# MT4614 Design of Experiments

## Course Description

This module introduces a wide range of features that occur in real comparative experiments.

The applications include

• trials of potential new medicines by the pharmaceutical industry;
• comparisons of new varieties of wheat for bread-making;
• evaluating different machine settings in industry.

Issues include

• whether and how to partition the experimental material into blocks (for example, do old and young people respond to this drug differently?);
• how much replication to use (too much experimental material may be a waste of resources, but too little will not give meaningful results);
• as well as type of design.

The module includes enough about the analysis of data from experiments to show what has to be considered at the design stage.

It also includes considerations of consultation with the scientist and interpretation of the results.

## Syllabus

1. Introduction to concepts in the design of real comparative experiments.

2. Randomization, replication, power.

3. Simple linear model, orthogonal subspaces, analysis of variance.

4. Blocking. Fixed effects or random effects. Orthogonal designs.

5. Factorial designs. Main effects and interactions. Control treatments.

6. Row-column designs. Latin squares.

7. Observational units smaller than experimental units. False replication.

8. Split-plot designs. Treatment effects in different strata.

9. Structures defined by families of orthogonal factors. Eigenspaces of highly structured variance-covariance matrices.

10. Showing factors on a Hasse diagram. Using the Hasse diagram to calculate degrees of freedom and allocate treatment effects to strata. Skeleton analysis of variance.

## Intended Learning Outcomes

On completion of the module, students should be able to do the following.

• Have a meaningful discussion with a non-mathematical scientist, including asking questions about the purpose of the experiment, the meaning of technical scientific words, any operational restrictions imposed by time or space or staffing or money, and later explain to the scientist the outcome of the data analysis in terms that (s)he can understand.

• Construct and appropriately randomize certain standard classes of design, including completely randomized designs, complete-block designs, orthogonal row-column designs and split-plot designs.

• Understand and state relevant definitions and theorems, and prove those theorems and also slight variants of those theorems.

• Understand and explain the concepts of main effects and interaction in factorial experiments, and their extension to factorial experiments with an extra control treatment.

• Write and understand the appropriate linear model, interpret this as a collection of expectation subspaces and a variance-covariance matrix with known eigenspaces, use these to sketch out how to do the data analysis by hand and then do it using the appropriate code in R, and verify that the output from R matches the format that is expected.

• Understand and use a Hasse diagram to show the relationships between factors, including finding out which stratum each treatment subspace is in and identifying any pseudoreplication.