STAT0023辅导、辅导R程序语言

日期：2023-04-13 07:33

STAT0023 Computing for Practical Statistics
In-course assessment 2, take-home component (2022–23 session)
Table of Contents
Rubric ........................................................................................................................................................................... 2
Background and overview ................................................................................................................................... 3
Detailed instructions .............................................................................................................................................. 4
Your tasks ............................................................................................................................................................... 4
Submission requirements ................................................................................................................................ 4
Marking criteria.................................................................................................................................................... 6
Hints on tackling the assessment ................................................................................................................. 7
Appendix: the ReferendumResults.csv data set..................................................................................10
Data sources .......................................................................................................................................................10
Martin Rosenbaum’s BBC article (data source MR1) ......................................................................10
Martin Rosenbaum’s data set from the 2011 UK census (data source MR2) ........................10
The UK Register of Geographic Codes (data source RGC) ...........................................................11
UK age structure by ward, from the 2011 UK census (data source ASW) ..............................11
Data processing .................................................................................................................................................11
Description of variables ..................................................................................................................................12
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Rubric
Your solutions should be your own work and are to be submitted electronically to the
course Moodle page by 12 noon on MONDAY, 24TH APRIL 2023.
You can work either alone or in pairs for this assessment. It is up to you to form your
own pairs. You MUST register your choices on Moodle by 12 noon on WEDNESDAY,
22ND MARCH 2023, even if you choose to work alone.
If you choose to work in a pair, you will be jointly responsible for the work that is
submitted and you will be awarded the same mark.
Ensure that you electronically ‘sign’ the plagiarism declaration on the Moodle page when
submitting your work. If you choose to work in a pair, both of you should check what has
been submitted before signing this declaration: if any plagiarism or collusion is identified
with anyone outside your pair, you will share responsibility for it.
Late submission will incur a penalty unless there are extenuating circumstances
(e.g. medical) supported by appropriate documentation and notified within one week of
the deadline above. Penalties, and the procedure in case of extenuating circumstances,
are set out in the latest editions of the Statistical Science Department student
handbooks which are available from the departmental web pages.
Failure to submit this in-course assessment will mean that your overall examination mark
is recorded as “non-complete”, i.e. you will not obtain a pass for the course.
Submitted work that exceeds the specified word count will be penalized. The penalties
are described in the detailed instructions below.
Your solutions should be your own work. When uploading your scripts, you will be
required to electronically sign a statement confirming this, and that you have read the
Statistical Science department’s guidelines on plagiarism and collusion (see below).
Any plagiarism or collusion can lead to serious penalties for all students involved, and
may also mean that your overall examination mark is recorded as non-complete.
Guidelines as to what constitutes plagiarism may be found in the departmental student
handbooks: the relevant extract is provided on the ‘In-course assessment 2’ tab on the
STAT0023 Moodle page. The Turn-It-In plagiarism detection system may be used to scan
your submission for evidence of plagiarism and collusion.
You will receive feedback on your work via Moodle, and you will receive a provisional
grade. Grades are provisional until confirmed by the Statistics Examiners’ Meeting
in June 2023.

Background and overview
On 23rd June 2016, a referendum was held in the UK to decide whether or not to remain
part of the European Union (EU). 72% of registered voters took part. Of those, 51.2% voted
to leave the EU, and 48.1% voted to remain.
This result was unexpected, and there was extensive commentary on the reasons for it at
the time. On 6th February 2017, the BBC News web site carried an article entitled “Local
voting figures shed new light on EU referendum” (the article is at
http://www.bbc.co.uk/news/uk-politics-38762034). The article is by Martin Rosenbaum, a
Freedom of Information specialist at the BBC. He obtained data from 1070 electoral wards,1
giving the numbers of ‘Leave’ and ‘Remain’ votes cast in each ward. The Appendix to these
instructions provides details of how he obtained the data.
In his article, Martin Rosenbaum calculated some statistical associations between the
proportion of ‘Leave’ votes in a ward, and some of its social, economic and demographic
characteristics according to the most recent UK census which was conducted in 2011. He
examined characteristics such as education, age and ethnicity taken individually. However,
he did not investigate them jointly. This could be important, because there may be other
variables that simultaneously influence (say) education level and the propensity to vote
‘Leave’, and which thereby create the illusion of a causal link between them.
The BBC web page provides the voting data that were used in Martin Rosenbaum’s analysis,
but not the census data. However, he very kindly shared his census data in response to a
request from us at the time — and, for this in-course assessment, they have been
supplemented with some additional information as well.
The data are provided to you in the CSV file ReferendumResults.csv in the ‘In-course
assessment 2’ section of the STAT0023 Moodle page. For each of the 1070 electoral wards,
this file provides values of around 45 variables that may be relevant in understanding why
people voted as they did (the Appendix to these instructions gives a full list of variables,
along with other metadata). In addition, for the first 803 wards in the data file, the numbers
of ‘Leave’ votes are provided, as well as the total number of votes for ‘Leave’ and ‘Remain’
combined. For the final 267 wards however, the numbers of ‘Leave’ votes are not provided
to you: they are given as -1 in the data file.
Your task in this assessment is to use the data on the first 803 wards, to build a statistical
model that will help you to:
Understand the social, economic and demographic characteristics that are associated
with the voting outcome for a ward; and
Estimate the proportion of ‘Leave’ votes in each of the 267 wards for which you don’t
have this information.

1 An electoral ward is the smallest administrative division for election purposes in the UK, typically with a
population of around 5500. There are almost 9500 electoral wards in the UK.
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Detailed instructions
You may use either R or SAS for this assessment.
Your tasks
1. Read the data into your chosen software package, and carry out any necessary recoding
(e.g. to deal with the fact that -1 represents a missing value).
2. Carry out an exploratory analysis that will help you to start building a sensible statistical
model to explain and predict the proportion of ‘Leave’ votes in a ward. This analysis
should aim to reduce the number of candidate variables to take into the subsequent
modelling exercise, as well as to identify any important features of the data that may
have some implications for the modelling. You will need to consider the context of the
problem to guide your choice of exploratory analysis. See the ‘Hints’ below for some
ideas.
3. Using your exploratory analysis as a starting point, develop a statistical model that
enables you to predict the proportion of ‘Leave’ votes in a ward, based on (a subset of)
the ward characteristics; and also to understand the variation in proportions of ‘Leave’
votes between different wards. To be convincing, you will need to consider a range of
models and to use an appropriate suite of diagnostics to assess them. Ultimately
however, you are required to recommend a single model that is suitable for
interpretation, and to justify your recommendation. Your chosen model should be either
a linear model, a generalized linear model or a generalized additive model.
4. Use your chosen model to predict the proportion of ‘Leave’ votes for each of the 267
wards with missing voting data, and also to estimate the standard deviation of your
prediction errors.
Submission requirements
Submission for this assessment is electronic, via the STAT0023 Moodle page. You are
required to submit three files, as follows:
1. A report on your analysis, not exceeding 2500 words of text plus two pages of graphs
and / or tables. The word count includes titles, footnotes, appendices, references etc. —
in fact it includes everything except the two pages of graphs / tables and, if present, the
separate page describing the contribution of each pair member (see below).
Your report should be in three sections, as follows:
I. Introduce the problem context and describe briefly what aspects you considered
at the outset, how you used these to start your exploratory analysis, and what
were the important points to emerge from this exploratory analysis.
II. Describe briefly (without too many technical details) what models you considered
in step (3) above, and why you chose the model that you did.
III. State your final model clearly, summarise what your model tells you about the
characteristics associated with the proportion of ‘Leave’ votes, and discuss any
potential limitations of the model.
Your report should not include any computer code. It should include some graphs and /
or tables, but only those that support your main points. Graphs and tables must
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appear on separate pages, or they will be not be marked and will contribute to your
word count.
In addition to your data analysis, if you are working as a pair then you must include
an additional page at the end of their report where each pair member briefly
describes their contribution to the project. You will need to agree this in your pairs
before submitting the report. If both pair members agree that they contributed equally
then it is sufficient to write a single sentence to that effect, or alternatively you are very
welcome to describe your own personal contribution to the project. Note that this page
will not be marked and does not contribute to the word count; nor will different marks
be allocated to different pair members based on this. The purpose is to encourage you
all to be mindful about contributing to this piece of group-work.
Your report should be submitted as a PDF file named as ########_rpt.pdf, where
######## is your group ID, with any spaces replaced by underscores
(IMPORTANT!!!). For example, if your group ID is ‘ICA2 Group C’, your report should be
named ICA2_Group_C_rpt.pdf.
2. An R script or SAS program corresponding to your analysis and predictions. Your script /
program should run without user intervention on any computer with R or SAS installed,
providing the file ReferendumResults.csv is present in the current working directory /
current folder. When run, it should produce any results that are mentioned in your
report, together with the predictions and the associated standard deviations. The script
/ program should be named ########.r or ########.sas as appropriate, where
######## is your group ID with underscores instead of spaces. For example, if your
group ID is ‘ICA2 Group C’ and you use R, your script should be named ICA2_Group_C.r.
You may not create any additional input files that can be referenced by your script; nor
should you write any code that requires access to the internet in order to run it. If you
use R however, you may use the following additional libraries if you wish (together with
other libraries that are loaded automatically by these): mgcv, ggplot2, grDevices,
RColorbrewer, lattice and MASS. You may not use any other add-on libraries: for
present purposes, an “add-on library” is one that requires a library() or require()
command or equivalent (e.g. the package::command syntax) before it can be used, if your
R system is installed using default settings.
3. A text file containing your predictions for the 267 wards with missing voting data. This
file should be named ########_pred.dat, where ######## is your group ID with
underscores instead of spaces. The file should contain three columns, separated by
spaces and with no header. The first column should be the ward identifier
(corresponding to variable ID in file ReferendumResults.csv); the second should be the
predicted proportion of ‘Leave’ votes for that ward, and the third should be the standard
deviation of your prediction error.
NOTE: if you work in pairs, both members of a pair must confirm their submission on
Moodle before the submission deadline.
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Marking criteria
There are 75 marks for this exercise. These are broken down as follows:
? Report: 40 marks. The marks here are for: displaying awareness of the context for the
problem and using this to inform the statistical analysis; good judgement in the choice
of exploratory analysis and in the model-building process; a clear and well-justified
argument; clear conclusions that are supported by the analysis; and appropriate choice
and presentation of graphs and / or tables. The mark breakdown is as follows:
– Awareness of context: 5 marks.
– Exploratory analysis: 10 marks. These marks are for (a) tackling the problem in a
sensible way that is justified by the context (b) carrying out analyses that are
designed to inform the subsequent modelling.
– Model-building: 10 marks. The marks are for (a) starting in a sensible place that is
justified from the exploratory analysis (b) appropriate use of model output and
diagnostics to identify potential areas for improvement (c) awareness of different
modelling options and their advantages and disadvantages (d) consideration of the
social, economic and demographic context during the model-building process.
– Quality of argument: 5 marks. The marks are for assembling a coherent ‘narrative’,
for example by drawing together the results of the exploratory analysis so as to
provide a clear starting point for model development, presenting the model-building
exercise in a structured and systematic way and, at each stage, linking the
development to what has gone before.
– Clarity and validity of conclusions: 5 marks. These marks are for stating clearly
what you have learned about the social, economic and demographic characteristics
that are related to the voting outcome in a ward, and for ensuring that this is
supported by your analysis and modelling.
– Graphs and / or tables: 5 marks. Graphs and / or tables need to be relevant, clear
and well presented (for example, with appropriate choices of symbols, line types,
captions, axis labels and so forth). There is a one-slide guide to ‘Using graphics
effectively’ in the Week 1 slides for the course. Note that you will only receive credit
for the graphs in your report if your submitted script / program generates and
automatically saves all of these graphs when it is run.
Word and page limits. You will be penalised if your report exceeds EITHER the specified
2500-word limit or the number of pages of graphs and / or tables. Following UCL
guidelines, the maximum penalty is 7 marks, and no penalty will be imposed that takes
the final mark below 30/75 if it was originally higher. Subject to these conditions,
penalties are as follows:
– More than two pages of graphs and / or tables: zero marks for graphs and / or tables,
in the marking scheme given above.
– Exceeding the word count by 10% or less: mark reduced by 4.
– Exceeding the word count by more than 10%: mark reduced by 7.
In the event of disagreement between reported word counts on different software
systems, the count used will be that from the examiner’s system. The examiners will use
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an R function called PDFcount to obtain the word count in your PDF report: this function
is available from the Moodle page in file PDFcount.r.
Coding: 15 marks. There are 3 marks here for reading the data and handling missing
values correctly; 7 marks for effective use of your chosen software (e.g. programming
efficiently and correctly); and 5 marks for clarity of your code — commenting, layout,
choice of variable / object names and so forth.
Prediction quality: 20 marks. The remaining 20 marks are for the quality of your
predictions. Note, however, that you will only receive credit for your predictions if
your submitted prediction file is the same as that produced by your submitted
script / program when it is run: if this is not the case, your predictions will earn
zero marks.
For these marks, you are competing against each other. Your predictions will be assessed
using the following score:
where:
– is the actual proportion of ‘Leave’ votes (which the examiners know) for the th
prediction;
– = () is your corresponding prediction;
– is your quoted standard deviation for the prediction error.
The score is an approximate version of a proper scoring rule, which is designed to
reward predictions that are close to the actual observation and are also accompanied by
an accurate assessment of uncertainty (this was discussed during the Week 10 lecture,
along with the rationale for using this score for the assessment). Low values are better.
The scores of all of the students in the class (and the lecturer) will be compared: students
with the lowest scores will receive all 20 marks, whereas those with the highest scores
will receive fewer marks. The precise allocation of marks will depend on the distribution
of scores in the class.
If you don’t supply standard deviations for your prediction errors, the value of ? will be
taken as 1/2 for all of your predictions: this is the largest possible standard deviation for
any random variable taking values between 0 and 1, and the value of will be
correspondingly large so that you will receive few if any marks for your predictions.
Hints on tackling the assessment
1. There is not a single ‘right’ answer to this assignment. There is a huge range of options
available to you, and many of them will be sensible.
2. You are being assessed not only on your computing skills, but also on your ability to
carry out an informed statistical analysis: material from other statistics courses (in
particular STAT0006, for students who have taken it) will be relevant here. To earn high
marks, you need to take a structured and critical approach to the analysis and to
demonstrate appropriate judgement in your choice of material to present.
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3. At first sight, the task will appear challenging to many of you. However, there is a lot that
we already know: Martin Rosenbaum’s article is an obvious starting point. You may also
want to search for other commentaries on the UK referendum result, to gain some
understanding of what kinds of relationships you might look for in the data.
4. When building your model, you have two main decisions to make. The first is: should it
be a linear, generalized linear or generalized additive model? The second is: which
covariates should you include? You might consider the following points:
– Linear, generalized linear or generalized additive? This is best broken down into
two further questions, as follows:
Conditional on the covariates, can the response variable be assumed to follow a
normal distribution with constant variance? In this assignment, the response
variable is a proportion and therefore cannot have exactly a normal distribution.
However, there are thousands of votes in each ward: the Central Limit Theorem
may apply, therefore, so that the response distribution has approximately a
normal distribution — in which case you may judge that the approximation is
adequate for your purposes.
The ‘constant variance’ assumption may also be suspect: given that the response
is a proportion, you might think that a binomial distribution would be appropriate,
but the variance of a binomial proportion is (1 ? )/ in an obvious notation.
Since this depends on , and varies between wards, the variance cannot be
constant. Whether this is a problem depends on how much the ‘Leave’ probability
varies: if it doesn’t vary much, then you may wish to claim that the variance is
approximately constant. If it varies a lot however, then you could probably
improve your predictions (and hence your score!) by accounting for it. You might
consider using your exploratory analysis to gain some preliminary insights into
this point.
Are the covariate effects best represented parametrically or nonparametrically?
Again, your exploratory analysis can be used to gain some preliminary insights
into this. You may want to look at the material from week 6, for examples of
situations where a nonparametric approach is needed.
– Which covariates? The data file contains many potential covariates, some of which
are more important than others. You have many choices here, and you will need to
take a structured approach to the problem in order to avoid running into difficulties.
The following are some potentially useful ideas:
Look at other published commentaries on the referendum result. What measures
are considered useful? Can these be linked to covariates for which you have
information? Obviously, if you do this then you will need to acknowledge your
sources in your report.
Define useful summary measures on contextual grounds, and work with these. For
example, 16 of the potential covariates in the data file are percentages of the
population in different age categories (0 to 4, 5 to 7, … , all the way up to ‘90
plus’). You may decide just to work with ‘young voters’ (18 to 29 — 18 is the
minimum voting age in the UK), ‘working age’ (30 to 64 say) and ‘retirement age’
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(65 and above). Or, indeed, to adopt your own categories — the results are unlikely
to be sensitive to the precise definitions. Similar comments apply to the potential
covariates representing ethnicity, household deprivation and so on.
Define new variables based on the correlations between the existing variables, and
work with these. If several continuous variables are highly correlated, then it is
difficult to disentangle their effects and it may be preferable to work with a single
‘index’ that combines all of them. This is the basis of techniques such as Principal
Components Analysis, that were discussed during the Week 10 lecture (along with
how to implement them in R and SAS).
You should not start to build any models until you have formed a fairly clear strategy
for how to proceed. Your decisions should be guided by your exploratory analysis, as
well as your understanding of the context.
5. Don’t forget to look for interactions! For example, one of the variables in the data set is
RegionName, which is a factor (i.e. categorical covariate) indicating the UK region in which
each ward is located. Possibly there is regional variation in the strength of dependence
between other characteristics and the proportion of ‘Leave’ votes. Look at the analysis of
the iris data from Workshop 2, for a similar kind of situation.
Sometimes people get confused about the difference between interactions and
collinearity. Reminder:
– An interaction describes the way in which covariates must be considered in
combination to characterise their relationship with the response variable.
– By contrast, collinearity is just about correlations between the covariates: this has no
reference to the response variable. Collinearity just makes it harder to identify which
covariates are genuinely associated with the response (recall the “sheep energy”
example from Week 9).
6. You probably won’t find a perfect model in which all the assumptions are satisfied:
models are just models. Moreover, you should not necessarily expect that your model
will have much predictive power: maybe the covariates in the data set just don’t provide
very much useful information. You should focus on finding the best model that you can,
therefore — and acknowledge any deficiencies in your discussion.
7. To obtain the standard deviations of your prediction errors, you need to do some
calculations. Specifically:
i. Suppose ? = ?() is your predicted probability of voting ‘Leave’ for the th ward,
and that is the corresponding actual value.
ii. Then your prediction error will be ? ? .
iii. and ? are independent, because ? is computed using only information from the
first 803 wards and relates to one of the ‘new’ wards.
iv. The variance of your prediction error is thus equal to Var() + Var(?).
v. You can calculate the standard error of ? in both R and SAS, when making predictions
for new observations — see Workshops 6 and 9. Squaring this standard error gives
you Var(?).
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vi. You can estimate Var() by plugging in the appropriate formula for your chosen
distribution — for example, if you’re using a binomial distribution then Var?() =
?(1 ? ?)/ , where is the number of votes for the th ward.
vii. Hence you can estimate the standard deviation of your prediction error as ? =
√Var?() + Var(?). In fact, for the case of linear models this is exactly the calculation
that is used in the construction of prediction intervals (see your STAT0006 notes or
equivalent).

Appendix: the ReferendumResults.csv data set
Data sources
The data provided in ReferendumResults.csv are from several different sources, as follows:
Martin Rosenbaum’s BBC article (data source MR1)
The article at http://www.bbc.co.uk/news/uk-politics-38762034 provides an Excel
spreadsheet, containing localised voting data supplied to the BBC by councils which
counted the EU referendum. Results are provided for all individual wards where data were
available at this level of detail: there are 1283 such wards, of a total of 9291 wards in the UK.
Reasons for the figures not being available at the remaining wards are:
Three councils did not respond to the BBC’s request.
Some councils refused to give the information to the BBC.
For some councils, ballot boxes were mixed before counting so it was not possible to
identify the precise numbers of votes in each ward.
Important caveat: in many wards, some postal votes were mixed in prior to counting. The
BBC spreadsheet states “Figures which include postal votes cannot be treated as exact.
However broad patterns can still be identified in the data.”
Martin Rosenbaum’s data set from the 2011 UK census (data source MR2)
The variables in this data set are those that form the basis for the analysis reported in
Martin Rosenbaum’s article. He provided the following information when supplying the data
to us:
‘All the 2011 census data was downloaded via selecting datasets at
https://www.nomisweb.co.uk/query/select/getdatasetbytheme.asp?opt=3&theme=&s
ubgrp=. (I calculated adult mean age from the raw counts of adults of each age in
each ward). Please note that some areas have seen boundary changes to wards since
2011, so some wards with referendum voting data do not figure in this list.’
This spreadsheet contains data for 8570 wards.
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The UK Register of Geographic Codes (data source RGC)
Geographical information for each ward was obtained from the UK ‘Register of Geographic
Codes’, downloaded on 14th March 2017 and located by searching at
http://geoportal.statistics.gov.uk/.
UK age structure by ward, from the 2011 UK census (data source ASW)
Martin Rosenbaum’s census data contains information on the mean adult age in each ward,
but it is possible that more detailed information on age profiles would be useful.
Percentages of population in different age bands were obtained for the 2011 census, from
https://www.nomisweb.co.uk/query/select/getdatasetbytheme.asp?collapse=yes under
Census 2011, Key Statistics and then Age Structure. This provides information on the same
8570 wards that are present in data source MR2.
Data processing
The data sources have been combined in the following way to create
ReferendumResults.csv:
1. The spreadsheets from sources MR1, MR2 and ASW were merged using the nine-digit
ward identification code (identified as WardCode in MR1). The Remain variable, giving the
number of ‘Remain’ votes in each ward, was replaced by an NVotes column giving the
total number of ‘Leave’ and ‘Remain’ votes. There are 1070 wards remaining after this
merge: this decrease from the original 1283 wards in MR1 is due to the exclusion of
wards for which the boundaries changed between the 2011 census and the 2016
referendum.
2. Source RGC was used to identify the administrative area type and region name for each
ward, again based on its nine-digit identification code. Some ward codes were found to
be duplicated in source RGC, but in all cases the administrative area type and region
name were identical for the duplicates.
3. The rows of the data table were randomly shuffled, so that the order of wards no longer
corresponds to that in any of the data sources. This was done in order to prevent
‘cheating’ when making predictions.
4. A subset of 267 wards was identified for the ‘prediction’ part of the assessment. This was
done in such a way that the distributions of all of the covariates in this subset is very
similar to the distribution in the remaining 803 wards. Specifically:
a. In each region, 25% of the wards were sampled at random as candidates for making
predictions. This sample will be referred to as ‘Group 2’ below, with ‘Group 1’
comprising the remaining wards.
b. For each of the numeric covariates in the data set, a Kolmogorov-Smirnov test was
performed to test the null hypothesis that the underlying distributions in Groups 1
and 2 are the same.
c. The prediction sample was accepted only if the -values for all of the Kolmogorov-
Smirnov tests were greater than 0.5 (this is not a typo). Otherwise, a new candidate
sample was drawn in step (a) and the procedure was repeated.
The Kolmogorov-Smirnov test is used here as a convenient way to measure whether two
distributions are similar: the use of a high -value threshold is chosen to ensure that the
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resulting Groups 1 and 2 are very well balanced with respect to all of the covariate
values. Note, however, that the voting numbers were not included in this balancing
exercise: this is because the performance of predictions would be artificially enhanced if
the voting numbers were included (for example, we would know that the distribution of
Group 2 voting proportions is similar to that of Group 1