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Manual to Chronic Neck Pain Markov Decision Tree_

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					      Decision analysis of neck pain treatments (Supplementary Appendix)




Decision Analysis of Neck Pain Treatments:

         Supplementary Appendix




                        Page 1 of 35
                                   Decision analysis of neck pain treatments (Supplementary Appendix)




                                                     TABLE OF CONTENTS

TABLE OF FIGURES ....................................................................................................................... 3
I. Modeling and estimating the baseline course of neck pain. ................................................... 4
II. Background incidence rate of strokes in the general population .......................................... 5
III. Systematic search strategies ................................................................................................... 7
IV. Meta-analysis of single incidence proportions ...................................................................... 8
V. Meta-analysis of single incidence rates..................................................................................10
    A. Poisson regression ..........................................................................................................10
    B. Negative binomial regression .........................................................................................10
VI. Risk of stroke associated with chiropractic visits ...............................................................13
VII. Modeling mortality rates in the decision-analytic Markov model .......................................14
VIII. Health descriptions used to elicit quality-of-life weights ...................................................15
    A. Temporary health state scenarios (4-week):..................................................................15
    B. Permanent health state scenarios (lifetime): .................................................................18
IX. Eliciting utilities for permanent health states using the standard gamble scaling method
........................................................................................................................................................21
X. Calculation of utilities for temporary health states................................................................22
XI. Adapting the chained standard gamble when the loss outcome health state is preferred
to a temporary health state ..........................................................................................................24
XII. Eliciting utilities for health states considered worse than Death .......................................26
    A. The assumption that utilities for permanent health states lie on a scale anchored by
    Death and Good Health ............................................................................................................26
    B. The utility scale for certain individuals is not anchored by Death and Good Health:
    health states considered worse than Death ...........................................................................26
    C. Addressing the problem of measuring permanent health state utilities when a health
    state is considered worse than Death: Step 1 – how to obtain responses from
    respondents ..............................................................................................................................27
    D. Addressing the problem of measuring utilities when a health state is considered
    worse than Death: Step 2 – how to rescale utilities onto the Death – Good Health scale ..28
XIII. Obtaining temporary health state utilities when a permanent health state other than
Death is used as the loss outcome .............................................................................................32
References .....................................................................................................................................35




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                               Decision analysis of neck pain treatments (Supplementary Appendix)




                                                   TABLE OF FIGURES

Figure 1: Representation of the course of neck pain in the Markov model. ....................... 4
Figure 2: Standard gamble for a permanent health state................................................... 21
Figure 3: First stage of Torrance’s chained standard gamble. .......................................... 22
Figure 4: Second stage of Torrance’s chained standard gamble. ...................................... 22
Figure 5: Example from our study: first stage of a chained gamble for the temporary
     health state ‘Mild Neck Pain Episode’...................................................................... 23
Figure 6: Example from our study: second stage of a chained gamble for the temporary
     health state ‘Mild Neck Pain Episode’...................................................................... 23
Figure 7: Adaptation of Torrance’s chained gamble when loss outcome is preferred to a
     temporary health state. .............................................................................................. 24
Figure 8: Example from our study: adaptation of chained gamble when loss outcome
     ‘Excruciating Pain’ is preferred to the temporary health state ‘Mild Stroke’........... 25
Figure 9: ‘Death’ used as loss outcome for permanent health state standard gambles. .... 26
Figure 10: ‘Quadriplegia’ as the loss outcome for permanent health state standard
     gambles. .................................................................................................................... 27
Figure 11: Obtaining the utility of ‘Death’ on the ‘Quadriplegia’ – ‘Good Health’ scale.
     ................................................................................................................................... 27
Figure 12: Rescaling utilities relative to the Death (0.0) – Good Health (1.0) utility scale.
     ................................................................................................................................... 28
Figure 13: ‘Quadriplegia’ as loss outcome for second stage of chained gamble.............. 32
Figure 14: Example from our study: second stage of the chained gamble where
     ‘Quadriplegia’ was used as the loss outcome. .......................................................... 33




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                         Decision analysis of neck pain treatments (Supplementary Appendix)


I. Modeling and estimating the baseline course of neck pain.

We modeled a baseline course of neck pain that could be adjusted with treatment
effectiveness estimates. A person with troublesome neck pain experienced ‘Resolution’
if the person transitioned from ‘Troublesome Neck Pain’ to ‘No Troublesome Neck Pain’
in a successive cycle, ‘Persistence’ if the person remained in ‘Troublesome Neck Pain’
and ‘Recurrence’ if the person transitioned from ‘No Troublesome Neck Pain’ to
‘Troublesome Neck Pain’ (Figure 1). A person was ‘Asymptomatic’ if the person
remained in the ‘No Troublesome Neck Pain’ state.

We estimated transition rates ‘λ’ for ‘Resolution’ and ‘Recurrence’ by fitting an
exponential frailty model to Saskatchewan Health and Back Pain Survey (Côté et al.,
2004) data using the Maximum Likelihood theory which assumes that a parameter (in
this case, ‘λ’) follows a Normal distribution:

0 ~ Normal (, SE )

These transition rates were then transformed to probabilities using the equation:

1  e *time

‘Persistence’ and ‘Asymptomatic’ were modeled as the complements (1- λ) of the
probabilities for ‘Resolution’ and ‘Recurrence’, respectively.



                 Resolution
  Persistence


        Troublesome       No Neck
          Neck Pain        Pain



                              Asymptomatic
                Recurrence




Figure 1: Representation of the course of neck pain in the Markov model
Dotted line ovals represent the temporary neck pain-related health states ‘Troublesome
Neck Pain’ and ‘No Troublesome Neck Pain’. Arrows represent possible transitions
within and between neck pain-related health states.




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                     Decision analysis of neck pain treatments (Supplementary Appendix)


II. Background incidence rate of strokes in the general population

We estimated a background incidence of strokes in the general population of Ontario,
Canada for persons 45 years of age using two sources of data.

We estimated a background incidence of strokes in the general population of Ontario,
Canada for persons 45 years of age using two sources of data. Rates were constructed
with a count of a number of events (in this case, all types of stroke) and a measure of
person-time at risk. The numerator (i.e., number of stroke events) was obtained from data
assembled for the study ‘Examining Vertebrobasilar Artery Stroke in Two Canadian
Provinces,’ [insert BOYLE et al., NPTF 2007]. The number of strokes ‘a’ was obtained
by extracting all acute care hospital admissions for stroke for individuals aged 45 years in
Ontario for the period April 1, 1993 to March 31, 2002. Stroke was defined based on the
following International Classification of Diseases, Ninth edition (ICD-9) codes: 430, 431,
432, 433.0, 433.2, 433.1, 433.3, 433.8, 433.9, and 434. Only the first stroke occurrence
was extracted. Individuals who were hospitalized between April 1, 1991 and March 31,
1993 for stroke were excluded. During this time period, there were 666 strokes in
persons aged 45 years in Ontario. The denominator (i.e., person-time ‘PT’) was derived
from estimates of the Ontario general population obtained from Statistics Canada census
data (www.statscan.ca) for the above period for persons aged 45 years (total person-time
over this period estimated from CANSIM series V4685736 / 73 was 1,505,262).

We estimated the incidence of stroke for persons aged 45 years to be:
666 / 1,505,262 = 0.44 per 1000 person-years.

We estimated the standard error of this estimate to be:
             a                   666
SE ( IR )     2
                 = SE ( IR)              = 0.00002
            PT                1,505,2622

There are two limitations with our method for deriving an estimate of the incidence rate
of stroke in the general population of Ontario. The first limitation relates to the
numerator (i.e., counts of stroke events) of our estimate. The Canadian Institute for
Health Information Discharge Abstract Database only included individuals who had a
stroke and were discharged by March 31, 2002. Therefore, if an individual had a stroke,
for example, on March 5, 2002 and were discharged on April 3, 2002, this individual
would not have been included in the database. Based on patterns from previous years, it
appears that there were approximately 5% of strokes missing in this last fiscal year
compared to previous fiscal years. The second limitation relates to the denominator (i.e.,
person-time). It is likely that using census data to determine the number of individuals in
Ontario at risk for strokes for the time period 1993 to 2002 could have slightly inflated
the denominator because it was not possible to determine how many individuals were
excluded from the denominator due to previous stroke. However, it is also possible that
using census data somewhat underestimated the number of persons in the denominator.
For example, individuals from the First Nations who live on reservations and immigrant
populations tend to be undercounted in censuses. Furthermore, a count of persons living



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                    Decision analysis of neck pain treatments (Supplementary Appendix)


in shelters was undertaken for the first time for the 2001 Canadian Census. It should be
noted further that this is not an estimate of the number of homeless individuals because
there are many homeless individuals who do not stay at shelters. We could not use the
Registered Persons Database (i.e., the database of persons registered with the Ontario
Health Insurance Plan) to estimate the number of persons at risk because the number
provided by this database is inaccurate by an estimated 1 million persons. There are
several reasons for this inaccuracy, including: 1) certain individuals have been assigned a
more than one health card number and therefore are double-counted in the Registered
Person Database, and 2) the database is not up to date with respect to the number of
individuals who are no longer alive.

However, since the same incidence rate of stroke was used as the background risk of
stroke in our decision-analytic model for all treatment options, there is no differential
misclassification - and therefore should not be any bias introduced into the comparison of
treatment options in the decision-analytic model.




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                     Decision analysis of neck pain treatments (Supplementary Appendix)


III. Systematic search strategies

We performed electronic searches of Medline (1996 to 2006) to identify primary studies
that estimated incidence rates and incidence proportions of selected adverse events in the
general population. General inclusion criteria were: the Canadian population and studies
published in English. General exclusion criteria were: populations other than Canadian
populations (however, if Canadian estimates were not available we included estimates
based on the United States or Western European population). We also excluded studies
in which we could not determine the denominator (i.e., number of individuals at risk for
an adverse event) and numerator (i.e., number of cases of adverse events) from reported
methods and results. Studies whose internal validity was judged to be inadequate (i.e.,
results of the study could not be accepted with reasonable confidence due to
methodological limitations) were rejected.

On order to decrease the probability of omitting important sources of evidence, we
contacted content experts that were identified by their peers. Content experts were
requested to provide key references for selected variable estimates within their field of
expertise.

We performed systematic searches for evidence on the following:
- Annual incidence rate of hospitalization for acute myocardial infarction
- Incidence proportion for 28-day mortality after index myocardial infarction
- Incidence proportion for 1-year mortality after index myocardial infarction
- Proportion of acute myocardial infarctions that result in heart failure
- Incidence proportion for 1-year mortality after index heart failure
- Incidence proportion for 28-day mortality after index stroke
- Incidence proportion for 1-year mortality after index stroke
- Annual incidence rate of hospitalization for serious upper gastro-intestinal bleed
- Incidence proportion for 28-day and for 1-year mortality after serious upper
   gastrointestinal bleed
- Proportion of serious upper gastrointestinal bleeds that are treated with surgery




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                      Decision analysis of neck pain treatments (Supplementary Appendix)


IV. Meta-analysis of single incidence proportions

We pooled individual incidence proportions of the risk of adverse events to calculate
specific background risk estimates for input into our decision-analytic model (e.g.,
percent of individuals who are hospitalized for acute myocardial infarction who die
within 30 days of hospitalization). All analyses were performed using SAS software,
version 9.1 (SAS Institute Inc., Carey, NC, United States).

We pooled individual study estimates of incidence proportions with a random effects
intercept-only model for proportions, weighting individual estimates by the inverse of
their variance (w = 1/variance). For proportions, the conventional standard error is:
           p(1  p)
           ˆ     ˆ
 var( p) 
      ˆ             . Our approach was to assume an normal random effects distribution
              n
                                                                                 p 
on a scale defined by a transformation, the logit of p ‘  ’, where   log            , and the
                                                                                 1 p 
                        1     1
variance of  is v            , resulting in the logistic-normal model. First, we
                        x nx
transformed single incidence proportions to  , Second, we calculated a pooled (i.e.
mean)  and its standard error, weighting individual estimates with the inverse of their
variance using a random effects model. Third, we back-transformed the pooled  based
                                                                                  p 
                                                                      exp  log       
                                                                              1 p 
on the formula: e 
                         p
                            , such that p 
                                               e
                                                    , that is: p                     .
                                                  
                       1 p                  1 e                                     
                                                                   1  exp  log p  
                                                                                1 p  
                                                                                      
The following is an example of statistical coding used to pool individual incidence
proportions in SAS software:
_______________________________________________________________________
data MI;
       input studyid estim x n; /* x=number of events; n=individuals at
       risk */
       varest = ((1/x)+(1/(n-x)));
       logestim = log(estim/(1-estim));
       studyid = _n_;
       row = _n_; col = _n_; value = varest;
cards;
1      0.0954      335   3513
2      0.094       166   1763
3      0.06        25821 430442
4      0.0829      116   1399
5      0.119       80801 679000
6      0.129       486   3776
7      0.117       131   1120
;
run;

proc mixed data=MI order=data;



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                Decision analysis of neck pain treatments (Supplementary Appendix)

     class studyid;
     model logestim = / outp=pred outpm=param cl;
     random studyid / gdata = MI s;
     repeated diag;
     make 'SolutionR' out=randv;
run;
_______________________________________________________________________




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                     Decision analysis of neck pain treatments (Supplementary Appendix)


V. Meta-analysis of single incidence rates

We pooled individual incidence rates of the risk of adverse events (e.g., incidence rate of
hospitalization for acute myocardial infarction in the Ontario general population) to
calculate specific background risk estimates for input into our decision-analytic model.
All analyses were performed using SAS software, version 9.1 (SAS Institute Inc., Carey,
NC, United States).

We pooled individual study estimates of single incidence rates ‘ i ’ using a fixed-effects
intercept-only negative binomial model, where ‘ log( ) ’ is the intercept ‘  ’ and there is
an additional ‘offset’ variable ‘ log( PTi ) ’. The offset variable was treated as a constant
(i.e. a coefficient is not estimated for it, the value of the coefficient was fixed at 1).
Individual study events (i.e. counts) ‘ xi ’ were assumed to follow a Poisson distribution.
Individual study rates ‘ i ’ were assumed to vary across studies following a Gamma
distribution. Thus, combining the Poisson distribution for counts and Gamma distribution
for rates, the final model assumed a Negative Binomial distribution which accounted for
heterogeneity in individual rate estimates and variation across individual rate estimates.

The Negative Binomial distribution, and how it was used to meta-analyse single
incidence rates, is explained in the text below:

 A. Poisson regression
Given the true mean for the ‘ith’ study ‘  i ’, the observed count of events ‘ xi ’ that occur
during an observation period ‘PTi’ (i.e. person-time) is a random variable assumed to
have a Poisson distribution, xi ~ Poisson( i ) where i  i * PTi - the product of the
underlying rate ‘ i ’ and the observation time ‘PTi’ – that is, xi ~ Poisson(i * PTi ) . In
SAS statistical software (SAS Institute Inc., Carey, NC, United States), Poisson
regression models can be fit using the GENMOD procedure using the following model
specification code:

        <model x= / offset=logpt d=poisson link=log>

The above code defines: a) a Poisson distribution for the observed data ‘ xi ’, b) an
intercept-only model, c) the log link (which relates the log of the mean ‘ log( i ) ’ to the
covariates, and d) an offset term ‘ log( PTi ) ’ which adds this term as a covariate with a
fixed (i.e. not estimated) coefficient, fixed at a value of 1. The intercept from this model,
‘  i ’ is the estimate of the log of the assumed common (i.e. pooled rate) ‘ log(i ) ’.

B. Negative binomial regression
If there is more variation in the counts of events than can be explained by the within-
study Poisson variation, we can think of a between-study component of variance or
heterogeneity in the underlying rates. The Negative Binomial model can be thought of as
a two-level model where there is a Poisson variation within a study around the true study


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                     Decision analysis of neck pain treatments (Supplementary Appendix)


mean and a between study variation in the true rates that follows a Gamma distribution.
Given ‘  i ’, the true mean for the ‘ith’ study, the observed count in the study is a random
variable with a Poisson distribution with  i :
                  xi ~ Poisson (  i )                          (A)

And, because the mean equals the true rate times the observation time, we have

                i  i * PTi                                 (B)

In the Negative Binomial model, we introduce an extra term to accommodate unobserved
heterogeneity. That is, i  i * PTi *I or log (i) = log(i) + log(PTi) + log(i). If we
assume i follows a Gamma distribution with parameter  = 1/k - that is
                              
                  g ( i )         i 1 exp(  i ) for  > 0
                             ( )
then integrating the distribution of i into the Poisson distribution gives a negative
binomial distribution for xi incorporating heterogeneity across studies: xi ~ Negative
Binomial (  , k), which has mean  and variance  + 2 k. Note, if k=0 then the variance
is equal to , the mean, and this model reduces to the Poisson model.

In SAS statistical software (SAS Institute Inc., Carey, NC, United States), this is
represented by the procedure GENMOD with the following code:

       <model x= / offset=logpt d=negbin link=log>

The above code defines: a) a negative binomial distribution for observed data’ xi ’, b) an
intercept-only model, c) the log link which relates the log of the mean to the covariates,
and d) an offset term of log( PTi ) which is added as a covariate with a fixed (i.e. not
estimated) coefficient with a value of 1. This tells SAS that the mean ‘  i ’ is to be
decomposed as ‘ i * PTi ’ and the Gamma distribution is to be applied to the rates ‘ i ’.
The intercept in this model will be the estimate ‘  ’, the log of the mean rate ‘ log( ) ’.
The dispersion parameter reported in the SAS output will be close to zero if there is little
variation in rates between studies.

The following is an example of statistical coding used to pool individual incidence
proportions in SAS software:
________________________________________________________________________
data MI;
       input studyid x pt; /* x = number of events; pt = person-time */
       logpt = log(pt);
cards;
1      271321      196000000
2      14842       3543570
3      395         1127409
4      3776        2995200



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5     10079       7356900
6     1015        353481
;
run;
proc genmod data=MI;
      model x = / dist=nb link=log offset=logpt;
run;




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                    Decision analysis of neck pain treatments (Supplementary Appendix)


VI. Risk of stroke associated with chiropractic visits
An estimate of the risk of stroke associated with chiropractic visits was derived from
‘Cassidy JD, Boyle E, Côté P, He H, Hogg-Johnson S, Silver F, Bondy S, Risk of
vertebrobasilar stroke and chiropractic care: results of a population-based case control
and case-crossover study.” [insert CASSIDY NPTF 2007] Data from this case cross-
over study conducted in the province of Ontario, Canada were used to estimate the risk of
all strokes associated with chiropractic visits in persons <45 years of age.

Selection of cases: All acute care hospital admissions for stroke for individuals aged
older than two years and less than 45 years were extracted for the period April 1, 1993 to
March 31, 2002. Stroke was defined based on the following International Classification
of Diseases, Ninth Edition (ICD-9) codes: 430, 431, 432, 433.0, 433.2, 433.1, 433.3,
433.8, 433.9, and 434. Individuals who were hospitalized between April 1, 1991 and
March 31, 1993 for stroke were excluded.

Exposure to chiropractors: All ambulatory chiropractic encounters during the one-month
period prior to the hospital admission were extracted from the Ontario Health Insurance
Plan administrative billing database. The number of visits during the month preceding
the stroke was calculated for chiropractors.

Analysis: Conditional logistic regression models were constructed to determine the odds
of having a stroke after having three or more visits to a chiropractor during the one month
prior to the hospital admission. Accelerated bias corrected bootstraps with 2000
replications using the variance-covariance method were calculated. All statistical
analyses were performed using STATA/SE version 9.2 (Statacorp LP, College Station,
TX, United States).




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VII. Modeling mortality rates in the decision-analytic Markov model

Age-specific mortality rates were obtained from Statistics Canada Life Tables
(www.statscan.ca). Specifically, the conditional probability ‘qx’ (defined as either: 1) the
probability that a person of exact age ‘x’ years will die before reaching exact age ‘x+n’,
or 2) the proportion of individuals dying between exact age ‘x’ and age ‘x+n’) was
converted to a rate, using the formula:
  ln(1  p )
              where p = probability and t = time.
      t
Statistics Canada only provides age-sex-specific Life Tables 2000-2002 (i.e., not age-
specific only) on its website. We needed age-specific estimates for our model. To
address this problem, the software programme TreeAge performed a table look-up for
‘qx’ based on the age of the cohort from the Female Life Table and the Male Life Table.
These were added together then divided by 2, to estimate the age-specific ‘qx’. The
underlying assumption is that the percent distribution of men and women in the Ontario
general population is 50% for each sex. We tested this assumption by obtaining the
population estimate of men and the population estimate of women in the province of
Ontario for the years 1992 through to 2002 from Statistics Canada census data. We then
calculate the percent distribution of women and men for each year. We determined that
the greatest difference across each year between men and women was 1.48% (i.e. women
50.74%; men 49.26%). We therefore concluded that our assumption was reasonable.

All probabilities were converted into rates using the formula:
  ln(1  Pr)
                 so that these rates could be multiplied by the model variable ‘time’ to adapt
     time
the yearly rates into four week rates, since the model cycle length was four weeks. Rates
were then converted to probabilities using the formula:
1  e rate*t , where t = time.




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VIII. Health descriptions used to elicit quality-of-life weights
 A. Temporary health state scenarios (4-week):

                                  Endoscopically-treated Upper Gastrointestinal Bleed
Narrative version:
You have stomach pains for 2 weeks, then cramping with loose bowel movements. You vomit bloody material and are
taken to hospital emergency. You are hospitalized for 4 days. [Please stop reading while I describe your hospital
stay.]
   In the hospital you are not allowed to eat or drink. A doctor performs an ‘endoscopy’ which involves looking down
into your stomach through a long flexible tube. You are told that you have an ulcer, which has now stopped bleeding.
Later you are allowed fluids and then allowed to eat. On the fourth day you are discharged from hospital. [Please read
with me again.]
   As a result of this, for the 1 week after hospital discharge you have no difficulty taking care of yourself. You have
mild difficulty with work and household chores because you are tired. You have mild difficulty with leisure, social,
and family activities because you are tired. You have mild emotional distress: you are worried about your stomach
problems.

Point form version:

          Stomach pains for 2 weeks, Cramping; Loose bowel movements
          Vomiting of blood; Hospitalized for 4 days (endoscopy)
          …and as a result of this, for 1 week after hospital discharge you have:
              No difficulty taking care of yourself
              Mild difficulty with work (school) and household chores
              Mild difficulty with leisure, social, and family activities
              Mild emotional distress due to risks with blood transfusions



                                    Surgically-treated Upper Gastrointestinal Bleed
Narrative version:
You have indigestion for three days. You suddenly have severe stomach pain and are taken to the hospital emergency.
You are hospitalized for 2 weeks. [Please stop reading while I describe your hospital stay.]
   In the hospital, a stomach x-ray shows that you have an ulcer that has created a hole in your stomach. You are
rushed to emergency surgery. You wake up in intensive care. You have pain controlled with medication. You cannot
eat or drink, nor can you get out of bed for several days. After a few days you are allowed fluids then soft foods. After
two weeks you are discharged from hospital. [Please read with me again.]
   As a result of this, for the 2 weeks after hospital discharge, you have no difficulty taking care of yourself. You have
moderate difficulty with work and household chores because you are tired. You have moderate difficulty with leisure,
social, and family activities because you are tired. You have mild emotional distress: you are worried about your
stomach problems.

Point form version:

          Indigestion for 3 days, Sudden severe stomach pain
          Hospitalized for 2 weeks (surgery)
          …and as a result of this, for 2 weeks after hospital discharge you have:
              No difficulty taking care of yourself
              Moderate difficulty with work (school) and household chores
              Moderate difficulty with leisure, social, and family activities
              Mild emotional distress about your stomach problems




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                           Decision analysis of neck pain treatments (Supplementary Appendix)


                                                 Myocardial Infarction

Narrative version:
You awake with severe chest pain which spreads to your left arm and neck. You are sweating and short of breath. You
are taken to hospital emergency and hospitalized for 7 days. [Please stop reading while I describe your hospital stay.]
   In the hospital a doctor orders blood tests, heart tests, and chest x-rays. On the second day in the hospital you
receive an ‘angioplasty’ to clear a blocked artery in your heart. This procedure involves inserting a needle into a vein
in your groin region and threading a long plastic tube from your groin to your heart. After 7 days you are discharged
from hospital. [Please read with me again.]
   As a result of this, for the three weeks following hospital discharge you have no difficulty taking care of yourself.
You have mild difficulty with work and household chores because you are tired. You have mild difficulty with leisure,
social, and family activities because you are tired. You have moderate emotional distress; you are worried about your
heart problem.

Point form version:

          Severe chest pain; Left arm and neck pain; Sweating; Short of breath.
          Hospitalized for 1 week (angioplasty)
          …and as a result of this, for 3 weeks after hospital discharge you have:
              No difficulty taking care of yourself
              Mild difficulty with work (school) and household chores
              Mild difficulty with leisure, social, and family activities
              Moderate emotional distress about your heart problem


                                               Sample: Own Health State

Narrative version:
This describes your own health state

Point form version:

          Your own health
          …and as a result of this, for 4 weeks you have:
              Mild difficulty taking care of yourself

              Mild difficulty with work (school) and household chores

              Moderate difficulty with leisure, social, and family activities

              Mild emotional distress because of your neck problem




                                                       Page 16 of 35
                            Decision analysis of neck pain treatments (Supplementary Appendix)


                                   Good Health (Standard Gamble Gain Outcome)

Narrative version:
This card describes a person in good health for 4 weeks.

Point form version:


          Good health
          …and as a result of this, for 4 weeks you have:
              No difficulty taking care of yourself
              No difficulty with work (school) or household chores
              No difficulty with leisure, social, or family activities
              No emotional distress because of your health




                                Excruciating Pain (Standard Gamble Loss Outcome)

Narrative version:
You have excruciating pain. Pain medication partly relieves the excruciating pain. You are hospitalized for 4 weeks
and confined to a bed. As a result of this, during the 4 weeks of hospitalization you have complete inability to take care
of yourself and are completely dependent on others to bathe or go to the toilet. You have complete inability to do any
work and household chores, as well as engage in leisure, social, and family activities. You have severe emotional
distress because of your health.

Point form version:

          Excruciating pain; Medication partly relieves excruciating pain
          Hospitalization for 4 weeks
          …and as a result of this, during 4 weeks of hospitalization you have:
              Complete inability to take care of yourself

              Complete inability to work (school) and do household chores

              Complete inability to engage in leisure, social, and family activities

              Severe emotional distress because of your health




                                                       Page 17 of 35
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B.      Permanent health state scenarios (lifetime):
                                                       Heart Failure

Narrative version:
You tend to have low energy. Occasionally your legs swell at the end of the day. Sometimes your lungs retain fluid
and you get short of breath when lying down or exercising. As a result of this, you have no difficulty taking care of
yourself. You have moderate difficulty with work and household chores because you limit more strenuous activities.
You have moderate difficulty with leisure, social, and family activities because you get weak or short of breath if you
walk too far or become too active. You have mild emotional distress; you wish your activities were not limited.

Point form version:

          Low energy; Occasionally legs swell; Sometimes lungs retain fluid and short of breath
          …and as a result of this, all your life you have:
              No difficulty taking care of yourself
              Moderate difficulty with work (school) and household chores
              Moderate difficulty with leisure, social, and family activities
              Mild emotional distress because of your health




                                                Minor Stroke Disability

Narrative version:
You have weakness in your [left, right] arm and leg. You are unsteady on your feet. You have double vision. You see
two of any object you look at so you wear a patch on one eye. As a result of this all your life you have mild difficulty
taking care of yourself such as dressing and bathing. You have mild difficulty with work and household chores
because of weakness in your arm and leg. You have mild difficulty with leisure, social, and family activities because
some of these activities require fine balance, such as climbing stairs or going uphill. You have moderate emotional
distress: you feel isolated and somewhat helpless.

Point form version:

          Weakness of right/left arm and leg; Unsteady on feet
          Double vision
          …and as a result of this, all your life you have:
              Mild difficulty taking care of yourself
              Mild difficulty with work (school) and household chores
              Mild difficulty with leisure, social, and family activities
              Moderate emotional distress because of your health




                                                       Page 18 of 35
                            Decision analysis of neck pain treatments (Supplementary Appendix)


                                                 Major Stroke Disability

Narrative version:
You have paralysis of both arms and legs. You have double-vision: you see two of any object you look at. You have
difficulty talking and being understood by other people. As a result of this all your life you live in an institution and all
your time is spent in bed or in a wheelchair. You have complete inability to take care of yourself and you rely on
nursing staff for bathing, dressing, feeding and toileting. You have complete inability to work and do household
chores. You have complete inability to engage in leisure, social and family activities. You rely completely on friends
and family for social and family activities. You have severe emotional distress: you feel lonely, helpless, and
depressed by your health situation.

Point form version:

          Paralysis of both arms and legs;
          Double vision; Difficulty talking and being understood;
          …and as a result of this, all your life you have:
              Complete inability to take care of yourself
              Complete inability to work (school) and do household chores
              Complete inability to engage in leisure, social, and family activities
              Severe emotional distress because of your health



                                   Good Health (Standard Gamble Gain Outcome)

Narrative version:
This card describes a person with no health problems.

Point form version:



          Good health

          …and as a result of this, for all your life you have:

              No difficulty taking care of yourself
              No difficulty with work (school) or household chores
              No difficulty with leisure, social, or family activities
              No emotional distress because of your health




                                                       Page 19 of 35
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                                       Death (Standard Gamble Loss Outcome)

Narrative version:
This card describes Immediate Death.

Point form version:




                                                  Immediate Death




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IX. Eliciting utilities for permanent health states using the standard
gamble scaling method

We used the standard gamble direct scaling method to measure neck pain patients’
preferences for permanent (chronic) health states. The standard gamble consists of
asking respondents to make a choice between a certain option and a gamble. Figure 2
illustrates the standard gamble that was used in our study for a permanent health state
preferred to ‘Death’. Respondents were provided with a description of a permanent
health state, for example, ‘Mild Stroke Disability’ and asked to imagine living in this
health state for the remainder of their life. Respondents were then presented with a
choice of accepting or not accepting a hypothetical treatment to treat ‘Mild Stroke
Disability’. The treatment (i.e., the gamble) had two possible outcomes: a ‘Lifetime of
Good Health’ (the so-called gain outcome) with a probability p or ‘Immediate Death’ (the
loss outcome) with the probability 1 - p. The decision not to accept treatment resulted in
remaining in the permanent ‘Mild Stroke Disability’ health state for the rest of one’s life
(i.e., the certain option). Probability p was varied until respondents were indifferent
between the gamble and the certain option, at which point the utility of the health state
‘Mild Stroke Disability’ was p.

Hence, the probability p, or risk, of ‘Death’ that respondents were willing to take to avoid
living with ‘Mild Stroke Disability’ for the remainder of their lives reflected respondents’
preferences for this permanent health state. Presumably, as the desirability of a
permanent health state decreased, a respondent was willing to accept an increased risk of
‘Death’ (1 – p) with the hypothetical treatment, in order to avoid the certain option of
living in the permanent health state. Therefore, as the desirability of permanent health
states decreased, so did p, that is, so decreased the utilities of these health states.

Figure 2: Standard gamble for a permanent health state

                 Certain Option                                      Gamble

                                                                     Lifetime Good Health
                                                              p
               Permanent Health State
                                                             1- p
                                                                       Immediate Death
u PHS = [(p) (u LGH) + (1 - p) (uID )]
where: u PHS = utility of a ‘Permanent Health State’
       u LGH = utility of a 'Lifetime of Good Health’ = 1.0
       u Death = utility of ‘Immediate Death’ = 0.0




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X. Calculation of utilities for temporary health states
Torrance (1986) proposed that utilities for temporary health states could be measured
relative to each other using the standard gamble method as shown in Figure 3. In Figure
3, an intermediate, ‘Temporary Health State’ (Temp) is measured relative to the best state
(‘Lifetime of Good Health’ (LGH)) and the ‘Worst Temporary Health State’ (Worst). In
order to obtain a utility for the ‘Temporary Health State’ on the 0.0 – 1.0 (‘Death’ –
‘Good Health’) scale, Torrance proposed that the ‘Worst Temporary Health State’ should
be re-defined as a short-duration chronic state followed by ‘Immediate Death’ and
measured on the 0.0 – 1.0 (‘Death’ – ‘Good Health’) scale by the standard gamble
technique used for permanent states (Figure 4). Torrance’s second stage provides the
utility for the ‘Worst Temporary Health State’, which is then used in the equation in
Figure 3 to solve for the utility of the ‘Temporary Health State’.

Figure 3: First stage of Torrance’s chained standard gamble


                Certain Outcome                                      Gamble

                                                                     Lifetime Good Health
                                                               p
   Temporary Health State followed by a
        Lifetime of Good Health
                                                              1- p
                                                                      Worst Temporary
                                                                     Health State followed
                                                                           by LGH
uTemp.LGH = [(p) (u LGH) + (1 - p) (u Worst.LGH)]
where: u Temp.LGH = utility of ‘Temporary Health State’ followed by Lifetime of Good
                      Health
        u LGH = utility of a ‘Lifetime of Good Health’ = 1.0
        u Worst.LGH = utility of ‘Worst Health State’ followed by a Lifetime of Good Health
Figure 4: Second stage of Torrance’s chained standard gamble


                Certain Option                                       Gamble

                                                                     Lifetime Good Health
                                                               p
  Worst Temporary Health State followed
          by Immediate Death
                                                              1- p
uWorst.ID = [(p) (u LGH) + (1 - p) (u ID)] = p                         Immediate Death
where: u Worst.ID = utility of ‘Worst Temporary Health State’ followed by Immediate
                     Death
         u LGH = utility of a ‘Lifetime of Good Health’ = 1.0
         u ID = utility of ‘Immediate Death’ = 0.0



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For example, in our study, the loss outcome used was ‘Excruciating Pain’: a health state
described as experiencing poorly controlled excruciating pain while hospitalized in a
helpless state for four weeks. Figures 5 and 6 provide an example of how the utility of
the temporary health state ‘Mild Neck Pain Episode’ (a health state described as
experiencing a mild neck ‘kink’ for four weeks) was elicited with a two stage chained
gamble.

Figure 5: Example from our study: first stage of a chained gamble for the temporary health state
‘Mild Neck Pain Episode’


                Certain Option                                             Gamble

                                                                           Lifetime Good Health
                                                              p
  ‘Mild Neck Pain Episode’ followed by a
         Lifetime of Good Health                                             ‘Excruciating Pain’
                                                            1- p           followed by a Lifetime
UMild NP.LGH = [(p) (u LGH) + (1 - p) (u Excruciating Pain.LGH)]               of Good Health
where: u Mild NP.LGH = utility of Mild Neck Pain followed by a Lifetime of Good Health
        u LGH = utility of a ‘Lifetime of Good Health’ = 1.0
        u Excruciating Pain.LGH = utility of ‘Excruciating Pain’ followed by a Lifetime Good
        Health


Figure 6: Example from our study: second stage of a chained gamble for the temporary health state
‘Mild Neck Pain Episode’

                Certain Option                                              Gamble

                                                                           Lifetime Good Health
                                                              p
      ‘Excruciating Pain’ followed by
             Immediate Death
                                                            1- p
                                                                              Immediate Death
uExcruciating Pain.ID = [(p) (u LGH) + (1 - p) (u ID)] = p
where: u Excruciating Pain..ID = utility of ‘Excruciating Pain’ followed by Immediate Death
         u LGH = utility of a ‘Lifetime of Good Health’ = 1.0
         u ID       = utility of ‘Immediate Death’ = 0.0




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XI. Adapting the chained standard gamble when the loss outcome
health state is preferred to a temporary health state
Torrance’s chained method is based on the assumption that respondents prefer the
temporary states to the gamble’s loss outcome. In his method, the loss outcome must
therefore be a temporary health state which has the least desirability, or lowest
preference, compared to all other temporary heath states under consideration. Since the
loss outcome defines the lower bound of the temporary health state utility scale, it is
assigned a utility of 0.0.

In our study, we chose to define the loss outcome as ‘Excruciating Pain’: a health state
described as experiencing poorly controlled excruciating pain while hospitalized in a
helpless state for four weeks. However, we noticed that some respondents (nT1 = 20; nT2
= 7) preferred ‘Excruciating Pain’ to other temporary health states. Since Torrance’s
assumption did not hold true in these respondents, we could not apply his methods to
measure their utility of the temporary health state they considered worse than our loss
outcome.

To address this situation, Jansen and colleagues (1998) proposed a variation Torrance’s
method, which consists of adapting the first stage of Torrance’s two stage chained
gamble (described in Section X). The first stage is adapted such that the loss outcome
becomes the certain option and the gamble becomes a gamble with two possible
outcomes: the risk of the ‘Temporary Health State’ considered worse than the loss
outcome, followed by a ‘Lifetime of Good Health’ versus a ‘Lifetime of Good Health’
(Figure 7). Jansen and colleagues derived an equation in order to calculate the utility of
the ‘Temporary Health State’ (uTemp..LGH) considered worse than the loss outcome (Figure
7).

Figure 7: Adaptation of Torrance’s chained gamble when loss outcome is preferred to a temporary
health state


               Certain Option                                            Gamble
                                                                         Lifetime Good Health
                                                            p
     Worst Health State followed by a
        Lifetime of Good Health                                           Temporary Health
                                                          1- p            State followed by a
                                                                         Lifetime Good Health
u Temp.LGH = (uWorst.LGH - p)/(1 - p)
where: u Temp.LGH = utility of ‘Temporary Health State’ followed by a Lifetime of Good
                      Health
        u Worst.LGH = utility of ‘Worst Health State’ followed by a Lifetime of Good Health




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Figure 8 provides an example from our study of how we elicited the utility of the
temporary health state ‘Mild Stroke’ when ‘Mild Stroke’ was considered worse than our
loss outcome, using Jansen et al.’s adaptation.
Figure 8: Example from our study: adaptation of chained gamble when loss outcome ‘Excruciating
Pain’ is preferred to the temporary health state ‘Mild Stroke’


               Certain Option                                            Gamble
                                                                         Lifetime Good Health
     ‘Excruciating Pain’ followed by a
                                                            p
         Lifetime of Good Health                                             ‘Mild Stroke’
                                                              1- p      followed by a Lifetime
u mild stroke.LGH = (uexcruciating pain.LGH - p)/(1 - p)                      Good Health
where: u mild stroke.LGH = utility of a ‘Mild Stroke’ followed by a Lifetime of Good Health
          u LGH = utility of a ‘Lifetime of Good Health’ = 1.0
          uexcruciat’ pain.LGH = utility of ‘Excruciating Pain’ foll’d by Lifetime of Good Health




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XII. Eliciting utilities for health states considered worse than Death

 A. The assumption that utilities for permanent health states lie on a
scale anchored by Death and Good Health
The health state utility scale ranges from ‘Death’ to a ‘Lifetime of Good Health’ where
‘Death’ is assigned a utility of 0.0 and a ‘Lifetime of Good Health’ a utility of 1.0. By
implication, ‘Death’ is the worst possible health state and a ‘Lifetime of Good Health’ is
the best possible health state. In this framework, the utilities of all other health states are
assumed to range between the utility of ‘Death’ and the utility of a ‘Lifetime of Good
Health’. This is why a ‘Lifetime of Good Health’ is used as the gain outcome and
‘Death’ is used as the loss outcome for permanent health state standard gambles (Figure
9).

Figure 9: ‘Death’ used as loss outcome for permanent health state standard gambles


                Certain Option                                            Gamble

                                                                          Lifetime Good Health
                                                              p
                 Permanent health state
                                                            1- p
                                                                                     Death
u permanent health state = [(p) (u LGH) + (1 - p) (u death)]
where: u permanent health state = utility of permanent health state
        u LGH = utility of a 'Lifetime of Good Health = 1.0
        u Death = utility of Death = 0.0

B. The utility scale for certain individuals is not anchored by Death
and Good Health: health states considered worse than Death
However, when a permanent health state is particularly terrible, certain respondents
consider such a health state to be worse than ‘Death’. We observed this in our study for
the permanent health state ‘Quadriplegia’. ‘Quadriplegia’ was described as being
quadriplegic and institutionalized for life due to complete dependence on others for all
activities of daily living. We observed that certain respondents rated ‘Quadriplegia’
lower than ‘Death’ in the preceding rating scale task. By rating ‘Quadriplegia’ lowest on
the rating scale, these respondents had indicated that they considered ‘Quadriplegia’ to be
worse than ‘Death’. Hence, their health state utilities were anchored on a ‘Quadriplegia’
– ‘Lifetime of Good Health’ scale, rather than on a ‘Death’ – ‘Lifetime of Good Health’
scale.




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C. Addressing the problem of measuring permanent health state
utilities when a health state is considered worse than Death: Step 1 –
how to obtain responses from respondents
In our study, our first step towards addressing the problem of measuring utilities when
‘Quadriplegia’ was considered worse than ‘Death’ was to replace ‘Death’ with
‘Quadriplegia’ as the loss outcome. Where respondents had indicated that ‘Quadriplegia’
was worse than ‘Death’, ‘Quadriplegia’ replaced ‘Death’ as the loss outcome for all
permanent health state standard gambles (Figure 10).

Figure 10: ‘Quadriplegia’ as the loss outcome for permanent health state standard gambles


                Certain Option                                              Gamble
                                                                            Lifetime Good Health
                                                               p
               Permanent health state
                                                               1- p
u permanent health state = [(p) (u LGH) + (1 - p) (u quadriplegia)]              Quadriplegia
where: u permanent health state = utility of permanent health state
        u LGH = utility of a 'Lifetime of Good Health’ = 1.0
        u quadriplegia = utility of ‘Quadriplegia’ = 0.0

Because the permanent health state ‘Quadriplegia’ was used as the loss outcome in these
respondents, a utility of 0.0 was effectively assigned to ‘Quadriplegia’. This necessitated
measuring the utility of ‘Death’ relative to the ‘Quadriplegia’ – ‘Good Health’ scale with
the standard gamble illustrated in Figure 11.

Figure 11: Obtaining the utility of ‘Death’ on the ‘Quadriplegia’ – ‘Good Health’ scale


                Certain Option                                              Gamble

                                                                            Lifetime Good Health
                                                               p
                        Death
                                                             1- p
                                                                                 Quadriplegia
u death = [(p) (u LGH) + (1 - p) (u quadriplegia)]
where: u death = utility of ‘Death’
         u LGH = utility of a 'Lifetime of Good Health’ = 1.0
         u quadriplegia = utility of ‘Quadriplegia’ = 0.0




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                         Decision analysis of neck pain treatments (Supplementary Appendix)


We observed that permanent health state utilities were systematically higher in subjects
whose utilities were elicited with the loss anchor ‘Quadriplegia’, compared to subjects
whose utilities were elicited with the loss anchor ‘Death’.

 D. Addressing the problem of measuring utilities when a health state
is considered worse than Death: Step 2 – how to rescale utilities onto the
Death – Good Health scale
Our second step towards addressing the problem of measuring utilities when
‘Quadriplegia’ was considered worse than ‘Death’ was to rescale (i.e. linearly transform)
the utilities obtained from these respondents onto the ‘Death’ (0.0) – ‘Good Health’ (1.0)
scale (Figure 12).

Figure 12: Rescaling utilities relative to the Death (0.0) – Good Health (1.0) utility scale

                          hwtd               Death                       hp              Good Health


Utilities as elicited:    0.0___________UD ______________Uhp ________________1.00

Utilities rescaled:       U’hwtd ________ 0.0______________ U’hp _______________ 1.00

Where: hwtd = health state worse than death; hp = permanent health state; D = Death

First, ‘Death’ was assigned the utility of 0.0. Then, in order to rescale the utilities of
subjects who considered ‘Quadriplegia’ worse than ‘Death’, two calculations were
needed. One calculation was required to rescale the utility of ‘Quadriplegia’ ( U hwtd ). A
second calculation was required to rescale the utilities of the set of remaining permanent
health states that did not include ‘Death’ and ‘Good Health’. We derived two equations
to perform these calculations.

The first equation, based on the relationship between the elicited utilities that was
illustrated in Figure 12, was derived follows:

                                           1UD      1 0
                                                  
                                           U D  0 0  U hwtd

                                            1UD     1
                                                 
                                             UD    U hwtd

                                                        UD
                                            1UD 
                                                       U hwtd

                                          U hwtd (1  U D )  U D


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                                   U hwtd  U hwtd U D  U D

                                     U hwtd (U D  1)  U D

                                                    UD
                                       U 'hwtd 
                                                   U D 1

Therefore, the rescaled or transformed utility of the health state considered worse than
                                                 UD
‘Death’ (‘Quadriplegia’), U 'hwtd , is equal to        , which is a negative utility.
                                                U D 1

The second equation, also based on the relationship between the elicited utilities
illustrated in Figure 12, was derived as follows:

                                     1  U hp          1  U 'hp
                                                   
                                    U hp  U D         U 'hp  0


                                 1  U hp  '
                                            U  1  U ' hp
                                 U h  U D  hp
                                 p         

                                          1  U h       ' 
                               U ' h p           p
                                                          U hp   1
                                            Uh UD
                                          p                  
                                                              

                                         1Uh              
                                 U 'hp 1        p
                                                               1
                                         p Uh UD         
                                                            

                                       U h  U D  1  U hp       
                               U ' hp  p                           1
                                           U hp  U D             
                                                                  

                                            1U 
                                    U ' hp      D
                                                      1
                                            Uh U D 
                                            p       

                                                 U hp  U D
                                      U ' hp 
                                                  1UD




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Therefore, the rescaled or transformed utility of a permanent health state, U 'hp , is equal to
U hp  U D
             .
 1UD

Using data from our study, we can illustrate how we rescaled a respondent’s utilities for
health states when a health state was considered worse than ‘Death’. Consider a
respondent who, during the rating scale task, placed ‘Quadriplegia’ at 0 on the 100 point
‘Feeling Thermometer’. ‘Good Health’ was placed at 100, since this was the best
possible health state. The respondent then placed ‘Death’ at some point on the ‘Feeling
Thermometer above ‘Quadriplegia’ and the remaining permanent health states between
‘Death’ and ‘Good Health’. It follows that since this respondent had placed
‘Quadriplegia’ at 0 on the ‘Feeling Thermometer, this respondent considered
‘Quadriplegia’ to be the least desirable (i.e., worst) health state. Since this was the case,
‘Quadriplegia’ was used as the loss outcome for the standard gambles that were then
performed to elicit the utilities for the remaining permanent health states. Effectively,
this respondent’s utility scale ranged from ‘Quadriplegia’ (0.0) to ‘Good Health’ (1.0).

We elicited the respondent’s utilities for all the permanent health on the ‘Quadriplegia’ –
‘Good Health’ scale by the standard gamble described in Figure 10. We elicited the
respondent’s utility for ‘Death’ on the ‘Quadriplegia’ – ‘Good Health’ scale by the
standard gamble described in Figure 11. In this way, we determined that this
respondent’s utilities for the permanent health states ‘Moderate Neck Pain’, ‘Severe Neck
Pain’ and ‘Congestive Health Failure’ were: 0.80, 0.72, and 0.68, respectively. We
determined that this respondent’s utility for ‘Death’ was 0.25.

The next step was to rescale this respondent’s health state utilities onto the conventional
0.0 – 1.0 (‘Death’ – ‘Good Health’). Using the first equation we described above, we
rescaled the utility of ‘Quadriplegia’ as follows:
                UD
U 'hwtd 
              U D 1
                    0.25
U 'hQuadriplegia           = -0.33. Hence, this respondent’s utility for ‘Quadriplegia’ was
                   0.25  1
negative, since the respondent considered it to be worse than ‘Death’, on the conventional
0.0 – 1.0 (‘Death’-‘Good Health’) utility scale.

Using the second equation described above, we rescaled the respondent’s utilities for the
permanent health states ‘Moderate Neck Pain’, ‘Severe Neck Pain’ and ‘Congestive
Health Failure’ as follows:
           Uh U D
U ' hp  p
              1UD
                    0.80  0.25
U 'hModerate _ NP               0.73
                     1  0.25




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               0.72  0.25
U 'hSevere _ NP            0.63
                1  0.25
            0.68  0.25
U 'hCHF                 0.57
             1  0.25

It is a question for debate whether or not these rescaled ‘utilities’ are true von Neumann
and Morgenstern utilities that follow the axioms of utility theory, since they have been
rescaled onto a - ∞ to 1.0 scale. Some authors do refer to rescaled utilities as utilities
(i.e., HUI and EQ-5D developers), despite the fact that these rescaled utilities might be
negative, while other authors prefer to refer to them as ‘quality-of-life weights’.




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                      Decision analysis of neck pain treatments (Supplementary Appendix)


XIII. Obtaining temporary health state utilities when a permanent
health state other than Death is used as the loss outcome

As we described in Section XII, the permanent health state ‘Quadriplegia’ replaced
‘Death’ as the loss outcome for all permanent health state standard gambles whenever a
respondent considered ‘Quadriplegia’ worse than the health state ‘Death’. Therefore, the
use of ‘Quadriplegia’ as the loss outcome had implications for the second stage of
Torrance’s chained gamble (Section X, Figure 4). To understand the implications, recall
that the second stage of the chained gamble was proposed as a method to obtain a utility
for a temporary health state on the 0.0 – 1.0 (‘Death’ – ‘Good Health’) scale (described in
Section X). The second stage of the chained gamble consists of re-defining the
temporary health state as a short-duration chronic state followed by ‘Immediate Death’
(uWorst.ID). The re-defined temporary health state utility is then measured on the 0.0 – 1.0
(‘Death’ – ‘Good Health’) scale by the standard gamble technique used for permanent
states, as was described in Figure 4. This utility of the short duration chronic state
followed by ‘Immediate Death’ (uWorst.ID) is then used to replace the term uWorst.LGH in the
equation related to the first stage of the chained gamble (Figure 2) to solve for the utility
of a temporary health state on a 0.0 – 1.0 (‘Death’ – ‘Good Health’) scale.

Recall that in our study ‘Quadriplegia’ was used to as the loss outcome for the permanent
health state standard gambles whenever a respondent considered ‘Quadriplegia’ worse
than the health state ‘Death’. Therefore, for the second stage of Torrance’s chained
gamble, the loss outcome ‘Death’ was replaced by ‘Quadriplegia’ (Figure 13).

Figure 13: ‘Quadriplegia’ as loss outcome for second stage of chained gamble


                Certain Option                                             Gamble

                                                                           Lifetime Good Health
                                                             p
  Worst Temporary Health State followed
          by Immediate Death
                                                           1- p
uWorst.ID = [(p) (u LGH) + (1 - p) (u quadriplegia)] = p                Quadriplegia
where: u Worst.ID = utility of ‘Worst Temporary Health State’ followed by Immediate
                      Death
         u LGH = utility of a ‘Lifetime of Good Health’ = 1.0
         u quadriplegia = utility of ‘Quadriplegia’ = 0.0

When ‘Quadriplegia’ was used as the loss outcome for a respondent that had indicated
that ‘Quadriplegia’ was the worst health state, this respondent’s temporary health state
utilities also needed to be rescaled onto the 0.0 – 1.0 (‘Death’ – ‘Good Health’) scale as
we did for the respondent’s permanent health state utilities (described in Section XII).




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                      Decision analysis of neck pain treatments (Supplementary Appendix)


Here is an example to illustrate how we rescaled temporary health state utilities in
individuals who indicated that ‘Quadriplegia’ was the worst health state. ‘Excruciating
Pain’ (a health state described as experiencing poorly controlled excruciating pain while
hospitalized in a helpless state for four weeks) was used as the loss outcome for the first
stage of the chained gamble described in Figure 3. To perform the second stage of the
chained gamble as Torrance proposed, we redefined ‘Excruciating Pain’ as a short-
duration chronic state followed by ‘Immediate Death’. During second stage of the
chained gamble, the utility for ‘Excruciating Pain’ followed by ‘Immediate Death’
(uWorst.ID) was obtained by having the respondent faced with the certain option
‘Excruciating Pain followed by Immediate Death’ or a gamble between a ‘Lifetime of
Good Health’ and ‘Quadriplegia’ (Figure 14).

Figure 14: Example from our study: second stage of the chained gamble where ‘Quadriplegia’ was
used as the loss outcome


                Certain Option                                           Gamble

                                                                        Lifetime Good Health
  ‘Excruciating Pain’ (4 weeks) followed                    p
           by Immediate Death
                                                           1- p
                                                                             Quadriplegia
u Excruciating Pain.ID = [(p) (u LGH) + (1 - p) (u quadriplegia)] = p
where: u Excruciating Pain.ID = utility of ‘Excruciating Pain followed by Immediate Death
         u LGH = utility of a ‘Lifetime of Good Health’ = 1.0
         u quadriplegia = utility of ‘Quadriplegia’ = 0.0

 A. Rescaling temporary health state utilities elicited by Torrance’s two-
stage chained standard gamble
We then proceeded to rescale the temporary health state utilities of respondents who had
indicated that ‘Quadriplegia’ was a health state worse then ‘Death’ onto the 0.0 – 1.0
(‘Death’ – ‘Good Health’) scale. Since we had redefined ‘Excruciating Pain’ as a short-
duration permanent state followed by ‘Immediate Death’ as Torrance proposed, we used
the equations described in Section XII to rescale the utilities of the permanent health
states to rescale this short duration permanent health state. For example, consider a
respondent who had indicated that ‘Quadriplegia’ was a health state worse than ‘Death’,
and whose utility for ‘Death’ and ‘Excruciating Pain’ followed by ‘Immediate Death’
was 0.25 and 0.10, respectively. Using the equation
        Uh U D
U ' hp  p         , the rescaled utility for ‘Excruciating Pain’ followed by ‘Immediate
         1UD
Death’ for this individual would be:




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                                        Decision analysis of neck pain treatments (Supplementary Appendix)


                      0.10  0.25
                                   0.20 . As we described in Section X, Figure 3,
U 'hExcru _ Pain _ followed _ Death 
                        1  0.25
Torrance proposed replacing the term uWorst.LGH found in equation associated with the
first stage chained gamble with this term (u Worst.ID) for the term uWorst.LGH to relate the
utilities of the remaining temporary health state onto the 0.0 – 1.0 (‘Death’ – ‘Good
Health’) scale. In this way, we were able to rescale the temporary health state utilities
that were elicited by Torrance’s two stage chained gamble.




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                    Decision analysis of neck pain treatments (Supplementary Appendix)


References

Côté P, Cassidy JD, Carroll L, Kristman V. The annual incidence and course of neck pain
in the general population: a population-based cohort study. Pain 2004; 112:267-73.

Jansen, SJ. Stiggelbout AM, Wakker PP, Vliet Vlieland TP, Leer,JW, Nooy MA, Kievit
J. Patients' utilities for cancer treatments: a study of the chained procedure for the
standard gamble and time tradeoff. Med Decis Making 1998; 18:391-9.

Torrance, GW. Measurement of health state utilities for economic appraisal. J Health
Economics 1986;5:1-30.




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