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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-35/4/f/solution

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Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 35 / 4 / f / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-06

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

μ, τ

and study odds ratios, checking Markov chain Monte Carlo convergence diagnostics and sensitivity to the finite intercept bounds and scale prior distribution. The fitted hierarchy combines binomial sampling uncertainty with between-study heterogeneity.

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