1.4: Design Theory

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Page ID: 70856

Joy Morris
University of Lethbridge

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Like graph theory, design theory is probably not what any non-mathematician might expect from its name.

When researchers conduct an experiment, errors can be introduced by many factors (including, for example, the timing or the subject of the experiment). It is therefore important to replicate the experiment a number of times, to ensure that these unintended variations do not account for the success of a particular treatment. If a number of different treatments are being tested, replicating all of them numerous times becomes costly and potentially infeasible. One way to reduce the total number of trials while still maintaining the accuracy, is to test multiple treatments on each subject, in different combinations.

One of the major early motivations for design theory was this context: given a fixed number of total treatments, and a fixed number of treatments we are willing to give to any subject, can we find combinations of the possible treatments so that each treatment is given to some fixed number of subjects, and any pair of treatments is given together some fixed number of times (often just once). This is the basic structure of a block design.

Example \(\PageIndex{1}\)

Suppose that we have seven different fertilisers and seven garden plots on which to try them. We can organise them so that each fertiliser is applied to three of the plots, each garden plot receives 3 fertilisers, and any pair of fertilisers is used together on precisely one of the plots. If the different fertilisers are numbered one through seven, then a method for arranging them is to place fertilisers 1, 2, and 3 on the first plot; 1, 4, and 5 on the second; 1, 6, and 7 on the third; 2, 4, and 6 on the fourth; 2, 5, and 7 on the fifth; 3, 4, and 7 on the sixth; and 3, 5, and 6 on the last. Thus,

\(123\) \(145\) \(167\)

\(246\) \(257\) \(347\)

\(356\)

is a design.

This basic idea can be generalised in many ways, and the study of structures like these is the basis of design theory. In this course, we will learn some facts about when designs exist, and how to construct them.

After studying design theory in this course, you should be able to solve problems such as:

Is it possible for a design to exist with a particular set of parameters?
What methods might we use in trying to construct a design?