2. Data on corn yield y and nitrogen x are analysed by Fuller (1987, p. 18) who applies the identifiability restriction 2 = 57

2. Data on corn yield y and nitrogen x are analysed by Fuller (1987, p. 18) who applies the identifiability restriction σ2

δ

= 57 in a normal linear measurement error model yi = β0 + β1Xi + εi , Xi = μX + ηi , xi = Xi + δi ,

εi ∼ N0, σ2

ε



, δi ∼ N0, σ2

δ



, ηi ∼ N0, σ2

η



.

Instead consider modelling the apparent clustering in x (and hence X) values by adopting a discrete mixture model for X. Consider the change in fit (e.g. deviance information criterion)

by using one, two and three groups. A two-group model with one possible informative prior on 1/σ 2

δ , namely 1/σ 2

δ

∼ Ga(10, 513) may be coded as follows, model { for (i in 1:11) {y[i] ∼ dnorm(mu[i],tau)

mu[i] <- beta[1]+beta[2]*X[i]

x[i] ∼ dnorm(X[i],tau.delta)

# discrete mixture for X X[i] ∼ dnorm(muX[G[i]],tauX)

G[i] ∼ dcat(pi[1:2])}

pi[1:2] ∼ ddirch(alpha[1:2])

# measurement error variance tau.delta ∼ dgamma(10,513)

tau ∼ dgamma(1,0.001); VarY <- 1/tau tauX ∼ dgamma(1,0.001); VarX <- 1/tauX # regression parameters beta[1] ∼ dnorm(60,0.0001); beta[2] ∼ dnorm(0,0.001)
# cluster means of X muX[1]∼ dnorm(60,0.00001) ; muX[2] <- nu[1]+del[1]
del[1] ∼ dnorm(0,0.00001) I(0,)}
The data are list(x=c(50,51,53,64,64,69,70,70,94,95,97), y=c(99,96,90,86,91,104,86,96,99,110,115),alpha=c(1,1)).