**
****
**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] ~ 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

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
}.