Orange Trees: Non-linear growth curve

This dataset was originally presented by Draper and Smith (1981) and reanalysed by Lindstrom and Bates (1990). The data Yij consist of trunk circumference measurements recorded at time xj, j=1,...,7 for each of i = 1,..., 5 orange trees. We consider a logistic growth curve as follows:

   Yij   ~   Normal(ηij, τc)
   
   ηij   =      φi1
            _______________
            1 + φi2 exp( φi3 xj )
            
   θ i1   =   log(φi1)   
   θi2   =   log(φi2 + 1)   
   θi3   =   log(-φi3)   

The BUGS code is as follows

   model {
      for (i in 1:K) {
         for (j in 1:n) {
            Y[i, j] ~ dnorm(eta[i, j], tauC)
            eta[i, j] <- phi[i, 1] / (1 + phi[i, 2] * exp(phi[i, 3] * x[j]))
         }
         phi[i, 1] <- exp(theta[i, 1])
         phi[i, 2] <- exp(theta[i, 2]) - 1
         phi[i, 3] <- -exp(theta[i, 3])
         for (k in 1:3) {
            theta[i, k] ~ dnorm(mu[k], tau[k])
         }
      }
      tauC ~ dgamma(1.0E-3, 1.0E-3)
      sigma.C <- 1 / sqrt(tauC)
      for (k in 1:3) {
         mu[k] ~ dnorm(0, 1.0E-4)
         tau[k] ~ dgamma(1.0E-3, 1.0E-3)
         sigma[k] <- 1 / sqrt(tau[k])
      }
   }

Data
list(n = 7, K = 5, x = c(118.00, 484.00, 664.00, 1004.00, 1231.00, 1372.00, 1582.00),
      Y = structure(
         .Data = c(30.00, 58.00, 87.00, 115.00, 120.00, 142.00, 145.00,
                  33.00, 69.00, 111.00, 156.00, 172.00, 203.00, 203.00,
                  30.00, 51.00, 75.00, 108.00, 115.00, 139.00, 140.00,
                  32.00, 62.00, 112.00, 167.00, 179.00, 209.00, 214.00,
                  30.00, 49.00, 81.00, 125.00, 142.00, 174.00, 177.00),
         .Dim = c(5, 7)))
Inits for chain 1
list(theta = structure(
      .Data = c(5, 2, -6,
               5, 2, -6,
               5, 2, -6,
               5, 2, -6,
               5, 2, -6),
      .Dim = c(5, 3)),
      mu = c(5, 2, -6), tau = c(20, 20, 20), tauC = 20)
   
Inits for chain 2
list(theta = structure(
      .Data = c(3.0, 1.0, -1.0,
               3.0, 1.0, -1.0,
               3.0, 1.0, -1.0,
               3.0, 1.0, -1.0,
               3.0, 1.0, -1.0),
      .Dim = c(5, 3)),
      mu = c(3.0, 1.0, -1.0), tau = c(2, 2, 2), tauC = 2)



Results

[otrees1]

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