Using ranking and DCE data to value
health states on the QALY scale
using conventional and Bayesian
methods
Theresa Cain
AQl-5D Data
• Used ‘warm’ DCE data set
• 168 individuals value 8 pairs of health states
• A sample of 32 pairs of AQL-5D health states
valued
AQL-5D Health State
• Define
x
to be a 21 element vector of dummy variables defining
an AQL-5D health state,
x   x1 ,... x21
• The health state perfect health is a vector of zeros.
• All other health states have at least one variable equal to 1
• 20 dummy variables correspond to attribute levels in the AQL-5D
classification system. Each dummy variable equals 1 if an attribute
is at the corresponding level or a higher level, and zero otherwise.
• The element
x21
is equal 1 if the health state is death.
Utility
• Define U to be the utility individual
ij
• The relationship between U and
ij
i
has for health state
xij is
U ij  g (xij )   ij
g (xij )
is a function of
xij with unknown parameters and
represents the population mean utility for health state
 ij
xij
.
represents the variation in preference from the population
mean utility.
xij
Utility of Death
• The utilities are assumed to be on a scale where perfect health has a
utility of 1 and death has a utility of 0.
• For the health state death,
individuals
i
g (xij )  0
and
 ij  0
for all
Pair-wise Probabilities
• An individual
i
considers the two health states,
x
i1
xi 2 .
• The probability of choosing health state xi1 is written as
P(xi1 )  P[ g (xi 2 )   i 2  g (xi1 )   i1 ]
• If xi1 is compared to the health state death the probability is
written as
P(xi1 )  P[0  g (xi1 )   i1 ]
Type 1 Extreme Value Distribution
• The error are often assumed to have a Type 1 Extreme value
distribution. The pdf is

    
     
f ( )  exp 
 exp   exp 
  -< <

  
  

1

is the scale parameter. If death is assumed to be fixed at 0, the
scale parameter is uncertain. An alternative method is to fix the scale
parameter at 1 and allow the utility of death to be uncertain.
Logit Model
• For the pair-wise choice
xi1 , xi 2  , if the errors are assumed to
have a type 1 extreme value distribution the probability of choosing
health state xi1 is
 g (xi1 ) 
exp 




P(xi1 ) 
 g (xi1 ) 
 g ( xi 2 ) 
exp 
  exp 

  
  
• For the pair-wise choice
health state xi1 is
xi1, death,
the probability of choosing

 g (xi1 )  0.5722
P(xi1 )  1  exp   exp 






Equation for mean utility
• Linear Model:
•

Uij  g (xij )   ij , g(xij )  1  x 
T
ij
is the vector of unknown parameters,
  1 ,...20 ,d 
• If the health state is perfect health g ( x ij )  1 and xTij  0
• If the health state is not perfect health, xTij
represents the
decrease in utility from perfect health to health state x ij
• If the health state is death,
d  1 and g (xij )  0
Parameter estimation
• Values for the parameters
1 ,... 20
need to be inferred
•
Two methods used
-Maximum likelihood estimation
-Bayesian Inference using MCMC
and the scale parameter

Bayesian Inference
The likelihood function f (x |  ) represents the probability of the
observed data x for a given value of the parameter 
Maximum Likelihood estimation finds the value of  which
maximises this probability. Must rely on large sample approximation
to get confidence intervals for parameter estimates. Difficult to
assess uncertainty in health state utilities.
In Bayesian inference we treat the parameters as uncertain and
describe uncertainty about the parameters (and consequently the
health state utilities) with probability distributions.
Bayes’ Theorem gives a joint probability distribution for the model
parameters given the observed data.
Bayes’ Theorem
f  x |   p  
p  | x  
f  x
• p   is the prior distribution, the probability distribution of
before the data
•
p  | x 

x is observed
is the posterior distribution, the probability distribution
of parameter

after the data
x
is observed
Posterior Distribution
• The posterior distribution represents the uncertainty about the
parameters  given the observed data
• Important to understand uncertainty in parameters and
therefore utilities
• The posterior distribution cannot be derived analytically. A
simulation method must be used to sample from the
distribution. The sample will converge to the posterior
distribution.
Markov Chain Monte Carlo
• Generates a Random walk that
converges to posterior
distribution
• MCMC continues until
equilibrium
• If equilibrium occurs at time t,
the value of the parameter is
•
t 1 ,t  2 ,......
from
p( | x)
t
will be a sample
Prior Distribution
• The prior distribution
p  
can be derived from information
from a previous study or be based on your own belief
• In this model utilities are assumed to be on a scale where death
has a utility of 0 and perfect health has a utility of 1. A health state
cannot have a utility greater than 1. Therefore the parameter
estimates cannot be less than 0.
• It would also not be expected that any parameter estimates are
greater than 1. Few asthma health states would be considered
worse than death.
Gamma(1,10) Prior
• Shape parameter and rate   1
parameter   10
• Assumes parameters are more
likely to be closer to zero and
have a small probability of
being greater than 0.4
Gamma(5,15) prior
• Shape parameter   5 and
rate parameter   15
• Assumes parameters are
likely to be close to zero and
have a larger probability of
being between 0.2 And 0.4
Uniform(0,1) Prior
• A uniform prior over the (0,1) scale is also used
• Assumes parameter values are equally likely
• Used to test if allowing higher probability of higher or lower
parameter values changes the posterior distribution
Comparison between Maximum Likelihood
and posterior distributions
• Maximum likelihood estimates
used to calculate mean utility
for 48 health states
• 10000 parameter vectors
sampled from MCMC. Mean
and 95% posterior intervals of
health state utilities calculated
for each prior distribution
Comparison of Priors
Posterior distribution of parameter
• Attribute 3: Weather and
pollution
• Level 5: experience asthma
symptoms as a result of
pollution all the time
Posterior Distribution of a Health State
• Posterior distribution of worst
health state defined by AQL5D. Each attribute is at level 5.
Conclusion
• Posterior distributions similar when Gamma(1,10) and Uniform(0,1)
prior
• If the prior distribution does not favour larger values the posterior
distribution is robust to the prior
• Posterior intervals might not be precise enough to use in an
economic evaluation.