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 to be a 21 element vector of dummy variables defining

an AQL-5D health state,

• 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 is equal 1 if the health state is death.

x

1 21,...x xx

21x

Utility

• Define to be the utility individual has for health state

• The relationship between and is

is a function of with unknown parameters and

represents the population mean utility for health state .

represents the variation in preference from the population mean

utility.

iijU

ijU ijx

( )ij ij ijU g x

( )ijg xijx

ijx

ij

ijx

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, and for all

individuals

( ) 0ijg x

i

0ij

Pair-wise Probabilities

• An individual considers the two health states, .

• The probability of choosing health state is written as

• If is compared to the health state death the probability is

written as

i 1 2i ix x

1 2 2 1 1( ) [ ( ) ( ) ]i i i i iP P g g x x x

1ix

1ix

1 1 1( ) [0 ( ) ]i i iP P g x x

Type 1 Extreme Value Distribution

• The error are often assumed to have a Type 1 Extreme value

distribution. The pdf is

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.

1( ) exp exp exp - < <f

Logit Model• For the pair-wise choice , if the errors are assumed to

have a type 1 extreme value distribution the probability of choosing

health state is

• For the pair-wise choice , the probability of choosing

health state is

1 2,i ix x

1ix

1

11 2

( )exp

( )( ) ( )

exp exp

i

ii i

g

Pg g

x

xx x

1,i deathx

1ix

11

( ) 0.5722( ) 1 exp exp ii

gP

xx

Equation for mean utility

• Linear Model:

• is the vector of unknown parameters,

• If the health state is perfect health and

• If the health state is not perfect health, represents the decrease in utility from

perfect health to health state

• If the health state is death, and

•

( ) , g( ) 1 Tij ij ij ij ijU g x x x

1 20,... , d

( ) 1ijg x 0Tij x

Tijx

ijx

1d ( ) 0ijg x

Parameter estimation

• Values for the parameters and the scale parameter

need to be inferred

• Two methods used

-Maximum likelihood estimation

-Bayesian Inference using MCMC

1 20,...

Bayesian Inference

The likelihood function represents the probability of the observed data 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.

( | )f xx

Bayes’ Theorem

• is the prior distribution, the probability distribution of

before the data is observed

• is the posterior distribution, the probability distribution

of parameter after the data is observed

p

||

f pp

f

xx

x

x

|p x

x

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

• will be a sample

from

t

( | )p x

1 2, ,......t t

Prior Distribution

• The prior distribution 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.

p

Gamma(1,10) Prior

• Shape parameter and rate

parameter

• Assumes parameters are more

likely to be closer to zero and

have a small probability of

being greater than 0.4

1 10

Gamma(5,15) prior

• Shape parameter and

rate parameter

• Assumes parameters are

likely to be close to zero and

have a larger probability of

being between 0.2 And 0.4

5 15

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 AQL-

5D. 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.