[blockers0]     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.


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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:

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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] ~ dnorm(d, tau)
       }
       d ~ dnorm(0.0,1.0E-6)
       tau ~ dgamma(0.001,0.001)
       delta.new ~ dnorm(d, tau)
       sigma <- 1 / sqrt(tau)
    }
   
Data ( click to open )

Inits for chain 1       Inits for chain 2 ( click to open )

Results

A 1000 update burn in followed by a further 10000 updates gave the parameter estimates

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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 .