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a. State the six Bayesian models listed in Table 9.6 on page 257 of CJBH \
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b. In R, create a new data subset of the FEV data set used in Study Guide #4a \
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fev.dat1 =list(n=nrow(fevstar.dat),FEV=FEV, Smoke = Smoke)
bugs.data(data=fev.dat1, \
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Provide the central 95% posterior predictive interval for the FEV of a new \
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Cell["\<\
c. Fit the second model in Table 9.6, involving Age as the only predictor, to \
the subset of FEV data set used in Study Guide #4a containing only FEV and \
Age, and answer the questions below:\
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Provide the central 95% posterior predictive interval for the FEV of a new \
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Cell[TextData[{
 "From the Burn-in and BGR plots, there is some evidence of non-convergence \
as in the burn-in plots the 3 chains of ",
 Cell[BoxData[
  FormBox[
   SubscriptBox["\[Beta]", "2"], TraditionalForm]],
  FormatType->"TraditionalForm"],
 " do not overlap each other very well and there are fluctuations with the \
values of capillaries and in the bgr plot the shrink factor keeps on \
decreasing up to 5500 iterations."
}], "TextNoIndent",
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Cell[CellGroupData[{

Cell["Appendix", "Title",
 CellChangeTimes->{{3.689299522613782*^9, 3.689299539787876*^9}, {
  3.689329160501581*^9, 3.689329161185987*^9}}],

Cell[CellGroupData[{

Cell["R2openbugs code", "Section",
 CellChangeTimes->{3.6893291668987017`*^9}],

Cell[CellGroupData[{

Cell["Model I", "Subsection",
 CellChangeTimes->{{3.689329184836796*^9, 3.6893291873575478`*^9}}],

Cell["\<\
## SDS384.7 Study Guide #4a, Fall 2016
## Pre (before drawing the MCMC samples) processing FEV data in R

# PART 1
## Perform EDA on FEV data set

fevstar.dat  = \
data.frame(read.table(\[OpenCurlyDoubleQuote]/Users/chenyinglong/Documents/\
Bayesian Statistics Methods/FEV-Data-CJBH.txt\[CloseCurlyDoubleQuote], \
header=T))

attach(fevstar.dat)
fevstar.dat[1:3,]

ystar1 = subset(FEV, Smoke == 0)
length(ystar1)

ystar2 = subset(FEV, Smoke == 1)
length(ystar2)


#Side-by-side Box-plots
data.list = list(Nonsmoker = ystar1, Smoker = ystar2)
#quartz(width=4,height=4,pointsize=8)
boxplot(data.list, main=\[CloseCurlyDoubleQuote]Side-by-side Box-plots of FEV \
Data\[CloseCurlyDoubleQuote], ylab=\[CloseCurlyDoubleQuote]FEV\
\[CloseCurlyDoubleQuote], xlab = \[OpenCurlyDoubleQuote]\
\[CloseCurlyDoubleQuote])

# Regression - Scatter Plots
#quartz(width=8, height=8,pointsize=8)
plot(fevstar.dat)
title(\[OpenCurlyQuote]Matrix of scatter plots - FEV Data\[CloseCurlyQuote])

## check on the propriety of the joint posterior
# An indirect check on the full-rank of the design matrix X in the regression \
model
fevstar.mat = cbind(rep(1, 345), Age, Smoke, Age*Smoke)
dim(fevstar.mat)

fevstar.mat[1:5,]

xprimex = crossprod(fevstar.mat)
dim(xprimex)

det(xprimex)

## end of EDA
## BDA begins

# PART 2

## Partial Remote processing to get MCMC samples

## Install first the R-package R2OpenBUGS

library(R2OpenBUGS)

setwd(\[OpenCurlyDoubleQuote]/Users/chenyinglong/Documents/Bayesian \
Statistics Methods\[CloseCurlyDoubleQuote])
getwd()

## Model I

fev.dat1 =list(n=nrow(fevstar.dat), FEV=FEV, Smoke = Smoke)
bugs.data(data=fev.dat1, \
data.file=\[CloseCurlyDoubleQuote]FEV-Regression-model1.txt\
\[CloseCurlyDoubleQuote])

# fev.dat =list(n=nrow(fevstar.dat),Age=Age, FEV=FEV, Smoke = Smoke)
# 
# bugs.data(data=fev.dat, \
data.file=\[CloseCurlyDoubleQuote]FEV-Regression.txt\[CloseCurlyDoubleQuote])
# 
# ## Now leave R and launch OpenBUGS to load data manually using
# # FEV-Regression.txt; Open this text file from OpenBUGS and change file \
type as .txt
# 

# PART 3

# Full Remote processing to get MCMC samples
# without leaving R
 
OpenBUGS.pgm=\[CloseCurlyDoubleQuote]/Users/chenyinglong/.wine/drive_c/\
Program\\ Files/OpenBUGS/OpenBUGS323/OpenBUGS.exe\[CloseCurlyDoubleQuote]
 
set.seed(100)

library(R2OpenBUGS)

setwd(\[OpenCurlyDoubleQuote]/Users/chenyinglong/Documents/Bayesian \
Statistics Methods\[CloseCurlyDoubleQuote])
getwd()

# fev.dat =list(n=nrow(fevstar.dat),Age=Age, FEV=FEV, Smoke = Smoke)
fev.dat1 =list(n=nrow(fevstar.dat), FEV=FEV, Smoke = Smoke)

#INITS - Chain 1
inits1 =list(tau=1.0, beta = c(0.0, 0.0))
#INITS - Chain 2
inits2 =list(tau= 5.0, beta = c(1.0, 2.0))
#INITS - Chain 3
inits3 =list(tau=10.0, beta = c(0.0, -2.0))


inits=list(inits1, inits2, inits3)


model.I.with.coda= \
bugs(data=fev.dat1,inits,OpenBUGS.pgm=OpenBUGS.pgm,model.file = \
\[OpenCurlyDoubleQuote]Model-I.txt\[CloseCurlyDoubleQuote],
                               n.burnin=500,parameters=c(\
\[OpenCurlyDoubleQuote]post.prob.beta2\[CloseCurlyDoubleQuote],\
\[CloseCurlyDoubleQuote]FEVs\[CloseCurlyDoubleQuote], \
\[OpenCurlyDoubleQuote]beta\[CloseCurlyDoubleQuote]),
                               n.chains = 3,n.iter=5500, codaPkg = TRUE, \
useWINE = TRUE)

## avoid this 
#print(model.fit.SG4a.with.coda,digits=4) #plot(model.fit.SG4a.with.coda)

## Post (after drawing the MCMC samples) processing in R to check for \
convergence of
## MCMC samples to the posteriors

library(coda)
library(lattice)
out.coda = read.bugs(model.I.with.coda)
class(out.coda)
str(out.coda)

#dev.off()
#quartz(width=4,height=4,pointsize=8)
xyplot(out.coda)
#quartz(width=4,height=4,pointsize=8)
densityplot(out.coda)


#Brooks-Gelman-Rubin-plot
#Work with samples from the posterior densities of nodes - ONE at a time
# beta4.sample.only = \
bugs(data=fev.dat1,inits,OpenBUGS.pgm=OpenBUGS.pgm,model.file = \
\[OpenCurlyDoubleQuote]OpenBUGS-txt-study-guide-4a-Fa16.txt\
\[CloseCurlyDoubleQuote],
#                          \
n.burnin=500,parameters=c(\[OpenCurlyDoubleQuote]beta[2]\
\[CloseCurlyDoubleQuote]),
#                          n.chains = 3,n.iter=5500, codaPkg = TRUE, useWINE \
= TRUE)
beta.sample.only= \
bugs(data=fev.dat1,inits,OpenBUGS.pgm=OpenBUGS.pgm,model.file = \
\[OpenCurlyDoubleQuote]Model-I.txt\[CloseCurlyDoubleQuote],
                        n.burnin=500,parameters=c(\[OpenCurlyDoubleQuote]beta\
\[CloseCurlyDoubleQuote]),
                        n.chains = 3,n.iter=5500, codaPkg = TRUE, useWINE = \
TRUE)
o1 = read.bugs(beta.sample.only)
#quartz(width=4,height=4,pointsize=8)
class(o1)
str(o1)
gelman.plot(o1)
# 
# ## Assuming that CODA analysis provides no serious concerns of \
non-convergence
# ## of MCMC samples to the posteriors we seek, call bugs function again in \
R2OpenBUGS
# ## with codaPkg = FALSE to produce node statistics, DIC, etc
# ## for ALL nodes defined in \[OpenCurlyDoubleQuote]parameters\
\[CloseCurlyDoubleQuote]
# 
model.I=bugs(data=fev.dat1,inits = inits,OpenBUGS.pgm=OpenBUGS.pgm,model.file \
= \[OpenCurlyDoubleQuote]Model-I.txt\[CloseCurlyDoubleQuote],
                    n.burnin=500,parameters=c(\[OpenCurlyDoubleQuote]post.\
prob.beta2\[CloseCurlyDoubleQuote],\[CloseCurlyDoubleQuote]FEVs\
\[CloseCurlyDoubleQuote],\[CloseCurlyDoubleQuote]beta\[CloseCurlyDoubleQuote]),
                   n.chains = 3,n.iter=5500, bugs.seed = 12, codaPkg = FALSE, \
useWINE = TRUE)
# 
print(model.I,digits=4)
# quartz(width=4,height=4,pointsize=8)
# plot(model.I)\
\>", "TextNoIndent",
 CellChangeTimes->{3.6893312263251667`*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Model II", "Subsection",
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## SDS384.7 Study Guide #4a, Fall 2016
## Pre (before drawing the MCMC samples) processing FEV data in R

# PART 1
## Perform EDA on FEV data set

fevstar.dat  = \
data.frame(read.table(\[OpenCurlyDoubleQuote]/Users/chenyinglong/Documents/\
Bayesian Statistics Methods/FEV-Data-CJBH.txt\[CloseCurlyDoubleQuote], \
header=T))

attach(fevstar.dat)
fevstar.dat[1:3,]

ystar1 = subset(FEV, Smoke == 0)
length(ystar1)

ystar2 = subset(FEV, Smoke == 1)
length(ystar2)


#Side-by-side Box-plots
data.list = list(Nonsmoker = ystar1, Smoker = ystar2)
#quartz(width=4,height=4,pointsize=8)
boxplot(data.list, main=\[CloseCurlyDoubleQuote]Side-by-side Box-plots of FEV \
Data\[CloseCurlyDoubleQuote], ylab=\[CloseCurlyDoubleQuote]FEV\
\[CloseCurlyDoubleQuote], xlab = \[OpenCurlyDoubleQuote]\
\[CloseCurlyDoubleQuote])

# Regression - Scatter Plots
#quartz(width=8, height=8,pointsize=8)
plot(fevstar.dat)
title(\[OpenCurlyQuote]Matrix of scatter plots - FEV Data\[CloseCurlyQuote])

## check on the propriety of the joint posterior
# An indirect check on the full-rank of the design matrix X in the regression \
model
fevstar.mat = cbind(rep(1, 345), Age, Smoke, Age*Smoke)
dim(fevstar.mat)

fevstar.mat[1:5,]

xprimex = crossprod(fevstar.mat)
dim(xprimex)

det(xprimex)

## end of EDA
## BDA begins

# PART 2

## Partial Remote processing to get MCMC samples

## Install first the R-package R2OpenBUGS

library(R2OpenBUGS)

setwd(\[OpenCurlyDoubleQuote]/Users/chenyinglong/Documents/Bayesian \
Statistics Methods\[CloseCurlyDoubleQuote])
getwd()

## Model II

fev.dat2 =list(n=nrow(fevstar.dat), Age=Age, FEV=FEV)
bugs.data(data=fev.dat2, \
data.file=\[CloseCurlyDoubleQuote]FEV-Regression-model2.txt\
\[CloseCurlyDoubleQuote])

# fev.dat =list(n=nrow(fevstar.dat),Age=Age, FEV=FEV, Smoke = Smoke)
# 
# bugs.data(data=fev.dat, \
data.file=\[CloseCurlyDoubleQuote]FEV-Regression.txt\[CloseCurlyDoubleQuote])
# 
# ## Now leave R and launch OpenBUGS to load data manually using
# # FEV-Regression.txt; Open this text file from OpenBUGS and change file \
type as .txt
# 

# PART 3

# Full Remote processing to get MCMC samples
# without leaving R

OpenBUGS.pgm=\[CloseCurlyDoubleQuote]/Users/chenyinglong/.wine/drive_c/\
Program\\ Files/OpenBUGS/OpenBUGS323/OpenBUGS.exe\[CloseCurlyDoubleQuote]

set.seed(100)

library(R2OpenBUGS)

setwd(\[OpenCurlyDoubleQuote]/Users/chenyinglong/Documents/Bayesian \
Statistics Methods\[CloseCurlyDoubleQuote])
getwd()

# fev.dat =list(n=nrow(fevstar.dat),Age=Age, FEV=FEV, Smoke = Smoke)
fev.dat2 =list(n=nrow(fevstar.dat), Age=Age, FEV=FEV)

#INITS - Chain 1
inits1 =list(tau=1.0, beta = c(0.0, 0.0))
#INITS - Chain 2
inits2 =list(tau= 5.0, beta = c(1.0, 2.0))
#INITS - Chain 3
inits3 =list(tau=10.0, beta = c(0.0, -2.0))


inits=list(inits1, inits2, inits3)


model.II.with.coda= \
bugs(data=fev.dat2,inits,OpenBUGS.pgm=OpenBUGS.pgm,model.file = \
\[OpenCurlyDoubleQuote]Model-II.txt\[CloseCurlyDoubleQuote],
                               n.burnin=500,parameters=c(\
\[OpenCurlyDoubleQuote]post.prob.beta2\[CloseCurlyDoubleQuote],\
\[CloseCurlyDoubleQuote]FEV15\[CloseCurlyDoubleQuote],\[CloseCurlyDoubleQuote]\
beta\[CloseCurlyDoubleQuote]),
                               n.chains = 3,n.iter=5500, codaPkg = TRUE, \
useWINE = TRUE)

## avoid this 
#print(model.fit.SG4a.with.coda,digits=4) #plot(model.fit.SG4a.with.coda)

## Post (after drawing the MCMC samples) processing in R to check for \
convergence of
## MCMC samples to the posteriors

library(coda)
library(lattice)
out.coda = read.bugs(model.II.with.coda)
class(out.coda)
str(out.coda)

#dev.off()
#quartz(width=4,height=4,pointsize=8)
xyplot(out.coda)
#quartz(width=4,height=4,pointsize=8)
densityplot(out.coda)


#Brooks-Gelman-Rubin-plot
#Work with samples from the posterior densities of nodes - ONE at a time
# beta.sample.only = \
bugs(data=fev.dat1,inits,OpenBUGS.pgm=OpenBUGS.pgm,model.file = \
\[OpenCurlyDoubleQuote]OpenBUGS-txt-study-guide-4a-Fa16.txt\
\[CloseCurlyDoubleQuote],
#                          \
n.burnin=500,parameters=c(\[OpenCurlyDoubleQuote]beta[2]\
\[CloseCurlyDoubleQuote]),
#                          n.chains = 3,n.iter=5500, codaPkg = TRUE, useWINE \
= TRUE)
beta.sample.only= \
bugs(data=fev.dat2,inits,OpenBUGS.pgm=OpenBUGS.pgm,model.file = \
\[OpenCurlyDoubleQuote]Model-II.txt\[CloseCurlyDoubleQuote],
                         n.burnin=500,parameters=c(\[OpenCurlyDoubleQuote]\
beta\[CloseCurlyDoubleQuote]),
                         n.chains = 3,n.iter=5500, codaPkg = TRUE, useWINE = \
TRUE)
o1 = read.bugs(beta.sample.only)
#quartz(width=4,height=4,pointsize=8)
class(o1)
str(o1)
gelman.plot(o1)

# ## Assuming that CODA analysis provides no serious concerns of \
non-convergence
# ## of MCMC samples to the posteriors we seek, call bugs function again in \
R2OpenBUGS
# ## with codaPkg = FALSE to produce node statistics, DIC, etc
# ## for ALL nodes defined in \[OpenCurlyDoubleQuote]parameters\
\[CloseCurlyDoubleQuote]
# 
model.II=bugs(data=fev.dat2,inits = \
inits,OpenBUGS.pgm=OpenBUGS.pgm,model.file = \
\[OpenCurlyDoubleQuote]Model-II.txt\[CloseCurlyDoubleQuote],
                    n.burnin=500,parameters=c(\[OpenCurlyDoubleQuote]post.\
prob.beta2\[CloseCurlyDoubleQuote],\[CloseCurlyDoubleQuote]FEV15\
\[CloseCurlyDoubleQuote],\[CloseCurlyDoubleQuote]beta\[CloseCurlyDoubleQuote]),
                    n.chains = 3,n.iter=5500, bugs.seed = 12, codaPkg = \
FALSE, useWINE = TRUE)
# 
print(model.II,digits=4)
#quartz(width=4,height=4,pointsize=8)
#plot(model.II)\
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Cell["Model I", "Subsection",
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Cell["\<\


# #SDS 384.7    HW6 Model-I     Fall 2016 

# Building a Bayesian Model in OpenBUGS

# Bayesian Inferences after Fitting a Normal Linear 

# Regression, with INTERACTION TERMS.

# Data Source: Forced Expiratory Volume (FEV) Data, 

# as introduced on page 162 of CJBH Text 



model{

for(i in 1:n){

\tFEV[i] ~ dnorm(mu[i],tau)

\tmu[i] <- beta[1] + beta[2]*Smoke[i]}



 # Diffuse prior

beta[1] ~ dnorm(0,0.001) 

beta[2] ~ dnorm(0,0.001)


tau ~ dgamma(0.01,0.001)



## Predict the FEV of a new kid who smokes

FEVs ~ dnorm(mus,tau)

mus <-  beta[1] + beta[2] 



## Test H0: beta2 > 0

post.prob.beta2 <- step(beta[2])



# Monitor DIC separately from Inference > DIC menu

}\
\>", "TextNoIndent",
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Cell["\<\
# #SDS 384.7    HW6 Model-II     Fall 2016 

# Building a Bayesian Model in OpenBUGS

# Bayesian Inferences after Fitting a Normal Linear 

# Regression, with INTERACTION TERMS.

# Data Source: Forced Expiratory Volume (FEV) Data, 

# as introduced on page 162 of CJBH Text 



model{

for(i in 1:n){

\tFEV[i] ~ dnorm(mu[i],tau)

\tmu[i] <- beta[1] + beta[2]*Age[i]}



 # Diffuse prior

beta[1] ~ dnorm(0,0.001) 

beta[2] ~ dnorm(0,0.001)


tau ~ dgamma(0.01,0.001)



## Provide the central 95% posterior predictive interval for the FEV of a new \
kid who is 

FEV15 ~ dnorm(mu15,tau)

 

mu15 <-  beta[1] + 15*beta[2] 



## Test H0: beta2 >0

post.prob.beta2 <- step(beta[2])



# Monitor DIC separately from Inference > DIC menu

}\
\>", "TextNoIndent",
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Cell["\<\
                    mean     sd    2.5%      25%      50%      75%    97.5%   \
Rhat n.eff
post.prob.beta2   0.9201 0.2711   0.000   1.0000   1.0000   1.0000   1.0000 \
1.0010 15000
FEVs              3.2937 0.7590   1.787   2.7850   3.2890   3.8070   4.7751 \
1.0010 15000
beta[1]           3.1506 0.0452   3.063   3.1200   3.1510   3.1810   3.2390 \
1.0009 15000
beta[2]           0.1460 0.1048  -0.058   0.0753   0.1448   0.2159   0.3514 \
1.0011 11000
deviance        786.1755 2.4569 783.400 784.4000 785.5000 787.3000 792.5000 \
1.0009 15000\
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                    mean     sd     2.5%      25%      50%      75%   97.5%   \
Rhat n.eff
post.prob.beta2   1.0000 0.0000   1.0000   1.0000   1.0000   1.0000   1.000 \
1.0000     1
FEV15             3.6416 0.6766   2.3080   3.1850   3.6400   4.1012   4.959 \
1.0011 12000
beta[1]           1.2008 0.2072   0.8021   1.0590   1.1960   1.3410   1.615 \
1.0066   450
beta[2]           0.1629 0.0168   0.1293   0.1515   0.1633   0.1745   0.195 \
1.0056   440
deviance        709.4863 2.4181 706.8000 707.7000 708.8000 710.6000 715.600 \
1.0021  2300
\
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