Dugongs: nonlinear growth curve
Carlin and Gelfand (1991) present a nonconjugate Bayesian analysis of the following data set from Ratkowsky (1983):
The data are length and age measurements for 27 captured dugongs (sea cows). Carlin and Gelfand (1991) model this data using a nonlinear growth curve with no inflection point and an asymptote as X i tends to infinity:
Y i ~ Normal( m i , t ), i = 1,...,27
m i = a - bg Xi a , b > 0; 0 < g < 1
Standard noninformative priors are adopted for a , b and t , and a uniform prior on (0,1) is assumed for g . However, this specification leads to a non conjugate full conditional distribution for g which is also non log-concave. The graph and corresponding BUGS code is given below

model
{
for( i in 1 : N ) {
Y[i] ~ dnorm(mu[i], tau)
mu[i] <- alpha - beta * pow(gamma,x[i])
}
alpha ~ dnorm(0.0, 1.0E-6)I(0,)
beta ~ dnorm(0.0, 1.0E-6)I(0,)
gamma ~ dunif(0.5, 1.0)
tau ~ dgamma(0.001, 0.001)
sigma <- 1 / sqrt(tau)
U3 <- logit(gamma)
}
Data ( click to open )
Inits ( click to open )
Results
A 1000 update burn in followed by a further 10000 updates gave the parameter estimates
model
{
for( i in 1 : N ) {
Y[i] ~ dnorm(mu[i], tau)
mu[i] <- alpha - beta * pow(gamma,x[i])
}
alpha ~ dnorm(0.0, 1.0E-6)
beta ~ dnorm(0.0, 1.0E-6)
logit(gamma) <- U3
tau ~ dgamma(0.001, 0.001)
sigma <- 1 / sqrt(tau)
U3 ~ dnorm(0, 1.0E-4)
}
list (alpha = 1, beta = 1, tau = 1, U3 = 0)
mean sd MC_error val2.5pc median val97.5pc start sample
U3 1.912 0.2609 0.01072 1.415 1.904 2.459 2001 9000
alpha 2.665 0.07564 0.002835 2.544 2.655 2.848 2001 9000
beta 0.9753 0.07757 0.00325 0.8274 0.9752 1.132 2001 9000
gamma 0.8684 0.02941 0.00123 0.8046 0.8704 0.9212 2001 9000
sigma 0.09871 0.01474 2.373E-4 0.07482 0.09716 0.1321 2001 9000