The concise yet authoritative presentation of key techniques for
basic mixtures experiments



Inspired by the author's bestselling advanced book on the topic,
A Primer on Experiments with Mixtures provides an introductory
presentation of the key principles behind experimenting with
mixtures. Outlining useful techniques through an applied approach
with examples from real research situations, the book supplies a
comprehensive discussion of how to design and set up basic mixture
experiments, then analyze the data and draw inferences from
results.

Drawing from his extensive experience teaching the topic at
various levels, the author presents the mixture experiments in an
easy-to-follow manner that is void of unnecessary formulas and
theory. Succinct presentations explore key methods and techniques
for carrying out basic mixture experiments, including:

* Designs and models for exploring the entire simplex factor
space, with coverage of simplex-lattice and simplex-centroid
designs, canonical polynomials, the plotting of individual
residuals, and axial designs

* Multiple constraints on the component proportions in the form of
lower and/or upper bounds, introducing L-Pseudocomponents,
multicomponent constraints, and multiple lattice designs for major
and minor component classifications

* Techniques for analyzing mixture data such as model reduction
and screening components, as well as additional topics such as
measuring the leverage of certain design points

* Models containing ratios of the components, Cox's mixture
polynomials, and the fitting of a slack variable model

* A review of least squares and the analysis of variance for
fitting data

Each chapter concludes with a summary and appendices with
details on the technical aspects of the material. Throughout the
book, exercise sets with selected answers allow readers to test
their comprehension of the material, and References and Recommended
Reading sections outline further resources for study of the
presented topics.

A Primer on Experiments with Mixtures is an excellent
book for one-semester courses on mixture designs and can also serve
as a supplement for design of experiments courses at the
upper-undergraduate and graduate levels. It is also a suitable
reference for practitioners and researchers who have an interest in
experiments with mixtures and would like to learn more about the
related mixture designs and models.

Inspired by the author's bestselling advanced book on the topic, A Primer on Experiments with Mixtures provides an introductory presentation of the key principles behind experimenting with mixtures. Outlining useful techniques through an applied approach with examples from real research situations, the book supplies a comprehensive discussion of how to design and set up basic mixture experiments, then analyze the data and draw inferences from results.

Drawing from his extensive experience teaching the topic at various levels, the author presents the mixture experiments in an easy-to-follow manner that is void of unnecessary formulas and theory. Succinct presentations explore key methods and techniques for carrying out basic mixture experiments, including:

• Designs and models for exploring the entire simplex factor space, with coverage of simplex-lattice and simplex-centroid designs, canonical polynomials, the plotting of individual residuals, and axial designs

• Multiple constraints on the component proportions in the form of lower and/or upper bounds, introducing L-Pseudocomponents, multicomponent constraints, and multiple lattice designs for major and minor component classifications

• Techniques for analyzing mixture data such as model reduction and screening components, as well as additional topics such as measuring the leverage of certain design points

• Models containing ratios of the components, Cox's mixture polynomials, and the fitting of a slack variable model

• A review of least squares and the analysis of variance for fitting data

Each chapter concludes with a summary and appendices with details on the technical aspects of the material. Throughout the book, exercise sets with selected answers allow readers to test their comprehension of the material, and References and Recommended Reading sections outline further resources for study of the presented topics.

A Primer on Experiments with Mixtures is an excellent book for one-semester courses on mixture designs and can also serve as a supplement for design of experiments courses at the upper-undergraduate and graduate levels. It is also a suitable reference for practitioners and researchers who have an interest in experiments with mixtures and would like to learn more about the related mixture designs and models.

Preface ix

1. Introduction 1

1.1 The Original Mixture Problem 2

1.2 A Pesticide Example Involving Two Chemicals 2

1.3 General Remarks About Response Surface Methods 9

1.4 An Historical Perspective 13

References and Recommended Reading 17

Questions 17

Appendix 1A: Testing for Nonlinear Blending of the Two Chemicals Vendex and Kelthane While Measuring the Average Percent Mortality (APM) of Mites 20

2. The Original Mixture Problem: Designs and Models for Exploring the Entire Simplex Factor Space 23

2.1 The Simplex-Lattice Designs 23

2.2 The Canonical Polynomials 26

2.3 The Polynomial Coefficients as Functions of the Responses at the Points of the Lattices 31

2.4 Estimating The Parameters in the {q,m} Polynomials 34

2.5 Properties of the Estimate of the Response, (x) 37

2.6 A Three-Component Yarn Example Using A {3 2} Simplex-Lattice Design 38

2.7 The Analysis of Variance Table 42

2.8 Analysis of Variance Calculations of the Yarn Elongation Data 45

2.9 The Plotting of Individual Residuals 48

2.10 Testing the Degree of the Fitted Model: A Quadratic Model or Planar Model? 49

2.11 Testing Model Lack of Fit Using Extra Points and Replicated Observations 55

2.12 The Simplex-Centroid Design and Associated Polynomial Model 58

2.13 An Application of a Four-Component Simplex-Centroid Design: Blending Chemical Pesticides for Control of Mites 60

2.14 Axial Designs 62

2.15 Comments on a Comparison Made Between an Augmented Simplex-Centroid Design and a Full Cubic Lattice for Three Components Where Each Design Contains Ten Points 66

2.16 Reparameterizing Scheffé's Mixture Models to Contain a Constant (₀) Term: A Numerical Example 69

2.17 Questions to Consider at the Planning Stages of a Mixture Experiment 77

2.18 Summary 78

References and Recommended Reading 78

Questions 80

Appendix 2A: Least-Squares Estimation Formula for the Polynomial Coefficients and Their Variances: Matrix Notation 85

Appendix 2B: Cubic and Quartic Polynomials and Formulas …