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 Y _ij consist of trunk circumference measurements recorded at time x _j , j=1,...,7 for each of i = 1,..., 5 orange trees. We consider a logistic growth curve as follows:

   Y _ij    ~   Normal( h _ij , t _c )
   
    h _ij     =       f _i1
            _______________
            1 + f _i2 exp( f _i3 x _j )
            
    q _i1     =   log( f _i1 )    
    q _i2     =   log( f _i2 + 1)    
    q _i3     =   log( -f _i3 )    

The BUGS code is as follows

   model {
      for (i in 1:K) {
         for (j in 1:n) {
            Y[i, j] ~ dnorm(eta[i, j], tau.C)
            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])
         }
      }
      tau.C ~ dgamma(1.0E-3, 1.0E-3)
      sigma.C <- 1 / sqrt(tau.C)
      var.C <- 1 / tau.C
      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 ( click to open )

Inits ( click to open )


Results

The hybrid Metropolis algorithm is used to sample the theta parameters in this model. The step length used for this algorithm adapts for the first 4000 iterations and these samples are discarded from the summary statistics. A further 1000 update burn-in followed by 10000 updates gave the following parameter estimates:


The current point Metropolis algorithm is used to sample the theta parameters in this model. The Gaussian proposal distribution used for this algorithm adapts for the first 4000 iterations and these samples are discarded from the summary statistics. A further 1000 update burn-in followed by 10000 updates gave the following parameter estimates:

      mean   sd   MC_error   val2.5pc   median   val97.5pc   start   sample
   mu[1]   5.254   0.1242   0.004513   5.002   5.258   5.488   5001   10000
   mu[2]   2.22   0.1252   0.007917   1.994   2.216   2.469   5001   10000
   mu[3]   -5.861   0.1143   0.008563   -6.098   -5.86   -5.657   5001   10000
   sigma[1]   0.2245   0.1235   0.00357   0.07706   0.1963   0.5306   5001   10000
   sigma[2]   0.1342   0.1219   0.005743   0.02447   0.1009   0.4428   5001   10000
   sigma[3]   0.1098   0.09349   0.005828   0.02354   0.08214   0.3591   5001   10000
   sigma.C   8.025   1.216   0.03895   6.03   7.89   10.77   5001   10000
   theta[1,1]   5.079   0.08832   0.007158   4.949   5.066   5.326   5001   10000
   theta[1,2]   2.134   0.1542   0.01001   1.823   2.136   2.423   5001   10000
   theta[1,3]   -5.851   0.149   0.0126   -6.19   -5.849   -5.583   5001   10000
   theta[2,1]   5.395   0.05096   0.003465   5.3   5.393   5.505   5001   10000
   theta[2,2]   2.207   0.1245   0.008209   1.962   2.205   2.46   5001   10000
   theta[2,3]   -5.825   0.1015   0.007943   -6.028   -5.828   -5.624   5001   10000
   theta[3,1]   5.079   0.09932   0.008296   4.945   5.06   5.356   5001   10000
   theta[3,2]   2.187   0.1351   0.008393   1.915   2.188   2.447   5001   10000
   theta[3,3]   -5.908   0.1494   0.01298   -6.286   -5.89   -5.666   5001   10000
   theta[4,1]   5.441   0.04836   0.003287   5.347   5.442   5.543   5001   10000
   theta[4,2]   2.269   0.1395   0.009928   2.024   2.256   2.566   5001   10000
   theta[4,3]   -5.816   0.1021   0.008087   -6.008   -5.825   -5.591   5001   10000
   theta[5,1]   5.291   0.06828   0.005157   5.174   5.284   5.438   5001   10000
   theta[5,2]   2.299   0.1351   0.009323   2.05   2.295   2.589   5001   10000
   theta[5,3]   -5.907   0.1075   0.008937   -6.125   -5.903   -5.7   5001   10000