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MTHM017 Advanced Topics in Statistics

创建日期：2024-05-20 16:29:33

所属分类：数学mathematics

标签： MTHM017

Advanced Topics in Statistics

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MTHM017 Advanced Topics in Statistics

Assignment
Please make sure that the submitted work is your own. This is NOT a group assignment,
therefore approaches and solutions shouldn’t be discussed with other students. Plagiarism
and collusion with other students are examples of academic misconduct and will be reported.
More information on academic honesty can be found here.
The assignment has two main parts. Part A involves (i) fitting a Poisson regression model to assess the
effect of using different priors, and (ii) fitting a fixed effects and a random effects model in order to compare
posterior distributions under different models. Part B involves using different methods for classification of
data into two groups.
A. Bayesian Inference [66 marks]
1. The first question of part A involves fitting a Poisson regression model using the Scotland dataset, which
contains the observed and expected counts of lip cancer cases for the local authority administrative areas in
Scotland between 1975 and 1986.
a. [3 marks] Calculate the Standard Mortality Ratios (SMR=observed/expected) for each administrative
area and plot the distribution of the SMRs. Then plot a map of the SMRs by administrative area. To
map the SMRs you may use the ScotlandMap function readily available on the ELE page.
The code below uses random numbers to demonstrate how the ScotlandMap function works. Note
that in order for this to run you need to add the shapefiles for Scotland (.shx, .shp, .prj, .dbf) to your
working directory. The ScotlandMap.R file and the shapefiles are available on the ELE page.
library(tidyverse)
library(RColorBrewer)
library(sf)
library(rgdal)
source("ScotlandMap.R") # need to read in the ScotlandMap function
testdat <- runif(56) # generate random numbers to use as observations
ScotlandMap(testdat,figtitle="Scotland random numbers")
We are interested in estimating the relative risk (RR) for each area. The relative risk is the parameter
that quantifies whether an area i has a higher (RRi > 1) or lower (RRi < 1) occurrence of cases than
what we would expect from the reference rates. In order to estimate the relative risk we are going to fit
a Poisson model of the following form:
Obsi ∼ Pois(µi)
log(µi) = log(Expi) + β0 + θi
RRi = exp(β0)exp(θi)
Where θi ∼ Normal(0, σ2). Here, the Exp(ected) numbers are an ‘offset’, which are treated as fixed
data (i.e. no coefficient is estimated for this term).
b. [3 marks] Describe the role (interpretation) of β0 and the set of θ’s in this model and how they contribute
to the estimation of RR. Focus on the meaning of these parameters in the estimation of the relative risk.
1
c. [14 marks] Code up this Poisson(-Lognormal) model in JAGS to analyse the Scotland data. For the
parameter β0 use a Normal prior with mean 0 and variance 1000. And for the precision parameter
τ(= 1/σ2) of θi use a prior τ ∼ Gamma(1, 0.05) (which is parametrised so as to have a mean of 20).
Initialise 2 chains and run the model with these two chains. You will have to decide on the appropriate
values of n.iter and burnin. Produce trace plots for the chains and summaries of all the model
parameters. Investigate whether the chains for all the parameters have converged.
d. [4 marks] Extract the posterior means for the RR and map them. Compare the posterior mean relative
risk to the SMR for each area. Note that we can think about the SMR as the maximum likelihood
estimate of the relative risk.
e. [7 marks] Now calculate the posterior probabilities that the relative risk in each area exceeds 1. Extract
these probabilities and map them. For which areas are you at least 95% certain that there is an excess
risk of lip cancer (i.e. relative risk > 1)?
f. [6 marks] Repeat the analysis with different priors for β0 and τ . The exact choice is yours, but explain
why you have chosen them and what they mean. Map the two sets of RRs and explain any differences
you see in the map and the summaries of the posteriors for the parameters of the model.
2. In this question we will analyse the effect different models have on the posterior distribution. We will use
the surgical.csv file, which contains data on 12 hospitals, where the column n gives the total number of
heart surgery operations carried out on children under 1 year old by each centre between April 1991 and
March 1995, and the column r gives the number of these operations where the patient died within 30 days of
the operation.
A binomial model seems reasonable for these data, so we will focus on comparing different models for the
hospital specific mortality rates.
We will first fit a logistic random effects model of the following form:
ri ∼ Binom(ni, θi),
logit(θi) ∼ N(µ, σ2).
This model assumes that failure rates (θi) across hospitals are similar in some way. This similarity is
represented by the assumption that the mortality rates for the hospitals are drawn from the same (parent)
distribution.
a. [9 marks] Code this model in JAGS to analyse the surgical data. Use vague priors for the parameters
of the shared normal distribution. In particular you can use the following model definition:
jags.mod <- function(){
for(i in 1:12){
r[i] ~ dbin(theta[i],n[i])
logit(theta[i]) <- logit.theta[i]
logit.theta[i] ~ dnorm(mu, tau)
}
mu ~ dnorm(0,1.0E-3)
tau <- 1/sigmaˆ2
sigma ~ dunif(0,100)
}
Run the model for 10,000 iterations, with 2 chains, discarding the first 5,000 as ‘burn-in’. Produce
traceplots for the chains and summaries for the fitted parameters. Comment on whether the chains for
all the parameters have converged.
b. [8 marks] We want to know whether the assumptions of the above random effects model are reasonable,
or whether there is any evidence that the mortality rates for one or more of the hospitals are not drawn
from the same parent distribution. We will assess this by comparing the observed and predicted number
of deaths in each hospital.

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FE 825 IERG 5130 STAT 8178 EE161 ADS2 G54FUZ CS326 113B ECN 226 QBUS6860 ELEC3202 COMP9334 MGT11B MET CS 777 STA238 00099F Python SMM306 CMT206 EC116 SP MAST90084 CS480/680 BCPM0091 CMPT 300 ECON2209 FY20 STAT 432 CVEN9625 MATH1712 CMP9771M CMP3750M CIS 325 AI506 MAT344H1S MGT7180 PSTAT 160B CSYS5010 1D STAT 453/558 ME152B STA304H1F CSCI-128 ME428 CO3002 FINE 7640 COMP3234 ECSE 223 EE497 – 3D MATH5905 ECS648U ENG5009 MATH10242 EC665 A CPI221 ISOM5220 MACE40402 MGT B456F COMP9315 AFM 113 STAT 778 CS 564 MA231 CS 230 CS 124 COMP5511 CSC317 ITEC1010 ELEC207 INFT 3019 7CCMFM06 EE 161 5G ECLT5810 MATH 7349 CS526 O2 COMP90054 ELE00063M ACCT5034 ARC1015 CSCI 40 CSCI E40 FIN 524B CS 462 389COM CMT309 MFIN703 COMP4250 STA304 STA304H1S STA 1003 FIN 448 BUFN 732 CS520 CVEN3701 ECMT6006 PHAS0100 CSIT314 MDIA 5028 CSCI 2134 MAT 451 CMPT 459 STAT 457 INFO20003 COMP4141 IELE8066 F540 ECM9 CSE 6321 FIN 430 MATH3911 BFC5916 COMP1011 EEE 6209 STATS 240 CW3 COMP206 COMP1003 4331W ISE 530 CMPT 295 MATH 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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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