BAYESIAN POSTERIOR DISTRIBUTIONS FOR PROBABILISTIC SENSITIVITY ANALYSIS Gordon B. Hazen and Min Huang, IEMS Department, Northwestern University, Evanston IL 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  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 Tsevat’s conclusion that the choice of rapid HIV testing followed by zidovudine prophylaxis is not a close call. Probabilistic Sensitivity Analysis Using the Bayesian Paradigm In decision or cost-effectiveness analyses, observations y 1,…,y n may be available from past studies concerning unknown parameters  that influence future observations, costs and utilities. In order to conduct a probabilistic sensitivity analysis on , one needs to assign a probability distribution to . Bayesian methods provide in principle a straightforward method for this: Use the posterior distribution f(  | y 1,…,y n ) of  given the observations. In practice there may be difficulties in obtaining posterior distribution f(  | y 1,…,y n ) A prior distribution for  must be specified The burden of computing a posterior distribution on  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). p 0 = probability of HIV transmission without zidovudine prophylaxis p 1 = probability of HIV transmission with zidovudine prophylaxis Example 1 (continued): The effect of zidovudine prophylaxis on HIV transmission 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 Results: Baseline analysis Using the standard value of $50,000/QALY, we found a net benefit of $ 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. 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. p 0 = probability of HIV transmission without zidovudine prophylaxis p 1 = probability of HIV transmission with zidovudine prophylaxis Here pooling is incorrect and leads to misleadingly tight posterior distribution (compare graph at left). Observations from a single controlled study (or several pooled controlled studies) Observations from several heterogeneous studies (no controls) Observations from several heterogeneous controlled studies 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, Ades AE, Lu G, Claxton K. Expected value of sample information calculations in medical decision modeling. Medical Decision Making. 24 (2004) 4, Note that pooling results in a tighter distribution than not pooling. Improperly pooling may lead to misleading overconfidence in a probabilistic sensitivity analysis.