## Design and Analysis of Experiments

 Unit code: MATH68082 Credit Rating: 15 Unit level: Level 6 Teaching period(s): Semester 2 Offered by School of Mathematics Available as a free choice unit?: N

#### Requisites

None

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.

#### Aims

To introduce the student to the principles and methods of statistical analysis of designed experiments.

#### Overview

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.

#### Learning outcomes

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

• construct standard experimental designs and describe what statistical models can be estimated using the data;
• given the description of how a set of data were collected, identify what design was followed and its features, describe what assumptions are appropriate in modelling the data and perform the appropriate statistical analysis;
• compare experimental designs with respect to the D- and G-optimality criteria and state the General Equivalence Theorem;
• construct experimental designs for estimating nonstandard linear and nonlinear models, using an algorithmic approach, when needed, using local and pseudo-Bayesian criteria of optimality.

#### Assessment methods

• Other - 7%
• Written exam - 93%

#### Assessment Further Information

• Coursework: weighting 7%
• End of semester examination: Three hours, weighting 93%

#### Syllabus

1. Basic concepts; Definitions. Treatment, factors, plots, blocks, precision, efficiency, replication, randomisation and design. [2]
2. Completely randomised design. Fixed and random effects, contrasts, ANOVA table. [4]
3. Factorial designs. General factorial experiment; fixed and random effects; interactions. [3]
4. Nested designs. [2]
5. Blocking. Orthogonal designs: Randomised complete block designs; Latin square designs; extensions of the Latin square design. Non-orthogonal designs: Balanced incomplete block designs. [6]
6. 2m Factorial experiments; Confounding; fractional replication; aliasing. [4]
7. Response surface designs [1]
8. Criteria for design optimality [3]
9. The General Equivalence Theorem and its applications; construction of D-optimal experimental designs. [5]
10. Designs for nonlinear models. [3]

• A. C. Atkinson, A. N. Donev, R. D. Tobias (2007). Optimum Experimental Designs, With SAS. Oxford University Press.
• G. Cassela (2008). Statistical Design. Springer.
• Lawson, J. (2015). Design and Analysis of Experiments with R. CHapman and Hall/CRC.
• D. C. Montgomery (1997). Design and Analysis in the Design of Experiments, (eight edition). Wiley.

#### Feedback methods

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.

#### Study hours

• Lectures - 33 hours
• Tutorials - 11 hours
• Independent study hours - 106 hours

#### Teaching staff

Alexander Donev - Unit coordinator