MTH6134 - Statistical Modelling II - 2023/24
Topic outline
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This assessment is intended to be completed within 3 hours. Although you are expected to finish the assessment within this time frame, you will be given an additional 30 minutes for scanning and submitting your handwritten solution in case you have not already done so.
IMPORTANT – If you run out of time during this assessment, we will not accept IT issues as an extenuating circumstance. This is because we have given you ample time to complete and submit your work. If your attempt is still in progress at the end of the set time slot, any file you have uploaded will be automatically submitted.
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Overview: This week is the initial set up of the course. We do a gentle introduction to the Module, including looking at the SEATTLE Heart Failure Model and the MAGGIC model as motivating examples. We then looked at specific distributions and link functions.
Lectures
Pages 1-2 of the typed notes.Tutorial
No tutorial.Exercises
No exercises.
Extras
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Overview: This weeks material concludes the overview of distributions and link functions to then start with the topic of likelihood. We study likelihood using the material of multivariate linear regression as guiding theme.
Lectures
Pages 2-5.Tutorial
This tutorial (R lab) is a refresher on linear regression.Exercises
Try Exercises 1,2, 8 (parts 1-3) and 4,5.Extras
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Overview: The study of linear regression continues. We look at it from the point of view of likelihood.
Lectures
Pages 6-9.Tutorial
This (R) lab is an exercise on multiple regression.Exercises
Try exercises 13 (regression through the origin), 18 (binomial) and 19 (Poisson).Extras
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Overview: Having completed our review of regression via likelihood, we look at tests based upon likelihood ratios. Time permitting, we will start looking at the family of exponential distributions.
Lectures
We look at likelihood ratio tests and at Wilk's theorem. Students are recommended to look in detail at Example 2.8.Tutorial
The lab has a first look at regression of binomial data, exploring different link functions.Exercises
We will look at Exercises 21, 23 and 25.Extras
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Overview: This week we concentrate on the core topic of the Module as well as distributions of the exponential family.
Lectures
We start the core topic of the Module which are Generalized Linear Models (GLMs).Exercises
We have a first look at exponential distributions, doing exercise 28 and 29 for binomial and negative binomial distributions.Extras
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Overview: This week we continue our study of GLMs, with view on estimation.
Lectures
The material continues with Pages 13-15.Tutorial
The tutorial has a look at empirical link-transformation of the response as a vehicle for checking the fit of model to data.Exercises
The emphasis is in computation of Fisher information matrix, exercises 31 and 34.Extras
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This week we have no lectures and the midterm test on Thursday 091123 between 1000-1100. You can read the information that has been posted in the announcements section, reachable from this link https://qmplus.qmul.ac.uk/mod/forum/discuss.php?d=500762
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Here is a sample 'quiz', based upon some exercises. Read the following very carefully:
We have practiced this material for quite some weeks now. If you have not put an equivalent time doing the usual Module activities (attending lectures, doing the exercises and the tutorial/labs), then you have quite a bit of ground to catch up before you are in a suitable condition for a decent result in the midterm. None of the course material is hard, but you need to be familiar with it.The 'mock exam' has (for now) no solutions, which I'll put tomorrow. -
are here. Of course, the true proof of knowing whether you know the topic is that you mark your own attempt.
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Overview: This week we complete the development of the topic of GLM with a second take on estimation and residuals.
Lectures
Pages 14-18 of notes.Tutorial
The lab is concentrated on a Poisson regression for the cloth data set.Exercises
We look at the GLM against maximal models, this is exercise 38.Extras
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Overview: With a change of timetable to Wednesday, this week we continue our examination of binary response data.
Lectures
We look at pages 19-22.Tutorial
We will analyze the data in file rat.csv.Exercises
We look at residuals in exercise 39 and modelling in exercises 40 and 43.Extras
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Overview: This week we conclude GLMs for binary data and start covering count data regression.
Lectures
We are covering pages 21-23 of the lecture notes.Tutorial
The tutorial of this week involved a Poisson model with two means, which was contrasted (compared) with the Poisson model with a single mean, that is, the null model.Exercises
Exercise #43 is concerned with Poisson regression and a two factor model.Extras
As an extra for exercise #43, what if you analize considering (incorrectly) numerical values for the two factors of interest? -
Overview: This week we continue the topic of contingency tables. Time permitting, we will look at exponential regression.
Lectures
We look at pages 24-27 of the notes.Tutorial
The tutorial continues with analysis of contingency tables in R.Extras
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The module builds on theory already taught in Statistical Modelling I and develops the general theory of linear models. You will be able to fully undertand the concept of the generalised linear models, which can be used in problems where a normal distribution is not appropriate, such as when working with binary or count data. In the lab sessions, you will be able to learn how to apply these modelling techniques in real-world problems by programming in R.
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At the end of this module, students should be able to:
- Derive the likelihood, the likelihood equations and the Fisher information matrix, for a given regression model.
- Carry out likelihood ratio tests.
- Identify probability distributions belonging to an exponential family and adapt a description as a generalised linear model.
- Describe numerical procedures for estimation in generalised linear models.
- Explain the important theorems in probability theory utilised in test procedures in generalised linear models.
- Analyse data sets following binomial or Poisson distributions.
- Estimate parameters and test hypotheses in generalised linear models by means of R.
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This Module is to be evaluated with two in term tests (weeks 7 and 12), each test worth 10% of the Module total. There will be a final exam (in January) which will count for 80% of the Module mark.
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This is the basic Module material, introducing to Generalized Linear Models (GLMs). There may be some (minor) amendments to the typed notes during the term.
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We provide a pdf document with exercises for the Module for you to practice and discuss in lectures. This file may be amended during the term so always look at the qmplus version.
We'll also provide here some selected solutions to exercises and labs. Note that a) you are still required to work on exercises before looking at the contents of this file, b) some solutions are still left for you as exercises and importantly c) the files is to be updated as weeks go along, so always come back here for a fresh copy. The file will be your only source of exercises and solutions, look no further!
We also store here the R lab (tutorial) material used every week.
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In this space there will be some exam samples. Be aware that any past exam material made available implies *no* promise nor indication that future exam material will be identical.
Note that MTH6134 exams in repositories with date prior to 2019 (inclusive of that year) were for a different syllabus. (If you look at the material of those exams closely, all are gaussian glms with categorical variables ... but in any case the material is relatively specialized to that model and therefore, not relevant to the glm syllabus).
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... with selected solutions.
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more in the style of the qmplus quiz. This includes solutions and it is very similar to the other sample provided here 2021-2022.
(This is all I have for now, I have no more sample exams)
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Overview: This week we conclude the Module looking at Survival data
Lectures
The lectures will cover part 6 of the notes.Tutorial
The tutorial looked at leukaemia data with exponential regression.Exercises
We looked at constrained estimation, concluding the material on contingency tables.Extras
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Here I give a sample. Recall that I make no promise that the test will be similar to this sample in terms of specific questions. It is only similar in the level of difficulty and topics considered.
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