= Solution
Represent the independent locally flat intercept <prior distributions> by broad finite <uniform priors>, for example on $(-10,10)$; this is proper and approximately constant over plausible mortality logits. With $A$ calibrated above, rough <BUGS> code is
``
model {
mu ~ dnorm(0,0.25)
tau ~ dunif(0,A)
invtau2 <- pow(tau,-2)
for (j in 1:J) {
alpha[j] ~ dunif(-10,10)
beta[j] ~ dnorm(mu,invtau2)
logit(thetaC[j]) <- alpha[j]-beta[j]/2
logit(thetaT[j]) <- alpha[j]+beta[j]/2
rC[j] ~ dbin(thetaC[j],nC[j])
rT[j] ~ dbin(thetaT[j],nT[j])
oddsRatio[j] <- exp(beta[j])
}
}
``
Use $J=6$ and supply treated death counts $(3,7,5,102,32,22)$ with totals $(38,114,69,1533,209,680)$, and control death counts $(3,14,11,127,40,39)$ with totals $(39,116,93,1520,218,674)$. In <BUGS>, the second `dnorm` argument is a <precision parameter>, so `0.25` corresponds to <variance> four. Initialize the positive scale away from zero. Monitor $\mu,\tau$ and study <odds ratios>, checking <Markov chain Monte Carlo convergence diagnostics> and sensitivity to the finite intercept bounds and scale <prior distribution>. \b[The fitted hierarchy combines binomial sampling uncertainty with between-study heterogeneity.]
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