BAYESIAN POSTERIOR DISTRIBUTIONS FOR PROBABILISTIC SENSITIVITY ANALYSIS Gordon B. Hazen and Min Huang, IEMS Department, Northwestern University, Evanston IL

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BAYESIAN POSTERIOR DISTRIBUTIONS FOR
PROBABILISTIC SENSITIVITY ANALYSISGordon B.
Hazen and Min Huang, IEMS Department,
Northwestern University, Evanston IL
Observations from several heterogeneous
controlled studies
An Illustrative Probabilistic Sensitivity
Analysis We conducted a probabilistic
sensitivity analysis of the choice of
interventions to reduce vertical HIV transmission
in pregnant women, using data from Mrus Tsevat.
Abstract Purpose In probabilistic sensitivity
analyses (PSA), analysts assign probability
distributions to uncertain model parameters, and
use Monte Carlo simulation to estimate the
sensitivity of model results to parameter
uncertainty. Bayesian methods provide convenient
means to obtain probability distributions on
parameters given data. We present large-sample
approximate Bayesian posterior distributions for
probabilities, rates and relative effect
parameters, and discuss how to use these in PSA.
Methods We use Bayesian random effects
meta-analysis, extending procedures summarized by
Ades, Lu and Claxton (2004). We outline
procedures for using the resulting posterior
distributions in Monte Carlo simulation. Results
and conclusions We apply these methods to
conduct a PSA for a recently published analysis
of zidovudine prophylaxis following rapid HIV
testing in labor to prevent vertical HIV
transmission in pregnant women (Mrus and Tsevat
2004). Zidovudine prophylaxis is cost saving and
has net benefit 553 per pregnancy compared to
not testing for HIV, assuming a cost of 50,000
per lost QALY (mother and child). We based a PSA
on data cited from Mrus and Tsevat on seven
studies of vertical HIV transmission, as well as
data for 5 other probability parameters. Given
this data, the two parameters (log Risk
population mean) and (log Risk Ratio population
mean) for vertical HIV transmission have
approximate bivariate normal posterior with
mean/sd equal to ?1.39/0.12 and ?1.02/0.23, and
correlation ?0.108. Using these and other
posterior distributions for all 5 remaining
probabilities in a PSA yields zidovudine
prophylaxis optimal 95.9 (?0.19) of the time,
and the expected value of perfect information on
all 7 relative effects and probabilities equal to
10.65 (?0.87) per pregnancy. These results
concur with Mrus and Tsevats conclusion that the
choice of rapid HIV testing followed by
zidovudine prophylaxis is not a close call.
Observations from several heterogeneous studies
(no controls)
Observations from a single controlled study (or
several pooled controlled studies)
Results Baseline analysis Using the standard
value of 50,000/QALY, we found a net benefit of
522.96 per pregnancy for rapid HIV testing
followed by zidovudine prophylaxis. This is
consistent with the results of Mrus and Tsevat.
Results Probabilistic sensitivity analysis.
Based on 40,000 Monte Carlo iterations
Zidovudine prophylaxis optimal 95.9 (0.19) of
the time. The expected value of perfect
information on all 8 relative effects and
probabilities is equal to 10.65 (0.87) per
pregnancy. Conclusion These numbers indicate
that the optimality of zidovudine prophylaxis is
insensitive to simultaneous variation in these
eight probability and efficacy parameters This
is consistent with the conventional sensitivity
analyses conducted by Mrus and Tsevat.
Probabilistic Sensitivity Analysis Using the
Bayesian Paradigm
Example 2 Specificity of rapid HIV testing The
following data is drawn from the 3 sources
referenced by Mrus Tsevat (2004). p
specificity of rapid HIV testing
Example 1 (continued) The effect of zidovudine
prophylaxis on HIV transmission
In decision or cost-effectiveness analyses,
observations y1,,yn may be available from past
studies concerning unknown parameters x that
influence future observations, costs and
utilities. In order to conduct a probabilistic
sensitivity analysis on x, one needs to assign a
probability distribution to x. Bayesian methods
provide in principle a straightforward method for
this Use the posterior distribution f(x
y1,,yn) of x given the observations. In
practice there may be difficulties in obtaining
posterior distribution f(x y1,,yn) A
prior distribution for x must be specified
The burden of computing a posterior distribution
on x may be excessive. However, both of these
difficulties disappear when the number of
observations is large. In this case, the
prior distribution has little effect on the
resulting posterior, and approximate
large-sample normal posterior distributions can
be inferred without extensive computation.
Example 1 The effect of zidovudine prophylaxis
on HIV transmission The following data is drawn
from the 7 sources referenced by Mrus Tsevat
(2004). p0 probability of HIV transmission
without zidovudine prophylaxis p1 probability
of HIV transmission with zidovudine prophylaxis
Reference Mrus, J.M., Tsevat, J.
Cost-effectiveness of interventions to reduce
vertical HIV transmission from pregnant women who
have not received prenatal care. Medical Decision
Making, 24 (2004) 1, 30-39. Ades AE, Lu G,
Claxton K. Expected value of sample information
calculations in medical decision modeling.
Medical Decision Making. 24 (2004) 4, 207-227.
Note that pooling results in a tighter
distribution than not pooling. Improperly
pooling may lead to misleading overconfidence in
a probabilistic sensitivity analysis.
p0 probability of HIV transmission without
zidovudine prophylaxis p1 probability of HIV
transmission with zidovudine prophylaxis Here
pooling is incorrect and leads to misleadingly
tight posterior distribution (compare graph at
left).