Statistics

The three month dissertation period (mid-June to mid-September) will involve the application of statistical methodology to a substantive problem. This dissertation is written by the student under the direction of a supervisor. Many projects are collaborative: recent collaborations include GlaxoSmithKline; AstraZeneca; Wrightington Hospital; Royal Lancaster Infirmary; Leahurst Veterinary Centre; and the Department of the Environment.

Students will gain a thorough understanding of advanced statistical methods which go beyond the scope of MSc taught components, and will learn about the development of original statistical methodology which will contribute to a fuller understanding of existing methodology. Students are required to make innovative use of the statistical method, leading to substantive findings which would not readily be obtainable by routine application of standard techniques.

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This module will only be completed by students who, depending on their mathematical background, require an introduction, at graduate level, to two core areas which are essential building blocks to further advanced study of statistical modelling, methodology and theory. Students will study either this module or the other core module titled ‘Statistical Foundations I' but not both.

The areas that will be covered are statistical inference using maximum likelihood and generalised linear models (GLMs). Building on an undergraduate level understanding of mathematics, statistics (hypothesis testing and linear regression) and probability (univariate discrete and continuous distributions; expectations, variances and covariances; the multivariate normal distribution), this module will motivate the need for a generic method for model fitting and then demonstrate how maximum likelihood provides a solution to this. Following on from this, GLMs, a widely and routinely used family of statistical models, will be introduced as an extension of the linear regression model.

This module will develop the core topic of maximum likelihood inference previously introduced in MATH501 Statistical Fundamentals I by expanding on numerical and theoretical aspects. Numerical aspects will include obtaining the maximum likelihood estimate using numerical optimisation functions in R, and using the profile likelihood function to obtain both the maximum likelihood estimate and confidence intervals. Theoretical elements covered will include derivation of asymptotic distributions for the maximum likelihood estimator, deviance and profile deviance.

The second half of the module will introduce Bayesian inference as an alternative to maximum likelihood inference. Building on existing knowledge of the likelihood function, the prior and posterior distributions will be introduced. For simple models, analytical forms for the posterior distribution will be introduced and point estimates for the parameter obtained. For more complex models, numerical methods of sampling from the posterior distribution will be demonstrated.

This module provides an introduction to statistical learning. General topics covered include big data, missing data, biased samples and recency. Likelihood and cross-validation will be introduced as generic methods to fit and select statistical learning models. Cross-validation will require an understanding of sample splitting into calibration, training and validation samples. The focus will then move to handling regression problems for large data sets via variable reduction methods such as the Lasso and Elastic Net. A variety of classification methods will be covered including logistic and multinomial logistic models, regression trees, random forests and bagging and boosting. Examination of classification methods will culminate in neural networks which will be presented as generalised linear modelling extensions. Unsupervised learning for big data is then covered including K-means, PAM and CLARA, followed by mixture models and latent class analysis.

The aim of this module is to provide students with a range of skills that are necessary for applied statistical work including team-working, oral presentation, statistical computing, and the preparation of written reports including statistical analyses. All students will obtain a thorough grasp of R (including R objects and functions, graphs, basic simulations and programming) and be given an introduction to a second statistical computing package.

Students will also learn how to utilise LaTex for writing a complex and structured scientific report that may include mathematical formulae, tables and figures, as well as learn the intricacies of effective scientific writing style such as grammar, referencing, and the presentation of results in appropriate tables and graphs. They will enhance their oral presentation technique using LaTex Beamer to create slides that include complex mathematical formulae, as well as embark on an in-depth team project using Git, R Markdown or iPython notebooks.

Clinical trials are planned experiments on human beings designed to assess the relative benefits of one or more forms of treatment. For instance, we might be interested in studying whether aspirin reduces the incidence of pregnancy-induced hypertension, or we may wish to assess whether a new immunosuppressive drug improves the survival rate of transplant recipients.

This module combines the study of technical methodology with discussion of more general research issues, beginning with a discussion of the relative advantages and disadvantages of different types of medical studies. The module will provide a definition and estimation of treatment effects. Furthermore, cross-over trials, issues of sample size determination, and equivalence trials are covered. There is an introduction to flexible trial designs that allow a sample size re-estimation during the ongoing trial. Finally, other relevant topics such as meta-analysis and accommodating confounding at the design stage are briefly discussed.

Students will gain knowledge of the basic elements of clinical trials. They will develop the ability to recognise and use principles of good study design, and will also be able to analyse and interpret study results to make correct scientific inferences.

This module introduces the expectation-maximisation algorithm, an iterative algorithm for obtaining the maximum likelihood estimate of parameters in problems with intractable likelihoods. Students will explore the use of Markov chain Monte Carlo (MCMC) methods, and will discover the features of the Metro-Hastings algorithm, with emphasis on the Gibbs sampler, independence sampler and random walk Metropolis. Whilst relating to this, students will consider how such methods are closely integrated with Bayesian modelling techniques such as hierarchal modelling, random effects and mixture modelling.

Data augmentation will receive recurring coverage over the course of the module. Students will also gain transferrable knowledge of the usefulness of computers in assisting statistical analysis of complex methods, in addition to experience with the computer statistical package R.

Extreme Value Theory is an area of probability theory which describes the stochastic behaviour of events occurring in the tail of a distribution (eg. block maxima). This course will cover both an overview of key theoretical results and the statistical modelling approaches which are motivated by these results. Theoretical results covered will include limiting distributions for block maxima and Peaks Over Threshold events in the case of both independent and time-series data. Modelling will involve the development of extreme value statistical models and their application to data sets taken from financial and environmental applications. The concepts of risk will be explored, leading to an understanding of return levels and Value At Risk measures. The concept of extremal dependence will be introduced.

Almost every set of data, whether it consists of field observations, data from laboratory experiments, clinical trial outcomes, or information from population surveys or longitudinal studies, has an element of missing data. For example, participants in a survey or clinical trial may drop-out of the study, measurement instruments may fail, or human error invalidate instrumental readings. Missingness may or may not be related to the information being collected; for instance, drop out may occur because a patient dislikes the side-effects of an experimental treatment or because they move out of the area or because they find that they no longer have the time to attend follow up appointments. In this module you will learn about the different ways in which missing data can arise, and how these can be handled to mitigate the impact of the missingness on the data analysis. Topics covered include single imputation methods, Bayesian imputation, multiple imputation (Rubin's rules, chained equations and multivariate methods, as well as suitable diagnostics) and modelling dropout in longitudinal modelling.

Hierarchical data arise in a multitude of settings, specifically whenever a sample is grouped (or clustered) according to one or more factors with each factor having many levels. For instance, school pupils may be grouped by teacher, school and local education authority. There is a hierarchical structure to this grouping since schools are grouped within local education authority and teachers are grouped within schools. If multiple measurements of a response variable, say test score, are made for each pupil across multiple measurement times, the data are also longitudinal. This module motivates the need for statistical methodology to account for these kinds of hierarchical structure. The differences between marginal and conditional models, and the advantages and disadvantages of each, will be discussed. Linear mixed effects models (LMMs) for general multi-level data will be introduced as an extension to the linear regression model. Longitudinal data will be introduced as a special case of hierarchical data motivating the need for temporal dependence structures to be incorporated within LMMs. Finally, the drawbacks of LMMs will be used to motivate generalised linear mixed effects models (GLMMs), with the former a special case of the latter. GLMMs broaden the scope of data sets which can be analysed using mixed-effects models to incorporate all common types of response variable. All modelling will be carried out using the statistical software package R.

Introducing epidemiology, the study of the distribution and determents of disease in human population, this module presents its main principles and statistical methods. The module addresses the fundamental measures of disease, such as incidence, prevalence, risk and rates, including indices of morbidity and mortality.

Students will also develop awareness in epidemiologic study design, such as ecological studies, surveys, and cohort and case-control studies, in addition to diagnostic test studies. Epidemiological concepts will be addressed, such as bias and confounding, matching and stratification, and the module will also address calculation of rates, standardisation and adjustment, as well as issues in screening.

This module provides students with a historical and general overview of epidemiology and related strategies for study design, and should enable students to conduct appropriate methods of analysis for rates and risk of disease. Students will develop skills in critical appraisal of the literature and, in completing this module, will have developed an appreciation for epidemiology and an ability to describe the key statistical issues in the design of ecological studies, surveys, case-control studies, cohort studies and RCT, whilst recognising their advantages and disadvantages.

This module addresses a range of topics relating to survival data; censoring, hazard functions, Kaplan-Meier plots, parametric models and likelihood construction will be discussed in detail. Students will engage with the Cox proportional hazard model, partial likelihood, Nelson-Aalen estimation and survival time prediction and will also focus on counting processes, diagnostic methods, and frailty models and effects.

The module provides an understanding of the unique features and statistical challenges surrounding the analysis of survival avant history data, in addition to an understanding of how non-parametric methods can aid in the identification of modelling strategies for time-to-event data, and recognition of the range and scope of survival techniques that can be implemented within standard statistical software.

General skills will be developed, including the ability to express scientific problems in a mathematical language, improvement of scientific writing skills, and an enhanced range of computing skills related to the manipulation on analysis of data.

On successful completion of this module, students will be able to apply a range of appropriate statistical techniques to survival and event history data using statistical software, to accurately interpret the output of statistical analyses using survival models, fitted using standard software, and the ability to construct and manipulate likelihood functions from parametric models for censored data. Students will also gain observation skills, such as the ability to identify when particular models are appropriate, through the application of diagnostic checks and model building strategies.

The course is designed to provide foundational knowledge in linear and non-linear time-series analysis through building awareness of various well used time-series models. While the module focuses on univariate analysis, students will have time to read around lecture notes and materials to extend their understanding of these methods. By the end of the course, the student should understand both the theoretical and practical foundations of time-series analysis, how to fit, and choose from a range of models. They will understand different methods of evaluating time-series model performance and how these models can be used to provide forecasts.