Blocker: random effects
meta-analysis of clinical
trials
Carlin (1992) considers a Bayesian approach to meta-analysis, and includes the following examples of 22 trials of beta-blockers to prevent mortality after myocardial infarction.

In a random effects meta-analysis we assume the true effect (on a log-odds scale) d i in a trial i is drawn from some population distribution.Let r C i denote number of events in the control group in trial i , and r T i denote events under active treatment in trial i . Our model is:
r C i ~ Binomial(p C i , n C i )
r T i ~ Binomial(p T i , n T i )
logit(p C i ) = m i
logit(p T i ) = m i + d i
d i ~ Normal(d, t )
``Noninformative'' priors are given for the m i 's. t and d. The graph for this model is shown in below. We want to make inferences about the population effect d, and the predictive distribution for the effect d new in a new trial. Empirical Bayes methods estimate d and t by maximum likelihood and use these estimates to form the predictive distribution p( d new | d hat , t hat ). Full Bayes allows for the uncertainty concerning d and t .
Graphical model for blocker example:
BUGS language for blocker example:
model
{
for( i in 1 : Num ) {
rc[i] ~ dbin(pc[i], nc[i])
rt[i] ~ dbin(pt[i], nt[i])
logit(pc[i]) <- mu[i]
logit(pt[i]) <- mu[i] + delta[i]
mu[i] ~ dnorm(0.0,1.0E-5)
delta[i] ~ dt(d, tau, 4)
}
d ~ dnorm(0.0,1.0E-6)
tau ~ dgamma(0.001,0.001)
delta.new ~ dt(d, tau, 4)
sigma <- 1 / sqrt(tau)
}
Data ( click to open )
Inits ( click to open )
Results
A 1000 update burn in followed by a further 10000 updates gave the parameter estimates
Our estimates are lower and with tighter precision - in fact similar to the values obtained by Carlin for the empirical Bayes estimator. The discrepancy appears to be due to Carlin's use of a uniform prior for s 2 in his analysis, which will lead to increased posterior mean and standard deviation for d, as compared to our (approximate) use of p( s 2 ) ~ 1 / s 2 (see his Figure 1).
In some circumstances it might be reasonable to assume that the population distribution has heavier tails, for example a t distribution with low degrees of freedom. This is easily accomplished in BUGS by using the dt distribution function instead of dnorm for d and d new .