MAST90125留学生作业代写、代做Bayesian课程作业、R程序设计作业调试、R语言作业代写
Assignment 2, Question 2 MAST90125: Bayesian
Statistical Learning
Due: Friday 20 September 2019
There are places in this assignment where R code will be required. Therefore set the random
seed so assignment is reproducible.
set.seed(123456) #Please change random seed to your student id number.
Question Two (20 marks)
In lecture 3, we discussed how a Bayesian framework readily lends itself to combining information from
sequential experiments. To demonstrate, consider the following data extracted from the HealthIron study.
Serum ferritin levels were measured for two samples of women, one of C282Y homozygotes (n = 88) and
the other of women with neither of the key mutations (C282Y and H63D) in the HFE gene, so-called HFE
‘wildtypes’(n = 242). The information available is
• idnum: Participant id.
• homc282y: Indicator whether individual is Homozygote (1) or Wildtype (0).
• time: Time since onset of menopause, measured in years.
• logsf: The natural logarithm of the serum ferritin in µg/L.
The data required to answer this question are Hiron.csv, which can be downloaded from LMS.
a) Fit a standard linear regression,
E(logsf) = β0 + β1time
with responses restricted to those who are homozygote (homc282y = 1). This can be done using the lm
function in R. Report the estimated coefficients βˆ, estimated error variance,
b) Fit a Bayesian regression using a Gibbs sampler to only the wildtype (homc282y=0) data. Use the
output from your answer in a) to define proper priors for β, τ . For help, refer to lecture 13. For the
Gibbs sampler, run two chains for 10,000 iterations. Discard the first 1000 iterations as burn-in and
then remove every second remaining iteration to reduce auto-correlation. When storing results, convert
τ back to σ2
. When running the Gibbs sampler, incorporate posterior predictive checking, using the
test statistic T(y, β) = Pn
, where ei
is the predicted residual for
observation i at simulation j and e
is the replicate residual for observation i at simulation j. Report
posterior means, standard deviations and 95 % central credible intervals for β0, β1, σ2
combining results
for the two chains.
c) Perform convergence checks for the chain obtained in b). Report both graphical summaries and
Gelman-Rubin diagnostic results. For the calculation of Gelman-Rubin diagnostics, you will need to
install the R package coda. An example of processing chains for calculating Gelman-Rubin diagnostics
is given below.
Processing chains for calculation of Gelman-Rubin diagnostics. Imagine you have 4 chains of
a multi-parameter problem, and thinning already completed, called par1,par2,par3,par4
1
Step one: Converting the chains into mcmc lists.
library(coda)
par1<-as.mcmc.list(as.mcmc((par1)))
par2<-as.mcmc.list(as.mcmc((par2)))
par3<-as.mcmc.list(as.mcmc((par3)))
par4<-as.mcmc.list(as.mcmc((par4)))
Step two: Calculating diagnostics
par.all<-c(par1,par2,par3,par4)
gelman.diag(par.all)
d) Fit a standard linear regression,
E(logsf) = β0 + β1time
to all the data using the lm function in R. Report βˆ, and associated 95 % confidence intervals. Comparing
these results to the results from b), do you believe that sequential analysis gave the same results as fitting
the regression on the full data.
e) Report the results of posterior predictive checking requested in b). Do you believe the postulated model
was plausible. If not, what do you think is a potential flaw in the postulated model.
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