**
**Surgical: Institutional ranking

This example considers mortality rates in 12 hospitals performing cardiac surgery in babies. The data are shown below.

r

We first assume that the true failure probabilities are

p

BUGS

model

{

for( i in 1 : N ) {

p[i] ~ dbeta(1.0, 1.0)

r[i] ~ dbin(p[i], n[i])

}

}

logit(p

b

Standard non-informative priors are then specified for the population mean (logit) probability of failure, m , and precision, t .

BUGS

model

{

for( i in 1 : N ) {

b[i] ~ dnorm(mu,tau)

r[i] ~ dbin(p[i],n[i])

logit(p[i]) <- b[i]

}

pop.mean <- exp(mu) / (1 + exp(mu))

mu ~ dnorm(0.0,1.0E-6)

sigma <- 1 / sqrt(tau)

tau ~ dgamma(0.001,0.001)

}

A burn in of 1000 updates followed by a further 10000 updates gave the following estimates of surgical mortality in each hospital for the fixed effect analysis

and for the random effects analysis

The figures below show the posterior ranks for the estimated surgical mortality rate in each hospital for the random effect models. These are obtained by setting the rank monitor for variable p (select the "Rank" option from the "Statistics" menu) after the burn-in phase, and then selecting the "histogram" option from this menu after a further 10000 updates. These distributions illustrate the considerable uncertainty associated with 'league tables': there are only 2 hospitals (H and K) whose intervals exclude the median rank and none whose intervals fall completely within the lower or upper quartiles.