辅导案例-STATS 3023-Assignment 4

STATS 3023 Computational Bayesian Statistics III
Assignment 4
2020
Assignment 4 is due by 23:59 Monday 2 November 2020.
Assignments are to be submitted online on MyUni. Your answers to data
analysis questions should include the relevant R output and the code you
used to produce it.
1. Dowlati et. al. (2010) 1, summarised 16 studies in which the blood concentration of
interleukin-6 in pg/ml, was compared between patients suffering major depression
and a group of non-depressed subjects. Each study was summarised by a 95%
confidence interval for the mean difference in IL-6 concentration. The data are
shown below and are available in the file IL6.csv.
Study Difference Confidence Interval
Berk (1997) 10.99 (9.76, 12.22)
Brambilla (1998) -1.38 (-2.54, -0.22)
Dhabhar (2009) 0.35 (0.18, 0.52)
Kagaya (2001) -0.13 (-0.88, 0.62)
Kubera (2000) 1.3 (0.05, 2.55)
Leo (2006) 1.3 (0.84, 1.76)
Maes (1995a) 3.02 (2.71, 3.33)
Maes(1995b) 0.25 (0, 0.5)
Maes (1997) 4.1 (1.65, 6.55)
Mikova (2001) 6 (-5.97, 17.97)
O’Brien (2007) 0.51 (0.45, 0.57)
Pavon (2006) 0.22 (0.07, 0.37)
Pike (2006) 0.8 (0.2, 1.4)
Simon (2008) 4.75 (0.41, 9.09)
Sluzewska (1996) 3.04 (2.42, 3.66)
Yang (2007) 2.87 (2.42, 3.32)
Perform a Bayesian meta-analysis of the data using the following steps.
(a) Express the data in terms of a point estimate and a standard error for each
study.
(b) Postulate a suitable hierarchical model for the data, assuming the study vari-
ances to be known.
(c) Use Stan to implement your model.
(d) Comment on the convergence and effective sample size for the MCMC esti-
mates.
(e) Plot the 95% Bayesian credible intervals for the difference in mean IL-6 and
also the between study standard deviation.
1Yekta Dowlati, Nathan Herrmann, Walter Swardfager, Helena Liu, Lauren Sham, Elyse K. Reim,
and Krista L. Lanctot (2010). A Meta-Analysis of Cytokines in Major Depression. Biological Psychiatry:
https://doi.org/10.1016/j.biopsych.2009.09.033
1
(f) Obtain 95% Bayesian credible intervals for the difference in mean IL-6 and
also the between study standard deviation.
(g) Summarise the conclusions of your analysis with respect to elevation of IL-6
in patients with major depression.
2. The file schools.csv contains the following data on 7185 high school students from
160 different schools.
Variable Description
School School Identifier
Size School Size
Sector Public/Catholic
Sex Male/Female
SES Student SES
MathAch Mathematics Achievement Score
The purpose of the analysis is to investigate differences in Mathematics Achieve-
ment between Public and Catholic schools and also differences between Males and
Females using a hierarchical model in stanarm.
(a) Read the data into R. Make sure that School is converted to a factor. Replace
Size with a suitably recoded factor with levels small, medium, large.
(b) Fit the hierarchical model
MathAch~Sex+Sector+Size+SES+(1|School)
in rstanarm using the default priors and save the resulting stanreg object.
(c) Display the model summary for all parameters except the school-level intercept
terms.
i. Comment on convergence of the chains.
ii. Comment on the adequacy of the effective sample sizes.
(d) Obtain plots of the Sex, Sector, Size, SES parameters showing the esti-
mated posterior densities and the 80% Bayesian credible intervals.
(e) Discuss the effects of each of Sex, Sector, Size, SES on the Mathematics
Achievement Score.
(f) Give a careful interpretation of the parameter sigma.
(g) Give a careful interpretation of the parameter Sigma[School:(Intercept),(Intercept)].
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Honours Questions
The following questions are compulsory for Honours and Masters students and may be
attempted for bonus marks by all students.
1. Continuing with Question 1 of the main assignment, suppose a new study is planned
on different randomly chosen sub-population.
(a) Obtain a sample from the posterior predictive distribution for the difference,
µ0, in population means for the new study.
(b) Generate a histogram of the the posterior predictive distribution and also a
90% posterior predictive interval.
2. Consider a regression model, in which A is a binary factor with levels 1 and 2.
(a) Compare the two parametrisations, y~A and y~A-1, in the context of ordinary
fixed effects regression modelling.
(b) Compare the same two parametrisations in the context of a Bayesian regression
model.
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