Pharmaceutical Experimental Design and

Completely revised and updated, Pharmaceutical Experimental Design and Interpretation, Second Edition explains the major methods of experimental design and evaluation such as multivariate, sequential, and principal components analysis. With new sections on neural networks, artificial intelligence, fractional designs, and optimization techniques, this source will prove invaluable to anyone involved in the design and execution of pharmaceutical research studies and the interpretation of study data.

N. Anthony Armstrong is Senior Lecturer in Pharmaceutics, Welsh School of Pharmacy, University of Wales, United Kingdom

Chapter 1 Introduction to Experimental Design

1.1 The Experimental Process

1

(1)

1.2 Computers and Experimental Design

2

(2)

1.3 Overview of Experimental Design and Interpretation

4

(5)

Chapter 2 Comparison of Mean Values

2.1 Introduction

9

(1)

2.2 Comparison of Means when the Variance of the Whole Population is Known

10

(2)

2.3 Comparison of Two Means when the Variance of the Whole Population is Not Known

12

(6)

2.3.1 Treatment of Outlying Data Points

15

(3)

2.4 Comparison of Means between More Than Two Groups of Data

18

(7)

2.4.1 Analysis of Variance (ANOVA)

19

(2)

2.4.2 The Least Significant Difference

21

(1)

2.4.3 Two-Way Analysis of Variance

22

(3)

Chapter 3 Nonparametric Methods

3.1 Introduction

25

(1)

3.2 Nonparametric Tests for Paired Data

25

(4)

3.2.1 The Sign Test

25

(2)

3.2.2 The Wilcoxon Signed Rank Test

27

(2)

3.3 Nonparametric Tests for Unpaired Data

29

(4)

3.3.1 The Wilcoxon Two-Sample Test

29

(4)

Chapter 4 Regression and Correlation

4.1 Introduction

33

(1)

4.2 Linear Regression

33

(8)

4.2.1 The Number of Degrees of Freedom (Cell B11 in Table 4.4)

37

(1)

4.2.2 The Coefficient of Determination (r²) (Cell A10 in Table 4.4)

38

(2)

4.2.3 The Standard Errors of the Coefficients (Cells A9 and B9 in Table 4.4)

40

(1)

4.2.4 The F Value or Variance Ratio (Cell All in Table 4.4)

40

(1)

4.2.5 The Two Regression Lines

41

(1)

4.3 Curve Fitting of Nonlinear Relationships

41

(3)

4.3.1 The Power Series

42

(1)

4.3.2 Quadratic Relationships

42

(1)

4.3.3 Cubic Equations

43

(1)

4.3.4 Transformations

44

(1)

4.4 Multiple Regression Analysis

44

(4)

4.4.1 Correlation Coefficients

47

(1)

4.4.2 Standard Error of the Coefficients and the Intercept

48

(1)

4.4.3 F Value

48

(1)

4.5 Interaction between Independent Variables

48

(1)

4.6 Stepwise Regression

49

(1)

4.7 Rank Correlation

50

(2)

4.8 Comments on the Correlation Coefficient

52

(3)

Chapter 5 Multivariate Methods

5.1 Introduction

55

(1)

5.2 Multivariate Distances

55

(4)

5.2.1 Distance Matrices

55

(4)

5.3 Covariance Matrices

59

(3)

5.4 Correlation Matrices

62

(1)

5.5 Cluster Analysis

63

(4)

5.5.1 Cartesian Plots

63

(2)

5.5.2 Dendrograms

65

(2)

5.6 Discrimination Analysis

67

(3)

5.7 Principal Components Analysis

70

(5)

5.8 Factor Analysis

75

(8)

Chapter 6 Factorial Design of Experiments

6.1 Introduction

83

(1)

6.2 Two-Factor, Two-Level Factorial Designs

84

(5)

6.2.1 Two-Factor, Two-Level Factorial Designs with Interaction between the Factors

86

(3)

6.3 Notation in Factorially Designed Experiments

89

(2)

6.4 Factorial Designs with Three Factors and Two Levels

91

(3)

6.5 Factorial Design and Analysis of Variance

94

(6)

6.5.1 Yates's Treatment

95

(3)

6.5.2 Factorial Design and Linear Regression

98

(2)

6.6 Replication in Factorial Designs

100

(3)

6.7 The Sequence of Experiments

103

(1)

6.8 Factorial Designs with Three Levels

104

(6)

6.9 Three-Factor, Three-Level Factorial Designs

110

(5)

6.9.1 Mixed or Asymmetric Designs

114

(1)

6.10 Blocked Factorial Designs

115

(3)

6.11 Fractional Factorial Designs

118

(3)

6.12 Plackett–Burman Designs

121

(1)

6.13 Central Composite Designs

122

(4)

6.14 Box–Behnken Designs

126

(1)

6.15 Doehlert Designs

127

(2)

6.16 The Efficiency of Experimental Designs

129

(6)

Chapter 7 Response-Surface Methodology

7.1 Introduction

135

(1)

7.2 Constraints, Boundaries, and the Experimental Domain

136

(1)

7.3 Response Surfaces Generated from First-Order Models

137

(6)

7.4 Response Surfaces Generated by Models of a Higher Order

143

(7)

7.5 Response-Surface Methodology with Three or More Factors

150

(7)

Chapter 8 Model-Dependent Optimization

8.1 Introduction

157

(1)

8.2 Model-Dependent Optimization

158

(5)

8.2.1 Extension of the Design Space

161

(2)

8.3 Optimization by Combining Contour Plots

163

(2)

8.4 Location of the Optimum of Multiple Responses by the Desirability Function

165

(3)

8.5 Optimization Using Pareto-Optimality

168

(5)

Chapter 9 Sequential Methods and Model-Independent Optimization

9.1 Introduction

173

(1)

9.2 Sequential Analysis

173

(4)

9.2.1 Wald Diagrams

173

(4)

9.3 Model-Independent Optimization

177

(7)

9.3.1 Optimization by Simplex Search

177

(7)

9.4 Comparison of Model-Independent and Model-Dependent Methods

184

(5)

Chapter 10 Experimental Designs for Mixtures

10.1 Introduction

189

(1)

10.2 Three-Component Systems and Ternary Diagrams

190

(3)

10.3 Mixtures with More Than Three Components

193

(2)

10.4 Response-Surface Methodology in Experiments with Mixtures

195

(7)

10.4.1 Rectilinear Relationships between Composition and Response

195

(2)

10.4.2 Derivation of Contour Plots from Rectilinear Models

197

(1)

10.4.3 Higher-Order Relationships between Composition and Response

198

(2)

10.4.4 Contour Plots Derived from Higher-Order Equations

200

(2)

10.5 The Optimization of Mixtures

202

(1)

10.6 Pareto-Optimality and Mixtures

203

(2)

10.7 Process Variables in Mixture Experiments

205

(4)

Chapter 11 Artificial Neural Networks and Experimental Design

11.1 Introduction

209

(10)

11.1.1 Pharmaceutical Applications of ANNs

212

(7)

Appendix 1 Statistical Tables

A1.1 The Cumulative Normal Distribution (Gaussian Distribution)

219

(1)

A1.2 Student's t Distribution

219

(2)

A1.3 Analysis of Variance

221

(2)

Appendix 2 Matrices

A2.1 Introduction

223

(2)

A2.2 Addition and Subtraction of Matrices

225

(1)

A2.3 Multiplication of Matrices

226

(3)

A2.3.1 Multiplying a Matrix by a Constant

226

(1)

A2.3.2 Multiplication of One Matrix by Another

226

(1)

A2.3.3 Multiplication by a Unit Matrix

227

(1)

A2.3.4 Multiplication by a Null Matrix

228

(1)

A2.3.5 Transposition of Matrices

228

(1)

A2.3.6 Inversion of Matrices

229

(1)

A2.4 Determinants

229

(4)

Index

233

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Pharmaceutical Experimental Design and Interpretation, Second Edition

Summary

Author Biography

Table of Contents

Supplemental Materials