MATH4/68052 Generalised Linear Models: Coursework
(2025)
The marks awarded for this coursework constitute 30% of the total assessment for the module.
Your solution to the coursework should be reasonably concise - about 10 pages with tables, plots, and code, but there is no penalty if you do exceed this. It should take, on average, about 9 hours to complete all the work, including preparing your final document to be submitted.
Please read all the instructions and advice given below carefully. The submission deadline is 10:00 am on Monday 10 March, 2025 .
Late Submission of Work: Any student’s work that is submitted after the given deadline will be classed as late, unless an extension has already been agreed via mitigating circumstances or a DASS extension.
The following rules for the application of penalties for late submission are quoted from the latest University policy on submission of work for summative assessment:
”The mark awarded will reduce by 10% of the maximum amount available per 24 hours (e.g. if the work is marked out of 100, this means a deduction of 10 marks per 24 hours late. If the work is marked out of 20, the deduction would be 2 marks each 24 hours late.) The penalty applies as soon as an assignment is late; a 10% deduction would be issued if an assignment is submitted immediately after the deadline, and the work would continue to attract further penalties for each subsequent 24 hours the work was late, until the assignment is submitted or no marks remain.”
Your submitted solutions should all be in one typed document. This should be prepared using LaTeX or R Markdown. For each question you should provide explanations as to how you com- pleted what is required, show your workings and also comment on computational results, where applicable.
When you include a plot, be sure to give it a title and label the axes correctly.
When you have written or used R code to answer any of the parts, then you should list this R code after the particular written answer to which it applies. This may be the R code for a function you have written and/or code you have used to produce numerical results, plots and tables. R code should also be clearly annotated.
Do not use screenshots of R code/output. Instead, to include R code, use the verbatim environ- ment.
Your file should be submitted through the module site on Blackboard to the Turnitin assessment under Assesment & Feedback entitled ’GLM’s Coursework (2025)’ by the above time and date. The work will be marked anonymously on Blackboard so please ensure that your filename is clear, but that it does not contain your name and student id number. Similarly, do not include your name and id number in the document itself.
Turnitin will generate a similarity report for your submitted document and indicate matches to other sources, including billions of internet documents (both live and archived), a subscription repository of periodicals, journals and publications, as well as submissions from other students. Please ensure that the document you upload represents your own work and is written in your own words. The Turnitin report will be available for you to see shortly after the due date.
This coursework should hopefully help to reinforce some of the methodology you have been study- ing, as well as the skills in R you have been developing in the module. Correct interpretation and meaningful discussion of the results (i.e. attempt to put the results into context) are important in order to achieve a high mark for the coursework.
1. Suppose that we have a random sample of data pairs (x1 , y1 ), . . . , (xn , yn ), where theyi’s are observed values of a response variable, Y , and the xi’s are associated observed values of a numerical covariate.
The hypothesised GLM for the data is:
yi jxi = µi + ∈i with Yi jxi ~ N (µi , σ 2 ) and E[∈i] = 0, i = 1, . . . , n
and
1/µi = ηi = β0 + β1 xi = xiT β, , i = 1, . . . , n
You are reminded that it is shown in Section 1.3 of the notes that the Normal distribution belongs to the exponential family of distributions (EFD). The natural parameter,θ, disper- sion parameter, φ, and the functions a(φ), b(θ), c(y, φ) appearing in the definition of the EFD pdf are all clear from there.
(i) Give expressions for the vector of partial derivatives ∂β/∂e and the normal equations which are solved to find the maximum likelihood estimates of the parameters in the above GLM. You should specifically define the weights, working responses, and any function of the dispersion parameter which appear in these equations.
What is the Fisher Information matrix? Give its particular form for the given GLM. [4]
(ii) Give the expression for the log-likelihood function of the parameters based on the random sample (x1 , y1 ), . . . , (xn , yn ). Find an expression for the estimated values of µi i = 1, ..., n for the saturated model, and hence
find an expression for the scaled deviance of the model with linear predictors ηi = β0 + β1 xi , i = 1, . . . , n. What is the approximate distribution of the scaled deviance here? [3]
A random sample of data such as described above with n = 300 is contained in the file cw-q1-dat . txt. As a preliminary to the next parts, read the data into R, print out the first six rows as a check, and then produce a scatter plot of the data. Please do do not include this plot in your report.
(iii) Fit the above GLM to the data using R. Produce a summary of the fitted model object commenting on all the results included. [4]
[To visualise your fitted regression model, superimpose the regression curve from your fitted model on to your scatter plot of the data as a check of the fit. However, you should not submit this plot as part of your report.]
(iv) Explaining the method you use, calculate an approximate 95% confidence interval for E[Yj x = 0.5].
[Note that the covariance matrix for the parameter estimates can be obtained by using the vcov() function on the fitted model object.] [4] [Total marks for Q1 = 15]
2. The data in this question are from a study in the Philippines looking at household sizes and which factors may be important in influencing them. The data is stored in the spreadsheet called PHH-data . csv which contains n = 1300 observations of households collected from five particular regions in the country. The values of the following variables are included:
total = the number of people in the household other than the head of the household.
(This is our response variable.)
location = where the house is located (Central Luzon, Davao Region, Ilocos Region, Metro Manila, or Visayas)
age = the age of the head of household
numLT5 = the number in the household under 5 years of age
roof = the type of roof in the household (either Predominantly Light/Salvaged Mate- rial, or Predominantly Strong Material, where stronger material can sometimes be used as a proxy for greater wealth). These are coded as ”PL/SM” and ”PSM”, respectively
Note that the column in the spreadsheet labelled household is just an index from 1 : 1300.
We will be modelling the data using Poisson GLMs with the canonical (log) link function and having the counts in total as the response variable for each one.
You can read the data into R and convert the variables location and roof to factor variables with the given reference levels using the following code.
phh<- read. csv(file="PHH-data. csv", header=TRUE)
names(phh) head(phh)
phh$location<- factor(phh$location) levels(phh$location)
phh$location<- relevel(phh$location, ref= "CentralLuzon")
phh$roof<- factor(phh$roof) levels(phh$roof)
phh$roof<- relevel(phh$roof, ref= "PL/SM")
(i) Fit a Poisson GLM with log link and a linear predictor including an intercept and the covariate age. ie. log(λi ) = β0 + β1 age, where λi is the mean number in household i = 1, ..., 1300 (other than the head) Comment on a summary of the fitted model object.
Produce an appropriate residual plot to check the fit of your model and include this in your report. (To help in the interpretation of this plot it can be helpful to add a local polynomial regression line which can be done using the function loess in R. Please see the extra provided notes if you are not familiar with this.) Comment, with reasons, on what your plot tells you about the fit of your model.
How can you interpret the value ofβ(ˆ)1 . the coefficient of age in your fitted model? [4]
(ii) Add age-squared to the linear predictor of the model in (ii) and fit this new model in R. Comment on a summary of the fitted model. As in part (ii), check the fit of your new model using an appropriate residual plot, which should again be included in your report. Comment on the results. [3]
(iii) Now add the variables location, numLT5, and roof to your linear predictor already containing age and age-squared. Summarise the fit of the model and comment on the results.
Show that the linear predictor can be reduced to that just containing age, age-squared, and numLT5. Use this fitted model to estimate the value of the dispersion parameter, φ . Comment on your estimated value of φ . [4]
(iv) Using your fitted model from part (iii) based on age, age-squared, and numLT5, cal- culate the age of the head when the estimated mean number in the household (other than the head) is a maximum, irrespective of the value of mumLT5..
Calculate an approximate 95% confidence interval for the maximum mean number in the household (other than the head) when numLT5 = 1. [4]
[Total marks for Q2 = 15]
[Total marks for the coursework = 30]
ILOs Tested:
ILO1: write down the fitted model, assess goodness-of-fit, test significance of parameters, compare models and use the chosen model to calculate various quantities of interest;
ILO2: write down a GLM with factors/covariates as appropriate, state the associated assumptions and constraints, derive the likelihood equation and algorithms for model fitting;
ILO6: use generalised linear models (GLMs), including logistic regression and log linear models with a Poisson response, to analyse data with dependence on one or more explanatory variables.