Bayesian Posteriors for Probabilistic Sensitivity Analysis
Hazen
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Large-Sample Bayesian Posterior Distributions for
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Probabilistic Sensitivity Analysis
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Gordon B. Hazen
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Min Huang
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IEMS Department
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Northwestern University
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August 2004
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Bayesian Posteriors for Probabilistic Sensitivity Analysis
Hazen
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Abstract
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In probabilistic sensitivity analyses, analysts assign probability distributions to uncertain
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model parameters, and use Monte Carlo simulation to estimate the sensitivity of model
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results to parameter uncertainty. Bayesian methods provide convenient means to obtain
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probability distributions on parameters given data. We present simple methods for
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constructing large-sample approximate Bayesian posterior distributions for probabilities,
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rates, and relative effect parameters, for both controlled and uncontrolled studies, and
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discuss how to use these posterior distributions in a probabilistic sensitivity analysis.
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These results draw on and extend procedures from the literature on large-sample
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Bayesian posterior distributions and Bayesian random effects meta-analysis. We apply
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these methods to conduct a probabilistic sensitivity analyses for a recently published
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analysis of zidovudine prophylaxis following rapid HIV testing in labor to prevent
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vertical HIV transmission in pregnant women. Key Words: decision analysis, cost-
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effectiveness analysis, probabilistic sensitivity analysis, Bayesian methods, random
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effects meta analysis, expected value of perfect information, HIV transmission,
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zidovudine prophylaxis.
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Introduction
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Sensitivity analysis is today a crucial element in any practical decision analysis, and can
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play any of several roles in the decision analysis process. Analysts have long recognized
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the dimensionality limitations of graphically based sensitivity analysis in portraying the
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robustness of a decision analysis to variations in underlying parameter estimates. If
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graphical methods allow at most 2- or 3-way sensitivity analyses, how can one be sure
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that a decision analysis is robust to the simultaneous variation of 10 to 20 parameters?
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Probabilistic sensitivity analysis was introduced to address this issue1,2. In a probabilistic
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sensitivity analysis, the analyst assigns distributions to uncertain parameters and can
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thereby compute as a measure of robustness the probability of a change in the optimal
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alternative due to variation in an arbitrary set of parameters, or alternately3,4, the expected
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value of perfect information regarding any set of parameters. This computation is most
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frequently done via Monte Carlo simulation.
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The task of fitting distributions to uncertain parameters prior to a probabilistic sensitivity
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analysis has been approached in several standard ways. Traditionally, distributions of
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unobservable parameters (such as probabilities or rates) are fitted with a combination of
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mean and confidence interval estimated from data. Distributions typically used are the
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beta distribution2,5,6, the logistic-normal distribution1,7,8,9 , the uniform distribution and
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the normal distribution10. Analysts also use bootstrap methods to obtain sampling
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distributions11, 9. For observable parameters (such as costs), these methods are also
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applicable. However, in this case, it may be more convenient to simply fit a theoretical
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distribution to the empirical distribution of observations. For example, in Goodman et
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Bayesian Posteriors for Probabilistic Sensitivity Analysis
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al.10, observations show a peaked distribution, and a triangular distribution was used.
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Lord, J. et al.11 use a piecewise linear approximation to the empirical distribution.
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Here we will be concerned with obtaining parameter distributions using Bayesian
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methods. Bayesian methods automatically yield posterior distributions for parameters
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given observations without any need for distribution fitting, and therefore seem the
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natural choice for selecting parameter distributions for probabilistic sensitivity analysis.
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They have not been used extensively for this purpose, however, due to the burden of
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computing posterior distributions and the inconvenience of specifying the prior
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distribution required for the Bayesian approach. However, both of these difficulties
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disappear when the number of observations is large. In this case, the prior distribution
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has little effect on the resulting posterior, and approximate large-sample normal posterior
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distributions can be inferred without extensive computation.
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In the following, we will develop a convenient methodology to obtain large-sample
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approximate Bayesian posterior distributions and use these distributions in probabilistic
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sensitivity analyses for probabilities, rates and relative effect parameters. As an example,
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we apply these methods to conduct a probabilistic sensitivity analysis for a recently
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published analysis by Mrus and Tsevat of zidovudine prophylaxis following rapid HIV
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testing in labor to prevent vertical HIV transmission in pregnant women12. This
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probabilistic sensitivity analysis yields zidovudine prophylaxis optimal 95.9% of the time,
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and the expected value of perfect information on all relative effect and probability
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parameters is equal to approximately $10.65 per pregnancy. These results concur with
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Mrus and Tsevat’s conclusion that the choice of rapid HIV testing followed by
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zidovudine prophylaxis is not a close call.
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Ades, Lu and Claxton13 have presented similar methods for computing approximate
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expected values of sample information, a topic we do not address here. The methods we
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discuss here overlap with Ades et al. in part, but differ in the use of random effects
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models for combining heterogeneous studies, where Ades et al. calculate a point estimate
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for the overall population mean, but we obtain an approximate normal posterior
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distribution. The latter more accurately reflects population-wide parameter variation.
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Our material also augments Ades et al. by addressing the issue of combining data from
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controlled and uncontrolled studies. More generally, the methods we present for
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obtaining approximate normal posteriors for (possibly transformed) parameters are
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standard in the Bayesian literature and have been summarized in many texts14,15 .
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However, the specific result we present for the relative efficacy situation in a random
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effects context has not, to our knowledge, appeared previously.
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Overview of Model Types
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The influence diagram Figure 1(a) shows the common situation in which observed data yi
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from each of n studies i = 1,…,n are influenced by an unknown parameter ξ. Typically, ξ
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is a probability or rate, and yi is a count of observed critical events in some subject
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population i. The observations y1,…,yn inform the choice of a decision or policy whose
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cost or utility for an individual or group is influenced by the not-yet-observed count y of
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critical events for that patient or group. The not-yet-observed count y is also influenced
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by the unknown ξ. Analysts can use the observations y1,…,yn to make statistical
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inferences about the unknown ξ, and can use these inferences to make predictions about
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the critical count y and to make recommendations concerning the optimal decision or
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policy. We will be interested in Bayesian procedures in which the analyst calculates a
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posterior probability distribution for ξ and uses it to form a predictive distribution for y,
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from which an expected-utility maximizing decision or policy can be computed.
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Many studies compare a treatment with a control, and for these, the underlying parameter
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ξ must be taken to be a vector ξ = (ξ0,ε), where ξ0 is the unknown probability or rate for
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the control group and ε is the unknown efficacy, some measure of the effectiveness of the
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treatment. The unknown probability or rate for the treatment group may be ξ1 = ξ0 + ε,
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or ξ1 = ξ0⋅ε, or some other function of ξ0 and ε, depending on model specifics. In this
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situation, the observations yi are also vectors yi = (yi0,yi1), or perhaps yi = (yi0,yi1−yi0), one
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component for control and one for treatment or treatment effect, and the decision or
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policy in question might be whether the treatment is cost-effective, or beneficial in terms
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of expected utility.
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Figure 1(a) assumes that the unknown parameter ξ has the same value for each
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population i, that is, the populations are homogenous. One may test this assumption
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statistically, as we discuss below. If homogeneity is confirmed, it is common to proceed
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with analyses that effectively pool the data as though they are from the same source, as
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shown in Figure 1(b).
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(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
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Figure 1: (a) An influence diagram depicting the cost-effectiveness or decision-analytic setting when
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data from homogenous studies is available to estimate a parameterξ. In a Bayesian approach, the
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posterior distribution on ξ given y1,…,yn can be used to determine the optimal decision or policy
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given the observations y1,…,yn. In this situation, observations are typically pooled as if they come
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from a single study, as shown in (b).
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If homogeneity fails, then the populations are heterogeneous, with a different value ξi of
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the unknown parameter for each population i. This situation is depicted in Figure 2,
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which shows a random effects model of heterogeneity, in which it is assumed that the
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values ξi are independent draws from a population of ξ-values with mean µ. Once again
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the observations y1,…,yn inform the choice of a decision or policy whose cost or utility is
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influenced by the not-yet-observed count y of critical events for a patient or group.
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However, now y is influenced by an unknown ξ drawn from the same population of ξ-
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values with mean µ. The analyst aims to use the observations y1,…,yn to make inferences
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about µ and in turn about ξ, from which predictions of the critical count y can be made.
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A Bayesian analyst would seek a posterior distribution on µ, from which could be
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inferred a predictive distribution on ξ. This in turn would allow the computation of a
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utility-maximizing decision or policy given the observations y1,…,yn.
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This situation also allows for controlled studies. Again the observations yi = (yi0,yi1) or
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perhaps yi = (yi0,yi1−yi0) would be for control and treatment groups, the unknown ξi =
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(ξi0,εi) would consist of an component ξi0 for the ith control group, and an efficacy
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parameter εi for the ith group, and µ = (µ0,µε) would also have components for control
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and efficacy.
Observations
from past
studies
µ
ξ1
y1
.
.
.
.
.
.
ξn
yn
Decision
or Policy
Future
observations
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ξ
y
Cost or
Utility
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Figure 2: An influence diagram depicting the cost-effectiveness or decision-analytic setting when data
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from heterogeneous studies is available to estimate a parameter ξ. Past studies aimed at estimating ξ
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are portrayed using a random effects model in which the corresponding parameter ξi for the ith study
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is sampled from a population with mean µ. In a Bayesian approach, the posterior distribution on µ
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given y1,…,yn is used to infer a predictive distribution on ξ. This predictive distribution can be used
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to determine the optimal decision or policy given the observations y1,…,yn.
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In the sequel, we examine in turn each combination of these two dichotomies
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(homogenous/heterogeneous, uncontrolled/controlled), summarizing how to obtain large-
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sample approximate Bayesian posterior distributions for ξ (in the homogeneous case) or
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µ (in the heterogeneous case), and how to use these posterior distributions to conduct a
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probabilistic sensitivity analysis using Monte Carlo simulation. In each situation we
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illustrate our results with data cited by Mrus and Tsevat12 in their analysis of zidovudine
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prophylaxis following rapid HIV testing in labor. We conclude, as mentioned above,
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with a complete probabilistic sensitivity analysis of all probability and efficacy
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parameters in the Mrus and Tsevat model.
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Observations from a Single Study
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If, as in Figure 1, there is only a single study, or if there are multiple studies that pass a
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homogeneity test and may be pooled, then the posterior distribution of the unknown
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parameter ξ can be calculated exactly using standard conjugate Bayesian methods. These
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are summarized in Table 1.1. Exact posterior distributions may be obtained when ξ is a
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probability p, a rate λ, or a mean. The beta, gamma and normal distributions mentioned
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may be found in standard probability texts. The table also illustrates how posterior
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distributions simplify when the prior is noninformative, reflecting absence of any prior
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data.
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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)
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If sample size is large, and parameters are appropriately transformed, then posterior
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distributions may be approximated by normal distributions in the manner depicted in
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Table 1.2. For probability parameters p, the appropriate transformation is the log
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transformation ξ = ln p, or the logit transformation ξ = logit(p) = ln (p/(1−p)); and for
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rate parameters λ, the log transformation ξ = ln λ is appropriate. The observations k are
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transformed similarly to approximately normal observations y. After transformation, the
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conjugate normal posterior from Table 1.1 can be applied. The large sample size implies
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that σ −2 is very large compared to σ0−2, which may be effectively assumed zero, so the
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noninformative case from Table 1.1 applies.
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The final column of Table 1.2 indicates how a random parameter value may be generated
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for Monte Carlo simulation: First generate a random ξ from its approximate posterior
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normal(y, σ2) distribution, then transform ξ back to obtain a random value of the original
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parameter p or λ. These large-sample approximate normal posteriors are unnecessary for
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this simple case, as the conjugate beta or gamma posteriors from Table 1.1 are exactly
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correct and easily computed. However, the large-sample approximations generalize to
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more complicated situations that we will discuss below, where conjugate prior models are
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no longer tractable. We present them here to illustrate our general approach in a simple
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situation.
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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
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Example 1: Acceptance rate for rapid HIV testing and treatment
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Mrus and Tsevat cite Rajegowda et al16, who observed that in n = 539 patients, k = 462
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(85.71%) were willing to accept rapid HIV test and treatment. To obtain a Bayesian
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posterior on the acceptance rate p, we may use the conjugate beta posterior in Table 1.1
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or either of the large sample approximate posteriors for binomial data in Table 1.2. The
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former yields a beta(k+1, n−k+1) = beta(463, 78) posterior. The normal log-odds model
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yields y = logit(k/n) = 1.7918 and σ = (1/k + 1/(n−k))1/2 = 0.1231 and hence a normal
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posterior with mean 1.7918 and standard deviation 0.1231 on ξ = logit(p). The lognormal
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model yields y = log(k/n) = −0.1542 and σ = (1/k − 1/n)1/2 = 0.0176 and hence a normal
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posterior with mean −0.1542 and standard deviation 0.0176 on ξ = log(p). (Here all
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logarithms are to the natural base e.) The beta posterior and the implied normal log-odds
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and lognormal posteriors on p are graphed in Figure 3, and are virtually identical,
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reflecting the fidelity of the large-sample normal approximation.
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To generate a random value of p for the purposes of Monte Carlo simulation, we may
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sample directly from the beta(463,78) posterior. Alternately, as indicated in Table 1.2,
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we may sample ξ from either of the normal posteriors just derived, and then transform via
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p = eξ/(1+eξ) or p = eξ as appropriate to obtain a random p.
0.75
0.8
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0.85
p
0.9
0.95
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Figure 3: Beta, normal log-odds, and lognormal approximate posterior distributions for the
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acceptance rate p for HIV testing and treatment, based on data from Rajegowda et al. The
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approximate normal log-odds and lognormal-posteriors are so close to the true beta posterior that it
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is difficult to distinguish them.
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Observations From a Single Controlled Study
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We consider now the case of observations from a single controlled study, or from several
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controlled studies that may be pooled. Table 2.1 shows the situation in which the
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observations y0,y1 from the control and treatment groups are normally distributed. Here
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the parameter vector (ξ0,ε) consists of the unknown mean ξ0 in the control group and the
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unknown efficacy ε. The mean in the treatment group is then ξ1 = ξ0 + ε. Table 2.1
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gives the posterior distribution of (ξ0,ε) under a noninformative prior when all likelihoods
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are normal. Note that ξ0 and ε are dependent a posteriori, and the table gives the
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marginal distribution of ξ0 and the conditional distribution of ε given ξ0. Generating
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parameter values ξ0 and ξ1 for Monte Carlo simulation is then an easy 3-step process, as
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summarized in the table.
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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..
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Unlike the case of a single uncontrolled study, controlled studies involving binomial or
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Poisson observations do not have tractable conjugate Bayesian updates. However, for
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suitable transformations of parameters and observations summarized in Table 2.2, large-
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sample normal approximations are valid, and results from Table 2.1 may be applied.
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Table 2.2 gives the resulting approximate normal posterior distributions. To generate
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random variates for a probabilistic sensitivity analysis, one may use the procedure from
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Table 2.1 and then transform back as indicated in Table 2.2 to obtain the desired
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probability or rate random variates.
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Table 2.2: Controlled studies involving binomial or Poisson observations and their large-sample
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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
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Example 2: The effect of zidovudine prophylaxis on HIV transmission
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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