Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 1 2 3 Large-Sample Bayesian Posterior Distributions for 4 Probabilistic Sensitivity Analysis 5 6 7 8 Gordon B. Hazen 9 10 Min Huang 11 12 IEMS Department 13 Northwestern University 14 15 16 August 2004 1 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 17 Abstract 18 In probabilistic sensitivity analyses, analysts assign probability distributions to uncertain 19 model parameters, and use Monte Carlo simulation to estimate the sensitivity of model 20 results to parameter uncertainty. Bayesian methods provide convenient means to obtain 21 probability distributions on parameters given data. We present simple methods for 22 constructing large-sample approximate Bayesian posterior distributions for probabilities, 23 rates, and relative effect parameters, for both controlled and uncontrolled studies, and 24 discuss how to use these posterior distributions in a probabilistic sensitivity analysis. 25 These results draw on and extend procedures from the literature on large-sample 26 Bayesian posterior distributions and Bayesian random effects meta-analysis. We apply 27 these methods to conduct a probabilistic sensitivity analyses for a recently published 28 analysis of zidovudine prophylaxis following rapid HIV testing in labor to prevent 29 vertical HIV transmission in pregnant women. Key Words: decision analysis, cost- 30 effectiveness analysis, probabilistic sensitivity analysis, Bayesian methods, random 31 effects meta analysis, expected value of perfect information, HIV transmission, 32 zidovudine prophylaxis. 2 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 33 Introduction 34 Sensitivity analysis is today a crucial element in any practical decision analysis, and can 35 play any of several roles in the decision analysis process. Analysts have long recognized 36 the dimensionality limitations of graphically based sensitivity analysis in portraying the 37 robustness of a decision analysis to variations in underlying parameter estimates. If 38 graphical methods allow at most 2- or 3-way sensitivity analyses, how can one be sure 39 that a decision analysis is robust to the simultaneous variation of 10 to 20 parameters? 40 Probabilistic sensitivity analysis was introduced to address this issue1,2. In a probabilistic 41 sensitivity analysis, the analyst assigns distributions to uncertain parameters and can 42 thereby compute as a measure of robustness the probability of a change in the optimal 43 alternative due to variation in an arbitrary set of parameters, or alternately3,4, the expected 44 value of perfect information regarding any set of parameters. This computation is most 45 frequently done via Monte Carlo simulation. 46 The task of fitting distributions to uncertain parameters prior to a probabilistic sensitivity 47 analysis has been approached in several standard ways. Traditionally, distributions of 48 unobservable parameters (such as probabilities or rates) are fitted with a combination of 49 mean and confidence interval estimated from data. Distributions typically used are the 50 beta distribution2,5,6, the logistic-normal distribution1,7,8,9 , the uniform distribution and 51 the normal distribution10. Analysts also use bootstrap methods to obtain sampling 52 distributions11, 9. For observable parameters (such as costs), these methods are also 53 applicable. However, in this case, it may be more convenient to simply fit a theoretical 54 distribution to the empirical distribution of observations. For example, in Goodman et 3 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 55 al.10, observations show a peaked distribution, and a triangular distribution was used. 56 Lord, J. et al.11 use a piecewise linear approximation to the empirical distribution. 57 Here we will be concerned with obtaining parameter distributions using Bayesian 58 methods. Bayesian methods automatically yield posterior distributions for parameters 59 given observations without any need for distribution fitting, and therefore seem the 60 natural choice for selecting parameter distributions for probabilistic sensitivity analysis. 61 They have not been used extensively for this purpose, however, due to the burden of 62 computing posterior distributions and the inconvenience of specifying the prior 63 distribution required for the Bayesian approach. However, both of these difficulties 64 disappear when the number of observations is large. In this case, the prior distribution 65 has little effect on the resulting posterior, and approximate large-sample normal posterior 66 distributions can be inferred without extensive computation. 67 In the following, we will develop a convenient methodology to obtain large-sample 68 approximate Bayesian posterior distributions and use these distributions in probabilistic 69 sensitivity analyses for probabilities, rates and relative effect parameters. As an example, 70 we apply these methods to conduct a probabilistic sensitivity analysis for a recently 71 published analysis by Mrus and Tsevat of zidovudine prophylaxis following rapid HIV 72 testing in labor to prevent vertical HIV transmission in pregnant women12. This 73 probabilistic sensitivity analysis yields zidovudine prophylaxis optimal 95.9% of the time, 74 and the expected value of perfect information on all relative effect and probability 75 parameters is equal to approximately $10.65 per pregnancy. These results concur with 76 Mrus and Tsevat’s conclusion that the choice of rapid HIV testing followed by 77 zidovudine prophylaxis is not a close call. 4 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 78 Ades, Lu and Claxton13 have presented similar methods for computing approximate 79 expected values of sample information, a topic we do not address here. The methods we 80 discuss here overlap with Ades et al. in part, but differ in the use of random effects 81 models for combining heterogeneous studies, where Ades et al. calculate a point estimate 82 for the overall population mean, but we obtain an approximate normal posterior 83 distribution. The latter more accurately reflects population-wide parameter variation. 84 Our material also augments Ades et al. by addressing the issue of combining data from 85 controlled and uncontrolled studies. More generally, the methods we present for 86 obtaining approximate normal posteriors for (possibly transformed) parameters are 87 standard in the Bayesian literature and have been summarized in many texts14,15 . 88 However, the specific result we present for the relative efficacy situation in a random 89 effects context has not, to our knowledge, appeared previously. 90 Overview of Model Types 91 The influence diagram Figure 1(a) shows the common situation in which observed data yi 92 from each of n studies i = 1,…,n are influenced by an unknown parameter ξ. Typically, ξ 93 is a probability or rate, and yi is a count of observed critical events in some subject 94 population i. The observations y1,…,yn inform the choice of a decision or policy whose 95 cost or utility for an individual or group is influenced by the not-yet-observed count y of 96 critical events for that patient or group. The not-yet-observed count y is also influenced 97 by the unknown ξ. Analysts can use the observations y1,…,yn to make statistical 98 inferences about the unknown ξ, and can use these inferences to make predictions about 99 the critical count y and to make recommendations concerning the optimal decision or 100 policy. We will be interested in Bayesian procedures in which the analyst calculates a 5 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 101 posterior probability distribution for ξ and uses it to form a predictive distribution for y, 102 from which an expected-utility maximizing decision or policy can be computed. 103 Many studies compare a treatment with a control, and for these, the underlying parameter 104 ξ must be taken to be a vector ξ = (ξ0,ε), where ξ0 is the unknown probability or rate for 105 the control group and ε is the unknown efficacy, some measure of the effectiveness of the 106 treatment. The unknown probability or rate for the treatment group may be ξ1 = ξ0 + ε, 107 or ξ1 = ξ0⋅ε, or some other function of ξ0 and ε, depending on model specifics. In this 108 situation, the observations yi are also vectors yi = (yi0,yi1), or perhaps yi = (yi0,yi1−yi0), one 109 component for control and one for treatment or treatment effect, and the decision or 110 policy in question might be whether the treatment is cost-effective, or beneficial in terms 111 of expected utility. 112 Figure 1(a) assumes that the unknown parameter ξ has the same value for each 113 population i, that is, the populations are homogenous. One may test this assumption 114 statistically, as we discuss below. If homogeneity is confirmed, it is common to proceed 115 with analyses that effectively pool the data as though they are from the same source, as 116 shown in Figure 1(b). 6 Bayesian Posteriors for Probabilistic Sensitivity Analysis (a) Hazen (b) Observations from past studies y1 Pooled observations from past studies . . . ξ yn Decision or Policy ξ Future observations y y pooled Cost or Utility Decision or Policy Future observations y Cost or Utility 117 Figure 1: (a) An influence diagram depicting the cost-effectiveness or decision-analytic setting when 118 data from homogenous studies is available to estimate a parameterξ. In a Bayesian approach, the 119 posterior distribution on ξ given y1,…,yn can be used to determine the optimal decision or policy 120 given the observations y1,…,yn. In this situation, observations are typically pooled as if they come 121 from a single study, as shown in (b). 122 If homogeneity fails, then the populations are heterogeneous, with a different value ξi of 123 the unknown parameter for each population i. This situation is depicted in Figure 2, 124 which shows a random effects model of heterogeneity, in which it is assumed that the 125 values ξi are independent draws from a population of ξ-values with mean µ. Once again 126 the observations y1,…,yn inform the choice of a decision or policy whose cost or utility is 127 influenced by the not-yet-observed count y of critical events for a patient or group. 128 However, now y is influenced by an unknown ξ drawn from the same population of ξ- 129 values with mean µ. The analyst aims to use the observations y1,…,yn to make inferences 130 about µ and in turn about ξ, from which predictions of the critical count y can be made. 131 A Bayesian analyst would seek a posterior distribution on µ, from which could be 7 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 132 inferred a predictive distribution on ξ. This in turn would allow the computation of a 133 utility-maximizing decision or policy given the observations y1,…,yn. 134 This situation also allows for controlled studies. Again the observations yi = (yi0,yi1) or 135 perhaps yi = (yi0,yi1−yi0) would be for control and treatment groups, the unknown ξi = 136 (ξi0,εi) would consist of an component ξi0 for the ith control group, and an efficacy 137 parameter εi for the ith group, and µ = (µ0,µε) would also have components for control 138 and efficacy. Observations from past studies µ ξ1 y1 . . . . . . ξn yn Decision or Policy Future observations 139 ξ y Cost or Utility 140 Figure 2: An influence diagram depicting the cost-effectiveness or decision-analytic setting when data 141 from heterogeneous studies is available to estimate a parameter ξ. Past studies aimed at estimating ξ 142 are portrayed using a random effects model in which the corresponding parameter ξi for the ith study 143 is sampled from a population with mean µ. In a Bayesian approach, the posterior distribution on µ 144 given y1,…,yn is used to infer a predictive distribution on ξ. This predictive distribution can be used 145 to determine the optimal decision or policy given the observations y1,…,yn. 146 In the sequel, we examine in turn each combination of these two dichotomies 147 (homogenous/heterogeneous, uncontrolled/controlled), summarizing how to obtain large- 148 sample approximate Bayesian posterior distributions for ξ (in the homogeneous case) or 8 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 149 µ (in the heterogeneous case), and how to use these posterior distributions to conduct a 150 probabilistic sensitivity analysis using Monte Carlo simulation. In each situation we 151 illustrate our results with data cited by Mrus and Tsevat12 in their analysis of zidovudine 152 prophylaxis following rapid HIV testing in labor. We conclude, as mentioned above, 153 with a complete probabilistic sensitivity analysis of all probability and efficacy 154 parameters in the Mrus and Tsevat model. 155 Observations from a Single Study 156 If, as in Figure 1, there is only a single study, or if there are multiple studies that pass a 157 homogeneity test and may be pooled, then the posterior distribution of the unknown 158 parameter ξ can be calculated exactly using standard conjugate Bayesian methods. These 159 are summarized in Table 1.1. Exact posterior distributions may be obtained when ξ is a 160 probability p, a rate λ, or a mean. The beta, gamma and normal distributions mentioned 161 may be found in standard probability texts. The table also illustrates how posterior 162 distributions simplify when the prior is noninformative, reflecting absence of any prior 163 data. 9 Bayesian Posteriors for Probabilistic Sensitivity Analysis 164 Hazen Table 1.1: Single-study models using conjugate prior distributions Parameters Probability p Rate λ Observations k events in n independent trials; k ~ binomial(n, p ) k events in duration ∆t; Prior distributions Posterior distribution p ~ beta(α ,β ) p ~ beta(α+ k, β+n− k) Noninformative uniform prior on p (α = 1, β = 1) p ~ beta(k+1, n− k+1) λ ~ gamma(r,θ) λ ~ gamma(r+k, θ+∆t) Noninformative prior (r = 0, θ = 0) on λ λ ~ gamma(k, ∆t) k~ Poisson(λ∆t) Mean ξ ξ ~ normal(ξ1, σ12) y ~ normal(ξ, σ2) σ2 known or ξ ~ normal(ξ0,σ02) estimated noninformative uniform prior on ξ (σ0−2 = 0) ξ1 = σ 0−2ξ0 + σ −2 y , σ1−2 = σ0−2 + σ−2 σ 0−2 + σ −2 ξ ~ normal(y, σ2) 165 166 If sample size is large, and parameters are appropriately transformed, then posterior 167 distributions may be approximated by normal distributions in the manner depicted in 168 Table 1.2. For probability parameters p, the appropriate transformation is the log 169 transformation ξ = ln p, or the logit transformation ξ = logit(p) = ln (p/(1−p)); and for 170 rate parameters λ, the log transformation ξ = ln λ is appropriate. The observations k are 171 transformed similarly to approximately normal observations y. After transformation, the 172 conjugate normal posterior from Table 1.1 can be applied. The large sample size implies 173 that σ −2 is very large compared to σ0−2, which may be effectively assumed zero, so the 174 noninformative case from Table 1.1 applies. 175 The final column of Table 1.2 indicates how a random parameter value may be generated 176 for Monte Carlo simulation: First generate a random ξ from its approximate posterior 177 normal(y, σ2) distribution, then transform ξ back to obtain a random value of the original 10 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 178 parameter p or λ. These large-sample approximate normal posteriors are unnecessary for 179 this simple case, as the conjugate beta or gamma posteriors from Table 1.1 are exactly 180 correct and easily computed. However, the large-sample approximations generalize to 181 more complicated situations that we will discuss below, where conjugate prior models are 182 no longer tractable. We present them here to illustrate our general approach in a simple 183 situation. 184 Table 1.2: Large-sample approximate single study models for probabilities and rates. Parameters Probability p Observations k events in n independent trials k ~ binomial(n, p) Transformation to an approximate normal model (Table 1.1) ξ = logit(p) y = logit(k/n) Generating new parameter values for MC simulation: First generate ξ ~ normal(y, σ2), then: Calculate p = σ2 = 1/k+ 1/(n− k) ξ = log(p) eξ . 1 + eξ Calculate p = eξ. y = log(k/n) σ2 = 1/k − 1/n Rate λ k events in duration ∆t k ~ Poisson(λ∆t) ξ = log(λ) y = log(k/∆t) Calculate λ = eξ. σ2 = 1/k 185 Example 1: Acceptance rate for rapid HIV testing and treatment 186 Mrus and Tsevat cite Rajegowda et al16, who observed that in n = 539 patients, k = 462 187 (85.71%) were willing to accept rapid HIV test and treatment. To obtain a Bayesian 188 posterior on the acceptance rate p, we may use the conjugate beta posterior in Table 1.1 189 or either of the large sample approximate posteriors for binomial data in Table 1.2. The 190 former yields a beta(k+1, n−k+1) = beta(463, 78) posterior. The normal log-odds model 191 yields y = logit(k/n) = 1.7918 and σ = (1/k + 1/(n−k))1/2 = 0.1231 and hence a normal 192 posterior with mean 1.7918 and standard deviation 0.1231 on ξ = logit(p). The lognormal 11 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 193 model yields y = log(k/n) = −0.1542 and σ = (1/k − 1/n)1/2 = 0.0176 and hence a normal 194 posterior with mean −0.1542 and standard deviation 0.0176 on ξ = log(p). (Here all 195 logarithms are to the natural base e.) The beta posterior and the implied normal log-odds 196 and lognormal posteriors on p are graphed in Figure 3, and are virtually identical, 197 reflecting the fidelity of the large-sample normal approximation. 198 To generate a random value of p for the purposes of Monte Carlo simulation, we may 199 sample directly from the beta(463,78) posterior. Alternately, as indicated in Table 1.2, 200 we may sample ξ from either of the normal posteriors just derived, and then transform via 201 p = eξ/(1+eξ) or p = eξ as appropriate to obtain a random p. 0.75 0.8 202 0.85 p 0.9 0.95 203 Figure 3: Beta, normal log-odds, and lognormal approximate posterior distributions for the 204 acceptance rate p for HIV testing and treatment, based on data from Rajegowda et al. The 205 approximate normal log-odds and lognormal-posteriors are so close to the true beta posterior that it 206 is difficult to distinguish them. 207 Observations From a Single Controlled Study 208 We consider now the case of observations from a single controlled study, or from several 209 controlled studies that may be pooled. Table 2.1 shows the situation in which the 12 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 210 observations y0,y1 from the control and treatment groups are normally distributed. Here 211 the parameter vector (ξ0,ε) consists of the unknown mean ξ0 in the control group and the 212 unknown efficacy ε. The mean in the treatment group is then ξ1 = ξ0 + ε. Table 2.1 213 gives the posterior distribution of (ξ0,ε) under a noninformative prior when all likelihoods 214 are normal. Note that ξ0 and ε are dependent a posteriori, and the table gives the 215 marginal distribution of ξ0 and the conditional distribution of ε given ξ0. Generating 216 parameter values ξ0 and ξ1 for Monte Carlo simulation is then an easy 3-step process, as 217 summarized in the table. 218 Table 2.1: Normally distributed observations from a single controlled study. Parameters ξ,ξ 0 1 ε = ξ1− ξ0 Prior distributions Observations 0 0 2 1 1 2 y ~ normal(ξ , σ0 ) y ~ normal(ξ , σ1 ) σ02,σ12 known or estimated from data. ξ ,ε are independent, each with a noninformative prior 0 Posterior distribution ξ0 ~ normal(y0, σ02) ε|ξ0 ~ normal(y1−ξ0,σ12) Generating new parameter values ξ0, ξ1 for Monte Carlo simulation Step 1. Generate ξ0 ~ normal(y0, σ02). Step 2. Generate ε ~ normal(y1−ξ0, σ12) Step 3. Calculate ξ1 = ξ0+ε.. 219 220 Unlike the case of a single uncontrolled study, controlled studies involving binomial or 221 Poisson observations do not have tractable conjugate Bayesian updates. However, for 222 suitable transformations of parameters and observations summarized in Table 2.2, large- 223 sample normal approximations are valid, and results from Table 2.1 may be applied. 224 Table 2.2 gives the resulting approximate normal posterior distributions. To generate 225 random variates for a probabilistic sensitivity analysis, one may use the procedure from 13 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 226 Table 2.1 and then transform back as indicated in Table 2.2 to obtain the desired 227 probability or rate random variates. 228 Table 2.2: Controlled studies involving binomial or Poisson observations and their large-sample 229 normal approximation. Parameters Observations Probabilities p0, p1 k0 events in n0 independent trials without intervention k1 events in n1 independent trials with intervention Transformation to an approximate normal model (Table 2.1) Generating a new parameter value for MC simulation: Steps 1, 2 and 3 are as in Table 2.1. Then ξ0 =logit(p0), Step 4. Calculate ξ1 =logit(p1), eξ0 p0 = , 1 + eξ0 y0 = logit(k0/n0), k0 ~ binomial(n0, p0) y1 = logit(k1/n1), k1 ~ binomial(n1, p1), σ0 = 1/ k0+ 1/(n0− k0), 2 p1 = σ12 = 1/ k1+ 1/(n1− k1). Probabilities p0, p1 k0 events in n0 independent trials without intervention RR=p1/ p0 k1 events in n1 independent trials with intervention eξ1 1 + eξ1 ξ0 = log(p0), Step 4. Calculate ξ1 = log(p1), p0 = y0 = log(k0/n0), 1 k0 ~ binomial(n0, p0) y = log(k1/n1), k1 ~binomial(n1, p1) σ02 = 1/ k0− 1/n0, eξ0 , ξ ε p1 = e 1 , or RR= e . σ12 = 1/ k1− 1/n1. Rates λ0, λ1 RR=λ1/ λ0 k0 events in duration ∆t0 without intervention ξ0 =log(λ0), Step 4. Calculate ξ1 =log(λ1), λ0 = e k1 events in duration ∆t1 with intervention y0 = log(k0/∆t0), k0 ~ Poisson(λ0∆t0) y1 = log(k1/∆t1), k1 ~ Poisson(λ1∆t1), ξ0 ξ ε λ1 = e 1 , or RR= e . σ02 = 1/ k0, σ12 = 1/ k1 230 Example 2: The effect of zidovudine prophylaxis on HIV transmission 231 Mrus and Tsevat cite seven controlled studies17,18,19,20,21,22,23 giving data for estimating the 232 effect of zidovudine prophylaxis on mother-to-infant HIV transmission risk. Observed 14 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 233 data are listed as follows, where k i0 of ni0 infants in the control group i are HIV infected, 234 and ki1 of ni1 infants in the prophylaxis group i are HIV infected, i=1,2,…7. zidovudine With zidovudine Group Without 0 0 0 0 ki ni k i / ni ki1 ni1 k i1 / ni1 i 1 47 152 0.309 19 416 0.0457 2 51 216 0.236 16 204 0.0784 3 30 95 0.316 21 336 0.0625 4 6 16 0.375 4 29 0.1379 5 30 115 0.261 19 115 0.1652 6 37 198 0.187 18 194 0.0928 7 1019 5571 0.18291 223 2269 0.09828 235 Assuming homogeneity tests are passed (we will discuss this further below), we can pool 236 the 7 study groups into one as follows. Without zidovudine With zidovudine 0 1220 k n0 6363 n1 3563 k 0 / n0 0.191733 k 1 / n1 0.089812 y 0 = log(k 0 / n 0 ) −1.65165 y 1 = log(k 1 / n1 ) −2.41004 σ 02 = 1 / k 0 − 1 / n 0 0.000663 σ 12 = 1 / k 1 − 1 / n1 0.002844 k 1 320 237 According to Table 2.1, the transformed log-risk parameter ξ0 = log p0 has a posterior 238 approximate normal distribution with mean −1.65165 and standard deviation 0.0006631/2 239 = 0.0257. Moreover, the efficacy parameter ε has, given ξ0, an approximate normal 240 distribution with mean 2.41004−ξ0 and standard deviation 0.0028441/2 = 0.0533. 241 Observations From Several Heterogeneous Studies 242 Consider now the random effects situation depicted in Figure 2, in which observations 243 relevant to a parameter ξ arise from several heterogeneous studies randomly drawn from 244 an overall population with mean µ. Let σ ξ2 be the variance of this overall population. If 15 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 245 σ ξ2 is positive, then the studies are heterogeneous, whereas if σ ξ2 = 0, then all ξi are 246 equal and the homogenous case of Figure 1 obtains. A popular test24 of the null 247 hypothesis σ ξ2 = 0 uses the statistic Q given by 248 n Q = ∑ ω i ( yi − y null null ) ω 2 i =1 null i = 1 σ i2 ∑ω = ∑ω null y null i i i yi null . (1) i 249 Q has an approximate χ 2 distribution with n−1 degrees of freedom under the null 250 hypothesis, so one would reject this hypothesis if Q > χα2 (n − 1) for some appropriate test 251 level α. However, it is known that this test has low power. If σ ξ2 = 0 is rejected, then a 252 moment estimator of σ ξ2 is24 253 σˆξ = ( Q − (n − 1) ) 2 n ∑ω i =1 null i + n n − ∑ (ω inull ) 2 / ∑ ω inull i =1 i =1 . (2) 254 Table 3.1 gives the posterior distribution for µ for heterogeneous studies when all 255 likelihoods are normal, and a two-step Monte Carlo procedure for probabilistic sensitivity 256 analysis. This two-step procedure may be collapsed into a single step, as the posterior 257 distribution of ξ is normal( y , σξ2 + 258 distribution. Table 3.2 gives large-sample approximate posterior distributions for 259 transformed probabilities and rates under binomial and Poisson sampling, drawing on the 260 results in Table 3.1. (∑ ω ) i i −1 ) and one may sample directly from this 16 Bayesian Posteriors for Probabilistic Sensitivity Analysis 261 Hazen Table 3.1: Normally distributed observations from heterogeneous groups (random effects model) Parameter for study i ξi Observations yi for study i Prior distributions Posterior distribution yi ~ normal(ξi,σi2) µ has a noninformative prior µ~ −1 normal( y , ( ∑ i ωi ) ) σi2 known or estimated. ξi|µ ~ normal(µ,σξ2) σξ2 known or estimated. Step 1. Generate µ ~ normal( y , where ωi = (σ i2 + σ ξ2 ) y= −1 (∑ ω ) ∑ ω y −1 i i i i Generating a new parameter value ξ for Monte Carlo simulation (∑ ω ) i −1 i ). Step 2. Generate ξ ~ normal(µ,σξ2). i 262 Table 3.2: Binomial and Poisson observations from heterogeneous groups and their large-sample 263 normal approximation. Para-meter for study i Observations for study i Probability pi ki events in ni independent trials. ki|pi ~ binomial(ni, pi ) Transformation to an approximate normal random effects model (Table 3.1) ξi = logit(pi), yi = logit(ki/ni), Generating new parameter value for MC simulation: Steps 1 and 2 are as in Table 3.1. Then: Step 3. Calculate p = eξ . 1 + eξ σi2 = 1/ki + 1/(ni−ki). ξi = log(pi), Step 3. Calculate p = eξ. yi = log(ki/ni), σi2 = 1/ki − 1/ni Rate λi ki events in duration ∆ti. ξi = log(λi), ki|λi ~ Poisson(λi∆ti). yi = log(ki/∆ti ), Step 3. Calculate λ = eξ. σi2 = 1/ki 264 Example 3: Specificity of rapid HIV testing 265 Mrus and Tsevat cite three studies25,26,27 providing data on the specificity of rapid HIV 266 testing, observing that in ni people who were not HIV-infected, there were k i having 267 negative test result, i = 1,2,3. The data and its logit transformation are as follows. 17 Bayesian Posteriors for Probabilistic Sensitivity Analysis 268 269 Hazen Group i ki ni pˆ i = ki/ni yi = logit(ki/ni) 1 2 3 446 783 486 451 790 496 0.988914 0.991139 0.979839 4.490881 4.717223 3.883624 σi2 = 1/ki + 1/(ni−ki). 0.202242 0.144134 0.102058 The test statistic (1) for the null hypothesis σ ξ2 = 0 of homogeneity yields Q = 3 ∑ω i =1 null i ( yi − y null ) 2 = 3.08, with p-value = 0.21, insufficient to reject σ ξ2 = 0. However, 270 due to the low power of this test, we may consider this insufficient evidence for 271 homogeneity and continue to assume heterogeneity. In this case, the moment estimator 272 (2) yields σˆξ2 = 0.07795. 273 Using this estimate and the results in Table 3.1 and Table 3.2 under the logit 274 transformation, we obtain (∑ i ωi ) −1 = 0.07338, 275 approximate posterior distribution of parameter µ is normal with mean 4.318, and 276 variance 0.07338. 277 Random variate generation is then a three-stage process, as indicated in Table 3.1 and 278 Table 3.2: First generate µ from this normal distribution, then generate ξ from a normal 279 eξ distribution with mean µ and variance σˆξ2 = 0.07795, and then transform via p = to ξ 280 obtain a random value of the specificity p. Alternately, we could generate ξ directly from 281 a normal distribution with mean 4.318 and standard deviation (0.07338 + 0.07795)1/2 = 282 0.389. 283 Had we accepted homogeneity as indicated by the Q statistic, we would have pooled the 284 three studies to obtain k = 1715 negative test results in n = 1737 HIV-negative individuals. 285 Using the logit transformation from Table 1.2, we obtain y = logit(k/n) = 4.356, σ2 = 1/k ∑ωy i i i = 58.84, y = 4.318. Hence the 1+ e 18 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 286 + 1/(n−k) = 0.04604 and a posterior distribution on ξ that is normal with mean 4.356 and 287 standard deviation 0.046041/2 = 0.215. This has virtually the same mean as the unpooled 288 case but roughly half the standard deviation. 289 The implied pooled and unpooled posterior distributions for the specificity p are graphed 290 in Figure 4. The effect of retaining heterogeneity is a wider posterior on p than would be 291 obtained by accepting homogeneity and pooling. Were the homogeneity hypothesis 292 correct and σξ2 really equal to zero, the two posteriors would be nearly identical† – here 293 the wider posterior for the unpooled case reflects residual uncertainty about homogeneity. 294 So in this light the decision to retain heterogeneity as a possibility seems prudent. 200 100 0 0.95 0.96 0.97 0.98 0.99 1 p Unpooled Pooled 295 296 Figure 4. Log-odds normal posterior distributions on the specificity p of rapid HIV test, one based on 297 pooling three studies under the assumption they are homogeneous, and the other unpooled result 298 based on combining studies using the random effects model. The latter gives a wider posterior 299 distribution, reflecting residual uncertainty regarding whether the studies are really homogeneous. † They would be exactly the same if instead of pooling studies before applying the normal approximation, as was done here, we pooled after applying a normal approximation to each study. 19 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 300 301 Observations From Several Heterogeneous Controlled Studies 302 For heterogeneous controlled studies, each element in Figure 2 is a two-dimensional 303 vector consisting of baseline and efficacy components: µ = ( µ0, µε)T, ξi=( ξi0, εi)T, where 304 the superscript T denotes vector transpose. If yi0, yi1 are the (possibly transformed) 305 control and treatment observations in study i, we adopt, for consistency, the notation yi = 306 (yi0 , yi1− yi0)T, which records the control observation and the treatment effect yi1− yi0. 307 Table 4.1 gives the posterior distribution for the population mean µ and a Monte Carlo 308 procedure for probabilistic sensitivity analysis when all likelihoods are normal. This 309 posterior distribution is bivariate normal with 2×2 covariance matrix equal to the matrix 310 inverse of the sum 311 determined by the variance estimates σ i20 , σ i21 , σ ξ20 , σ ε2 as indicated in the table. The 312 unknown population mean µ has posterior mean vector y given by the product of the 313 covariance matrix 314 The Monte Carlo procedure described in the table requires the generation of a bivariate 315 normal vector µ. We discuss how to accomplish this in Appendix C. ∑ Ω , where each Ωi is itself the matrix inverse of the 2×2 matrix i i (∑ Ω ) i i −1 with the sum of matrix products Ωiyi. 20 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 316 Table 4.1: Normally distributed observations from heterogeneous controlled studies (random effects 317 model) Parameters for study i Observations yi for study i Prior distributions ξi0, ξi1 yi0 ~ normal(ξi0, σ i20 ). µ = (µ0, µε)T has a noninformative prior. µ = (µ0, µε)T ~ −1 normal( y , (∑i Ωi ) ) ξi0|µ0 ~ 2 normal(µ0, σ ξ0 ). σ i20 + σ ξ20 −σ 2 i0 εi= − ξi1 ξi0 ξi = (ξi0,εi)T yi1 ~ 2 normal(ξi1, σ i1 ). σ i20 , σ i21 known or εi|µε ~ 2 estimated. normal(µε, σ ε ). σ ξ2 , σ ε2 known 0 or estimated. Generating new parameter values ξ0,ξ1 for Monte Carlo simulation Posterior distribution where Ωi = σ − σ i20 2 i0 + σ i21 + σ ε2 y = (∑i Ω i ) (∑i Ω i y i ) , Step 1. Generate µ = (µ0, µε)T ~ −1 normal( y , (∑i Ω i ) ) −1 Step 2. Generate ξ0 ~ 0 σ2 normal(µ , ξ ). 0 −1 yi = (yi0 , yi1− yi0)T Step 3. Generate ε~ 2 normal(µε, σ ε ). Step 4. Calculate ξ1 = ξ0+ε. 318 319 Table 4.2 summarizes the corresponding large-sample normal approximations for 320 binomial and Poisson observations in heterogeneous controlled studies. 21 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 321 Table 4.2: Binomial and Poisson observations from heterogeneous controlled studies, and their large- 322 sample normal approximation. Parameters for study i Observations for study i Probabilities pi0, pi1 ki0 events in ni0 independent trials without intervention ki1events in ni1 independent trials with intervention ki0 ~ binomial(ni0,pi0 ) ki1 ~ binomial(ni1,pi1) Transformation to an approximate normal random effects model (Table 4.1) Generating new parameter value: Steps 1,2,3and 4 are as in Table 4.1. Then: ξi0 =logit(pi0) Step 5. Calculate ξi1 = logit(pi1) yi0 logit(ki0/ni0) = p0 = yi1 = logit(ki1/ni1) σ = 2 i0 1/ki0 + − 1/(ni0 ki0) eξ0 , 1 + eξ0 eξ1 p1 = . 1 + eξ1 σ i21 = 1/ki1 + 1/( ni1− ki1) ξi0 =log(pi0) Step 5. Calculate ξi1 = p0 = e 0 , log(pi1) yi0 = log(ki0/ni0) yi1 = log(ki1/ni1) ξ ξ ε p1 = e 1 , or RR= e . σ i20 = 1/ki0− 1/ni0 σ i21 = 1/ki1 − 1/ni1 Rates λi0,λi1 ki0 events in duration ∆ti0 without intervention ξi0 =log(λi0) ξi1 = log(λ 1 i ) ki1 events in duration ∆ti1 with intervention yi0 = log(ki0/∆ti0) ki0 ~ Poisson(λi0∆ti0) yi1 = log(ki1/∆ti1) ki1 ~ Poisson (λi1∆ti1) σ i20 = 1/ki0 Step 5. Calculate λ0 = eξ0 , ξ ε λ1 = e 1 , or RR= e . σ i21 = 1/ki1 323 Example 4: The effect of zidovudine prophylaxis on HIV transmission (continuation of 324 Example 2) 325 In Example 2, we pooled the 7 studies estimating HIV transmission risk with and without 326 zidovudine prophylaxis. In fact, if we calculate the test statistic (1) with the control 327 observations yi0, we obtain Q0 = 39.83 with p-value 4.9×10−7. The corresponding statistic 328 Qε using the treatment effects yi1 − yi0 gives Qε = 38.6 with p-value 8.7×10−7. The null 22 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 329 hypothesis of homogeneous studies is therefore untenable. The corresponding estimates 330 (2) are σˆξ20 = 0.06911 and σˆ ε2 = 0.2563. A random effects analysis using the log 331 transformation from Table 4.2 may be performed as follows. Group i 1 2 3 4 5 6 7 332 yi0 = log(ki0 / ni0 ) -1.174 -1.443 -1.153 -0.9808 -1.344 -1.677 -1.6988 σ i20 = 1/ ki0 − 1/ ni0 0.0147 0.015 0.0228 0.10417 0.0246 0.022 0.0008 yi1 = log(ki1 / ni1 ) -3.086 -2.546 -2.773 -1.981 -1.8 -2.377 -2.3199 σ i21 = 1/ ki1 − 1/ ni1 0.05 0.058 0.045 0.216 0.044 0.05 0.004 yi1 − yi0 -1.913 -1.102 -1.62 -0.94 -0.457 -0.7 -0.621 We obtain 0.013 −0.0028 333 (∑i Ω i ) −1 = −0.0028 0.0474 334 ∑Ωy 335 y = −1.02 i i i −113.2 = −28.32 −1.39 336 We conclude that µ = (µ0, µε)T has approximate bivariate normal posterior distribution 337 with mean vector and covariance matrix . −1.02 −0.0028 0.0474 338 We may compare the implied joint posterior distribution on p0,p1 with the implied 339 posterior obtained by (incorrectly) pooling the seven studies as we did in Example 2, and 340 the result is in Figure 5. The posterior distributions are quite different, which is 341 consistent with the strong rejection of the hypothesis of homogeneity that we noted above. 342 When studies are pooled, the large sample size of study 7 overwhelms the remaining 343 studies, tightens the posterior variance and drags the posterior mean down towards the 344 smaller study-7 mean. Under the heterogeneity assumption, however, study 7 serves as −1.39 0.013 −0.0028 23 Bayesian Posteriors for Probabilistic Sensitivity Analysis 345 only one of seven studies, although with a little higher weight due again to its large 346 sample size. (a) Hazen (b) 347 Figure 5. Joint log-normal approximate posterior distributions of HIV vertical transmission risk (p0) 348 without intervention and vertical transmission risk (p1) with zidovudine prophylaxis based on data 349 from seven studies. In (a), the studies are treated as heterogeneous and not pooled. In (b) the studies 350 are (incorrectly) pooled. In addition to different dispersions, note that these posterior distributions 351 have quite different modes, as the scale of graph (b) is only half that that of graph (a). 352 The accuracy of the approximate posterior distribution depends on having large sample 353 sizes. However, the control arm in study 4 appears to fail this requirement, as it contains 354 only ni = 16 observations with ki = 3 transmissions, giving pˆ i = 0.375. Nevertheless, as 355 Figure 6 shows, we may be reassured that the normal approximation to the binomial with 356 these parameters is fairly good even though the sample size is small. 24 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 1 0.5 0 0 5 10 Binomial Normal Approximation 357 358 Figure 6: Distribution functions of the discrete binomial with n = 16, p = 0.375, and its normal 359 approximation Normal(6,3.75). 360 Combining heterogeneous controlled and uncontrolled studies 361 It may happen that in addition to controlled studies, there may be additional studies that 362 have no treatment arm. The methods from Table 4.1 and Table 4.2 still apply in this case 363 if they are modified as follows. For studies i without treatment arm, in Table 4.2 set the 364 missing observation yi1 to an arbitrary value, and omit the calculation of σ i21 . Then in 365 Table 4.1 set 366 1 σ 2 +σ 2 Ωi = i 0 ξ0 0 0 0 367 (which is the form taken by Ωi when σ i21 is infinitely large). Then apply the procedures 368 in Table 4.1 and Table 4.2 as before. The zero entries in Ωi ensure that the arbitrary value 369 of yi1 does not affect the computation of y given in Table 4.1. 25 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 370 An Illustrative Probabilistic Sensitivity Analysis 371 We now apply these methods to conduct a probabilistic sensitivity analysis for all 372 probability and rate parameters in Mrus and Tsevat’s recently published analysis of 373 zidovudine prophylaxis following rapid HIV testing in labor to prevent vertical HIV 374 transmission in pregnant women12. Figure 7 shows a decision tree depicting this problem, 375 and in Table 5 we list all probability and relative risk parameters used in this model. In 376 the examples above, we have already calculated posterior distributions for four of these 377 parameters. We discuss four remaining parameters here. 378 26 Bayesian Posteriors for Probabilistic Sensitivity Analysis (a) Hazen $208 52.27323601 . Test+, treat $208 52.27323601 $2,613,435 $242 52.27354212 . 1-Pd Do nothing Accept Pac $39 52.3 Offer rapid testing (b) $22,623 49.44215 Pd Deliver before prophylaxis adequate Test− $208 52.27324 1-Pac Prophylaxis adequate Refuse Infant HIV + Mother HIV + $52 Infant HIV - $419 52.3 Mother HIV - 379 Figure 7: The decision to offer rapid testing in labor followed by zidovudine prophylaxis, as modeled 380 by Mrus and Tsevat12. In (a), a mother who accepts rapid HIV testing receives zidovudine upon a 381 positive test, unless delivery occurs before treatment can take effect. The probability of a positive 382 test depends on sensitivity, specificity and HIV prevalence. The subtree (b), in which mother and 383 infant HIV status is revealed, follows each of the terminal nodes in (a). Branch probabilities in (b) 384 depend on the path taken in (a). 27 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 385 Table 5: Probability and relative risk parameters in the Mrus and Tsevat model12. The baseline 386 values listed are values we calculated by transforming posterior means obtained using techniques 387 from Table 1 through Table 4. Variable Description Baseline Value Pp Prevalence of HIV infection in pregnant women without prenatal care 0.0048 Pt0 Vertical transmission risk without intervention (Example 4) 0.2491 rr Relative reduction in vertical transmission risk of zidovudine prophylaxis (Example 4) 0.6401 Pac Test and treatment acceptance rate (Example 1) 0.8558 Pd Percentage of women delivering before preventative therapy can take effect 0.2746 Psen Rapid HIV antibody test Sensitivity 0.9962 Pspe Rapid HIV antibody test Specificity (Example 3) 0.9871 388 389 Prevalence of HIV infection in pregnant women without prenatal care: Mrus and Tsevat 390 estimate the prevalence of HIV infection in pregnant women in the United States to be p0 391 = 0.0017 according to a completed survey for the year 1991. They assign a relative risk 392 (RR) in the range 2−4 to women without prenatal care. The resulting prevalence of HIV 393 infection without prenatal care is p = p0⋅RR. 394 Because the estimate p0 is based on a very large sample, we take the value p0 = 0.0017 as 395 fixed and include only the relative risk parameter RR in our probabilistic sensitivity 396 analysis. Mrus and Tsevat cite studies by Lindsay et al.28 and Donegan et al.29 on the 397 relative risk of HIV infection without prenatal care. We obtain a posterior on RR from 398 these studies using Table 4.1 and the log transformation in Table 4.2. The relevant 399 observations are as follows. 28 Bayesian Posteriors for Probabilistic Sensitivity Analysis Study i 1 2 With prenatal care ki0 ni0 ki0 / ni0 26 7356 0.0035 82 3891 0.0211 Hazen Without prenatal care ki1 ni1 ki0 / ni0 12 834 0.0144 11 254 0.0433 400 Given this data, the approximate posterior distribution of parameter µ = (µ0,µε)T is 401 multivariate normal with mean vector and covariance matrix −1.0398 −0.0122 402 We are only interested in the efficacy parameter µε, the marginal distribution of which is 403 approximate normal with mean −1.0398 and variance 0.1165. The conditional 404 distribution of ε = ln RR given µε is normal with mean µε and variance σ ε2 = 0.1240, 405 from which it follows that the marginal distribution of ε is normal with mean −1.0398 406 and standard deviation (0.1165 + 0.1240)1/2 = 0.4904. The resulting 95% credible 407 interval −1.0398 ± 0.9612 on ε translates to a 95% credible interval (1.08, 7.39) for RR, 408 much wider than the 2−4 range used by Mrus and Tsevat. 409 Percentage of women delivering before preventative therapy can take effect: Mrus and 410 Tsevat estimate the percentage p of women delivering before preventative therapy can 411 take effect as the product p = p1⋅a of the percentage p1 of women without prenatal care 412 who deliver within 4 hours of presentation and the percentage a of women of such 413 women who would deliver before preventative therapy can take effect. 414 Mrus and Tsevat take a = ½ based on an estimated 2 hours needed to test, treat, and 415 derive benefit from the drug. Our re-analysis uses a = ½ as baseline but for probabilistic 416 sensitivity analysis we take a to have a beta distribution with mean ½ and 95% credible 417 interval 0.25−0.75, that is, a beta(7,7) distribution. Regarding p1, Mrus and Tsevat cite 418 Donegan et al30 as observing k = 306 of n = 557 women without prenatal care delivering −4.727 0.7966 −0.0122 . 0.1165 29 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 419 within 4 hours of presentation. This yields a beta(307,252) posterior on p1, assuming a 420 noninformative prior (see Table 1.1). 421 Rapid HIV antibody test sensitivity: Mrus and Tsevat cite three studies25,26,27 providing 422 data on rapid HIV test sensitivity. These studies, which we pool, observe k = 262 423 positive test results out of n = 262 HIV-infected individuals (100% sample sensitivity). 424 Assuming a noninformative uniform prior, this yields a beta(263,1) posterior, (again, see 425 Table 1.1). 426 Results of probabilistic sensitivity analysis 427 In our base case analysis, we used the transformed means of the normal posterior 428 distributions for the eight probability and efficacy parameters discussed in this paper, 429 along with Mrus and Tsevat’s estimates for the remaining cost and quality-of-life 430 parameters. This analysis and the subsequent probabilistic sensitivity analysis was 431 conducted in Microsoft Excel using the Stotree software written by one of the 432 authors31,32,33,34. Using the standard value of $50,000/QALY, we found a net benefit of 433 $522.96 per pregnancy for rapid HIV testing followed by zidovudine prophylaxis. This is 434 consistent with the results of Mrus and Tsevat. 435 We performed a probabilistic sensitivity analysis jointly on the eight probability and 436 efficacy parameters discussed in this paper, leaving the remaining cost and quality 437 parameters at the levels specified by Mrus and Tsevat. Our Monte Carlo simulation takes 438 random draws for parameters from their posterior distributions and estimates the 439 probability of decision change and the expected value of perfect information. 30 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 440 Based on 40,000 Monte Carlo iterations, we estimate zidovudine prophylaxis optimal 441 95.9% (±0.19%) of the time, and the expected value of perfect information on all 8 442 relative effects and probabilities equal to $10.65 (±$0.87) per pregnancy. These numbers 443 indicate that the optimality of zidovudine prophylaxis is insensitive to simultaneous 444 variation in these eight probability and efficacy parameters, a conclusion consistent with 445 the conventional sensitivity analyses conducted by Mrus and Tsevat. We note, however, 446 that the uncertainty surrounding the cost and quality parameters in this model is 447 substantial, and these should also be included in a probabilistic sensitivity analysis. For 448 the purposes of this paper, however, we confine our analysis to the probability and 449 efficacy parameters. 450 We also investigated the sensitivity of the optimal decision to variation in each parameter 451 individually, setting all other parameters equal to their transformed normal means. In 452 10,000 iterations of Monte Carlo simulation for each of the eight parameters, we 453 observed no instances when zidovudine prophylaxis was suboptimal, so both the 454 probability of decision change and the expected value of perfect information are zero for 455 each parameter to within the accuracy of the simulation. 456 Conclusion 457 We have presented large-sample Bayesian methods for obtaining approximately normal 458 posterior distributions for transformations of probability, rate, and relative-effect 459 parameters given data from either controlled or uncontrolled studies. Means, variances 460 and covariances of these approximating normal distributions can be quickly calculated 461 from study data. Analysts can then generate random variates from these distributions and 31 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 462 reverse transform to obtain probability, rate or relative-effect random variates for the 463 purposes of probabilistic sensitivity analysis. This variate generation and transformation 464 procedure is simple enough to be conducted on a spreadsheet, as we have done. Because 465 these are large-sample approximations, caution must be exercised in applying these 466 techniques to studies with small sample sizes. The results we present may also be used 467 for the calculation of expected value of sample information using Monte Carlo simulation, 468 much in the manner of Ades et al.13 469 Our results assume that population variances within each study − the quantities σ2 in 470 Table 1.1, σ02, σ12 in Table 2.1, σi2 in Table 3.1, σi02, σi12 Table 4.1 – are well estimated 471 by sample variances, and this is very likely to be so for large samples. We also assume 472 for heterogeneous studies that the variances across studies − the quantity σξ2 in Table 3.1, 473 and the pair of quantities σ ξ20 , σ ε2 in Table 4.1 − are known or well estimated. However, 474 because the number of studies is typically not large, this is unlikely to be so. A fully 475 Bayesian approach would calculate posterior distributions of these parameters given 476 study data, but such posteriors do not take on a simple form, and may even be improper 477 when a noninformative prior distribution is employed.35 Because we use point estimates 478 for these cross-study variances instead of calculating posterior distributions, the 479 predictive distributions from which we sample during probabilistic sensitivity analysis 480 are too tight to some unknown degree, much in the same way that a normal distribution 481 based on known variance is tighter than the corresponding t-distribution that includes 482 uncertainty about variance. The resulting probabilistic sensitivity analysis may therefore 483 underestimate problem sensitivity. A fully Bayesian approach to combining 484 heterogeneous studies would require the specification of a proper prior distribution for 32 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 485 cross-study variance and the use of Markov chain Monte Carlo techniques to estimate 486 posterior distributions. 33 Bayesian Posteriors for Probabilistic Sensitivity Analysis 487 Hazen Appendix A: Proofs for Tables 488 489 Lemma14,36: If a random vector y has a multivariate normal distribution y|µ ~ 490 Normal ( µ , Σ) , and µ is parameterized as Normal ( µ 0 , Σ 0 ) , then 491 i) The posterior on µ given y is also normal distributed: µ| y1,..., yn ~ Normal(µn, Σ n ). 492 Where 493 494 µn = (Σ −1 + Σ −1 ) −1 (Σ −1 y + Σ −1 µ0 ), Σ n = (Σ −1 + Σ −1 )−1 0 0 0 ii) The marginal distribution of y is Normal ( µ 0 , Σ 0 + Σ) . 495 496 Lemma15: (Delta-Method) If Xn is approximately Normal(µ,σn2) , where σn is a 497 sequence of constants tending to zero, µ is a fixed number, and g is a differentiable 498 function, then for large n, g(Xn) is approximately normal with mean g(µ) and standard 499 deviation σn|g′ (µ)|. 500 501 Theorem 136: Suppose y1,..., yn are n random variables, with yi|ξi ~ Normal(ξi, σi2 ), and 502 y1,..., yn are independent given ξ1,..., ξn, where ξ1,..., ξn given µ are i.i.d. random variables, 503 each is Normal(µ,σξ2). 504 505 i) If µ has prior distribution Normal(µ0, σ02), then the posterior on µ given y1,..., yn is also normal distributed: µ| y1,..., yn ~ Normal(µn, σn2). 34 Bayesian Posteriors for Probabilistic Sensitivity Analysis 506 If µ has noninformative uniform prior distribution, i.e., f ( µ ) ∝ µ 1 , then µ| ii) y1,..., yn ~ Normal( y , 507 1 ∑ω i 508 509 Here µn = ∑ i ωi y + τ 0 µ0 ∑ i ωi + τ 0 τ0 = Hazen 1 σ 02 , σ n2 = 1 τn , ). i τ n = ∑ i ωi + τ 0 , and ω i = 1 , y= σ + σ ξ2 2 i ∑ω y ∑ω i i i i , i . 510 511 Theorem 2: Suppose y1,... yn are n (d-dimensional) random vectors, with yi|ξi ~ 512 Normal(ξi, Σi ), and y1,..., yn are independent given ξ1,..., ξn, where ξ1,..., ξn . given µ are 513 i.i.d. (d-dim) random vectors, each is Normal(µ, Σ ξ 0 ). 514 i) yn is also normal distributed: µ| y1,..., yn ~ Normal(µn, Σn ). 515 516 517 If µ has prior distribution Normal(µ0, Σ0 ), then the posterior on µ given y1,..., ii) If µ has noninformative uniform prior distribution, i.e., f ( µ ) ∝ µ 1 , then µ| y1,..., yn ~ Normal( y , (∑i Ω i ) −1 ). 518 Here µn = (∑ i Ωi + Σ −1 ) −1 (∑ i Ωi yi + Σ −1 µ0 ), Σ n = (∑ i Ωi + Σ −1 )−1 , and Ω i = (Σ i + Σ ξ 0 ) −1 , 519 y = ( ∑i Ω i ) −1 ( ∑i Ω i y i ) . 0 0 0 520 35 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 521 Pf. Let y=( y1,..., yn)T, ξ=(ξ1,..., ξn)T, Σ=diag{ Σ1,…, Σ n} , Σ ξ = diag{ Σ ξ 0 ,…, Σ ξ 0 } , 522 then y|ξ ~ Normal(ξ, Σ), and ξ | µ~ Normal(µ, Σ ξ ), µ=( µ,… µ)T is an dn-dim vector. 523 So we have y| µ ~ Normal(ξ, Σ+ Σ ξ ), with density 1 f ( y | µ ) ∝ µ exp − ( y − µ )T (Σ + Σξ ) −1 ( y − µ ) 2 1 = exp − ∑i( yi − µ )T (Σ i + Σξ 0 ) −1 ( yi − µ ) 2 524 1 T = exp − ∑i( yi Ω i yi − 2µ T Ω i yi + µ T Ω i µ 2 1 ∝ µ exp − ∑i( µ T Ω i µ − 2µ T Ω i yi ) 2 1 = exp − ( µ T ∑iΩ i µ − 2µ T ∑iΩ i yi ) 2 1 = exp − ( µ T ∑iΩ i µ − 2µ T ∑iΩ i y ) 2 1 ∝ µ exp − ( µ − y )T ∑iΩ i ( µ − y ) 2 525 If µ has prior distribution Normal(µ0, Σ0 ), the posterior on µ given y is f (µ | y) = f ( y | µ ) f (µ ) 526 1 1 ∝ µ exp − ( µ − y )T ∑iΩ i ( µ − y ) ⋅ exp − ( µ − µ 0 )T Σ −01 ( µ − µ 0 ) 2 2 1 ∝ µ exp − ( µ − µ n )T Σ −n1 ( µ − µ n ) , where 2 µ n = (∑iΩ i + Σ −1 ) −1 (∑iΩ i y + Σ −1µ 0 ) = (∑iΩ i + Σ −1 ) −1 (∑iΩ i yi + Σ −1µ 0 ), 0 0 0 0 Σ n = (∑iΩ i + Σ 0 ) . −1 −1 527 If µ has noninformative uniform prior distribution, then f (µ | y) = f ( y | µ ) 528 1 ∝ µ exp − ( µ − y ) T ∑i Ω i ( µ − y ) 2 36 Bayesian Posteriors for Probabilistic Sensitivity Analysis 529 Hazen so µ| y1,...,yn ~ Normal( y , (∑i Ω i ) −1 ). 530 531 Proofs for tables 532 Since the models in Table 1.1,Table 1.2 and Table 2.1,Table 2.2 can be considered as 533 special and simple case (when n=1, and µ= ξ) of the models in Table 3.1,Table 3.2, and 534 Table 4.1,Table 4.2, we need only give proofs for the latter Tables. 535 Proofs for Table 3.1,Table 3.2 536 Table 3.1 is immediately from Theorem 1, since y1,..., yn are n random variables, with 537 yi|ξi ~ Normal(ξi, σi2 ), and y1,..., yn are independent given ξ1,..., ξn, where ξ1,..., ξn given µ 538 are i.i.d. random variables, each is Normal(µ,σξ2). So, if µ has noninformative uniform 539 prior distribution, then µ| y1,..., yn ~ Normal( y , 1 ∑ω i ). i 540 In Row 1 of Table 3.2, we have n study groups, and each study i, i=1,2…n, has ki 541 successes in ni independent trials: ki ~ binomial(ni, pi), where pi is the ‘true’ probability 542 estimated by study i. And yi = logit(ki/ni), ξi = logit(pi). 543 We want to show that each observed yi has approximately normal distribution about their 544 expected value ξi, i.e, yi|ξi is approximately Normal(ξi, σi2), moreover, σi2 is estimated by 545 σi2 = 1/ki + 1/(ni-ki). 546 Since ki| pi ~ binomial(ni ,pi ), by the normal approximation of binomial distribution 547 (when ni is large), ( ki / ni )| pi approximately Normal(pi, 548 ni), by the Delta-Method above, straightforward computations show that yi|ξi is 1 pi (1 − pi ) ). For yi = logit(ki/ ni 37 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 1 1 pi (1 − pi ) ⋅ (logit’(pi ))2 = ). By replacing ni ni pi (1 − pi ) 549 approximately Normal(ξi, σi2 = 550 pi by its estimate ki /ni , we use 1/ki + 1/(ni−ki) as an estimator of σi2. 551 Row 2 of Table 3.2 is a similar model as in Row 1, and the only difference is yi = log(ki/ 552 ni), ξi = log(pi) (instead of yi = logit(ki/ ni), ξi = logit(pi)). We get by Delta-Method that 553 σi2 = 1/ki − 1/ni (instead of σi2 = 1/ki + 1/(ni−ki)). 554 In Row 3 of Table 3.2, each study i, i=1,2…n, has ki occurrences in duration ∆ti, where 555 ki~ Poisson(λi∆ti). And yi = log(ki/∆ti ), ξi = log(λi). We want to show that yi|ξi is 556 approximately Normal(ξi, σi2 ). Here, σi2 = 1/ki . 557 Since ki | λi ~ Poisson(λi∆ti). by the normal approximation of Poisson distribution (when 558 ∆ti is large), (ki /∆ti)| λi approximately Normal(λi , λi/∆ti). For yi = log(ki/∆ti), using delta- 559 Method (the Lemma above), yi|ξi is approximately Normal(ξi, σi2 =1/(λi∆ti)). Since ki/∆ti 560 is an estimator of λi , we use 1/ki as an estimator of σi2. 561 Proofs for Table 4.1,Table 4.2 562 Table 4.1 is immediate from Theorem 2: Since y1,... yn are n (2-dim)random vectors, with 563 σ2 − σ i20 , and y1,..., yn are independent given yi|ξi ~ Normal(ξi, Σi ), where Σi = i 02 2 2 − σ i 0 σ i 0 + σ i1 564 ξ1,..., ξn, where ξ1,..., ξn . given µ=( µ0, µε)T are i.i.d. (2-dim) random vectors, each is 565 σ 2 Normal(µ, Σ ξ 0 ), with Σ ξ 0 = ξ 0 0 0 . So if µ has noninformative uniform prior σ ε2 38 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 566 distribution, then µ| y1,..., yn ~ Normal( y , (∑i Ω i ) −1 ),where Ω i = (Σ i + Σ ξ 0 ) −1 = 567 σ i20 + σ ξ20 −σ 2 i0 568 In Row 1 of Table 4.2, we have n study groups, and each study i, i=1,2…n, has ki0 569 successes in ni0 independent trials without intervention, and ki1 successes in 570 ni1independent trials with intervention. ki0 ~ binomial (ni0,pi0 ), ki1 ~ binomial (ni1,pi1), ki0, 571 ki1are independent given pi0, pi1.And yi0 = logit(ki0/ ni0), ξi0 = logit(pi0), yi1 = logit(ki1/ ni1), 572 ξi1 = logit(pi1), xi= yi1- yi0, εi= ξi1 -ξi0. 573 We want to show that each observed yi0 , xi have approximately normal distributed about 574 their expected value ξi0, εi. Let yi=(yi0 , xi)T, ξi=(ξi0, εi)T then we want to show that yi|ξi is 575 σ i20 approximately Normal(ξi, Σi), moreover, Σi is estimated by Σi = 2 − σ i0 576 = 1/ ki0 + 1/( ni0- ki0), σ i21 = 1/ ki1 + 1/( ni1- ki1). 577 By Row 1 of Table 3.2, we have yi0 | ξi0 is approximately Normal(ξi0, σ i20 ), and yi1 | ξi1 is 578 approximately Normal(ξi1, σ i21 ), where σ i20 , σ i21 are estimated by σ i20 = 1/ ki0 + 1/( ni0- ki0), 579 σ i21 = 1/ ki1 + 1/( ni1- ki1). 580 We have assumed that yi0 , yi1 are independent given ξi0 , ξi1 .Note that xi= yi1- yi0 and yi0 581 σ2 − σ i20 . are negative linearly dependent, Cov(xi, yi0)= − σ i20 . So we get Σi = i 02 2 2 − σ i 0 σ i 0 + σ i1 582 Row 2 of Table 4.2 is a similar model as in Row 1, and the only difference is yi0 = log(ki0 / 583 ni0), ξi0 = log(pi0), yi1 = log(ki1/ ni1), ξi1 = log(pi1), (instead of yi0 = logit(ki0/ ni0), ξi0 = − σ i20 −1 . σ i20 + σ i21 + σ ε2 − σ i20 , σ i20 2 2 σ i 0 + σ i1 39 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 584 logit(pi0), yi1 = logit(ki1/ ni1), ξi1 = logit(pi1)). Using the result in Row 2 of Table 3.2, and 585 by the same argument as above, we get Σi = 586 σ i21 = 1/ ki1 - 1/ ni1. 587 In Row 3 of Table 4.2, each study i, i=1,2…n, has ki0 occurrences in duration ∆ti0 without 588 intervention, and ki1 occurrences in duration ∆ti1with intervention .ki0~Poisson (λi0∆ti0), 589 ki1~ Poisson (λi1∆ti1), ki0, ki1are independent given λi0, λi1. And yi0 = log(ki0/∆ti0), ξi0 = 590 log(λi0), yi1 = log(ki1/∆ti1), ξi1 = log(λi1), xi= yi1- yi0, εi= ξi1 -ξi0. 591 Also we want to show that each observed yi0 , xi are approximately normal distributed 592 about their expected value ξi0, εi. Let yi=(yi0 , xi)T, ξi=(ξi0, εi)T , then we want to show that 593 σ i20 yi|ξi is approximately Normal(ξi, Σi), moreover, Σi is estimated by Σi = 2 − σ i0 594 σ i20 = 1/ ki0 , σ i21 = 1/ ki1 . The argument is also the same as above, by using the result in 595 Row 3 of Table 3.2. 596 Appendix B: Models with Unbalanced Data 597 If there are n control groups and m treatment groups, with n>m, using the same notations 598 as before, the model could be explained as follows: There are n-m study groups (which 599 do not contain a treatment arm) giving data yi0 which estimate parameters ξi0, i=1,2,…n-m. 600 For these studies, yi0 | ξi0 are independent and have approximately normal distributions 601 with mean ξi0, and variance σi02. And ξ10,..., ξn-m0 given µ0 are i.i.d. random variables, each 602 is Normal(µ0,σξ02). In the other m study groups with treatment arms, we have yi=(yi0 , 603 xi)T, ξi=(ξi0, εi)T, where xi= yi1- yi0, εi= ξi1 − ξi0. Here yi|ξi is approximately Normal(ξi, σ i20 − σ i0 2 − σ i20 and σ i20 = 1/ ki0- 1/ ni0, σ i20 + σ i21 − σ i20 , σ i20 + σ i21 40 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 604 σ2 − σ i20 . And ξn-m+1,… ξn given µ = ( µ0, µε)T Σi), where Σi is estimated by Σi = i 02 2 2 σ σ σ − + i0 i0 i1 605 σ 2 are i.i.d. (2-dim) random vectors, each is Normal(µ, Σ ξ 0 ), with Σ ξ 0 = ξ 0 0 606 A Bayesian approach to obtain the posterior for µ =( µ0, µε)T would be as follows: Under 607 the assumption of noninformative uniform prior distribution for µ =( µ0, µε)T, update the 608 distribution for µ0 (using the methods developed in Table 3.1 and Table 3.2) according to 609 the data from the (n-m) groups which do not contain studies for treatment. Then use this 610 distribution for µ0 and noninformative distribution for µε as the prior to obtain the 611 posterior distribution for µ =( µ0, µε)T given the data from the remaining m control groups 612 and m treatment groups (using the conclusion in Theorem 2). 613 This approach leads to the same result as the simpler method we presented in the main 614 part of this paper, which will be shown in the following Theorem. 0 . σ ε2 615 616 Theorem 3: For a random effects model with unbalanced data, which is discussed above, 617 suppose µ=( µ0, µε)T has noninformative uniform prior distribution, then µ| y1,..., yn ~ 618 Normal( y , (∑ Ω i ) −1 ), where n i =1 n n y = (∑ Ω i ) −1 (∑ Ω i yi ) , Ω i = ω i 0 , ω i = 2 1 2 , σ i0 + σ ξ i =1 i =1 0 0 619 i=1,2,…n-m, 0 620 621 and 2 2 Ω i = (Σ i + Σ ξ 0 ) −1 = σ i 0 + σ ξ −σ 2 i0 0 − σ i20 σ i20 2 2 + σ i1 + σ ε −1 i=n-m+1,…n. 41 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 622 623 Pf: First using Theorem 1, we get the posterior of µ0 given data yi0, i=1,2,…n-m, which n−m 624 1 is Normal( y (0n − m ) , n−m ∑ω i =1 ), where y (0n − m ) = i ∑ω y i =1 n−m i ∑ω i =1 0 i , ωi = i 1 . Then we use this to σ + σ ξ20 2 i 625 update the prior of parameter µ=( µ0, µε)T. Since µε has a noninformative distribution, or 626 Normal(0, σ2) with σ2 ≈ ∞, the distribution of µ = ( µ0, µε)T is multivariate normal(µ0,Σ0), 627 1 n−m y ( n−m ) where µ0= Σ0= ∑ ω i 0 i =1 0 628 Applying Theorem 2, the posterior on µ given yn-m+1,..., yn is normally distributed: µ| y1,..., 629 yn ~ Normal(µn, Σn ), where 0 , hence Σ0-1= 2 σ 0 Σn = ( 630 631 n−m n i = n − m +1 i = n − m +1 i =1 i =1 0 n−m = ∑ ωi 1 i =1 0 σ2 0 n−m = ∑ Ωi . 0 i =1 ∑ Ω i + ∑ Ω i ) −1 = (∑ Ω i ) −1 and µn = ( n ∑ Ω i + Σ −01 ) −1 ( i = n − m +1 n ∑Ω y i = n − m +1 i i + Σ −01 µ 0 ) n−m 0 n−m 0 n n y ω ω 0 y −1 −1 ∑ ∑ i i ( n − m ) ( n − m ) = (∑ Ω i ) ( ∑ Ω i y i + i =1 ) 0 ) = (∑ Ω i ) ( ∑ Ω i y i + i =1 0 i =1 i = n − m +1 i =1 i = n − m +1 0 0 n−m n n n−m n n ∑ ω i y i0 −1 −1 ) = ( Ω ) ( Ω y + Ω i yi ) = (∑ Ω i ) ( ∑ Ω i y i + i =1 ∑ ∑ ∑ i i i 0 i =1 i = n − m +1 i =1 i =1 i = n − m +1 n 632 n n ∑ Ω i + Σ −01 ) −1 = ( n−m ∑ ωi i =1 0 n n n = (∑ Ω i ) (∑ Ω i y i ) i =1 633 −1 i =1 This completes the proof. 42 Bayesian Posteriors for Probabilistic Sensitivity Analysis Hazen 634 Appendix C: Bivariate Normal Random Variate Generation 635 x Proposition37: (From joint to conditional) If the joint distribution of random vector y 636 mx is multivariate normal with mean and covariance matrix m y 637 conditional distribution of y given x is multivariate normal with conditional mean (my|x = 638 E[y|x]) 639 640 641 Σ xx Σ yx Σ xy , then the Σ yy my|x = my + ΣyxΣxx−1(x − mx) and conditional covariance matrix (Σyy|x = Cov(y,y|x) = E[(y − my|x)(y − my|x)T | x]) Σyy|x = Σyy − ΣyxΣxx−1Σxy. 642 643 Therefore, when µ= ( µ0, µε)T is bivariate normal with mean (m1, m2) T and covariance 644 σ 2 σ 12 , then we can generate a bivariate normal vector µ by 2 steps: matrix 1 2 σ 21 σ 2 645 Step1: Generate µ0 from N(m1, σ 12 ), i.e., normal distribution with mean m1 and 646 variance σ 12 . 647 Step2: Generate µε from N( m2 + ( µ 0 − m1 )σ 21 / σ 12 , σ 22 − σ 12σ 21 / σ 12 ), which is the 648 conditional distribution of µε when µ0 is given. 43 Bayesian Posteriors for Probabilistic Sensitivity Analysis 649 Hazen References 1 Doubilet, P., Begg, C.B., Weinstein, M.C., Braun, P., McNeil, B.J. 1985. 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