Sensitivity to prior distributions:
                        application to Magnesium
                        meta-analysis


   model
   {
   #   j indexes alternative prior distributions
      for (j in 1:6) {
         mu[j] ~ dunif(-10, 10)
         OR[j] <- exp(mu[j])
      
   #   k indexes study number
          for (k in 1:8) {
             theta[j, k] ~ dnorm(mu[j], inv.tau.sqrd[j])
             rtx[j, k] ~ dbin(pt[j, k], nt[k])
             rtx[j, k] <- rt[k]
             rcx[j, k] ~ dbin(pc[j, k], nc[k])
             rcx[j, k] <- rc[k]
             logit(pt[j, k]) <- theta[j, k] + phi[j, k]
             phi[j, k] <- logit(pc[j, k])
             pc[j, k] ~ dunif(0, 1)
          }
       }
      
   #   k again indexes study number
       for (k in 1:8) {
          # log-odds ratios:
          y[k] <- log(((rt[k] + 0.5) / (nt[k] - rt[k] + 0.5)) / ((rc[k] + 0.5) / (nc[k] - rc[k] + 0.5)))
    #    variances & precisions:
          sigma.sqrd[k] <- 1 / (rt[k] + 0.5) + 1 / (nt[k] - rt[k] + 0.5) + 1 / (rc[k] + 0.5) +
                   1 / (nc[k] - rc[k] + 0.5)
          prec.sqrd[k] <- 1 / sigma.sqrd[k]
       }
       s0.sqrd <- 1 / mean(prec.sqrd[1:8])

   # Prior 1: Gamma(0.001, 0.001) on inv.tau.sqrd

       inv.tau.sqrd[1] ~ dgamma(0.001, 0.001)
       tau.sqrd[1] <- 1 / inv.tau.sqrd[1]
       tau[1] <- sqrt(tau.sqrd[1])

    # Prior 2: Uniform(0, 50) on tau.sqrd

       tau.sqrd[2] ~ dunif(0, 50)
       tau[2] <- sqrt(tau.sqrd[2])
       inv.tau.sqrd[2] <- 1 / tau.sqrd[2]

    # Prior 3: Uniform(0, 50) on tau

       tau[3] ~ dunif(0, 50)
       tau.sqrd[3] <- tau[3] * tau[3]
       inv.tau.sqrd[3] <- 1 / tau.sqrd[3]

    # Prior 4: Uniform shrinkage on tau.sqrd

       B0 ~ dunif(0, 1)
       tau.sqrd[4] <- s0.sqrd * (1 - B0) / B0
       tau[4] <- sqrt(tau.sqrd[4])
       inv.tau.sqrd[4] <- 1 / tau.sqrd[4]

    # Prior 5: Dumouchel on tau
       
       D0 ~ dunif(0, 1)
       tau[5] <- sqrt(s0.sqrd) * (1 - D0) / D0
       tau.sqrd[5] <- tau[5] * tau[5]
       inv.tau.sqrd[5] <- 1 / tau.sqrd[5]

    # Prior 6: Half-Normal on tau.sqrd

       p0 <- phi(0.75) / s0.sqrd
       tau.sqrd[6] ~ dnorm(0, p0)C(0, )
       tau[6] <- sqrt(tau.sqrd[6])
       inv.tau.sqrd[6] <- 1 / tau.sqrd[6]

   }


Data     ( click to open )

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

Results for 6 different prior distributions on the random-effects variance: