MMath Mathematics

Experiments are carried out by researchers in many fields including biology, medicine, chemistry, physics, engineering and agriculture. In such experiments the results are affected both by the choice of factors to study and experimental error (such as measurement error or inherent randomness between experimental units). Choosing a good experimental design ensures that the aim of the study where it is used is achieved. Moreover, the statistical analysis of data collected from such designed experiments is simple, easier to interpret and the experimental resources are spent most efficiently. The main principles for designing and analyzing experiments will be introduced. Various standard experimental designs and the analysis of data obtained using them are covered. Criteria for optimality of experimental designs will be introduced. Methods for constructing nonstandard designs when the model is linear or nonlinear in the parameters will be presented.

This course is suitable for any Year 3 student with average Year 2 marks of 55%.

Students are not permitted to take, for credit, MATH48082 in an undergraduate programme and then MATH68082 in a postgraduate programme at the University of Manchester, as the courses are identical.

To introduce statistical principles and methods for the design of experiments and analysis of the resulting data.  

On successful completion of this course unit students will be able to: 

• Construct both standard and non-standard experimental designs
• Analyse data from designed experiments using a model-based approach, both by hand and using R
• Assess the statistical properties of a proposed design, such as which model(s) can be estimated and/or its performance in terms of D- or G-optimality, and compare these properties with those of other potential designs
• Explain key ideas and assumptions underpinning the statistical design and analysis of experiments and their relationship to specific examples
• Prove results in the statistical design and analysis of experiments using relevant theoretical techniques, for example the General Equivalence Theorem

1. Basic concepts: treatments and experimental units, factors, randomization.
2. Completely randomized design.
3. Block designs: randomized complete block design, balanced incomplete block design.
4. Factorial experiments: two-way layout, two-level full factorials and fractional factorials. Confounding with blocks. Construction and analysis.
5. Response surface methodology.
6. Random effects. Consequences for ANOVA tests, estimation of variance components. Nested designs.  
7. Optimal design for linear models: D-optimality and G-optimality, algorithmic approaches, General Equivalence Theorem.
8. Nonlinear models and optimal design for nonlinear models: nonlinear least squares, locally D-optimal design, Bayesian designs. 

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.

• Take-home Coursework: weighting 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.

• A. C. Atkinson, A. N. Donev, R. D. Tobias (2007). Optimum Experimental Designs, With SAS. Oxford University Press.
• G. Casella (2008). Statistical design. Springer.
• A. Dean, D. Voss, D. Draguljić (2017). Design and analysis of experiments, 2nd edn, Springer.
• P. Goos and B. Jones (2011). Optimal design of experiments: a case study approach, Wiley.  
• J. Lawson (2015). Design and Analysis of Experiments with R. Chapman and Hall/CRC.
• D. C. Montgomery (2019). Design and Analysis of Experiments, 10th edn. Wiley.
• G. Oehlert (2010) A first course in Design and Analysis of Experiments. http://users.stat.umn.edu/~gary/Book.html
• N. Taback (2022). Design and Analysis of Experiments and Observational Studies using R, Chapman and Hall/CRC. https://designexptr.org
• C.-F. J. Wu and M. S. Hamada (2021). Experiments: planning and optimization, Wiley.

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.