Experiments with Mixtures Designs, Models, and the Analysis of Mixture Data

ISBN-10: 0471393673

ISBN-13: 9780471393672

Edition: 3rd 2002 (Revised)

Authors: John A. Cornell

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This text teaches readers how to design and set up mixture experiments and then analyse the data and draw inferences from the results.
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Book details

List price: $200.00
Edition: 3rd
Copyright year: 2002
Publisher: John Wiley & Sons, Incorporated
Publication date: 2/7/2002
Binding: Hardcover
Pages: 680
Size: 6.25" wide x 9.25" long x 1.25" tall
Weight: 2.288
Language: English

Preface to the Third Edition
Preface to the Second Edition
Introduction
The Original Mixture Problem
General Remarks About Response Surface Methods
A Factorial Experiment or a Mixture Experiment?
An Historical Perspective
References and Recommended Reading
Questions
The Original Mixture Problem: Designs and Models for Exploring the Entire Simplex Factor Space
The Simplex-Lattice Designs
The Canonical Polynomials
The Polynomial Coefficients as Functions of the Responses at the Points of the Lattices
Estimating the Parameters in the {q, m} Polynomials
Properties of the Estimate of the Response, y(x)
A Three-Component Yarn Example Using a {3, 2} Simplex-Lattice Design
The Analysis of Variance Table
Analysis of Variance Calculations of the Yarn Elongation Data
The Plotting of Individual Residuals
Testing the Degree of the Fitted Model: A Quadratic Model or Planar Model?
Some Comments on the Use of Check Points for Testing Model Lack of Fit
A Numerical Example Illustrating the Use of Check Points for Testing Lack of Fit
The Simplex-Centroid Design and the Associated Polynomial Model
An Application of a Four-Component Simplex-Centroid Design. Blending Chemical Pesticides for Control of Mites
Axial Designs
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
Reparameterizing Scheffe's Mixture Models to Contain a Constant ([beta subscript 0]) Term: A Numerical Example
Questions to Consider at the Planning Stages of a Mixture Experiment
Summary
References and Recommended Reading
Questions
Least-Squares Estimation Formulas for the Polynomial Coefficients and Their Variances: Matrix Notation
Cubic and Quartic Polynomials and Formulas for the Estimates of the Coefficients
The Partitioning of the Sources in the Analysis of Variance Table When Fitting the Scheffe Mixture Models
The Use of Independent Variables
Transforming from the q Mixture Components to q-1 Mathematically Independent Variables
A Numerical Example: Sensory Flavor Rating of Fish Patties
Defining a Region of Interest Inside the Simplex: An Ellipsoidal Region
A Numerical Illustration of the Inverse Transformation from the Design Variables to the Mixture Components
Enlarging the Unit Spherical Region of Interest
Some Discussion on Design Strategy When Fitting Response Surfaces
Rotatable Designs
A Second-Order Rotatable Design for a Four-Component System
Defining a Cuboidal Region of Interest in the Mixture System
Summary
References and Recommended Reading
Questions
An Alternative Transformation from the Mixture Component System to the Independent Variable System
A Form of the Orthogonal Matrix T
Multiple Constraints on the Component Proportions
Lower-Bound Restrictions on Some or All of the Component Proportions
Introducing L-Pseudocomponents
A Numerical Example of Fitting an L-Pseudocomponent Model
Upper-Bound Restrictions on Some or All of the Component Proportions
An Example of the Placing of an Upper Bound on a Single Component: The Formulation of a Tropical Beverage
Introducing U-Pseudocomponents
The Placing of Both Upper and Lower Bounds on the Component Proportions
Formulas for Enumerating the Number of Extreme Vertices, Edges, and Two-Dimensional Faces of the Constrained Region
Some Procedures for Calculating the Coordinates of the Extreme Vertices of a Constrained Region
Multicomponent Constraints
Some Examples of Designs for Constrained Mixture Regions: CONVRT and CONAEV Programs
The Use of Symmetric-Simplex Designs for Fitting Second-Order Models in Constrained Regions
Multiple Lattices for Major and Minor Component Classifications
Categorizing the Mixture Components: An Ellipsoidal Region of Interest
A Numerical Example of a Categorized Component Experiment
Summary
References and Recommended Reading
Questions
An Orthogonal Matrix for the Categorized-Components Problem
The Relationship Between the Coefficients of the Terms in the Double-Scheffe Model (4.82) and the Interaction Model (4.83)
The Analysis of Mixture Data
Techniques Used in the Analysis of Mixture Data
Test Statistics for Testing the Usefulness of the Terms in the Scheffe Polynomials
Model Reduction
An Example of Reducing the System from Three to Two Components
A Criterion for Selecting Subsets of the Terms in the Scheffe Models
A Numerical Example Illustrating the Integrated Mean-Square Error Criterion
Screening Components
A Seven-Component Octane-Blending Experiment: An Exercise in Model Reduction
Other Techniques Used to Measure Component Effects
The Slope of the Response Surface Along the Component Axes
A Numerical Example Illustrating the Slope Calculations for a Three-Component System: Studying the Flavor Surface Where Peanut Meal Is Considered a Substitute for Beef in Sandwich Patties
Leverage and the Hat Matrix
A Three-Component Propellant Example
Summary
References and Recommended Reading
Questions
The Derivation of the Moments of the Simplex Region
Other Mixture Model Forms
The Inclusion of Inverse Terms in the Scheffe Polynomials
Fitting Gasoline Octane Numbers Using Inverse Terms in the Model
An Alternative Model Form for Modeling the Additive Blending Effect of One Component in a Multicomponent System
A Biological Example on the Linear Effect of a Powder Pesticide in Combination with Two Liquid Pesticides Used for Suppressing Mite Population Numbers
Other Models That Are Homogeneous of Degree One
The Use of Ratios of Components
Cox's Mixture Polynomials: Measuring Component Effects
An Example Illustrating the Fits of Cox's Model and Scheffe's Polynomial
Log Contrast Models
A Numerical Example Illustrating the Testing of Inactivity and Additivity Effects of the Components in a Three-Component System Using Log Contrast Models
Octane Blending Models
A Numerical Example Illustrating the Calculations Required for Obtaining the Research and Motor Octane Prediction Equations for a Group of Blends
Fitting a Slack-Variable Model
A Numerical Example Illustrating the Fits of Different Reduced Slack-Variable Models: Tint Strength of a House Paint
Summary
References and Recommended Reading
Questions
The Form of the Multiplier Matrix B[subscript 2] for Expressing the Parameters in Cox's Quadratic Model as Functions of the Parameters in Scheffe's Model
Estimation Equations for Coefficients That Are Subject to Linear Restrictions
The Inclusion of Process Variables in Mixture Experiments
Designs Consisting of Simplex-Lattices and Factorial Arrangements
A Numerical Example of a Fish Patty Experiment: Studying Blends of Three Fish Species Prepared with Three Processing Factors
Testing the Component Blending Properties and the Effects of the Process Variables When the Set of Mixture Blends Is Embedded in the Processing Conditions
A Numerical Example of a Three-Component by Two-Process Variable Split-Plot Experiment: Fitting a Quadratic Mixture Model in the Presence of Interacting Process Variables
A Reparameterization of the Combined Model Form for Measuring the Effects of the Process Variables: An Example of Model Reduction
The Use of Fractional Designs in the Process Variables
A Numerical Example of the Fit of a Combined Model to Data Collected on Fractions of the Fish Patty Experimental Design
Computer-Aided Fractionation of Lattice Designs
Mixture-Amount Experiments
Process Variables and q-1 Mixture-Related Variables
A Numerical Example Involving Three Mixture Components and One Process Variable
Questions Raised and Recommendations Made When Fitting a Combined Model Containing Mixture Components and Other Variables
Summary
References and Recommended Reading
Questions
A Generalized Least-Squares Solution for Fitting the Mixed Model in the Mixture Components and Process Variables to Data from a Split-Plot Experiment
Additional Topics
Block Designs for Mixture Experiments
Symmetric-Simplex Block Designs for Fitting the Scheffe Second-Order Model
An Example of Orthogonal Blocking Using a Symmetric-Simplex Design
Constructing Orthogonal Blocks Using Latin Squares
Weighted Versus Unweighted Least-Squares Estimates of the Parameters in the Scheffe Models
Some Comments on Design Criteria and Some Results Using the ACED Program
Constant Prediction Variance on Concentric Triangles for Three-Component Systems
Altering the Terms in the Scheffe-Type Models to Improve the Accuracy and/or Stability of the Coefficient Estimates
Collinearity Problems Resulting from Performing Experiments in Highly Constrained Regions
A Numerical Example Illustrating the Fitting of Segmented Scheffe Models to Freezing-Point Data from a Two-Component System
Biplot Displays for Multiple Responses
A Five-Response Plastics-Compounding Example
Optimizing Several Responses Simultaneously
Recalling the Three-Component Propellant Example of Section 5.13
Summary
References and Recommended Reading
Questions
The Modified L-Pseudocomponent Model and the Centered and Scaled Intercept Model
Expressing the Coefficients in the Scheffe Quadratic Model As Functions of the Coefficients in the L-Pseudocomponent, Modified L-Pseudocomponent, and Centered and Scaled Intercept Models
Matrix Algebra, Least Squares, and the Analysis of Variance
Matrix Algebra
Some Fundamental Definitions
A Review of Least Squares
The Analysis of Variance
A Numerical Example: Modeling the Texture of Fish Patties
The Adjusted Multiple Correlation Coefficient
The Press Statistic and Studentized Residuals
Testing Hypotheses About the Form of the Model: Tests of Significance
References and Recommended Reading
Data Sets from Mixture Experiments with Partial Solutions
Experiment One: Fruit Punch Experiment
Experiment Two: Chick Feeding Experiment
Experiment Three: Concrete Batches
Experiment Four: Surface Resistivity of Paper Coatings
Experiments Five, Six, and Seven: Estimating Solubilities of Multisolvent Systems
References and Recommended Reading
CONVRT Program Listing for Calculating the Coordinates of the Extreme Vertices of a Constrained Region
CONAEV Program Listing for Calculating the Coordinates of the Centroids (Approximate) of the Boundaries of a Constrained Region
Listings of Subroutines Called by CONVRT and CONAEV
Bibliography and Index of Authors
Answers to Selected Questions
Appendix
Index
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