there is a health
state s with f qs > 0. Then, for any , > 0, there exists
a health state s, with 0 < f qs, < f qsD1−1 ,. And
by condition (E), there exists a duration such that the
health–duration pair (s is less preferred than the health–
duration pair (s, 1). Hence, Df qs < D1f qs, ), and
thus D < ,. Hence, Dt tends to zero as t tends to
zero because Dt is an increasing function. The argument
is essentially the same if there is a health state s with
f qs < 0.
Proof of the converse implications consists of verifying each of the conditions (A)–(E). These verifications are
straightforward.
Harvey and Østerdal: Cardinal Scales for Health Evaluation
Decision Analysis 7(3), pp. 256–281, © 2010 INFORMS
It remains to show that the functions f q and Dt as
described are unique up to positive multiples. First, assume
that f ∗ q = af f q and D∗ t = aD Dt, where af > 0 and
aD > 0. Then, w ∗ s t = D∗ tf ∗ qs = aD af Dtf qs) is a
proper cardinal scale for , and the functions f ∗ q and
D∗ t have the properties (i), (ii). Conversely, assume that
w ∗ s t = D∗ tf ∗ qs is a proper cardinal scale for where f ∗ q and D∗ t have the properties (i) and (ii). Then,
f ∗ q is a proper cardinal scale for s , and D∗ t is a proper
cardinal scale for T . By Lemma 1, it follows that f ∗ q =
af f q+bf and D∗ t = aD Dt+bD , where af > 0 and aD > 0.
But f ∗ qs0 = f qs0 = 0 and D∗ 0 = D0 = 0, and thus
bf = 0 and bD = 0. Proof of Theorem 4. By assumption, has a proper cardinal scale, ws t = Dtf qs, where f q and Dt have
the properties in parts (i) and (ii) of Theorem 3. In particular, f q and Dt are proper cardinal scales for the
induced relations s and T . Theorem 3 implies that a function w ∗ s t = D∗ tf ∗ qs is a proper cardinal scale for ,
where f ∗ q and D∗ t have the properties in parts (i) and
(ii) of Theorem 3, if and only if f ∗ q = af f q and D∗ t =
aD Dt for some constants af > 0 and aD > 0. It follows that
the implications to be shown here for the function f q are
independent of those to be shown here for the function Dt.
We will prove only the forward implications because the
converse implications are straightforward to verify.
To show part (i), assume that the relation has cardinal
neutrality for qs. Then, qs is a proper cardinal scale for
S , and thus f ∗ qs = qs − q0 is a proper cardinal scale
for s . Hence, f ∗ q is a positive linear transformation of
the cardinal scale f q mentioned above. But f ∗ qs0 = 0
and f qs0 = 0, and thus f ∗ q = af f q for some constant
af > 0.
The proof of the forward implications in part (ii) is parallel to the above argument with a proper cardinal scale of
the form (1) in the role of the function qs.
To show part (iii), assume that the relation has cardinal timing neutrality. Because the scale, ws t = Dtf q,
is proper, there exists a health state s with f qs = 0.
But cardinal timing neutrality implies that Dt2 f qs −
Dt1 f qs = Dt2 + tf qs − Dt1 + tf qs for any
durations in the interval T . By dividing by f qs, this
equation implies that Dt2 − Dt1 = Dt2 + t − Dt1 + t.
Choosing t2 = t1 + t, it follows that Dt satisfies Jensen’s
functional equation. Thus (as noted in the proof of Theorem 1), Dt is linear, i.e., Dt = at + b, for some constants
a b. But Dt is strictly increasing with D0 = 0, and thus
Dt = at where a > 0.
To show part (iv), assume that has cardinal timing
aversion. By using arguments similar to those above, it follows that Dt2 − Dt1 > Dt2 + t − Dt1 + t for t1 < t2
and t > 0. In particular, with t2 = t1 + t, we have Dt2 >
1
Dt2 − t + 12 Dt2 + t. Hence, Dt is strictly concave.
2
To show part (v), first assume that the relation has constant cardinal timing aversion (I). Then, Dt1 + t f qs1 −
Dt1 f qs1 = Dt2 + t f qs2 − Dt2 f qs2 , for
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t1 < t2 t > 0, implies Dt1 + t + tf qs1 − Dt1 +
tf qs1 = Dt2 + t + tf qs2 − Dt2 + tf qs2 ,
for t > 0. But there exists a health state s such that
f qs = 0, and thus for any R > 0 there are health states
s1 s2 with f qs2 /f qs1 = R. Choosing these health
states, it follows that
Dt1 + t − Dt1 Dt1 + t + t − Dt1 + t
=
Dt2 + t − Dt2 Dt2 + t + t − Dt2 + t
(14)
for any t1 < t2 t > 0, and t > 0. In Harvey (1998a,
Theorem 5.2), it is shown that if a strictly increasing, continuous function Dt is a solution of this functional equation,
then it is a linear-exponential function. By part (iv), Dt is
strictly concave, and thus it is a negative-exponential function. Because D0 = 0, this implies that Dt = a1 − e−rt for some r > 0 and a > 0.
Next assume that the cardinal relation has constant cardinal timing aversion (II). Then, Dt2 f qs −
Dt1 f qs = Dt4 f qs − Dt3 f qs, for s t1 < t2 , and
t3 < t4 , and thus, Dt2 + tf qs − Dt1 + tf qs =
Dt4 + tf qs − Dt3 + tf qs, for t > 0. Choose a
health state s such that f qs = 0, and choose durations
such that t2 = t3 and t2 − t1 = t4 − t3 . Define t = t1 t̂ =
t2 t = t4 . Then, the above implication states that Dt̂ =
1
Dt + 12 Dt implies Dt̂ + t = 12 Dt + t + 12 Dt + t.
2
Therefore, the function Dt is a solution of the functional
relationship (13), and thus it is a linear-exponential function. By the argument used in the previous case, it follows
that Dt = a1 − e−rt for some r > 0 and a > 0. Proof of Theorem 5. We will obtain Theorem 5 from
the social welfare model in Harvey (1999, Theorem 1), to
be called the SW model. More accurately, we will prove
Theorem 5 by using the proof of the SW model, because
Theorem 5 generalizes of the SW model in an important
respect. We can omit proofs of the converse implications
because they are straightforward to verify.
Suppose that the set C of social consequences in the SW
model is chosen as the product set H1 × · · · × Hn . As noted
in §6, an individual cardinal relation i for a set Hi corresponds to a cardinal relation, to be denoted by SW
i , for the
set C in the SW model. And a population cardinal relation
P for H is the same as a population cardinal relation SW
P
for C. The condition of Pareto agreement here is equivalent
to the conditions (A) and (B) in the SW model.
Consider condition (C) in the SW model. It states in part
that there exist continuous cardinal scales for the relations
i and P . It also states that the set C is connected, a condition that we do not require here. Instead, we assume that
the relations i and P are proper. Thus, we simply assume
the properties in Lemma 1 of the SW model, which state
that SW
and SW
are represented by cardinal scales that
i
P
are continuous with interval ranges. It follows by Lemma 1
in §3 that any cardinal scales wi hi and wP h) have these
properties.
Next, consider condition (D) in the SW model. It states
(in our terminology) that for any health distributions
hi i = 1 n, there is a health distribution h such that
h is indifferent to hi according to i i = 1 n. Here,
this condition is redundant. The reason is that for the
given distributions hi we can define a distribution h =
1
n
i
h1 hn where hi is the ith component of hi . Then,
for each i = 1 n, the health distribution h has the same
ith component as hi , and thus it is indifferent to hi according to i .
In summary, the assumptions in Theorem 5 imply that
the cardinal relations i and P have the properties of the
and SW
that are stated in the Lemma 1 of
relations SW
i
P
the SW model. As the reader can verify, it is these properties rather than conditions (C) and (D) that are used in the
proof of the SW model. Hence, by using that proof of the
SW model, it follows that for any cardinal scales wi hi and
wP h there exist constants ai > 0 i = 1 n, and b such
that
wP h = a1 w1 h1 + · · · + an wn hn + b
Therefore, for any given individual cardinal scales
wi hi i = 1 n, the function, vP h = wP h − b =
a1 w1 h1 + · · · + an wn hn , is a population cardinal scale of
the form (5).
Now we show that the constants ai in (5) are unique up
to a common positive multiple. For any 0 > 0, the function, vP0 h = 0vP h = 0a1 w1 h1 + · · · + 0an wn hn , is a cardinal scale for the relation P . Conversely, suppose that
a function vP0 h = a∗1 w1 h1 + · · · + a∗n wn hn is a cardinal
scale for P . Then, vP0 h = 0vP h + 1 for some constants
0 > 0 and 1, and thus, (a∗1 − 0a1 w1 h1 + · · · + a∗n −
0an wn hn = 1. The cardinal scales wi hi have nonpoint
ranges because the relations i are proper, and thus a∗i =
0ai i = 1 n (and then 1 = 0) because the terms on the
left-hand side can vary independently of one another. Proof of Theorem 6. Because the individual cardinal
relations i i = 1 n, are equal, they have the same cardinal scales. By choosing the scales wi hi i = 1 n, as
such a common scale, Theorem 6 follows as a special case
of Theorem 1. Proof of Corollary 1. Suppose that wi hi ) are individual scales. Then, wi h1 > wi h0 for each i = 1 n
because h1 is preferred to h0 by each individual. Thus,
the functions vi hi = wi hi − wi h0 /wi h1 − wi h0 )
are also individual scales. But vi h0 = 0 vi h1 = 1 for
i = 1 n, and cardinal uniqueness implies that vi hi are
the only scales with such values.
Suppose that vP h = a1 v1 h1 + · · · + an vn hn is a population scale associated with the individual scales vi hi . Then,
for a health distribution h such that hi = h1 for i in Sh)
and hi = h0 otherwise, we have vP h = i∈Sh ai as was to
be shown. Proof of Theorem 7. Properness and cardinal uniqueness imply that there are individual scales wi hi such
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280
that wi h0 = 0 i = 1 n. Suppose that wP h = a1 w1 h1 + · · · + an wn hn is an associated population scale. Then,
wP h0 h0 = 0.
First, assume condition (a) or condition (c). Then, for
any health outcome h, the indifference (8) or the indifference (9) implies that wP hi = h; hk = h0 k = i = wP hj = h;
hk = h0 k = j, and thus ai wi h = aj wj h. For j = 1, it follows that wi h = a1 /ai w1 h. Thus, w1 h is an individual
scale for every individual cardinal relation. Suppose that
wI h denotes the scale w1 h and that vP h = 01 wI h1 +
· · ·+0n wI hn denotes an associated population scale. By the
previous argument, 0i wI h = 0j wI h for any i j. Hence,
0i = 0j for any i j.
Second, assume condition (b) or condition (d). Then, by
essentially the above argument there is a common individual scale wI hi such that wI h0 = 0. Suppose that wP h =
a1 wI h1 + · · · + an wI hn is an associated population scale.
For any i j, there exists a health outcome h such that h
is nonindifferent to h0 (and thus wI h = 0), and either
the indifference (8) or the indifference (9) is satisfied. Then,
wP hi = h; hk = h0 k = i = wP hj = h; hk = h0 k = j, and
thus ai wI h = aj wI h. Because wI h = 0, it follows that
ai = a j .
It is straightforward to verify the converse implications
that in an equals weights model the conditions (a)–(d) are
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