WinBUGS: Nonlinear regression model

Revisiting Example 7.3 (pages 251-2) on non-linear regression models

Data presented by Ratkowski (1983) on the temporal evolution of the dry weight of onion bulbs.

One of the nonlinear models considered by Gelfand, Dey and Chang (1992) is

y ~ N(a-bg^x, t ^-2)

with p(a,b,g,t²) = p(a) p(b) p(g) p(t²) and

a ~ N(0.0,1000000)

b ~ N(0.0,1000000)

g ~ U(0,1)

t² ~ G(0.01,0.01)

WINBUGS CODE FOR THE NONLINEAR MODEL

# NONLINEAR MODEL

model{

for( i in 1 : N ) {

y[i] ~ dnorm(mu[i], tau2)

mu[i] <- alpha - beta * pow(gamma,x[i])

}

alpha ~ dnorm(0.0, 1.0E-6)

beta ~ dnorm(0.0, 1.0E-6)

gamma ~ dunif(0.0, 1.0)

tau2 ~ dgamma(0.01, 0.01)

}

# DATA

list(x=c(1.0,1.5,1.5,1.5,2.5,4.0,5.0,5.0,7.0,8.0,8.5,9.0,9.5,

9.5,10.0,12.0,12.0,13.0,13.0,14.5,15.5,15.5,16.5,17.0,22.5,

29.0,31.5),y=c(1.80,1.85,1.87,1.77,2.02,2.27,2.15,2.26,2.47,

2.19,2.26,2.40,2.39,2.41,2.50,2.32,2.32,2.43,2.47,2.56,2.65,

2.47,2.64,2.56,2.70,2.72,2.57),N=27)

# INITIAL VALUES

list(alpha=1,beta=1,tau2=1,gamma=0.9)