MMath Mathematics

 Longitudinal and spatial data analysis are essential tools in modern research, particularly in fields such as public health, epidemiology, environmental science, and the social sciences. As the availability of such data rapidly increases, statisticians play a vital role in developing sophisticated models to address complex scientific questions and provide meaningful insights.

The primary objective of longitudinal data analysis is to investigate how a response variable changes over time in relation to explanatory variables while accounting for within-subject correlations. This involves identifying and quantifying random variations from different sources and characterising the within-subject correlation structure. Longitudinal data is essential for studying individual trajectories, temporal trends, and treatment effects over time, making it highly relevant in clinical trials and behavioural research.

Spatial data analysis, on the other hand, focuses on data observed across different geographical locations. The goal is to model and understand spatial dependence, identify spatial patterns, estimate the effects of exposures or interventions across regions, and make predictions for unobserved locations. This is essential in fields like environmental monitoring, disease mapping, and urban planning, where spatial variability plays a key role in decision-making.

Both longitudinal and spatial data analysis deal with correlated data, where observations are not independent, which distinguishes these methods from standard statistical techniques. This course provides students with the necessary skills to analyse and model such data, enabling them to apply these techniques to real-world problems. 

Students are not permitted to take, for credit, MATH48132 in an undergraduate programme and then MATH68132 in a postgraduate programme at the University of Manchester, as the courses are identical.

This unit aims to:

Introduce students to advanced statistical methods for analysing correlated data, with a particular focus on models for both longitudinal and spatial datasets. The course will cover key theoretical concepts and practical applications, equipping students with the skills to handle complex data structures that arise when observations are collected over time (longitudinal) or across geographical or spatial locations.  

On successful completion of this course unit students will have a good understanding of: 

• apply advanced statistical models, including general linear models with correlated random errors, linear mixed models, generalise linear mixed models and generalised estimating equations, to analyse longitudinal data;
• describe the theory behind the parameter estimation and model selection criteria for these models;
• distinguish between various types of spatial data and choose appropriate statistical methods for their analysis;
• identify and model spatial autocorrelation in spatial data to capture spatial dependence;  
• implement the statistical methods in statistical software R for real-data analysis.

1. Introduction to Longitudinal data

2. Linear mixed models: Fixed effects, random effects,  various covariance models variance components,  restricted maximum likelihood estimation, prediction of random effects

3. Non-normal longitudinal data models

4. Introduction to spatial data: types of spatial data, motivating examples from environmental epidemiology and social sciences. Geostatistical theory and modelling: stationarity and isotropy, Gaussian processes, covariance function, Variograms, Model adequacy and spatial prediction.  

5. Area-level data

6. Point process data 

Teaching is composed of two hours of lectures and one tutorial class per week. One week is reserved for coursework. Teaching materials will be made available online for reference and review. 

• Coursework 30%.
• End of semester examination: weighting 70%

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

• Davis, C. S. (2002). Statistical methods for the analysis of repeated measurements. Springer, New York
• Diggle, P. J., Heagerty, P., Liang, K Y. and Zeger, S. L. (1994). Analysis of longitudinal data. 2nd Edition. Oxford University Press
• Fitzmaurice, G. M., Laird, N. M., and Ware, J. H. (2004). Applied longitudinal analysis. New York, Wiley.
• Gelfand, A. E., Diggle, P., Guttorp, P., & Fuentes, M. (2010). Handbook of spatial statistics. CRC press.
• Moraga, P. (2019). Geospatial health data: Modeling and visualization with R-INLA and shiny. Chapman and Hall/CRC.
• Diggle, P. J., & Ribeiro, P. J. (2007). An overview of model-based geostatistics. Model-based Geostatistics, 27-45.
• Baddeley, A., Rubak, E., & Turner, R. (2015). Spatial point patterns: methodology and applications with R. CRC press.

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

· You will normally have approximately 75-120 minutes of video content. Normally you would spend approximately 2.5-4 hrs per week studying this content independently
· You will normally have exercise or problem sheets, on which you might spend approximately 2-2.5hrs per week
· There may be other tasks assigned to you on Blackboard, for example short quizzes, short-answer formative exercises or directed reading
· In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 9 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.

The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.