Design of Experiments

Abstract

This chapter presents the framework for the design and analysis of experiments. First, the general principles of design, including confounding, signal-to-noise ratio, randomisation, and blocking, are considered. Next, the commonly encountered factorial and fractional factorial designs are analysed in detail. Both analysis and design of such experiments, including the topics of model determination, replicates, confounding patterns, and resolution, are explored. Appropriate methods, including the development of orthogonal and orthonormal bases, for the analysis of such experiments using computers are presented. Although the results focus on 2-factorial design, higher-order design experiments are also considered, and the procedure for their analysis is explained. Detailed examples and cases are given. Third, methods for analysis of curvature, or quadratic terms, in a model are examined using factorial design with centre point replicates. Finally, the idea behind response surface methodologies, such as central composite design and optimal design, is briefly explored. Examples drawn from a wide range of different examples are considered. By the end of this chapter, the reader should be able to design and analyse factorial and fractional factorial experiments and curvature experiments and perform basic response surface methodologies using appropriate computational assistance.

Appendix A4: Nonmatrix Approach to the Analysis of 2k-Factorial Design Experiments

It will be assumed that a 2^k-factorial experiment has been designed with n _R full replicates. Furthermore, it will be assumed that all the factors have been coded so that −1 and 1 represent the upper and lower levels in the experiment. The same notation as presented in Chap. 4 will be used. Thus, instead of calculating inverses and transposes, the following simplifications work for a 2^k-factorial experiment:

$$ {\mathcal{A}}^T\mathcal{A}={2}^k{\mathcal{I}}_k, $$

(4.104)

where $\mathcal{I}$ _k is the k × k identity matrix,

$$ {\left({\mathcal{A}}^T\mathcal{A}\right)}^{-1}={2}^{-k}{\mathcal{I}}_k $$

(4.105)

$$ \widehat{\overrightarrow{\beta}}={2}^{-k}{\mathcal{A}}^T\overrightarrow{y} $$

(4.106)

If $\overline{\mathcal{A}}$ is used, then the results are

$$ {\overline{\mathcal{A}}}^T\overline{\mathcal{A}}={2}^k{n}_R{\mathcal{I}}_k $$

(4.107)

$$ {\left({\overline{\mathcal{A}}}^T\overline{\mathcal{A}}\right)}^{-1}={2}^{-k}{\left({n}_R\right)}^{-1}{\mathcal{I}}_k $$

(4.108)

The sum of squares due to errors, SSE, can be computed using the following formula:

$$ SSE=\left({n}_R-1\right){\displaystyle \sum_{i=1}^{2^k}{s}_i^2}, $$

(4.109)

where s _i is the standard deviation for the replicates for treatment i. Thus the standard deviation, $\widehat{\sigma}$, can be determined as

$$ \widehat{\sigma}=\sqrt{\frac{SSE}{l^k\left({n}_R-1\right)}}=\sqrt{\frac{{\displaystyle \sum_{i=1}^{2^k}{s}_i^2}}{l^k}} $$

(4.110)

The effect due to each variable can be determined from

$$ \mathrm{Effect}=2\widehat{\beta} $$

(4.111)