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Application of Multi-Attribute Utility Theory to Measure Social Preferences for Health States

by

GEORGE W. TORRANCE, Ph.D. Faculty of Business

MICHAEL H. BOYLE, M.Sc. Dept. of Pediatrics

SARGENT P. HORWOOD, M.D. Dept. of Pediatrics

FACULTY OF BUSINESS

����"'�Wtl�R�

McMASTER UNIVERSITY HAMIL TON, ONTARIO

Research and Working Paper Series No. 189 June, 1982

APPLICATION OF MULTI-A'ITRIBUTE UTILITY THEORY

'ID MEASURE SOCIAL PREFERENCE'S FOR HEALTH STATES

George W. Torrance, Ph.D.; Michael H. Boyle, M .Sc . ;

Sargent P . Horwood, M.D. .

From the Faculty of Business and the Departments of Pediatr ics and Clinical

Epidemiology and Biostatistics of the Faculty of Health Sciences, McM as ter

Univers ity, Hamilton, Ontario.

Address : Dr. George w. Torrance Faculty of Business McMaster Univers ity Hamilton, Ontario, Canada LBS 4 M4 (416) 5 25-9140, Ext. 4434

Running Head: Preferences for Health States

4226 New Street Burlington, Ontario Canaaa L7L 1T4

(416) 634-0329

Keywords : multi-attribute utility theory social preferences health states health services

Subject Classification for the OR/MS index :

# 271 (health care), soc ial preferences for multi-attr ibute ·

health states.

#852 (util ity/preference, mu lti-attr ibute), measurement of

social preferences for health states.

June 1982

APPLICATION OF MULTI-A'ITRIBUI'E UTILITY THEORY

'IO MEASURE SOCIAL PREFERENCES F'OR HEALTH STATF.S - --

George W. Torrance, Michael H. Boyle, and Sargent P. Horwood

McMaster Univers ity, Ham ilton, Ontario, Canada

ABS TRACI'

A four-attr ibute health s tate class ification sys tem des igned to

uniquely categorize the health status of all individuals two years

of age and over is presented. A social preference function defined

over the health state class ification system is required. Standard

mu lti-attr ibute utility theory is inves tigated for the tas k,

problems are identified and modifications to the standard method

are proposed. The modif ied method is f ield tes ted in a survey

res earch project involving 11 2 home interviews. Results are

presented and discussed in detail for both the soc ial preference

function and the performance of the modified method. A recommended

social preference function is presented, complete with a range of

uncertainty. The mod ified method is found to be applicable to the

tas k -- no i ns u r mountable di f f icu l t ies a r e e ncounter ed.

Recommendations are presented, based on our experience, for other

inves tigators who may be interes ted in reapplying the method in

other studies.

INTRODUCrION.

The applicability of cost-effectiveness analysis as a technique for the

evaluation and comparison of alternative health care programs is limited by

the di fficulty of measuring progr am effects (outcomes , consequences) i n

commensurable units across programs o f different types. One approach to

overcoming this limitation is to determine the ultimate impact of each program

on the health states of each individual affected by the program and to use a

social preference function defined over the relev ant health states as the

common unit of measure (for example, see Fanshel and Bush [1970] , Tarrance et

al. [ 197 2 ] , Torrance [ 197 6a] , Weinstein and Stason [ 1977 ] , Rosser and Watt

[1978] ) .

The general approach in determining such a social preference function is

to define a set of health states of interest, to identify a group of subjects

(judges) , to measure each subject's preferences for the health states, and to

aggregate ac�oss the subjects to determine the overall social preference

function. Within this general approach a number of methods are available for

the measurement of the subj ects' preferences -- these methods have been

summarized and compared by Fischer [1979]. Using Fischer's terminology , past

work in the measurement of social preferences for health states has generally

used holist ic utility assessment methods (examples include Patrick et al.

[ 197 3a] , Patrick et al. [ 1973b] � Torrance et al. [ 197 3 ] , Torrance [ 1976b] ,

Rosser and Kind [1978] , and Sackett and Tarrance [1978] ) , although there has

been one application of a statistically inferred model using the functional

measurement approach (Kaplan et al. [ 1976] ) . This paper reports the results

of a study which uses a third method -- the explici tly.decomposed multi­

attribute utili ty method using the conditional utility function-based

procedure. This is , in fact, the classical or standard multi- attribute

l

utility (MAU) method (Keeney and Raiffa [197 6 ) , Farquhar [1977 ) ) and , for

brevi ty, it will be referred to as the MAU method. It is an appropriate and

relatively efficient method when the health states are described by a multi­

attr ibute classi fication system. The purpose of this paper is twofold:

first, to repor t the .modifications made to the MAU method to adapt i t to a

survey research project for the measurement of collec tive social preferences,

and second, to repor t the results of a field study to test the modified MAU

method and to develop a specific social preference function.

The paper i s divided into fiv e sections. Secti on 1 contai ns a

description of the multi-attribute health state classi fication system for

which a social preference function is desired. In Section 2 we review the

standard MAU method and describe the modifications made to it for this study.

The field work and results are presented in Sec tion 3. Section 4 contains a

di scussion and interpretation of the results , while our conclusions and

recommendations are given in Section 5.

2

1 . MULTI-ATI'RIBUTE HEALTH STATE CLASSIFICATION SYSTEM

The health states in this study are defined according to the four­

attribute health state classification system shown in Table I. The system was

developed to classify and to follow for life the health outcomes of randomly

selected infants in an evaluation of neonatal intensive care (Boyle et al.

[1982] ) . A ll selec ted children (age range 2-1 5 years) have had their current

health status classi fied , their past health pattern reconstructed and their

future health pattern forecast using the health state classification system.

The children represent a wide variety of disabilities, mostly chronic.

Each attribute in the Health State Classi fication System is subdivided

into a number o� levels such that each person can be classified at every point

in time into one level on each attribute. A social preference function

defined over the health states described by Table 1 is required. Since each

feasible combination of attribute levels defines a unique health state, the

system implicitly includes a large number of different states; too many to

measure preferences explicitly using holi stic uti li ty assessment methods.

Thus an approach based on MAO theory was selected.

--TABLE I ABOUT HERE--

2. MULTI-ATI'RIBUTE UTILITY THEORY FOR SOCIAL PREFERENCES

2 .1 MAU Methoo

Multi-attribute utility (MAU) theory (Keeney and Raiffa [1976] , Farquhar

[1977 ] ) i s concerned with expr essing the utili ties of multiple-attribute

outcomes or consequences as a function of the uti li ties of each attribute

taken singly. The theory specifies several possible functions (addi tive ,

multiplicative and multilinear) and the conditions (independence conditions to

be met) under which each would be appropriate. As a practical matter Keeney

and Raiffa [197 6 , p. 298 ] suggest that for four or more attributes the

3

reasonable models to consider are the additive and the multiplicative. Since

our problem contains four attributes, we restrict our attention to these two

forms.

Standard MAU theory has been developed for the case of a single decision

maker or a single decision making unit. We will first rev iew ·briefly how this

standard theory applies to our multi-attribute problem, and then describe the

modifications requ ired to adapt the method for a survey research project to

measure collective social preferences. ReaderS' unfamiliar with standard MAU

theory who f ind the r emainder of this subsection too condensed may wish to

refer to one of the following : Keeney and Raiffa [197 6 ] , Farquhar [1977 ] ,

Fischer [1979) . A ll notation used throughout the paper is summarized for easy

reference in Table II .

-TABIE II ABOUT HERE-

The conventional method for measuring utilities is the standard gamble

technique , a lottery-based procedure f irst proposed by von Neumann and

Morgenstern [ 1953]. In this method two reference outcomes are estal?lished,

for example '!i. * and 2.0, and the utility of the other outcomes are measured

relative to these two. For an intermediate outcome ! the subject is asked to

determine the probability p such that s/he is indifferent between the lottery

<p!* , (1-p) '!i_o> and '!i. for cer tain. Then u (2.) = p on the utility scale where

U (!* ) = 1 and U (!O ) = 0�

Additive Form

Additive independence exists if each attribute is additive independent of

the other attributes. A ttribute j is additive independent of attributes j if

the subject is indifferent between lottery L1 and lottery Lz for all values of

x'j and x."j , where L1 = <.5!' , .5!"> and Lz = <.5{x'j , x"j ) , .5 (x"j , x'j»·

If additive independence exists , the multi-attribute utility function is

4

additive .

* u (�)

where

4 = 2:. k · u*

· (x · ) j=l J J J

4 :!. kj = 1 j=l

and kj = u* (x*j ,x0j)

. '

j = 1 , 2., 3 , 4

(1)

(2)

(3) One simple method to determine the function is to measure each s ingle­

attr ibute utility function u*j (Xj ) separately relative to u*

j (x*j ) = 1 and

u*j (x0j) = O, while holding the levels of all other attributes constant. Then

the k's are determined by measur ing the utility of the specific multi­

attr ibute health states in (3) relative to u* (_�{) = 1 and u* �) = O.

Alternatively and equivalently, one can define and measure the additive

multi-attribute disutility function 4

u* (x) = .z · c ·li'* · (x · ) - j=l J J J

4 w�ere f cj = 1

J=l

(4)

(5)

and cj = \i* (x0j , x*5) , j = 1 , 2 , 3 , 4 (6)

Again , the function can be determined by measur ing each single-attr ibute

disutility function u* j (xj ) separately, relative to u.*

j (x*

j ) = 0 and

li*j (x0j) = 1, while holding the levels of all other attributes constant. The

cj can be determined from (6) by measur ing the disutility of the specif ied

1 . . b 1 . -* ( * -* 0 1 mu t1-attr 1 ute health states re ative to u � ) = 0 and u (� ) = . • Note ,

that for the additive case cj = kj all j , but the cj notation is introduced

here for convenience later.

Multiplicative Form

Mutual utility independence exists if every subset of {x1 ,x:z.,x3 ,x4 } is

utility independent of its complement. A subset of attr ibutes is utility

independent of its complementary set if the conditional preference order for

5

lotter ies involv ing only changes in the levels of attr ibutes in the subset

does not depend on the levels at which the attributes in the complementary set

are held f ixed. A dd it ive independence implies mutual utility independence,

but not v ice versa.

I f mutual utility independence exists , the m ulti-attr ibute utility

function is additive , or multiplicative of the form

* ' 1 4 * U (X) = - ( (1 + kk•U · (X · ) ) - 1) -

k j=l J J J (7)

One method to determine the function is to measure each u*j<xj) , determ ine the

four kj v alues from (3) , and find the k v alue by iteratively solv ing (8) ,

which is (7 ) for x = x*

.

4 1 + k = . (1 + kkj)

j=l

Parameter k is related to parameters kj as follows:

4 if kj > 1 , then -1 < k < O,

j=l

4 if kj = 1 , then k = O and the additive rrodel holds , and

j=l

4 if kj < l; then k > 0

j� '

(8) .

(9a)

(9b)

(9c)

The three cases can be distinguished in terms of the multivar iate risk posture

which they represent {Richard (1975] ) . Case (9a) represents multivariate r isk

aversion , case' (9b) multivar iate-r isk neutrality and case (9c) multivariate

r isk seeking behav ior. The attr ibutes in case (9a) can be character ized as

"substitutes" while those in (9c) are "complements" (Keeney and Raif fa [ 1976,

p. 240 ) ). The intuitive interpretation of this characte r i zation is that

substitute attr ibutes are such that an improvemment in one is relatively

6

satisfying , while an improvement on two or more i s not that much better .

Conversely, with complementary attributes an improvement on any one alone is

not very useful, while a simultaneous improvement on several· is much be tter.

·As an alternative to (7 ) one can define and measure the mu ltiplicative

mult i-attribute disutility function

u* (�) 1

= -J:. -k [ // (1 + cc · u · (x ·)) - l]

. 1 J J J c J=

,..4.. where (1 + c) = JI (1 + ccj)

j=l

4

(10)

(11)

and if z cj > 1 , then -1 < c < O, (12a) j=l

4 if Z:. cj = 1 , then c = O and the additive rrodel holds , and (12b)

j=l

4 if £. cj < 1 , then c > 0 ( 12c)

j=l

The function can be found by measuring the li* j (xj) , determining the Cj from

(6) and c from (11).

Case (1 2a) represents multivariate r isk seeking behavior , case (1 2b)

multivariate risk neutrality and case (1 2c) multivariate risk aversion. In

case (1 2a) the disatt r ibutes can be character i zed as subst itutes, while in

case (1 2c) they are complements. The intui tive notion her e is that in case

(1 2a) a reduction on any one attribute is bad , while a reduction on two or

more is not that.much worse. Conversely, in case (12c) a reduction on any one

attribute alone is not so bad, whi le a simultaneous reduction on several is

very serious.

7

preferences. The problems and the modifications we adopted to overcome them

are described in the next subsection.

2. 2. Modifications to MAU Method ------ -· - ---

Independence Condition

The establishment and veri fication of the independence conditions is

normally a tedious, exacting and time consuming task requiring extensive

interviewer-subject interaction (see, for example, Keeney and Raiffa [1976]).

This approach i s only feasible for s tudies with a small number of subj ects.

As an alternative to step l we elected to assume the existence of mutual

utility independence (note that this assumption is fully consistent with our

pr evious action of restricting the investigation to the additive and the

multiplicative MAU models} and then to test this assumption later with data

obtained by measuring the disutility of additional multi-attribute health

s tates. These additional "test" states are not used in the construction of

the mult�-attribute disuti lity functions, but are used later to test their

fit.

Measurement Technigues

A second problem for a survey research proj ect with many subjects

selected randomly from the general public is the complexity and difficulty of

administering the usual lottery-based utility measurement techniques like the

von Neumann-Morgenstern standard gamble (Torrance [ 1976b], Wolfson et al.

[ 1981 ] }. We circumvented this problem in our study by using simpler

measurement methods not involving probabilities. All our single-attribute

measur ements were made using a category scaling method, while our multi-

attribute measurements used the time trade-off procedure.

The category scaling method used a visual analog device called a "feeling

thermometer." I t is a thermometer-shaped 0-100 scale on a felt board with 0

9

labelled "least desirable" and 100 labelled "most desirable." The "levels" of

·the attribute being measured are printed on narrow foam sticks pointed at each

end which can be placed on the felt board beside the thermometer. The subject

was asked to think of himself as being in these situations for a lifetime with

everything else normal, or average. He was asked to place the most desirable

level x* j at 100 , the least desir able x0j at O, and the others in between in

order of desir ability, with ties allowed, and spaced such that the relative

distance between the l evels cor responds to hi s feelings about the relative

differ ences in des ir ability. Then for any level x'j the value of the level,

* * * * 0 . v j (X'j ), on the value scale where v j (x j ) = 1 and v j (x j ) = 0 is the

thermometer reading beside the level divided by 100. Later in the interview,

after considerable intervening material the subject was asked to remeasure one

of the a ttributes (randomly determined) as a reliability check.

Because uncertainty is not used in the category scaling method, i t

meas ures a va lue funct ion v*j<xj) as opposed to a utili ty function u

*j (xj)

(Keeney and Raiffa [ 197 6 ] ) . To conver t these values to uti li ties we need to

know the relationship, for health state preferences, be tween value functions

as measured by category scaling and utility functions as measured by a lottery

technique like the von Neumann-Morgenstern standard gamble. Two studies have

i nvestigated this relationship. Our previous wor k (Note 1) found for

population means the following power curve relationship be tween disvalue and

disutility

u = v-i. 6 (13)

while recent work by Wolfson and his colleagues (Note 2) provides independent

confirmation of the general nature of this relationship although not of the

specific par ameter value. In the wor k reported here we have used (13) to

convert the single-attribute measures from values to utilities. It should be

10

noted that (13) repr esents a par ticular attitude toward uncer tainty. By

adop ting (13) we do not measur e these subj ects' uncertainty atti tudes, but

assume they are the same as those of previous subjects. Later, as part of the

sensitivity analysis, we investigate the impact of this assumption.

The time trade-off procedure as developed by Torrance et al. [1972] and

later modified to handle states worse than death (Torrance [ 1982]) was used

for all the multi-attribute measurements. For chronic state � preferred to

death the subject was asked to determine the time t such that s/ he is

indifferent between (i) a lifetime (70 years) in the chronic state and (ii) a

healthy but shor ter life of t year s. Then vh (�) = t/70 on the value scale

where vh (�h) = 1 and vh (�d) = O. For chronic state � dispr eferr ed to death

the subject was asked to determine the time t such that s/he is indifferent

between (i ) a healthy life for t year s followed by the remainder of li fe (to

age 7 0 ) i n � and (ii ) to die i n hospi tal shor t ly a f ter bir th. Then

vh (� ) =o<t/ ( t-70 ) wher e o\ is a par ame ter of the i n s tr ument and the

interviewing procedur e such that the lower limit of the instrument is -1.0 .

This maintains symmetry between the value scale for states pref erred to death

which runs from O to 1, and the scale for states dispreferred to death which

runs from O to -1. In all cases the time trade-off procedure was supplemented

with visual aids. Later in the interview, after intervening material, . the

subject was asked to remeasure one of the multi-attribute health states using

the time trade-off method as a reliability check.

Since lotteries are not used in the time trade-off procedure, the results

do not i ncorpor ate the subj ect's atti tude toward uncer tainty. However,

previous work (Torrance et al. [1973], Torrance [1976b]) has demonstrated that

for states preferred to death population means measured by this method are

empirically equivalent to those obtained from a lottery-based technique, the

11

von Neumann-Morgenstern standard gamble, which does incorporate attitude

toward uncertainty. Consequently, in this study the time trade-off results

are assumed to represent utilities. For states dispreferred. to death no such

previous work exists and, in the absence of any, the scaled (to -1.0} results

of the time trade-off technique are tentatively accepted as utilities.

Extreme Levels

A problem can arise in measuring the disutility of the "corner" states in

l'> if the subjects differ in the levels which they rate as best and worst on

the individual attributes. The corner states for a subject are defined by the

best and worst levels as perceived by that par.ticular subject. This can

complicate the interview since the corner states, therefore, are not known

until the interview is in progress and the corner states may differ from

subject to subject. One solution is to cope with the situation during the

interview by designing a flexible interviewing format. Another approach is to

ensure that each attribute contains one extremely good and one extremely bad

level, so· that most if not all subjects would agree on the best and worst. A

variation on this approach used by Krischer [197 6] is to prespecify to the

subject which is the best and which is the worst level, and ask the subject to

rate the others relative to these two extremes. A third alternative (which we

used} is to prespecify for each attribute a good level, xaj, and a bad level,

xb j, but not tell the subject. In this way the subject is unconstrained in

providing his true preferences; rather the necessary corrections are handled

later in the calculations. The prespecified levels, xaj and xhj, are used in

(6) to create a set of common corner states for measurement on all subjects,

as follows:

cj = lfl(xbj, xa

j}, j = 1, 2, 3,4 (14}

Then, for those subjects for whom the measured states are not true corner

12

states, the calculation method requires a slight modification. In step 2

before equation (1 3) is applied the single-attribute disvalue functions

v*j(Xj)r are transformed to the Va

j(Xj) scale by the following positive linear

transformation

::a -'* -* a -* b -* a v · {x · ) = [ v · (x · ) - v · (x · ) ] / [ v · (x · ) - v · (x · ) ] , . J J J J J J J J J J

-* b -* a v · (x · ) > v · (x · ) J J J J (15)

The remainder of the method is unaffected except that the resulting multi­

attribute disutility function is i:iC-(�) as opposed to u*(�).

Social Aggregation

A significant issue in applying multi-attribute utility theory to measure

collective social preferences is the question of how one should aggregate

individual preferences into social preferences. For example, the methods

described above will produce individual multi-attribute disutility functions

UC1(?9. The question of aggregating such individual cardinal preferences into

collective social preferences has been addressed by a number of authors, and

different sets of assumptions lead to different results -- for example, ·

Hildreth [1953], Harsanyi [1975] and Keeney [1976) all argue that aggregation

is valid while Kalai and Schmeidler [1977] argue the converse. ·We agree with

those who favour aggregation, and we would underscore the point made by

Harsanyi [1975] and by Dyer and Sarin [1979] that such comparisons of

individual preferences are common in practice -- indeed, in order to make

social decisions, and in the very process of making those decisions,

indiviqual preferences must be and are compared. The question, then, is not

whether to make such comparisons but how to make them. In the spirit of

Hildreth (1953) we establish two clearly defined outcomes, one gCXJd and one

bad, as anchor points (but not necessarily end points) for the utility scale.

The good outcome is a normal healthy life (defined as birth to age 70), �h,

13

and is given a utility of 1 for each individual. The bad outcome is death in

hospital shortiy after birth, .!d, and is given a utility of O for each

individual. The central basis for the aggregation is that the difference in

utility between these two outcomes is set equal across people. The

aggregation method is the arithmetic mean. The mean is the method recommended

by Harsanyi [1975], and is also the method we obtain from the models of

Hildreth [1953] or Keeney [1976] by assigning each individual equal weight.

The overall method is also consistent with the recommended practice in health

program evaluation in which for each individual, regardless of who sjhe is,

immediate death contributes zero and one healthy year contributes one unit to

the effectiveness of the program .(Weinstein and Stason [1977] ). To implement

the method we measure, for each individual, the utility of .!a and .!b relative

to uh(!d) = O and uh(!h) = 1 and transform the individual's multi-attribute

disu tility function to the uh u tili ty sc ale by the nega tive linear

transformation

uh (.!) = uh (!a) _ {uh (,!a) _ uh (,!b) 1-ua (,!) (16)

If uhiQ9 is the resulting utility function for individual i of m individuals,

the social utility function is simply

1 m h Ul (_!) = - £. u i (!) m i=l

(17)

where the reference states of healthy and dead have values 1 and 0

respectively1 that is, u1 (,!h) = 1, u1 (_!d) = o.

u1 (!) is obtained by performing the social aggregation at the final level

o:e the process after each individual's unique uh(!) has been determined. An

alternative approach is to perform the social aggregation at the basic level

on the measured data and develop a group solution all the way through the

process. This alternative approach is introduced as a convenient approxima-

14

tion although it does not have the theoretical underpinning of the first

approach. The method simply consists of measuring the fundamental utilities

on a common scale right from the beginning and using the arithmetic mean as

the group measure. Specifically, the single-attribute disvalue functions for

each individual v* j (xj) as measured by the category scaling method are first

converted to the common -va scale using (15), and are then aggregated using the

arithmetic mean into the group functions V�j(Xj)• These group disvalue

I functions V�j (Xj} are converted to group disutility functions uaj (xj) by (13).

The individual utilities for the corner states in (14) and for states !a and

!b are measured by the time trade-off method on the uh utility scale, and the

means are used to represent the group preferences uhG(!)· These are converted

to disutilities on the ua scale using

ua (x) = [uh (xa) - uh (x)]/[uh (xa) - uh (xb)] G- G_ G- G- G- (18)

which is.the inverse o f (16} at the group level, and the disutilities u.a

G(�)

of the four corner states in (14) are the cj values for the group. The group

disutil:i.ty function uaG(�_) is specified using (4) and (5) or (10) and (11) as

appropriate. Finally, (16) is used at the group level to specify the desired

social utili�y function as follows:

U (x) = uh (xa) - [uh (xa) - uh {xb)]U8 (x) 2- G- G- G- G- (19)

where again the reference states of healthy and dead have values 1 and 0

respectively1 that is, u2(�) = 1, u2(!d) = 0.

2.3 Summary of Method

To summarize, the method we used was to measure individual single­

· attribute value functions v*j using the category scaling method and individual

·utilities uh for multi-attribute states using the time trade-off technique.

The v*j values were converted tov*

j disvalues usingv*j = 1 - v*

j and then to

vaj disvalues using (15). For model U1 the individual Vaj disvalues are

15

converted to uaj disutilities using (13) , the individual uh utilities are

converted toua disutilities using the inverse of (16) , the cj values are

found from {14), (4) and (5) or (10 ) and (11) as appropriate are used to

determine the individual ua(�) functions, and (16) and (17) are used to

determine the social preference function. For model u2 the individual va j

disvalues and the individual uh utilities are aggregated using the arithmetic

mean before being converted by (13) and the inverse of (16) respectively, the

cj values a�e found from (14), (4) and (5) or (10) and (11) are used to

determine the group uaG(�) function, and (16) is used to specify the social

preference function.

Field Work

3. FIELD MFASUREMENTS AND RESULTS

Parents of school children were used as subjects. A random sample of 148 names was drawn by the Research Department of the Hamilton Board of Education.

Interviews were conducted in the subject's home by the professional

interviewing staff of a local survey research firm. (Copies of the

interviewer's manual and instructions are available upon request.) For that

part of the interview reported here (chronic states) subjects were told to

imagine that they are in the health situation described and that it lasts for

a lifetime (birth to age 70 ) .

Each subject provided single-attribute value functions v* j (xj) for each

attribute using the category scaling method (feeling thermometer), and

utilities uh�) for the seven multi-attribute health states shown in Table IV

using the time trade-off method. States A-E in Table IV are used in the

construction of the multi-attribute utility function, while states F and G are

used later to test the fit. States A-D are the four "corner" states required

in (14) to determined the cj values, while states E is �b required in (16) to

16

conver t between ua and uh. The states in Table IV were presented to the

subjects on printed cards containing the descriptions of the attribute levels

shown in the table. The subj�cts were asked to interpr et the cards as

follows: "Each situation lasts a lifetime, let's say all the way from birth

to age 70. Please think of yourself as being in these situations but in every

other way as healthy as possible."

Completed interviews were obtained from 76% (112/148) of the names drawn

in the random sample. The remai�ing 24% can be divided into those who could

not be contacted (1%), those contacted but ineligible due to language barrier

(8%), those who r efused to par ticipate (14%) and those who broke-off the

interview (1%).

seventy-eight percent (87/112) of completed interviews produced usable

data. In the r emaining cases the subject gave at least one r esponse that

indicated confusion with regard to the measurement task. On the single-

attribute measurement task a respondent was categorized as "confused" if s/he

ranked any level as more desir able than the prespecif ied level 1. On the

multi-attribute measurement task a respondent was categorized as "confused" if

s/he·ranked any state as less desirable than state E. Ten respondents

displayed confusion on the one-attribute task, 11 on the multi-attribute task

and 4 on both. Confused responders were eliminated from the data; all results

are calculated from the remaining 87 oases.

In measuring the single-attribute functions, the modification involving

xa j and xb

j was used. Based on a pilot study xa j was established as Pl, Rl,

Sl and Hl and xbj as PG, R5, 84 and H8 for the four attr ibutes respectively.

These were the levels most frequently designated as best and worst in the

pilot study. All 87 respondents selected each xaj as the best, i.e. x*

j =

x8j, all j. (Recall, those who did not were ruled out as "confused"). On the

17

other hand, not all agreed that xbj was the worst on each attribute. Twenty­

one of the 87 subjects (_24%) disagreed with one or more xb

j, 16 of these

disagreed on only one attribute while 5 disagreed on 2 attributes. Thus there

were 26 disagreements out of a total of 3 48 xb j 's (7.5%). The disagreements

were primarily in attribute 4 as would be expected, the distribution of

disagreements by attribute was ( 5 , 3, 4 ,14). Since x* j = xa

j, all j, for all

resp::mdents, v*

j (xa j) = 0 and (15) simplifies to

-a -k '""* b v · (x · ) = v · (x · ) /v · (x · ) J J J J J J (20)

Note the simplicity of (20) in that it can be applied to all the measurements

and where it is not required it will be made inoperative automatically because .

-*

h in that case v j CX-j) = 1.

Results

Four single-attribute disvalue functions were determined for each of the

87 subjects using transformation (20). These were converted to value

functions and are reported in aggregate form in Table III.

· ---TABLE III ABOUT HERE---

The utility to each individual for each of the seven multi-attribute

health states was measured directly on the uh utility scale with the time

trade-off technique. The findings are reported in aggregate form in Table IV.

---TABLE DI ABOUT HERE---

The reliability of the measurement methods is reported in Table v.

Reliability is based on replicated measures taken during the same interview.

The correlation coefficient r is the product moment (Pearson's) correlation

coefficient between the original measure and the repeated measure. The

precision6'e is the standard error of measurement and is calculated from

O-e = Jf a2;2N, where d is the difference between the original measure and the

repeated measure.

18

--TABLE V ABOUT HERE--

For the case of social aggregation at the final level, leading to U1 �), each ind iv idual's mult i-att r i bute d i su t i l ity function ua(�) must be

determined. As part of this determination, the cj parameters from (14) and,

if appropr iate , the c parameter from (11) must be calculated. A frequency

distribution of the resulting parameters is given in Table VI.

--TABLE VI ABOUT HERE--

For the additive model to be appropriate an individual's four Cj values

should sum to 1.0 yielding a c value of O. As can be seen from Table VI this

was not the case for any of the subjects. In fact, very few c values were

anywhere close to 01 the closest being .11 and this was the only one that fell

in the interval 0 ±. .251 while only 7 fell in the interval 0 ±. .50. Thus, the

multiplicative model was selected as the more appropr iate model for all

subjects. Each individual's multiplicative multi-attr ibute disutility

function was used in (16) and the results of this in (17) to g ive the des ired ·

social utility function u1 (!) · The function i s not repor ted here as i t i s

only available in tabular form and the table has 960 entries. However , copies

are available upon request.

The alternative approach to soc ial aggregation leads to the function

u2(�). The group single-attribute value functions �Gj (xj) are given in Table

III , "Mean Value" column. The group utilities for the multi-attribute states

uhG (!9 are shown in Table IV, "Mean utility" column. Four of these, A-D, are

transformed to disutilities using (18) with uhG�a) = l because in our study

!a= !h and with uhG (!b) = - .39 because !b = E. It follows from (14) that

these disutilities are the group cj parameters; the values are (c1,c2,c3,c4) = 4

(.7 2 , .7 3 , .40 , .44) . Since £ cj >l , we know from (12a) that -l<c<O , and j=l

19

solving (11) gives c = -.97. Thus the u2 social preference function developed

in this study can be specified in total as follows:

-ua · (x · ) = GJ J

1 u:a (x) = -G - c

[ 1 - V�j (Xj ) ] l . 6

[fr (1+ccjuaj <xj ) ) -11 J=l

U2 (�9 = 1 - 1.39 uaG(�)

(2la)

(2lb)

(2lc)

The complete u2 social preference function for each of the 960 health states

in Table I can be determined using equation set (21) in conjunction with the

data in Table III and the Cj and c values specified al:x>ve .

How do the results compare between the two methods of aggregation?

Utility values were calculated for each of the 960 states using each method of

aggregation, u1 � and u2 �) , and the two sets of data were compared. The two

sets of utilities are highly correlated (r = .995) , however, there is a small

but statistically significant (p < .0001) bias: u1 (�) - u2 (�) h_as a range of

-.09 to .09 with a mean value of .01. An analysis of the differences shows no

particular systematic pattern.

How well does the model fit the two test points F and G? The mean

measured utility for these two test points is .67 and .31 (Table IV) , while

the nodel results for the same states are . 50 and . 01 .

4 . DI SCUSSI ON

Field Work

Respondent eligibility and participation rates are consistent with our

previous work in the f ield (Torrance [ 1976b] , Sackett and Torrance [ 1978]) .

The current study reconfirms our former findings that a high proportion of

eligible subj ects will participate in these studies and that very few

participants will break-off the interview. We take this as evidence that the

general thrust of the study as well as the specific measurement tasks are

2 0

found to be acceptable by the general public.

On the other hand , the level of respondent confusion is disappointing ,

although it is consistent with other studies in which interviewer intervention

was prohibited . For ex ample, Krischer [19 7 6 , Table l] reports even higher

levels of r e spond e n t confusion in a s tudy using self-admini s te r ed

questionnaires1 while Kaplan and his colleagues [1979] noted somewhat lower ,

but still substantial, confusion rates (17% ) in a study using structured

interviews. To ameliorate this problem in future work we recommend that

interviewer intervention be allowed in clearly defined cases of respondent

confusion like those in our study. The interviewer's role would be to

identify and explain the apparent inconsistency to the sub j ect but not to

insist on its rectification. In addition respondent confusion on the single-

attribute task could be reduced by avoiding attributes with a double content

like our att ributes x1, x2 and x3 (Table I ) . I t i s instructive to note that

all of ou r single-attribute respondent confusion came from these attributes,

and none came from attribute x4 which contains only one construct. It would

appear that the double content of attributes x1 , x2 and x3 overloaded the ·

information processing capability of some of the respondents.

Extreme Levels

The technique of measuring a common set of corner points involving xaj

and xbj on all r espo ndents regardless of their par ticular x

*j or x0

j

preferences proved useful in practice, but what are the full implications of

doing thi s? I s it always suitable? Does it require the sati sfaction of

addition al assumptions regarding the underlying utility structure? Does it

introduce additional measurement error? First, the method as describ ed in

th �s paper is not suitable for any individual who prefers xb j to xa

j, on any

attribute1 fortunately, this did not happen in our study. Second, the method

2 1

can be viewed as simply determining the subject's multi-attr ibute disutility

function u* �) on the reduced multi-attribute space obtained by omitting on

each attribute any xj level preferred to xaj and any xj levels dispreferred to

�j· Given these omissions, xaj = x*

j and �j = x0j , on each attributes, and

u* (�) = ua(�) can be found in the conventional way.·

Then, assuming that the

independence conditions necessary for u*�) on the restricted space also apply

over the enlarged space, the ua �) for the enlarged space is simply the same

function with the uaj (Xj ) scales now extended beyond the range 0 - 1 to

incorporate the omitted levels. Thus , no new or additional independence

assumptions are required. Finally, the method requires the same number and

type of measurements and so does not introduce any new measurement er ror;

however , it may be more sens itive to existing measurement error . That is,

since xa j and xb j are closer together in preference than x*

j and x0 j, their

difference may have greater proportional error and this in turn would lead to

greater er ror in the final uh (�) values. A measure of the extent to which

this may be a problem in any par ticular application is the extent of

extrapolation beyond the reduced multi-attribute space required to incorporate

the enlarged space. In our study, this is the v�lue of ua (�0) for the

individual. For the 21 cases in our study this value ranged from 1.00 to 1.30

with a mean of 1.05. Thus, the average extrapolation was 5% indicating that

even when xbj was not the worst level, it was a near worst level and

consequently there was little impact on model error . Although the common

corner points were no problem in this study, they could pose a problem in

another study if the extrapolation is greater , both qecause of the magnifying

effect on measurement error and because the independence conditions are most

apt to be violated at the extremes of the ranges.

2 2

States Worse Than Death

One of the mote unusual and interesting results to come from our study is

the frequency with which states are perceived as worse than death (Table IV). Eighty percent (70/87) of the subjects identified one or more health states as

wor se than deathr and every health state was identified by some subjects as

worse than death. This is the third time, to our knowledge, that socia l

preference measurement techniques have discovered chronic health states widely

considered as worse than death. Rosser and Kind [1978, Tab le 2 ] report

similar findings for severe chronic dysfunctional states in adults, and

Lathrop and Watson [ 198 2 ] report similar findings for mutations. The

implications of these findings of states worse than death are discussed in

detail elsewhere (Torrance [ 1982] ) •

Reliabili ty

The reliability (reproducibility) of the fundamental measurements (Table

V) i s consistent with our p revious wor k in the field -- correlation

coefficients are relatively high, and yet so are the standard errors of

measurement. The correlation coefficients range from .86 to .94 and compare

favourably with previous studies where they ranged from .77 to .96 (Torra.pee

[ 1976b ] , Torrance et al. [ 1973 ] ) . Precision a s a propo r tion of scale length

ranges from .09 3 to .15 3 compared to previous work where the range is from

.081 to .139 (Torrance [ 1976b ] ) . The high correlation coefficients suggest

that in repeated measures of health state preferences on the same individual

good states remain good and bad states remain bad, while the sizeable standard I

errors of measurement remind us that the numerical quantification of these

preferences at the individual level is not particularly precise. On the other

hand, at the group level the precision is quite satisfactory, as demonstrated

by the small standard errors of the mean (Table III and IV).

2 3

Model FQrm and Multivariate Risk Attitude

The parameter values in Table VI show that the add itive form of the

multi-attribute disutility function is inappropriate for all of our subjects.

This is consistent with other work in the health field. Giauque and Peebles

[1976] assert that the additive model is inappropriate for their application

while Krischer [1976] found that only 10 % of h is 100 subjects met the

conditions for additive independence.

Ta ble VI a lso sho w s tha t for 91 % (79/ 8 7) of th e subj ects, th e

disattributes are substitutes (attr ibutes are complements). The intuitive

notion of th is is that each d isattribute -- P6, R5, S 4 and H8 -- is bad and

two or more·

together is not that much worse. This finding is consistent with

the assertion used by Giauque and Peebles [1976] to rule out add itive

independence. At f irst glance, the f ind ing appears to be contrary to

Kr i sch e r's resu l t [19 7 6] wh ere he found speech and cosme tics to be

substitutable attributes (complementary d�sattributes) in 87% (78/90) of the

subjects with a multiplicative model. On further examination, however, both

results can be seen to be examples of multivariate risk s�king behavior for

losses ?md multivariate risk aversion for gains (Fischer and Kamlet [1981) ) .

This comes about because of the different reference level or status quo in the

two stud ies. In our study the subjec ts were healthy and were viewing the

various (unhealthy) outco�es as losses; in Krischer's work the reference level

was a ch!ld born with a cleft lip and palate and the var ious outcomes of

different treatment approaches were seen as gains.

Social Asgregation

We investigated two methods of social aggregation -- aggregation at the

f ina l level lead ing to u1 (�) and aggrega tion at the basic level lead ing to

u2 (!!_). The former method, lead ing to U1 (�) , is considered the "correct"

2 4

I!

method for a number of reasons1 i t is consistent with the usual method of

determining the utility of a health state for each individual and aggregating

these, it is consistent with classical multi-attribute utility theory which is

developed for the case of a single decision maker {or a single decision making

unit}, and it allows different multi-attribute utility functions {additive,

multiplicative) to be used for different people as appropriate to their

underlying utili ty structure. The other method, leading to u2{�), is

considered a convenient approximation method. It is convenient because it is

far less wor k, the results can be displayed much more compactly, and it is

better suited to sensitivity analysis, to modification (li ke adding a new

level to an attribute) and to interpretation. The question then is, how well

does u2 (�) approximate u1 (�)? In our s tudy the approximation is relatively

good; the correlation is excellent, there is a small consistent underestimate

which cou ld be added to u2 (�} as a correc tion factor, and the remaining error

aJ?pears to be non-systematic.

Inde�ndence Condititions

The results from the MAU model underestimate the mean measured utility

for the two central test points F and G. There are a number of possible

explanations for the underestimate, and unfortunately, this stud:y is unable to

discriminate among them. The lack of fit may represent erro:rs in the model, I

errors in the measurement of the test states, or both. Errors in the model,

in turn, may be due to violations of mutual uti lity independence or to

systematic errors in the measurements of the model's parameters. Violations

of mutual utility independence would imply that a model more complicated than

the multiplicative should be used. Alternatively, a similar result can be-

achieved by adding more curvature in (13). In this study the fit is improved

as the exponent in {1 3} is increased. This suggests that perhaps the

25

•parameter value of 1.6 in (13), taken from previous work, understates the true

parameter value for this study .

Systematic errors in the measurement of the model's parameters could also

accoun t for the differen ce between the model and the test states. A

systematic error may be in troduced during the measuremen t of the multi-

attribute health states A-D Table IV. These states are described to the

subject by one dysfunction level and tl)ree blanks -- the blanks are intended

to represent "no dysfunction ", but are described to the subject as being "in

every other way as healthy as possible". To the ex ten t that responden ts

men tally fill in the blan ks with other dysfunc tion s, the cj values would be

overstated. This would lead to a model which overstates disutility

(under states utility ) , as observed . Also, the preferen ce measuremen t

procedure for states dispreferred to death is new and its validity is yet to

be established. If it contributes measurement error, it would affec t ooth the

test state F and G and the MAU model.

Finally, the lack of fit may represent errors in the measurement of the

test states. Fischer's r eview of the relevan t psychological literature

suggests that at the individual level discrepancies be tween direct holistic

assessments (like states F and G) and decomposed evaluation procedures (like '

our MAU model) are, at least in part, due to the unreliabili ty of the holistic

assessments, especially when the number of value-relevant attributes exceeds

six. (Fischer { 1979 pp. 474-475 ] ) . This explan ation may apply in our study

since, al though n ominally there are on ly four attributes, it can be argued

that because three of these contain a double content, there are actually seven

value-relevant characteristics which the subject must con sider in making a

holistic assessmen t. Fischer's review suggests that in this case, the MAU

model for the individual may indeed be the more valid. On the other hand, the

2 6

force of this explanation is mitigated bY the fact that we pool results across

87 subjects and, in this way, substantially reduce the unreliability problem.

In summary, the discrepancy between the test states and the model may be

due to errors in the measurement of the test states, errors in the model, or

both . The curren t research cannot discrimina te among these possibili ties,

a l though it seems likely tha t i t i s caused by some of both. On the other

hand, the current work does allow us to estima te a range of un�er tain ty for

the utility values.

Sensitivity Analysis

Uncertainty about the correct utility value to assign to any health state

can be attributed to two sources, methodology and measurement. Methodology

refer s to our uncertainty abou t the true multi-a ttribute utili ty model for

this situa tion. A range of methodologic uncer tainty is shown in Figure 1 .

Curve B i s the uti li ty function u1 (,!! ) given i n (17) , which includes

relationship (13) . Curv e C is the same u ti lity function, but calculated

without (1 3); i.e., assuming u = v. Fina lly, curve A is ba sed on the

assumption that the test states are correct and is a power curve fitted to the

points. Measurement uncertainty includes both sampling error and measurement

imprecision . These are combined in the standard error Sx figures shown in

Table IV. The mean of the seven values, .06, is used as a represen ta tive Sx

value. In our study of neonatal intensive care (Boyle et al. [1982)) the two

sources of uncertainty are added in order to test our findings over an extreme

range of u tili ty va lues. The upper limit for the sensitivity analysis is

curve A + 2Sx {not to exceed 1.0 ) while the lower limit is curve C - 2Sx.

--FIGURE 1 ABOUT HERE--

27

5 .. CONCLUSIOOS

In th is study a mod if ied MAU method has been u sed to measure social

preferences for a system of health states. The utilities so measured are for

use in a cost-effectiveness (cost-utility) analysis of neonatal intensive

care. Conclusions from the study are of two types: conclusions with respect

to the measured utilities and their applicability in the cost-effectiveness

study, and conclusions with respect to the method and its applicability by

other investigators in other studies.

The m easured utilities are based on responses from a random sample of

parents � losses from the sample (inelig ibl e subjects, nonpar ticipants and

confused responders) , although always undesirable, are consistent with

previous studies in the field. The reliability of the basic measurements is

satisfactory and comparable to previous work. Tpe technique of prespecifying

good and bad r eference levels for each attr ibute (xaj and xb

j ) and using a

common set of "corner" states was successfu l and created only a m inimal

"extrapolation" of th e multi-attribute. utility functions. The additive form

of the multi-attribute utility function is not appropriate, the multiplicative

form is recommended . The two aggregation methods u1 <.�9 and u2 (!) produce

comparable results - u1 {!) is the theoretically correct model, and u2 {!) is a

convenient and close approx imat ion. The utility correction relationship given

by (13) i s found to be d irectionally cor rect. u1 (_2) with the utility

correction relationsh ip still underestimates the two test states, and an

analysi s of th is underestimate lead s to the inclusion of the test states in

the specif ication of a range of uncer tainty on the f inal social pr efer ence

function . The final recommended soc ial preference function is u1� developed

us ing (13) and applied in conju nction with sens itivity ana lysis over a

spec if ied range of uncerta inty. Th is recommended function is used in our

28

evaluation of neonatal intensive care (Boyle et al. [ 1982] ) . The overall conclusion with respect to the modified �U method i s that it

looks promising as an approach to the measurement of social preferences for

health s tates that are defined by a multi-attr ibute classification system.

Except for very small classification systems, i t is a relatively eff icient

method compared to other approaches. It showed good po tential in this study

and no insurmountable di fficulties were encountered. However, a number of

lessons were learned, and these are summarized below as recommendations to

other investigators who may be interested in reapplying the method in other

studies. F irst, the health s tate classification system should be designed

such that each attr ibute contain only one concept (to minimize respondent

confusion), and such that the attributes can be conceived of by the subjects

as being independent. It is not essential that the attributes actually be

independ�nt in reality all the time, but simply that the subjects are able to

visuali ze them as independent. This is required because each measurement

question must speci fy what the subject is to assume about the other

attributes, and this assumption must be plausible. The preference measurement

instruments were acceptable to the subjects and showed satisfactory

reli ability. They can be recommended for reuse in other stud ies with two

provisions: first, it would be advisable to recalibrate the u = v1·6

relationship for the particular situation under study and second, for states

d i spreferred to death, i t would be useful to measure the utili ty by sever al

techni ques to investigate validity.. The technique of prespecifying common

"corner" states was quite successful in this study and can be recommended for

reuse. With respect to the use of test states to check the fit of the model

our recommendation is twofold. For problems with many attributes (about seven

or more) the use of test states is infeasible because of the unreliability of

29

the holistic utility assessments of the test states themselves. In such cases

the necessary independence conditions must be investigated directly. For

problems with few attributes (about six or less) the use of test states is a

feasible option, but they should be randomly selected from the entire multi­

attr ibute space and there shou ld be su fficient of them to be properly

representative. Although our study found the multiplicative model to be more

appropriate than the additive model for all subjects, this cannot be

generalized. Each study will have to determine from its own data which model

is more appropriate for each subject and overall. In our study the two

methods of aggregation produced comparable results and, although we suspect

that this will apply to other studies, we recommend, for the small extra work

involved, that other researchers investigate this in their own situation.

Finally, we hope that other investigators interested in measuring social

preferences for health states which can be defined by a multi-attribute

classification system will consider and use the MAU method along with those

modifications and recommendations that they find appropriate.

3 0

NOrES

1. Relationship (13) was not reported directly in that form in our previous

wor k. The ac tual repor ted functions in the original notation (C =

category scaling, T = time trade-off, S = standard gamble) are C = l -

(l-T)0.62 (Torrance [1976b] ) and T = s (Torrance [1976b] , Torrance et al.

[197 3 ] ) which combine to form (1-S) = ( l-c) l.61. In addition, our

pr evious wor k (To r r ance [19 7 6 b ] ) found but did not r epor t th e

relationship (1-T) = (l-C) l.58 which would lead to the same relationship

(13) •

2. Table 2 in Wolfson et al. [1982) provides 35 data points relating S to c.

Although Wolfson fits a linear function to this data, a power curve like

(13) can also be fitted giving (1-S) = (l-C) 2.16. Thus, the Wolfson data

confirms the same gene:i;:al shape of the relationship although certainly

not the specific parameter value.

ACKNOOLEDGMENTS

This research was suppor ted in par t by the Ontario Ministry of Health

(DM366) and by the Natural Sciences and Engineering Research Council (A4129 ) .

The authors also thank John Sinclair, Kathy Bennett, Jim Julian and an

anonymous referee for their help.

3 1

TABLE I

HEALTH STATE CLASSIFICATION SYSTEM --

(AGE � 2 YEARS) X1 PHYSICAL FUNCTION: r.'DBILITY AND PHYSICAL AcrNITY"'°

· Level Code Description Xl

1 Pl

2 P2

3 P3

4 P4

5 PS

6 PG

Being able to get around the house, yard, neighbourhood or community WITHOUT HELP from another person; AND having NO limit�tion in physical ability to lift, walk, run, jump or bend. Being able to get around the house, yard, neighbourhood or communi ty WITHOUT HELP from another person; AND having SOME limitations in physical ability to lift, walk, run, jump or bend. Being able to get around the house, yard, neighourhood or community WITHOUT HELP from another person; AND NEEDING mechanical aids to walk or get around. NEEDING HELP from another person in order to get around the house, yard, neighbourhood or community; AND having SOME limitations in physical ability to lift, walk, run, jump or bend. NEEDING HELP from another person in order to get around the house, yard, neighbourhood or community; AND NEEDING mechanical aids to walk or get around. NEEDING HELP from another person in order to get around the house, yard, neighbourhood or community; AND NOT being able to use or control the arms and legs.

X2 ROIB FUNCTION: SELF CARE .AND ROIB ACTIVITY4

Level Code Description X2

1 Rl

2 R2

3 R3

4 R4

5 R5

Being able to eat, dress, bathe and go to the toilet WITHOUT HELP; AND having NO limitations when playing, going to school, working or in other activities.

·

Being able to eat, dress, bathe and go to the toilet WITHOUT HELP; AND having SOME limitations when working, going to school, playing or in other activities. Being able to eat, dress, bathe and go to the toilet WITHOUT HELP; AND NOT being able to play, attend school or work. NEEDING HELP to eat, dress, bathe or go to the toilet; AND having SOME limitations when working, going to school, playing or in other activities. NEEDING HELP to eat, dress, bathe or go to the toilet; AND NOT being able to play, attend school or work.

3 2

X3 SOCIAL-EMJI'IONAL FUNCI'IOO: EM)I'IONAL WELL-BEING AND SOCIAL AcrIVITY

Level Code Description xf Sl

2

3

4

82

83

S4

Being happy and relaxed most or all of the time , AND having an average number of friends and contacts with others. Being happy and relaxed most or all of the time , AND having very few friends and little contact with others. Being anxious or depressed some or a good bit of the time , AND having an average number of friends and contact with others. Being anxious or depressed some or a good bit of the time , AND having very few friends and little contact with others.

X4 HEAL'IH PROB� Level Code Description

xi 2

3 4

5

6 7 8

Hl H2

H3 H4

H5

H6 H7 H8

Having no health problem. Having a minor physical deformity or disfigurement such as scars on the face. Needing a hearing aid. Having a medical problem which causes pain or discomfort for a few days in a row every two months. Needing to go to a special school because of trouble learning or remembering things. Having trouble seeing even when wearing glasses. Having trouble being understood by others. Being blind OR deaf OR not able to speak.

+Multiple choices within each descr iption are applied to individuals as appropr iate for the ir age. For example , a three years old child is not expected to be able to get around the community without help from another person.

::J:rndividuals with more than one health problem are classified according to the problem they consider the rrost serious .

3 3

TABLE II . NOrATION

xj , j = 1 , 2 , 3 ,4 represents attribute j (Table I )

xj represents the level on attribute j (Table I )

x*j represents the best (most preferred) level on attribute j according to the

individual.

x0j represents the worst (least preferred) level on attribute j according to the individuals

xaj represents a pr�specif ied good level (the level most preferred by most

people) on attribute j .

xbj represents a prespecified bad level (the level least preferred by most

people} on attribute j .

v*j <xj ) is the individual' s singte-a}tr ibute val�e function for attribute j ,

on the value scale with v j (x j ) = 1 and v j (x0 j ) = 0 .

vaj (xj ) is the individual' s single-attr ibute value f1511ction for attribute j ,

on the value scale with va j (x

a j ) = 1 and va

j (x j > = O.

VG is the group single-attribute value function (the arithmetic mean of the individual values) .

v = 1 - v is the associated disvalue function.

u* j cxj > , ua i (x;) , liG and u have parallel def initions but for util ity rather ' thafi value.

x = (x1,x2,x3,x4) represents a multi-attribute health state with the specified levels on each attribute.

!* , !o, �a ,, !b , !' r epresen\ spec�f ic irultk-atjf ibute health states formed

according to the rule ! = (x 1 , x 2 , x 3 , x 4) , where k can be * , O ,a, • • �

"!{1 represents the specific .health state , full healthy life .

!d represents the specific health state , death at birth •

. (xk j , x Pj) represent specific multi-attr ibut� health states formed accoring to

the rule that attribute j is at level x and all other attributes are at level x.Q..

u* (x} is the individual's multi-attr ibute utility function , on the utility -scale with u* <!*> = l and u* (!,0} = O .

ua (�} is the individual's multi-at1r ibute utility function, on the utility scale with ua (_!f} = 1 and ua (!, ) = O .

uh (x) i s the individual' s multi-attr ibute utility fuction, on the utility -scale with uh (�h) = 1 and uh (!d) = O.

3 4

u = 1 - u is the associated disutility function .

u1 (�) is S\e group multi-aJ=:tribute utility function on the utility scale with u1 (x ) = 1 and u1 (x ) = O, formed by aggregating at the f inal level. i .e:-u1 <.�) is the mean uh (�) .

U2 (�) is �e group multi-�ttribute utility function on the utility scale with u2 (� ) = 1 and u2 (� ) = O, formed by aggregating at the basic level. (This is defined more precisely later.)

L = <p�' , p�"> is a lottery (an uncer tain event) with probability p1 that the outcome will be �· and probability p2 that it will be �"·

3 5

Table III . Single-Attribute Value Functions (N=87)

Physical Function Role Function Social Em::>tional Function

ME:!an Std. Mean Std . Mean Std. Level Value Error Level Value Error Level Value Error

Pl 1 .00 . OOO Rl 1.00 . OOO Sl 1 .09 . OOO

P2 . 62 .082 R2 • 71 .021 S2 . 65 .027

P3 . 38 . 101 R3 • 32 .019 S3 • 25 .026

P4 . 37 .021 R4 . 30 .022 S4 .00 . OOO

PS . 10 .085 R5 . oo . OOO

P6 . oo . OOO

Table IV. Utility of Multi-Attribute Health States (N=87)

Attribute Levels Mean

Health Problem

Mean Std . Level Value Error

Hl 1 .00 . OOO

H2 . 49 .040

H3 . 47 .047

H4 . 46 .037

HS . 30 .062

H6 . 2S . OS4

H7 . 22 .074

H8 . oo . OOO

Standard State described on card

j % of times rated worse Utility Error than death

x11 No card - healthy - 1 .00 . OOO

A (P6 ) 61% .01 .069

B ( R5 ) S9% - .01 .063

c ( S4 ) 28% . 4S . OS3

D ( H8) 26% . 39 .06S

E (P6 , RS , 84, H8) 80% - . 39 .064

F (P2 , R2, H4) 13% . 67 .037

G (PS , R2 , H5) 31% . 31 .064

xd No c(:lrd - death I -- .00 . OOO

3 6

Table V. Reliability of Measures

Scale Replicated Measure length measures

L N --------------------- ------ ------ ----

Single-attribute measures (category scaling)

Physical function o , l

Role function o, l

Social-:Errotional function o , l

Health problem o , l

Multi-attribute measures (time trade-off)

Chronic states -1 , 1

Table VI . Parameters for ua {�)

Parameter cl --

Frequency dist.

Cj = 1 29

. 5 _:: Cj < l 36

0 < Cj < .5 22

Cj = 0 0 --

Total frequency 87 --

\ Median . 81

132

105

84

184

87

C2 C3

29 6

43 27

15 54

0 0

87 87

. 78 . 30

3 7

Precision as a Correlation proportion of Coefficient Precision scale length

r Ge oefL - --------- - -------- -----------

. 93 . 095 . 095

. 94 . 093 . 093

. 86 . 153 . • 153

. 87 . 122 . 122

. 88 . 240 . 120

C4 Parameter c

Frequency dist.

15 . .

19 c > 0 8

53 c = 0 0

0 c < 0 79

87 Total frequency 87

. 30 Median -1 . 00

8£

OO"L 6£'0-

:I

OO'L

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Faculty o f Bus iness McMaster Univers ity

WORKING PAPER SERIES

101 . Torran ce , Geo rge W . , "A Generalized Cos t-effectivenes s Model for the Evaluation o f Health Programs , " Novemb er , 19 7 0 .

102 . Isbes ter , A . Fraser and Sandra C . Cas t le , " Teachers and Collective Bargaining in Ontario : A Means to What End? " November , 1971 .

103 . Thomas , Arthur L . , "Trans fer Pri ces o f the Multinational Firm : When Will They be Arbi trary ? " ( Reprin ted from : Abacus , Vol . 7 , No . 1 , June , 1971) .

104 . Szendrovits , Andrew Z . , "An Economic Production Quantity Model with Holding Time and Costs of Work-in-process Inventory , " March , 1974 .

111 . Bas u , s . , " Inves tment Performance o f Coilllllon S tocks in Relation to their Price-earnings Ratios : A Text of the Effi cient Market Hypothesis , " March , 19 75 .

112 . Trus cott , William G . , "Some Dynamic Extens ions of a Dis crete Location­Allocation Problem , " March , 1 9 7 6 .

113 . Basu , S . and J . R . Hanna , "Accounting for Changes in the General Purchasing Power of Money : The Impact on Financial S tatements of Canadian Corporations for the Period 1967-74 , " April 19 7 6 . ( Reprinted from Cos t and Manage�ent , January-February , 1 9 7 6 ) .

114 . Dea l , K . R . , "Verification o f the Theoretical Consis tency of a Dif ferential Game in Advertising , " March, 1 9 7 6 .

114a. Deal , K . R . , "Op timizing Advertising Expenditures in a Dynamic Duopoly , " March , 1 97 6 .

1 1 5 . Adams , Roy J . , "The Canada-United S tates Labour Link Under Stress , " [ 1 9 76 ] .

116 . Thomas , Arthur L . , "The Extended App roach to Joint-Cos t Allocation : Relaxation of Simplify ing Assumptions , " June , 1 9 7 6 .

117 . Adams , Roy J . and C . H. Rummel , "Worker ' s Participation in Management in Wes t Germany : Impac t on the Work , the Enterprise and the Trade Unions , " Sep tember , 19 7 6 .

1 1 8 . S zendrovits , Andrew Z . , " A Comment o n ' Optimal and Sys t em Myopic Polic ies for Multi-echelon Produc tion/ Inventory As sembly Sys tems ' , " [ 1 9 7 6 ] ,

119 . Meadows , Ian S . G . , "Organic S t ructure and Innovation in Small Work Groups , " October , 1 9 7 6 .

Continued on Page 2 • . .

- 2 -

1 2 0 . Basu , S . , "The Effect o f Earnings Yield o n Ass es sments o f the Associa tion Between Annual Accounting Income Numbers and Security Prices , " Octob er , 1 9 7 6 .

121 . Agarwal , Naresh C . , "Labour Supply Behaviour o f Marri ed Women - A Model with Permanent and Transi tory Variab les , " October , 19 7 6 .

122 . Meadows , Ian S . G . , "Organic S t ructure , Satis faction and Personali ty , " October , 1 9 7 6 .

12 3 . Banting , Peter · M. , "Cus tomer S ervice in Indus trial Marketing : A Comparative S tudy , " October , 1 9 7 6 . (Reprinted from : European Journal of Marketing , Vol . 10 , No . 3 , Sunnner , 1 9 7 6 ) .

124 . Aivazian , V . , "On the Comparative-S tat;i.cs o f Asset Demand , " August , 1 9 7 6 .

125 . Aivazian , V . , " Contamination b y Risk Reconsidered , " October , 1 9 7 6 .

12 6 . Szendrovit s , Andrew z . and George O . Wesolowsky , "Variation in Op timizing Serial Mul ti-State Produc tion/ Inventory Sys tems , March , 1 9 7 7 .

1 2 7 . Agarwal , Naresh C . , " S i ze-S tructure Relationship : A Further Elaboration , " March , 197 7 .

1 2 8 . Jain , Harish C . , "Minority Workers , the S t ructure of Labour Marke ts and Anti-Discrimination Legislation , " March , 1 9 7 7 .

14 9 . Adams , Roy J . , "Employer Solidarity , " March , 1 9 7 7 .

130 . Gould , Lawrence I . and S tanley N . Laiken , "The Ef fect o f Income Taxation and Inves tment Priori ties : The RRSP , " March , 1 9 7 7 .

131 . Callen , Jeffrey L . , "Financial Cos t Allocations : A Game-Theoretic App roach, " March, 1 9 7 7 .

1 3 2 . Jain , Harish C . , "Race and Sex Discrimination Legislation in North America and Britain : Some Lessons for Canada , " May , 1 9 7 7 ..

133 . Hayashi , Kichiro . " Corporate Planning Practi ces in Japanese Multinationals . " Accepted for pub lication in the Academy o f Management Journal in 19 78 .

1 3 4 . Jain , Harish C . , Neil Hood and Steve Young , "Cross-Cultural Aspects of Personnel Policies in Mul ti-Nationals : A Cas e S tudy of Chrysler UK" , June , 1 9 7 7 .

135 . Aivazian , V . and J . L . Callen , "Inves tment , Market S tructure and the Cos t of Capi tal " , July , 19 7 7 .

Continued on Page 3 . . .

- 3 -

136 . Adams , R . J . , " Canadian Industrial Relations and the German Example " , October , 1 9 7 7 .

137 . Callen , J . L . , "Production , E f f ic iency and Welf are in the U . S . Natural Gas Transmiss ion Indus t ry " , October , 1 9 7 7 .

138 . Richards on , A . W . and Weso lowsky , G . O . , " Co s t-Volume-Profit Analys is and the Value o f Informat i on " , November , 1 9 7 7 .

139 . Jain , Harish C . , "Labour Marke t P roblems o f Native Peop le in Ontario " , De cember , 19 7 7 .

140 . Gordon , M . J . and L . I . Gould , " The Cos t o f Equity Capital : A Reconsid­eration " , January , 1 9 7 8 .

141 . Gordon , M . J . and L . I . Gould , "The Cos t o f Equity Cap ital with Pers onal Income Taxes and Flotation Co s t s " , January , 1 9 7 8 .

142 . Adams , R . J . , "Dunlop Af ter Two Decades : Sys t ems Theory a s a Framework For Organi z ing the Field o f Indus trial Re lations " , January , 1 9 7 8 .

143 . Agarwal , N . C . and Jain , H . C . , "Pay Dis crimination Agains t Woinen in Canada : Issues and Policies " , February , 1 9 7 8 .

144 . Jain , H . C . and S loane , P . J . , "Race , Sex and Minority Group Di s crimination Legislation in North Ameri ca and Britain " , March , 19 7 8 .

145 . Agarwal , N . C . , "A Labour Market Analysis o f Executive Earnings " , June , 1 9 7 8 .

146 . Jain , H . C . and Young , A . , "Racial Dis crimination i n the U . K . Labour Market : Theory and Eviden ce " , June , 1 9 7 8 .

147 . Yagil , J . , "On Al ternative Methods o f Treating Ris k , " Sep t ember , 1 9 7 8 .

148 . Jain , H . C . , "At t i tudes t oward Communication Sys tem : A Comparison o f Anglophone and Francophone Hosp ital Employees , " S ep tember , 19 78 .

14 9 . Ros s , R. , "Market ing Through the Japanes e Dis t ribution Sys tem" , November , 1 9 7 8 .

150 . Gould , Lawrence I . and S t anley N . Laiken , "Dividends vs . Cap ital Gains Under Share Redemp tions , " December , 19 7 8 .

151. Gould , Lawrence I . and S tanley N . Laiken , "The Impact o f General Averaging on Income Reali zation Decis ions : A Caveat on Tax Def erral , " Decemb er , 19 7 8 .

152 . Jain , Harish C . , Jacques Normand and Rabindra N . Kanungo , "Job Mo tivation of Canadian Anglophone and Francophone Hospital Emp loyees , April , 1 9 7 9 .

153 . S t idsen , Bent , "Communi cat ions Re lat ions " , Apri l , 1 9 7 9 .

1 5 4 . S z endrovits , A . Z . and Drezner , Zvi , "Op timizing N- S tage Production / Inventory Sys tems by Transporting Dif ferent Numb ers o f Equal-Sized Ba tches at Various S tages " , April , 1 9 7 9 . C t ' d p 4 on inue on age • . .

- 4 -

155 . Trus co t t , W . G . , "Allocation Analysis of a Dynamic Dis tribut ion Prob lem" , June , 1 9 7 9 .

156 . Hanna , J . R . , "Measuring Capital and Income" , November , 1 9 7 9 .

1 5 7 . Deal , K . R . , "Numeri cal So lut ion and Multiple S cenario Inves tigat ion o f Linear Quadrat i c Di f ferential Games " , November , 1 9 7 9 .

158 . Hanna , J . R . , "Pro fessional Accounting Education in Canada : P roblems and Prospec ts" , November , 1 9 7 9 .

15 9 . Adams , R . J . , "Towards a More Competent Labor Forc e : A Training Levy S cheme for Canada" , December , 1 9 7 9 .

1 6 0 . Jain , H . C . , "Management o f Human Res ources and P roduc tivity " , February , 1980 .

161 . Wens ley , A . , "The E f fici ency of Canadian Foreign Exchange Markets " , February , 1 980 .

162 . Tihanyi , E . , "The Marke t Valuation o f Deferred Taxes " , March , 1980 .

163 . Meadows , I . S . , "Quality o f Working Life : P ro gres s , P rob lems and Prospec ts " , March , 1 980 .

164 . S z endrovit s , A . Z . , " The Ef fect o f Numbers of S tages on Multi- S tage Produc t ion/Inventory Models - An Emp irical S tudy" , April , 1980 .

165 . Laiken , S . N . , " Current Action to Lower Future Taxes : General Averaging and Anti cipated Income Models " , April , 1980 .

1 6 6 . Love , R . F . , "Hull Properties in Location Prob lems " , Apri l , 1980 .

1 6 7 . Jain , H . C . , " Disadvantaged Groups on the Labour Marke t " , May , 1980 .

1 6 8 . Adams , R . J . , · "Training in Canadian Indus try : Res e arch Theory and Policy Impl ications" , June , 1980 .

1 6 9 . Joyner , R . C . , "App lication o f Process Theories to Teaching Uns tructured Managerial Decis ion Making " , Augus t , 1980 .

1 7 0 . Love , R . F . , "A S t opping Rule for Facili ties Location Algori thms " , Sep tember , 1980 .

1 7 1 . Abad , . Prakash L . , "An Op timal Control App roach to Marke ting - Produc tion Planning" , October , 1980 .

1 7 2 . Abad , Prakash L . , "Decentralized Planning With An iI:nterdependent Marketing-Product ion Sys tem" , October , 1980 .

1 7 3 . Adams , R . J . , " Indus trial Relations Sys tems in Europe and North America" , October , 1980 .

·

Continued on Page 5 . . .

1 7 4 .

1 7 5 .

1 7 6 .

1 7 7 .

1 7 8 .

1 7 9 .

180 .

181 .

182 .

183 .

184 .

185 .

1 8 6 .

1 8 7 .

188 .

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Gaa , James C . , " The Role o f Cen tral Rulemaking In Corporate Financial Rep orting" , February , 1 981 .

Adams , Roy J . , "A Theory of Emp loyer At ti tudes and Behaviour Towards Trade Unions In Wes tern Europe and North America" , February , 1 9 8 1 .

Love , Rob ert F . and Jsun Y . Wong , "A 0-1 Linear Program To Minimize Interac tion Cos t In S cheduling" , May , 1 9 8 1 .

Jain , Harish , "Employmen t and Pay Di s criminat ion in Canada : Theories , Evidence and Policies " , June , 1 9 8 1 .

Bas u , S . , "Marke t Reaction to Accoun ting Policy De liberation : The In flation Accoun ting Case Revisited" , June , 1 9 8 1 .

Bas u , s . , "Risk Information and Financial Lease Di sclosures : S ome Empirical Evidence" , June , 1981 .

Basu , S . , "The Relationship be tween Earnings ' Yield , Market Value and Return for NYSE Common S t o cks : Further Evidence " , Sep tembe r , 19 81

Jairi. , H . C . , "Race and Sex Dis crimination in Emp loyment in Canada : Theorie s , evidence and p o li cie s " , July 1981 .

Jain , H . C . , "Cross Cultural Management of Human Res ources and the Multinati onal Corporations " , October 1981 .

Meadows , Ian , "Work System Charact eristics and Emp loyee Response s : An Exp loratory S tudy " , October ; 1981 .

Svi Drezner , S z endrovit s , Andrew Z . , Wes olowsky , George 0 . ' 'Mul ti-s t age Product ion wi th V ariab le Lot S izes and T ranspo r t at ion of P art ial Lot s " , J anuary , 1982 .

Bas u , S . , "Res idual Risk, Firm Size and Returns f o r NYS E Common Stocks : Some Empirical Evidence " , February , 1982 .

Jain , Harish C . and Muthuchidambram, S . "The Ontario Human Rights Code : An Analys is o f the Public Poli cy Through Selected Cas e s o f Discriminat ion In Employment " , March , 1982 .

Love Robert F . , Dowling , Paul D . , "Op timal Wei ghted Q, Norm Parame ters For . Facilities Layout Dis tance Charac terizat ions " , PApr i l , 1982 ,

S teine r , G . , " S ingle Machine S cheduling with Precedence Constraints of Dimension 2 " , June , 1982 ,

.'