WinBUGS: Linear hierarchical model

Revisiting Example 5.6 (pages 169-60) on simple linear hierarchical regression on time

Souza (1997) entertain several hierarchical and dynamic models to describe the nutritional pattern of pregnant women.

She entertains the following linear hierarchical model

y_ij ~ N(α_i+β_ix_ij,1/τ²)

α_i ~ N(α_c,1/τ_α)

β_i ~ N(β_c,1/τ_β)

α_c ~ N(0,1000)

β_c ~ N(0,1000)

τ_α ~ G(0.001,0.001)

τ_β ~ G(0.001,0.001)

τ² ~ G(0.001,0.001)

for i = 1,...,I=68, j=1,...,n_i and n = n₁ +...+ n_I = 427.

WINBUGS CODE FOR THE LINEAR HIERARCHICAL MODEL

# MODEL

model{

for (i in 1:I){

for (j in 1:J){

mu[i,j] <- alpha[i]+beta[i]*x[i,j]

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

}

}

for (i in 1:I){

alpha[i] ~ dnorm(alphac,taua)

beta[i] ~ dnorm(betac,taub)

}

alphac ~ dnorm(0.0,0.001)

betac ~ dnorm(0.0,0.001)

taua ~ dgamma(0.001,0.001)

taub ~ dgamma(0.001,0.001)

tau2 ~ dgamma(0.001,0.001)

}

# FIRST SET OF INITIAL VALUES

list(alpha=c(0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00), beta=c(0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00), alphac=0.00000E+00, betac=0.00000E+00, taua=1.00000E+00, taub=1.00000E+00, tau2=1.00000E+00)

# SECOND SET OF INITIAL VALUES

list(alpha=c(0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00), beta=c(0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00), alphac=0.00000E+00, betac=0.00000E+00, taua=1.00000E+00, taub=1.00000E+00, tau2=1.00000E+00)

# DATASET

list(x= structure(.Data= c(8.57100E+00, 1.30000E+01, 1.75710E+01, 2.08570E+01, 2.45710E+01, 3.54290E+01, 4.01430E+01, 1.44290E+01, 1.87140E+01, 2.30000E+01, 2.70000E+01, 3.10000E+01, 3.74290E+01, 3.78570E+01, 1.34290E+01, 1.88570E+01, 2.31430E+01, 3.41430E+01, 3.70000E+01, 3.80000E+01, 3.92860E+01, 1.25710E+01, 1.70000E+01, 2.14290E+01, 3.24290E+01, 3.42860E+01, 3.85710E+01, 4.05710E+01, 1.37140E+01, 2.35710E+01, 2.77140E+01, 3.24290E+01, 3.52860E+01, 3.95710E+01, 4.18570E+01, 1.12860E+01, 1.48570E+01, 2.12860E+01, 2.50000E+01, 3.00000E+01, 3.44290E+01, 3.90000E+01, 1.31430E+01, 1.71430E+01, 2.30000E+01, 2.80000E+01, 3.14290E+01, 3.60000E+01, 3.80000E+01, 1.04290E+01, 1.44290E+01, 1.78570E+01, 2.18570E+01, 2.57140E+01, 3.04290E+01, 3.60000E+01, 8.85700E+00, 1.44290E+01, 1.77140E+01, 2.34290E+01, 2.80000E+01, 3.77140E+01, 3.84290E+01, 1.20000E+01, 1.50000E+01, 1.98570E+01, 2.42860E+01, 2.67140E+01, 3.10000E+01, 3.72860E+01, 1.14290E+01, 1.51430E+01, 1.94290E+01, 2.35710E+01, 2.78570E+01, 3.21430E+01, 3.45710E+01, 9.14300E+00, 1.48570E+01, 1.91430E+01, 2.60000E+01, 2.85710E+01, 3.40000E+01, 3.90000E+01, 1.01430E+01, 1.55710E+01, 2.62860E+01, 2.92860E+01, 3.32860E+01, 3.72860E+01, 3.75710E+01, 9.14300E+00, 1.50000E+01, 2.10000E+01, 2.38570E+01, 2.80000E+01, 3.10000E+01, 3.58570E+01, 1.20000E+01, 2.24290E+01, 2.64290E+01, 3.22860E+01, 3.55710E+01, 3.80000E+01, 4.14290E+01, 6.85700E+00, 1.10000E+01, 1.55710E+01, 1.98570E+01, 2.40000E+01, 2.87140E+01, 3.60000E+01, 1.14290E+01, 1.58570E+01, 2.08570E+01, 2.58570E+01, 3.04290E+01, 3.45710E+01, 3.92860E+01, 8.57100E+00, 1.41430E+01, 1.85710E+01, 2.32860E+01, 2.91430E+01, 3.32860E+01, 3.72860E+01, 1.18570E+01, 1.68570E+01, 2.07140E+01, 2.48570E+01, 2.88570E+01, 3.32860E+01, 3.70000E+01, 1.31430E+01, 1.74290E+01, 2.18570E+01, 2.62860E+01, 3.02860E+01, 3.54290E+01, 3.91430E+01, 1.40000E+01, 1.82860E+01, 2.28570E+01, 2.70000E+01, 3.12860E+01, 3.67140E+01, 4.12000E+01, 1.32860E+01, 1.71430E+01, 2.15710E+01, 2.61430E+01, 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3.11430E+01, 3.40000E+01, 4.00000E+01, 1.14290E+01, 1.54290E+01, 2.04290E+01, 3.10000E+01, 3.64290E+01, 3.80000E+01, 4.00000E+01, 1.14290E+01, 1.60000E+01, 1.92860E+01, 2.48570E+01, 2.94290E+01, 3.40000E+01, 4.05710E+01, 1.48570E+01, 1.95710E+01, 2.42860E+01, 3.15710E+01, 4.21430E+01, 4.30000E+01, 4.40000E+01, 1.44290E+01, 2.02860E+01, 2.50000E+01, 3.01430E+01, 3.31430E+01, 3.72860E+01, 4.00000E+01, 8.00000E+00, 1.50000E+01, 2.27140E+01, 2.45710E+01, 3.04290E+01, 3.48570E+01, 3.90000E+01, 1.30000E+01, 1.72860E+01, 2.18570E+01, 2.55710E+01, 3.25710E+01, 3.57140E+01, 3.90000E+01, 8.42900E+00, 1.34290E+01, 1.72860E+01, 2.34290E+01, 2.74290E+01, 3.14290E+01, 3.94290E+01, 6.71400E+00, 1.25710E+01, 1.65710E+01, 2.12860E+01, 2.65710E+01, 3.45710E+01, 3.85710E+01, 1.12860E+01, 1.61430E+01, 2.24290E+01, 2.72860E+01, 3.07140E+01, 3.75710E+01, 3.95710E+01, 1.17140E+01, 1.58570E+01, 2.02860E+01, 2.51430E+01, 2.88570E+01, 3.77140E+01, 3.91430E+01, 8.85700E+00, 1.15710E+01, 1.60000E+01, 2.38570E+01, 2.82860E+01, 3.57140E+01, 3.97140E+01, 1.35000E+01, 1.98570E+01, 2.50000E+01, 2.91430E+01, 3.38570E+01, 3.61430E+01, 4.20000E+01, 8.42900E+00, 1.15710E+01, 1.65710E+01, 2.08570E+01, 2.54290E+01, 2.97140E+01, 3.97140E+01, 8.28600E+00, 1.31430E+01, 1.82860E+01, 2.25710E+01, 2.71430E+01, 3.15710E+01, 4.14290E+01, 1.50000E+01, 2.07140E+01, 2.50000E+01, 2.91430E+01, 3.68570E+01, 4.10000E+01, 4.20000E+01, 9.42900E+00, 1.58570E+01, 2.04290E+01, 2.52860E+01, 2.98570E+01, 3.57140E+01, 4.12860E+01, 7.42900E+00, 1.14290E+01, 1.65710E+01, 2.04710E+01, 2.74290E+01, 3.58570E+01, 3.95710E+01, 1.22860E+01, 1.68570E+01, 2.20000E+01, 2.75710E+01, 3.20000E+01, 3.72860E+01, 4.14290E+01, 1.22860E+01, 1.65710E+01, 2.12860E+01, 2.55710E+01, 3.04290E+01, 3.45710E+01, 3.61430E+01, 1.14290E+01, 1.75710E+01, 2.21430E+01, 2.71430E+01, 3.24290E+01, 3.71430E+01, 4.15710E+01, 1.21430E+01, 1.64290E+01, 2.08570E+01, 2.51430E+01, 2.92860E+01, 3.34290E+01, 3.91430E+01, 1.47140E+01, 1.91430E+01, 2.35710E+01, 2.77140E+01, 3.21430E+01, 3.81430E+01, 4.07140E+01, 8.14300E+00, 1.28570E+01, 1.62860E+01, 2.18570E+01, 2.51430E+01, 3.08570E+01, 4.07140E+01, 8.28600E+00, 1.57140E+01, 2.62860E+01, 3.05710E+01, 3.77140E+01, 3.90000E+01, 4.01430E+01, 1.35710E+01, 1.85710E+01, 2.31430E+01, 3.14290E+01, 3.44290E+01, 3.70000E+01, 3.91430E+01, 1.48570E+01, 1.88570E+01, 2.35710E+01, 3.05710E+01, 3.22860E+01, 3.77140E+01, 3.82860E+01, 1.28570E+01, 1.71430E+01, 2.18570E+01, 2.65710E+01, 3.10000E+01, 3.55710E+01, 3.87140E+01, 6.85700E+00, 1.37140E+01, 1.97140E+01, 2.38570E+01, 3.00000E+01, 3.42860E+01, 4.12860E+01, 1.01430E+01, 1.51430E+01, 1.94290E+01, 2.41430E+01, 2.95710E+01, 3.30000E+01, 3.91430E+01, 8.57100E+00, 1.30000E+01, 1.81430E+01, 2.30000E+01, 2.81430E+01, 3.52860E+01, 4.08570E+01, 1.22860E+01, 1.65710E+01, 2.15710E+01, 3.08570E+01, 3.34290E+01, 3.50000E+01, 3.77140E+01, 7.00000E+00, 9.00000E+00, 1.38570E+01, 1.68570E+01, 2.58570E+01, 3.00000E+01, 3.70000E+01, 8.85700E+00, 1.10000E+01, 1.58570E+01, 2.00000E+01, 2.68570E+01, 3.25710E+01, 4.12860E+01, 1.00000E+01, 1.20000E+01, 1.47140E+01, 2.17140E+01, 2.65710E+01, 2.97140E+01, 4.07140E+01, 9.42900E+00, 1.37140E+01, 1.77140E+01, 2.20000E+01, 2.62860E+01, 3.05710E+01, 3.67140E+01), .Dim=c(68, 7)), y= structure(.Data= c(3.30000E+00, 6.10000E+00, 8.50000E+00, 7.90000E+00, 1.18000E+01, 1.48000E+01, NA, 2.00000E+00, 4.30000E+00, 8.00000E+00, 1.10000E+01, 1.22000E+01, 1.56000E+01, NA, 4.50000E+00, 6.80000E+00, 8.10000E+00, 1.35000E+01, 1.50000E+01, NA, NA, 5.50000E+00, 5.50000E+00, 8.00000E+00, 1.32000E+01, 1.39000E+01, 1.51000E+01, NA, 2.00000E+00, 9.50000E+00, 1.22000E+01, 1.82000E+01, 1.93000E+01, 2.20000E+01, NA, 1.80000E+00, 2.00000E+00, 4.50000E+00, 7.10000E+00, 9.50000E+00, 1.26000E+01, 1.35000E+01, 2.00000E-01, 1.40000E+00, 3.60000E+00, 5.80000E+00, 8.30000E+00, 9.00000E+00, NA, 3.80000E+00, 3.40000E+00, 5.40000E+00, 8.80000E+00, 1.04000E+01, 1.23000E+01, 1.53000E+01, 1.90000E+00, 5.90000E+00, 6.00000E+00, 1.06000E+01, 1.40000E+01, 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