Design and Analysis of Response Surface Designs with Restricted Randomization Wayne R

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2006 Design and Analysis of Response Surface Designs with Restricted Randomization Wayne R. Wesley

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COLLEGE OF ENGINEERING

DESIGN AND ANALYSIS OF RESPONSE SURFACE DESIGNS WITH

RESTRICTED RANDOMIZATION

Wayne R. Wesley

A Dissertation submitted to the Department of Industrial Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Degree Awarded: Summer Semester, 2006

______James R. Simpson Professor Directing Dissertation

______Anuj Srivastava Outside Committee Member

______Peter A. Parker Outside Committee Member

______Joseph J. Pignatiello, Jr. Committee Member

Approved:

______Chuck Zhang, Chair, Department of Industrial and Manufacturing Engineering

______Ching-Jen Chen, Dean, College of Engineering

The Office of Graduate Studies has verified and approved the above named committee members.

I dedicate this dissertation to my son and daughter Jowayne and Jowaynah Wesley.

iii ACKNOWLEDGEMENTS

I must first give thanks to the almighty God for good health, strength and wisdom to successfully complete my dissertation. It is with a deep sense of gratitude that I express my sincere appreciation to Dr. James Simpson and Dr. Peter Parker for their invaluable advice and contributions in directing the dissertation. Special thanks to Dr. Joseph Pignatiello and Dr. Anuj Srivastava for their meaningful comments and suggestions that helped to improve the quality of the dissertation. Additionally, I would like to give special recognition to my wife Joy Wesley for her unwavering support and commitment throughout my endeavors at The Florida State University. Special thanks to my mother Mary White for her tremendous sacrifice in giving me the opportunity to develop and pursue my career goals. To my fellow students and friends, Michelle, Lisa, Francisco, Rupert and Young-Guo thanks for your support and encouragement. Finally thanks to all those who supported me with their prayers, financially and otherwise.

iv TABLE OF CONTENTS

LIST OF TABLES...... viii

LIST OF FIGURES ...... xi

ABSTRACT...... xii

CHAPTER 1 ...... 1

1.0 INTRODUCTION ...... 1 1.1 Background...... 1 1.2 The Split Plot Model...... 4 1.3 Design Optimality...... 6 1.4 Problem Statement...... 9 1.5 Research Objectives...... 10 CHAPTER 2 ...... 11

2.0 REVIEW OF LITERATURE ...... 11 2.1 Introduction...... 11 2.2 First-Order Split Plot Designs...... 12 2.3 Second-Order Split Plot Designs ...... 14 2.4 Design Optimality Criteria...... 16 2.5 Summary...... 18 CHAPTER 3 ...... 19

3.0 Impact of Restricted Randomization on the Structure of the Information Matrix 19 3.1 Abstract...... 19 3.2 Introduction...... 20 3.3 The Split Plot Model...... 20 3.4 Structure of Second-Order SPD...... 24 3.5 Critical Functions of the Variance Ratio and Whole Plot Sizes ...... 28 3.6 Information Matrix for CCD and BBD Structures ...... 29 3.7 Impact of Design Structure on the Application of the Critical Functions 33 3.8 Outline of Analytical Characterization Scheme ...... 38 −1 3.9 Characterization of (XR' −1 X) for Split Plot CCD...... 40 −1 3.10 Characterization of (XR' −1 X) for Split Plot BBD...... 45 3.11 Generalized Variance of Parameter Estimates...... 46 3.11.1 Determinant for split plot CCD...... 47

v 3.11.2 Determinant for split plot BBD...... 47 3.12 Application...... 48 3.13 Conclusion ...... 54 CHAPTER 4 ...... 56

4.0 Prediction Variance Properties and G-Criterion Location for Second-Order Split Plot Designs ...... 56 4.1 Abstract...... 56 4.2 Introduction...... 57 4.3 The Split Plot Model...... 58 4.4 Prediction Optimality Criteria...... 60 4.5 Second-Order Split Plot Designs ...... 61 4.6 Critical functions of the variance ratio and whole plot sizes...... 62 4.7 Information Matrix for the Split Plot CCD...... 63 4.8 Information Matrix for the Split Plot BBD...... 67 4.9 Minimum and Maximum Prediction Variance ...... 70 4.10 Vminρ and Vmaxρ for Spherical Regions ...... 72 4.11 Vmin and Vmax for Whole Plot Design Space for Spherical Regions ρz ρz ...... 73 4.12 Vmin and Vmax for Subplot Design Space for Spherical Regions .... 74 ρx ρx

4.13 Vminρ and Vmaxρ for Combined Design Space for Spherical Regions.... 74 4.14 Vminρ and Vmaxρ for Cuboidal Regions...... 75 4.15 Vmin and Vmax for Whole Plot Design Space for Cuboidal Regions 76 ρz ρz 4.16 Vmin and Vmax for Subplot Design Space for Cuboidal Regions.... 76 ρx ρx

4.17 Vminρ and Vmaxρ for the Combined Design Space for Cuboidal Regions77 4.18 Delta G-Criterion Measures for Whole Plot and Subplot Design Spaces 79 4.19 Prediction Variance Assessment of Second-Order SPD...... 79 4.19.1 CCD with one whole plot factor and two subplot factors...... 79 4.19.2 BBD with two whole plot factors and two subplot factors...... 85 4.19.3 FCC with two whole plot factors and three subplots factors...... 88 4.20 Conclusion ...... 91 CHAPT ER 5 ...... 93

5.0 Integrated Prediction Variance for Response Surface Designs ...... 93 5.1 Abstract...... 93 5.2 Introduction...... 94 5.3 The Split Plot Model...... 95 5.4 IV – Optimality Criterion...... 97 −1 5.5 Analytical Characterization of (XR' −1 X) ...... 100 5.5.1 Critical functions of the variance ratio and whole plot sizes...... 100 5.5.2 Information Matrix for the Split Plot CCD...... 101 5.5.3 Information Matrix for the Split Plot BBD...... 105

vi 5.6 Matrix of Region Moments...... 108 5.7 Analytical Determination of Integrated Prediction Variance ...... 110 5.8 Integrated Variance for the Split Plot CCD ...... 111 5.9 Integrated Variance for the Split Plot BBD ...... 112 5.10 Integrated Prediction Variance for Split Plot Designs...... 113 5.10.1 SPD with one whole plot factor and three subplot factors...... 114 5.10.2 SPD with two whole plot factors and four subplot factors ...... 117 5.10.3 SPD with three whole plot factors and four subplot factors ..... 122 5.11 Integrated Prediction Variance for Completely Randomized Designs... 126 5.11.1 Hybrid and Small Composite Designs...... 127 5.11.2 Equiradial Designs...... 129 5.12 Comparison of Selected Cuboidal Designs...... 132 5.13 Conclusion ...... 134 CHAPT ER 6 ...... 136

6.0 CONCLUSION...... 136 6.1 Research Objective 1 ...... 137 6.2 Research Objective 2 ...... 137 6.3 Research Objective 3 ...... 138 6.4 Future Work...... 139 APPENDIX A...... 140 A1: Analytical Characterization of (X'R-1X)-1 for Split Plot CCD ...... 140 A2: Analytical Characterization of (X'R-1X)-1 for Split Plot BBD ...... 152 APPENDIX B ...... 163 B1: Determinant of the Information Matrix...... 163 B2: Derivation of the Determinant for Split Plot CCD...... 164 B3: Derivation of the Determinant for Split Plot BBD...... 166 APPENDIX C ...... 168 C1: Analytical Determination of v(z, x) for Split Plot CCD...... 168 C2: Analytical Determination of v(z, x) for Split Plot BBD...... 170 APPENDIX D...... 172 D1: Determination of Critical Points ...... 172 D2: Evaluation of Vminρ and Vmaxρ for Spherical Regions...... 174 D3: Evaluation of Vminρ and Vmax ρ for Cuboidal Regions ...... 177 APPENDIX E ...... 181

E1: Derivation of trace (Q11) and trace (Q22) for Split Plot CCD...... 181 E2: Derivation of trace (Q11) and trace (Q22) for Split Plot BBD...... 185 REFERENCES ...... 188

BIOGRAPHICAL SKETCH ...... 193

vii

LIST OF TABLES

Table 1: The X matrix for a CCD with one whole plot and two subplot factors (VKM D12)...... 26 Table 2: The X matrix for a balanced BBD with one whole plot and two subplot factors (VKM D12) ...... 27 Table 3: The X matrix for an unbalanced BBD with one whole plot and two subplot factors (VKM D12) ...... 27 Table 4: Table of notations for the split plot CCD ...... 30 Table 5: Table of notations for the split plot BBD ...... 31 Table 6: A resolution V CCD with = α = 1, two whole plots and three subplot factors (VKM) ...... 35

Table 7: A CCD with = α = 5 , one whole plot factor and three subplot factors (MWP) ...... 36 Table 8: A CCD with = α = 1, one whole plot factor and three subplot factors (MWP)37 Table 9: Table of scalar quantities for the split plot CCD ...... 44 Table 10: Table of constant quantities for the split plot CCD ...... 44 Table 11: Table of scalar quantities for the split plot BBD ...... 45 Table 12: Table of constant quantities for the split plot BBD ...... 46 Table 13: Modified VKM D23 with = α = 1, two whole plots and three subplot factors – Unbalanced...... 48 Table 14: Design parameters for the split plot CCD...... 49 Table 15: Constants, functions and D-value results for CCD across the different variance ratios ...... 50 Table 16: Relative efficiencies...... 51 Table 17: Design parameters for the BBD...... 52 Table 18: Constants, functions and D-value results for BBD across the different variance ratios ...... 53 Table 19: Table of notations for the split plot CCD ...... 64 Table 20: Table of scalar quantities for the split plot CCD ...... 66 Table 21: Table of constant quantities for the split plot CCD ...... 66 Table 22: Table of notations for the split plot BBD ...... 67

viii Table 23: Table of scalar quantities for the split plot BBD ...... 69 Table 24: Table of constant quantities for the split plot BBD ...... 69 Table 25: Table of coefficients of the variance function v(z, x) for CCD and BBD...... 72 Table 26: Critical Points for the evaluation of v(z, x)...... 73

Table 27: Standard CCD with = α = k and N=17...... 80 Table 28: Prediction variance properties and G-criterion location...... 80 Table 29: Relative measures of whole plot and subplot design spaces ...... 82

Table 30: Modified CCD with = α = k and N=16 ...... 82 Table 31: Prediction variance properties and G-criterion location for the modified CCD83 Table 32: Relative measures of whole plot and subplot design spaces for modified CCD ...... 85 Table 33: MWP_BBD with two whole plot factors and two subplot factors ...... 85 Table 34: Prediction variance properties and G-criterion location for the MWP BBD with two whole plot factors and two subplot factors...... 86 Table 35: Relative measures of whole plot and subplot design spaces MWP BBD with two whole plot factors and two subplot factors...... 88 Table 36: A resolution V VKM FCC with = α = 1, two whole plots and three subplot factors ...... 88 Table 37: Prediction variance properties and G-criterion location for the VKM FCC with two whole plot factors and three subplot factors...... 89 Table 38: Relative measures of whole plot and subplot design spaces for the VKM FCC ...... 91 Table 39: Table of notations for the split plot CCD ...... 102 Table 40: Table of scalar quantities for the split plot CCD ...... 104 Table 41: Table of constant quantities for CCD ...... 104 Table 42: Table of notations for the split plot BBD ...... 105 Table 43: Table of scalar quantities for the split plot BBD ...... 107 Table 44: Table of constant quantities for the split plot BBD ...... 107 Table 45: Spherical second and forth order region moments for 26≤ k ≤ ...... 110 Table 46: Split plot designs selected for evaluation ...... 114 Table 47: Design matrix for a balanced VKM BBD with one whole plot factor and three subplot factors ...... 115 Table 48: Design matrix for a balanced MWP BBD with one whole plot factor and three subplot factors ...... 115

ix Table 49: Design parameters for VKM BBD and MWP BBD with one whole plot factor and three subplot factors...... 116 Table 50: Integrated prediction variance for VKM BBD and MWP BBD with one whole plot factor and three subplot factors ...... 117 Table 51: Design matrix for a balanced VKM CCD with = α = 2.4495, two whole plot factors and four subplot factors ...... 118 Table 52: Design matrix for a balanced MWP CCD with = α = 2.4495, two whole plot factors and four subplot factors ...... 119 Table 53: Design parameters for VKM CCD and MWP CCD with = α = 2.4495, two whole plot factors and four subplot factors ...... 120 Table 54: Integrated prediction variance for VKM CCD and MWP CCD with = α = 2.4495, two whole plot factors and four subplot factors ...... 121 Table 55: Design matrix for a balanced VKM BBD with two whole plot factors and four subplot factors ...... 122 Table 56: Integrated prediction variance for VKM BBD with two whole plot factors and four subplot factors...... 122 Table 57: Design matrix for a balanced VKM CCD with = α = 1, three whole plot factors and four subplot factors ...... 123 Table 58: Design matrix for a balanced MWP CCD with = α = 1, three whole plot factors and four subplot factors ...... 124 Table 59: Design parameters for VKM CCD and MWP CCD with = α = 1, three whole plot factors and four subplot factors...... 125 Table 60: Integrated prediction variance for VKM CCD and MWP CCD with = α = 1, 3 whole plot factors and 4 subplot factors...... 126 Table 61: Integrated prediction variance for standard CCD and BBD spherical designs...... 127 Table 62: Comparison of IV values for near-saturated designs over spherical regions for k = 3 and 4...... 128 Table 63: Comparison of IV values for Hybrid and CCD designs over spherical regions for k = 6 and 7...... 129 Table 64: Integrated variances of selected equiradial designs...... 131 Table 65: Maximum prediction variance properties of selected equiradial designs for k = 2 factors ...... 131 Table 66: Integrated prediction properties of selected cuboidal designs...... 133

x LIST OF FIGURES

Figure 1: Typical assignment of subplot factors to whole plots ...... 3 Figure 2: Analytical Characterization Scheme ...... 40 Figure 3: 3D VDGs for the standard CCD with one whole plot factor and two subplot factors ...... 81 Figure 4: Comparison of whole plot and subplot spaces for the standard CCD with one whole plot factor and two subplot factors ...... 81 Figure 5: 3D VDGs for the modified CCD with one whole plot factor and two subplot factors ...... 83 Figure 6: Comparison of whole plot and subplot spaces for modified CCD with one whole plot factor and two subplot factors ...... 84 Figure 7: 3D VDGs the MWP BBD with two whole plot factors and two subplot factors...... 86 Figure 8: Comparison of whole plot and subplot spaces for the MWP BBD with two whole plot factors and two subplot factors...... 87 Figure 9: 3D VDG the VKM FCC with two whole plot factors and three subplot factors...... 89 Figure 10: Comparison of whole plot and subplot spaces for the MWP BBD with two whole plot factors and two subplot factors...... 90

xi ABSTRACT

Many industrial experiments are conducted under various conditions which do not facilitate complete randomization of all the experimental factors. In response surface methodology whenever there are restrictions on randomization the experimental procedure usually follows the split plot design approach. Split plot designs are used when there are factors which are difficult or costly to change or adjust during an experiment. Split plot designs are currently generating renewed interest because of their usefulness and practical application in industrial settings. Despite the work accomplished through various research efforts, there is still a need to understand the optimality properties of these designs for second-order response surface models. This dissertation provides the development of an analytical approach for the computation of various optimality properties for the assessment of second-order split plot designs. The approach involves a thorough investigation of the impact of restricted randomization on the information matrix, which characterizes much of the relationship between the design points and the proposed response surface model for split plot designs. Several important insights are presented for the construction of second-order split plot designs. In addition, the analytical equations reported compute exact design optimality values and are more efficient than currently available methods. A particular feature of these analytical equations is that they are functions of the design parameters, radius and variance ratio. Further, a significant result is the ability to efficiently compute the exact value of the integrated prediction variance for both split plot designs and completely randomized designs. The functionality of the computational procedures presented provides easy evaluation of the impact of changes in the design structure and variance ratio on the optimality properties of second-order split plot designs.

xii CHAPTER 1

1.0 INTRODUCTION

1.1 Background The three fundamental principles in design and analysis of experiments are randomization, replication and blocking. However, in most industrial experiments it becomes a challenge to adhere to these principles at all times. In situations where it is difficult, expensive or dangerous to adjust certain factors during experimentation it becomes necessary to reduce the frequency of adjusting this factor(s). The reduction in the frequency of adjusting certain factors during experimentation is referred to as restricted randomization. In statistical analysis, designs with restricted randomization are commonly referred to as split plot designs (SPD). Wooding (1973) presented a detailed description on the structure and use of SPD. According to Myers et al (2004), the foundation for response surface methodology was laid by Box and Hunter (1951). Response surface methodology (RSM) is a collection of statistical and mathematical techniques used for the characterization and optimization of systems or processes. In traditional response surface methodology experimentation is generally expected to be completely randomized. Therefore, design optimality criteria were developed based on the assumption that the design will be completely randomized. Unfortunately, there are situations which arise in industrial experimentation that will not facilitate complete randomization of the experimental runs or resetting of the factor levels for each experimental run. For example the magnetic alignment of carbon nanotubes is dependent on several factors. These factors are: 1. Strength of magnetic field 2. Temperature 3. Nanotube dispersion 4. Filtration flow rate 5. Aspect ratio (length/diameter)

1 The strength of the magnetic field can not be adjusted quickly due to the dangers associated with rapidly increasing or decreasing the electrical voltage required. Therefore, the magnetic field will have to follow a different randomization procedure than the remaining four factors. The experiment will have to be conducted such that the levels of the magnetic field are adjusted less frequently than the other four factors. A consequence of restricted randomization is a complicated error structure resulting from a split plot design. Therefore, established design optimality criteria for evaluating response surface designs have to be altered to accommodate the impact of the two error sources on the information matrix. Box and Hunter (1951) defined the information matrix as the sums of squares and products of the independent variables. Deeply embedded in the information matrix is the design of the experiment. The issue of randomization restriction in response surface methods (RSM) was investigated by Letsinger et al (1996). Based on a simulation study Letsinger et al (1996) recommended that the restricted maximum likelihood method (REML) is better for parameter estimation when compared to ordinary least squares (OLS) and iterated reweighed least squares (IRLS). Recently, the split plot design has been given much attention by researchers and continues to generate much interest as it is becoming increasingly popular in industrial experimentation. Most industrial experiments follow the split plot structure but in many cases are incorrectly analyzed as completely randomized experiment. SPDs are used when there are factors which are difficult or costly to change or adjust during an experiment. The experimental approach is to separate the design into two design spaces known as the whole plot, for hard to change factors, and the subplot, for easy to change factors. Applying this experimental approach to the magnetic alignment example, the classification of factors would be as follows: 1. Whole plot factor – Dangerous to change levels z → Magnetic field 2. Subplot factors – Safe and easy to change

x1 → Temperature

x2 → Filtration flow rate

x3 → Suspension concentration

2 x4 → Aspect ratio

Therefore if there are three levels of the whole plot factor (z) then the subplot factors (x1

x2, x3 and x4) would be assigned to each of the factor level settings for the whole plot factor. Each whole plot can only have one whole plot factor level setting. Figure 1 shows a typical assignment of the subplot factors to the whole plots.

Subplot x1 Whole plot 1 z - level 1 x2 x3 x4

Subplot x1 Whole plot 2 z - level 2 x2 x3 x4

Subplot x1 z - level 3 x Whole plot 3 2 x3 x4

Figure 1: Typical assignment of subplot factors to whole plots

The analysis of this experiment would therefore involve two different error terms to reflect the two separate experimental units. The experimental error associated with the whole plot partitioning is referred to as the whole plot error. Similarly, the experimental error associated with the subplot partitioning is referred to as the subplot error. These error terms are used to develop empirical models for characterizing the system or process being studied. The following section will present the details of the split plot model.

3 1.2 The Split Plot Model The general form for split plot models in matrix form is given as

yX= Θ++ where y is an N x 1 vector of expected responses, N is the total number of observations, Θ is a p x 1 vector of unknown model parameters, p is the number of model parameters, X is the N x p matrix of the levels of the independent variables, and and are N x 1 vectors of random variables for whole plot and subplot errors respectively. N and It is assumed that + ~ N (0, Σ). The structure of the variance-covariance matrix Σ is

⎡⎤Σ1 00...... ⎢⎥ 0 Σ2 0... 0 ⎢⎥ (1.1) Σ = ⎢0 ⎥ ⎢⎥ ⎢⎥0 ⎢⎥ ⎣⎦00 0Σa

22' where a is for the number of whole plots, Σiwww=+σσδε11×11×× Iw and w is the number of th subplot runs (whole plot size). In matrix form the structure of Σ i for the i whole plot is

222 2 ⎡⎤σσσδεδ+ σ δ ⎢⎥ σ 2 Σ = ⎢δ ⎥ (1.2) i ⎢⎥2 σδ ⎢⎥2222 ⎣⎦⎢⎥σδ σσσδδε+

2 2 2 where σ δ + σ ε represent the variance of individual observations, σ δ is the covariance

2 and whole plot error variance while σ ε is the subplot error variance. Given the structure of the two error terms, estimation of the model parameters is accomplished using the generalized least squares (GLS) method of estimation given by

4 −1 Θˆ = (X''Σ−11XX) Σ− y

−1 ˆ ' −1 2 2 where Var ()Θ = ()X Σ X . The two variance components σ δ and σ ε are needed to obtain parameter estimates and standard errors. Letsinger et al. (1996) point out that design optimality criteria, which are functions of the Var ()Θˆ , depend on the structure of the design particularly the information matrix (A). In the case of completely randomize designs (CRD) the information matrix is given as X' X . However, for SPD alphabetic optimality is now dependent on an information matrix of the form X'Σ−1X , where Σ is the variance- covariance matrix described in Equation (1.1). This situation complicates the process because now the optimal design will depend on the variance ratio or degree of correlation

2 2 η = σ δ σ ε , given by the correlation matrix (R). The correlation matrix (R) of the

22 observations is derived by dividing the variance covariance matrix, Σ, by σ δ +σ ε . In the computation of the information matrix the correlation matrix is used because it provides a unitless measure of the information matrix. An information matrix without units is desirable especially for design comparison purposes. Therefore, the information matrix without units is given as

⎛⎞−1 ''⎛⎞Σ −1 AX==⎜⎟⎜⎟ X XRX ⎜⎟σσ22+ () ⎝⎠⎝⎠δε where, R denotes the correlation matrix of the observations and its structure is of the form

5 ⎡ ηη⎤ 1 ⎢ 11++η η ⎥ ⎡⎤R01 ...... 0 ⎢ ⎥ ⎢⎥⎢ η ⎥ 0R2 0... 0 ⎢⎥⎢1+η ⎥ ⎢ ⎥ R0==⎢⎥ where Ri . ⎢⎥⎢ η ⎥ 0 ⎢⎥⎢ 1+η ⎥ ⎢⎥00 0R ⎢ ⎥ ⎣⎦a ⎢ ηη⎥ ⎢ 1 ⎥ ⎣11++ηη⎦

In terms of design optimality, the effect of the correlation matrix on the information matrix will give different results for SPD when compared to CRD. According to Goos and Vandebroek (2001 and 2004) the difference in design optimality values should not be surpri sing because desig ning a completely randomized experiment consists of determining the design points, while designing a split plot experiment requires simultaneously choosing the number of whole plots and the number of subplots within each whole plot and the design points for each plot type. The next section presents some commonly used design optimality criteria to assess the performance of response surface designs.

1.3 Design Optimality The following discussion presents a brief review of the design optimality criteria used to evaluate response surface designs. For details on these design criteria see Montgomery (2001) and Myers and Montgomery (2002). The main purpose of this section is to illustrate how the impact of the variance ratio, through the correlation matrix, is incorporated into the functions of the design optimality criteria. The A-optimality criterion focus is to minimize the ave rage variance of the parameter estimates based on a pre-specified model. The focus is on individual variances of the coefficients of the model parameters and does not take into account the covariance among the coefficients of the model parameters. The A-optimality criterion deals with minimizing the trace of the inverse of information matrix and is given as

6 Min trace[]A −1 −1 Min trace ⎡XR' −1 X⎤ ⎣ ⎦

−1 where trace is the sum of the diagonal elements of (XR' −1 X) which are the variances of the estimator of the coefficients of the model parameters. D-optimality focuses on maximizing the determinant of the information matrix and is expressed as

1/ p Max XR' −1 X

or equivalently

1/ p −1 Min ()XR' −1 X .

where, p is the number of parameters being estimated in the model. The use of this criterion results in the minimization of the generalized variance of the parameter estimates. The G-optimality criterion is associated with the prediction variance v(z, x) and seeks to minim ize the maximum prediction variance over the region of interest (Ξ). The region of interest characterizes the boundary of the operating conditions for the system (s) under consideration. In response surface methodology the two most commonly encountered regions are the spherical and cuboidal regions. G-optimality emphasizes the use of design points for which the maximum v(z, x) in the region of the design is minimized. Thus a G optimal design (ξ ) is one in which we have

⎡ ⎤ Min Max v()zx, ξ ⎣⎢ x∈Ξ ⎦⎥ where v(z, x) is given as

Var⎡⎤yˆ (zx , ) −1 vfzx,,==⎣⎦ zx' XR' −1 Xf zx, () 22 ()()() σσδε+

7 where f ()zx, is the model vector which gives the location in the design space and reflects the nature of the model under consideration. The general form of the second- order model vector used is given as

f z,x ' = () ⎡ 2222⎤ ⎣1,,z1… zwp x11,,……………… xsp zz21 ,, z wp−− z wp zx11 ,, z wp x sp xx 121 ,, x sp x sp z1 ,, z wp x 1 ,, x sp ⎦ where, z and x are whole plot and subplot factors respectively. The model vector is partitioned into linear, interaction and quadratic terms for the respective whole plot and subplot factors. The integrated variance criterion (IV-optimality) also denoted as either I or Q and is used to compute a single measure of prediction performance through an average process. That is, the average prediction variance v(z, x) is determined over some region of interest, Ξ, by integrating over Ξ. The IV-optimality criterion for a design (ξ ) can be expressed as follows:

−1 ⎧⎫1 ' ' −1 Min⎨⎬ f()zx,, XR X f() zx d zx d ξ ∫ () ⎩⎭Ω Ξ

⎧⎫1 Min⎨⎬ v()zx, d zx d ξ ∫ ⎩⎭Ω Ξ

where Ω= ddzx is the volume of the region Ξ. The IV-optimality can be further ∫Ξ simplified to

−1 ⎧⎫1 ' −1 ' Min⎨⎬ traceXR X f()() z,, x f z x d z d x ξ ()∫ . ⎩⎭Ω Ξ

In contrast to the IV-optimality where the focus is on the entire design region, V- optimality focuses on the average prediction variance associated with a set of points which are of particular interest to the practitioner. This optimality criterion, in many

8 cases, is implemented as the spherical (average) prediction variance of the surface area. The surface area used is based on whether or not the region is spherical or cuboidal.

1.4 Problem Statement The information matrix is the single most useful source of information for evaluating experimental design. All aspects of model building and analysis are based on this powerful source of information. It is therefore logical that if one is performing design analysis or design construction that consideration be given to the nature and structure of the information matrix. Such consideration is particularly important for split plot designs due to the fact that its information matrix takes on an added dimension brought about by restricted randomization. Another issue of concern is that the D-optimality criterion has been more popular than other optimality criteria primarily because of its simplistic computation. Welch (1984) argues that the heavy focus on D-optimality is inappropriate, because in practice many experiments are aimed at predicting the response over the region of interest rather than parameter estimation. Haines (1987 ) suggests that the apparent lack of attention given to the other design optimality criteria, such as G-optimality and IV-optimality in particular, is due to the computation al time required to compute these values. According to the Kiefer-Wolfowicz (1959 and 1961) equivalence theorem D- optimal designs are also G-optimal, in the limit of an infinite number of trials for completely randomized designs. However, this equivalence theorem will not hold for split-plot experiments because th e information matrix is no longer the same due the restriction on randomization. Goos and Vandebroek (2004) stated that there is a need to find good standard designs for split plot experimentation. They suggest that investigations be conducted to establish the relationship between the D-optimality and G- optimality criteria in the context of split plot experimentation. Despite the work accomplished through various research efforts, there are still areas of concern that require further investigation to facilitate a deeper understanding of the impact of restricted randomization on response surface designs. It is anticipated that the results of such an investigation would help to create a general approach that addresses

9 the design and analysis of split plot experiments. It is the intention of this research to advance the knowledge required to achieve such an approach through the characterization of the information matrix and its inverse for split plot designs.

1.5 Research Objectives The purpose of this research is to: 1. Investigate the impact of restricted randomization on the structure of the information matrix. 2. Develop a computationally efficient method to compute maximum and minimum prediction variances and perform an assessment of the impact of changes in the variance ratio on these variance properties for second-order split plot designs. 3. Develop a computationally efficient method to compute integrated prediction variance and perform a comparative assessment of the integrated prediction variance properties of response surface designs.

The dissertation consists of the five remaining chapters. Chapter 2 provides a general review of literature covering pertinent issues relating to split plot designs. Chapters 3 through 5 are stand-alone chapters detailing the new methods developed for evaluating and assessing design optimality properties of second-order response surface split plot designs. Each chapter contains its own literature review, problem statement, research objective, methodology description, conclusion section. The final chapter of this dissertation is Chapter 6 which summarizes the conclusions drawn from the three previous chapters.

10 CHAPTER 2

2.0 REVIEW OF LITERATURE

2.1 Introduction Myers et al. (2004) reviewed the status of response surface methodologies. They summarized work published on split plot designs. They suggested that industrial experimentation in the presence of restrictions on randomization follow the split plot design approach. They concluded that attention must be given to the way in which these experiments are analyzed and that the basic split plot analysis techniques must be extended to different types of industrial experiments. The attention should be focused on providing proper analysis techniques that accounts for the multiple error terms associated with the split plot design. Such analysis techniques are crucial because if split plot designs are incorrectly analyzed as completely randomizes designs then the analysis of the model is subjected to Type 1 and Type 2 errors. Further, incorrect analyses may also result in incorrect variance estimates which will affect the accuracy of the results. Simpson et al. (2004) present a practical approach for the implementation of the split plot design in a factorial setting. They suggest that the combination of carefully structured split-plot designs and mixed model analysis can be a fairly effective scheme for investigations in randomization restricted environment. They further suggest that the development of a general approach to split plot must focus on four important issues when conducting designed experiments particularly for the split plot design. The issues are adequate replication, empirical model building capability, lack of fit and the ability to testing for curvature. The test for lack of fit requires the replication of both whole plot and subplot units.

11 2.2 First-Order Split Plot Designs The design and construction of split plot experiments has been studied extensively in the literature. Bingham and Sitter (2001) present some design issues with fractional factorial split plot experiment. A fractional factorial experiment is a design in which only a subset of the experimental runs are conducted. They discuss the impact of the split plot structure on estimation, precision and the use of resources. They point out that the two main issues that arise when selecting competing designs are identifiability and precision. Identifiability is the ability to estimate as many of the main effects and two factor interactions as possible and precision is the ability to detect significant effects with as much power as possible. Among the issues of concern are costs and run size. They conclude that the decision of where to fractionate comes down to a compromise between minimizing the aberration of the design, maximizing the precision for the effects of interest, and selecting the most economical design. In addition, Huang et al. (1998) and Bingham and Sitter (1999) present methods to construct minimum aberration two-level and two-level fractional factorial split plot designs respectively. Using these methods they have generated an extensive list of minimum aberration designs to assist in comparing these designs. Goos and Vandebroek (2001) also develop an exchange algorithm for constructing D-optimal split plot designs. The resulting split plot designs allow substantial increases in design efficiency. Goos and Vandebroek (2001) argue that a variance ratio of 1 works fairly well over a wide range, however the designs are only D-optimal for the variance ratio specified. Attention has also been given to the analysis techniques for split plot designs. Each split plot design presents its own unique challenges. The fact that split plot designs have two sources of error as a result of the restricted randomization requires the use of the appropriate error term when performing the analysis to detect significant model terms. Bingham and Sitter (2001) summarize the work of Bisgaard (2000) which gave specific guidelines for the use of the respective error terms associated with the split plot design. The guidelines are as follows: 1. Whole plot main effects and interactions involving only whole plot factors are compared to the whole plot error.

12 2. Subplot main effects or interactions that are aliased with whole plot main effects or interactions involving only whole plot factors are compared to the whole plot error. 3. Subplot main effects and interactions involving at least one subplot factor that are not aliased with whole plot main effects or interactions involving only whole plot factors are compared to the subplot error.

The analysis of unreplicated split plot designs has also been studied. Unreplitcated split plot designs do not provide any source of error for the whole plot and therefore testing for significance of the whole plot factors is difficult. Loeppky and Sitter (2002) present two methods for analyzing unreplicated fractional factorial split plot designs. They consider the modification of the Lenth’s Methods by removing the block effect and using a perturbation method which accounts for restriction on randomization within blocks. In recent times attention has been given to applying standard RSM techniques to split plot design and analysis. Attempts have been made to apply method of steepest ascent and second-order models to split plot designs. Kowalski et al. (2005) investigate the issue of applying the method of steepest ascent to the split plot design. They utilize two equations involving Lagrange multipliers to handle the different types of factors, whole plot and subplot. They proposed three methods of performing steepest ascent, two of which involve sequential experimentation by either: 1. Fixing a whole-plot setting first, varying the subplot settings, and then varying the whole-plot settings or, 2. Varying the whole-plot settings with fixed subplot settings and then varying the subplot settings in a fixed whole-plot setting. The third method runs one experiment by varying both the whole-plot settings and subplot settings together. The issue of robust parameter design involving split plot designs has also received attention. Kowalski (2002) applies the principle of semifolding to split-plot experiments and stated that the semifolding on subplot factors must remain inside a given whole plot and explained that this leads to a more restrictive setting where semifolding is not as

13 powerful as in the general case. The work of Kowalski (2002) also extended some of the concepts of partial confounding proposed by Bisgaard (2000). The author suggests that the method works well, in the split-plot case because the aliasing problems are typically with only one or two factors. Kowalski (2002) demonstrates this method using a 24 run design. The author suggests that for all situations involving less than four whole plot factors, a general method can be used to construct 24 run designs. First, one constructs a 16 run design that uses four whole plots with four subplots in each whole plot. Further conditions given are: 1. If there are two whole plot factors, say A and B, then one uses the complete factorial design in the whole plot factors. 2. If there are three whole plot factors, say A, B, and C, then one uses one of the two half fractions found using the defining contrast I = ABC. The designs presented by Kowalski (2002) are based on augmenting the whole plots rather than increasing the number of whole plots. Further, the runs are less expensive than whole plots therefore adding them to the design to break the specific alias chains is cost-effective.

2.3 Second-Order Split Plot Designs Draper and John (1998) introduce the concept of constructing RSM designs within a split plot structure. They point out that second-order composite designs might be regarded as unsuitable for the split plot structure for the following two reasons: 1. Randomization of the run order of the hard to change factor will require it to be changed repeatedly. 2. Also that the central composite design (CCD) would require that the hard to change factor is set at five (5) levels. However, it can be argued that the use of five (5) will provide addition benefits such as degrees of freedom for lack-of-fit test. According to the authors these conditions make the CCD impractical and therefore unsuitable for split plot designs. As a result Draper and John (1998) focused their

14 attention on cube and star designs that will require less experimental levels of the hard to change factors. They illustrate how the designs can be augmented to fit the required quadratic model. They used the rotatability criteria to determine acceptable or optimal designs. Trinca and Gilmour (2001) consider the issue of randomization restrictions and outline the construction of near orthogonal multistratum designs. The term “multistratum” refers to the number of restrictions on randomization within an experimental unit. Each restriction within the experimental unit is referred to as a stratum. Therefore, having more that one stratum is a multistratum design. The two strata designs presented are considered to be a split plot design. Goos and Vandebroek (2003) stated that a major drawback with the method of Trinca and Gilmour (2001) is that the choice of their design points does not take into account the split plot error structure of the experiment. Vining et al. (2005) also address the issue of applying second-order models to split plot designs. They have demonstrated how second-order models such as CCD and Box-Behnken Designs (BBD) can be developed within a split plot structure. Their work provides a detailed proof of the conditions that guarantee the equivalence OLS and GLS coefficient estimates. The conditions outlined are: 1. Balanced design - each whole plot contains the same number of subplots. 2. Orthogonal subplot designs (not necessarily the same design). 3. Axial runs for the subplot factors are run in a single whole plot. Parker et al. (2005, 2006) generalize the conditions given by Vining et al. (2005) for the equivalence of OLS and GLS. Parker et al. (2005, 2006) used the results of their generalization to illustrate the construction of both balanced and unbalanced equivalent estimation designs for second-order split plot designs and focus on the minimum number of whole plots as an additional requirement.

15 2.4 Design Optimality Criteria Box and Draper (1975) outline some properties of “good” response surface designs which were developed by Box (1968). According to the authors all or any combination of these properties might be used for different circumstances. They stress that good designs must consider and satisfy multiple objective criteria. However, satisfying multiple criteria may require trade-offs and comprises among the criteria being considered. According to Myers et al. (1989) much of the work during the 1970s and 1980s resulted in the development of new second-order designs which were based on the concept of D-optimality. The development of optimal design theory in the field of experimental design emerged following World War II. Myers et al. (1989) further explain that optimal design theory has clearly become an important component in the general development of experimental design for the case of regression models. The optimal design theory was developed around variance reduction of the parameter estimates with regards to the fitted model. However, they warn that one major drawback with the optimal design theory is that its application is based on a set of assumptions that may not be very realistic. These assumptions include model assumptions, number of design points for a given set of factors, and estimation is the sole criteria. Atkinson (1988) gave a lengthy discussion on the optimally design theory and the general equivalence theorem. The most prominent design criterion is D-optimality which focuses on minimizing the variance of estimates of the model coefficients. Goos and Vandebroek (2003) provided an efficient algorithm to generate D-optimal designs when the number and sizes of whole plots are predetermined. Goos and Vandebroek (2004) constructed D-optimal first-order and second-order split plot designs and showed that these designs outperform completely randomized designs in terms of both D and G optimality criteria. However, the assumption was made that complete randomization is possible and does not represent the hard-to-change context. Mitchell (1974 and 2000) also presented an algorithm for the construction of D-optimal experimental designs. The decision to use a single criterion for the evaluation of an optimal design is not a good approach because there are other conditions that contribute to the performance of

16 a design. Using only a single criterion such as the D-optimal criterion finds an optimal design for the variance of the estimated coefficients and neglects the issue of variance of prediction which is also an important criterion. Myers et al. (1989) state that many standard designs have prediction variances which increase dramatically as one gets close to the design perimeter. As a result, any conclusions drawn concerning the response near the design boundry are suspect. They lament that very little work has been done that deals with prediction variances in the design assessment or comparison of designs. They explain that too often designs are evaluated on the basis of a single criterion when the important aspects of behavior are multidimensional. However, since 1989 graphical techniques have been developed to include variance dispersion graphs (VDG) by Giovannitti-Jensen and Myers (1989) and fraction of design space (FDS) plots by Zahran et al. (2003). Borkowski and Valeroso (2001) did a comparison of design optimally criteria of reduced models for response surface designs in the hypercube. Four design optimally criteria were evaluated based on their performance with respect to the design optimality criteria. The central composite, computer generated and small central composite response surface designs were used in the study. For each of the designs they considered, robustness against model misspecification was quantified by calculating A, D, G, and IV optimality criteria for the reduced second-order models. They present some interesting conclusion on how the inclusion or non-inclusion of linear, cross-product and quadratic terms affects the behavior of the alphabetic optimality criteria. They conclude that the CCD is robust with respect to the set of reduced models as well as for the four optimality criteria. However, the computer-generated designs did not perform very well across the four criteria and across the set of reduced models. In general, D, A, G, and IV are not remain the same across reduced models. They vary primarily in the following ways:

1. With the removal of pure quadratic terms and, to a lesser extent, the cross product terms. 2. When models contain differing numbers of design variables. 3. For asymmetric designs because of the dependence on the assignment of design factors to the variable labels.

2.5 Summary Despite the various research efforts the application and analysis of split plots have several areas for further research. While the value of the contributions made by researchers cannot be belittled, there are too many restrictions and assumptions that limit the general application to industrial experiments. Optimal design theory has been used in the development of split plot experiments but its application is mainly confined to the use of D-optimally and G-optimally to a lesser extent. The point was made that designs that are optimal with respect to one criterion may be poor with respect to the other criteria. It is purpose of this research work to extend the body of knowledge available for the development of split plot designs particularly in the area of second-order split plot designs.

18 CHAPTER 3

3.0 Impact of Restricted Randomization on the Structure of the Information Matrix

3.1 Abstract A thorough investigation is conducted on the impact of restricted randomization on the information matrix. The information matrix is the single most useful source of information for evaluating experimental designs. All aspects of model building and data analysis such as estimation and prediction are based on this one powerful source of information. Therefore, understanding how the information matrix is affected by restricted randomization is very important in developing optimal designs. Several important insights and implications are presented for the construction of second-order split plot designs (SPD) within a response surface structure. Specifically, the study focuses on the effect of the variance ratio of the whole plot error variance to the subplot error variance and the structure of the design, for both balanced and unbalanced SPD. The investigation reveals several critical functions of the variance ratio and whole plot size(s). It was observed that, depending on the assignment of subplot axial points, the pure quadratic terms of the subplot factor will have similar correlation influence as the whole plot pure quadratic terms. This correlation influence has a tendency to negatively affect the optimality of the design. In addition, the study provides the analytical characterization of the inverse of the information matrix for second-order SPD. A particular feature of these explicit expressions is that they are functions of the design parameters. Finally, the application of these analytical expressions is demonstrated using the generalized variance of the parameter estimates.

19 3.2 Introduction The three fundamental principles in design and analysis of experiments are randomization, replication and blocking. However, in most industrial experiments it becomes a challenge to adhere to these principles at all times. In situations where it is difficult, expensive or dangerous to adjust certain factors during experimentation it becomes necessary to reduce the frequency of adjusting this factor(s). The reduction in the frequency of adjusting certain factors during experimentation is referred to as restricted randomization. The issue of restricted randomization in response surface methods was the focus of attention in Letsinger et al. (1996). In statistical analysis, designs with restricted randomization are commonly referred to as split plot designs (SPD). Wooding (1973) presented a detailed description on the structure and use of SPD. The use of SPD is becoming increasingly popular in industrial experimentation. The split plot approach is to separate the experiment into two design spaces known as the whole plot, for hard to change factors, and subplot, for easy to change factors. Therefore, the challenge is to combine the two design spaces which will result in overall efficient experimentation according to some stated design optimality criteria such as D, G and IV optimality criteria.

3.3 The Split Plot Model The general form for split plot models in matrix form is given as

20 ⎡⎤Σ1 0...... 0 ⎢⎥0 00... ⎢⎥Σ2 Σ = ⎢⎥0 (1.3) ⎢⎥ ⎢⎥0 ⎢⎥ ⎣⎦00 0Σa

22' where a is for the number of whole plots, Σiwww=+σσδε11×11×× Iw and w is the number of th subplot runs (w hole plot size). In matrix form the structure of Σ i for th e i whole plot is

222 2 ⎡⎤σσσδεδ+ σ δ ⎢⎥ σ 2 Σ = ⎢⎥δ (1.4) i ⎢⎥2 σδ ⎢⎥2222 ⎣⎦⎢⎥σδ σσσδδε+

2 2 2 where σ δ + σ ε represents the variance of individual observations, σ δ is the covariance

2 and whole plot error variance while σ ε is the subplot error variance. Given the structure of the two error terms estimation of the model parameters is accomplished using the generalized least squares estimation approach given by

−1 Θˆ = (X''Σ−11XX) Σ− y

−1 Var ˆ X' −1X 2 2 where ()Θ = ()Σ . The two variance components, σ δ and σ ε , are needed to obtain parameter estimates and standard errors. Letsinger et al. (1996) point out that design optimality criteria, which are functions of the Var ()Θˆ , depend on the structure of the design, particularly the information matrix (A). In the case of completely randomize designs (CRD) the information matrix is given as X' X . However, for SPD alphabetic optimality is now dependent on an inform ation m atrix of the form X'Σ−1X , where Σ is the variance- covariance matrix described in Equation (1.1). This situation complicates the process

21 because now the optimal design will depend on the variance ratio or degree of correlation

2 2 η = σ δ σ ε , given by the correlation matrix (R). The correlation matrix (R) of the

22 observations is derived by dividing the variance covariance matrix, Σ, by σ δ +σ ε . In the actual computation of the information matrix the correlation matrix is used because it provides a unitless measure of the information matrix. An information matrix without units is desirable especially for design comparison purposes. As a result the information matrix would be without units and is given as

⎛⎞−1 ''⎛⎞Σ −1 AX==⎜⎟⎜⎟ X XRX ⎜⎟σσ22+ () ⎝⎠⎝⎠δε where R denotes the correlation matrix of the observations and its structure is of the form

⎡ ηη⎤ 1 ⎢ 11++η η ⎥ ⎡⎤R01 ...... 0 ⎢ ⎥ ⎢⎥⎢ η ⎥ 0R2 0... 0 ⎢⎥⎢1+η ⎥ . ⎢ ⎥ R0==⎢⎥ where Ri ⎢⎥⎢ η ⎥ 0 ⎢⎥⎢ 1+η ⎥ ⎢⎥00 0R ⎢ ⎥ ⎣a ⎦⎢ ηη⎥ ⎢ 1 ⎥ ⎣11++ηη⎦

Equation (1.5) shows the inverse of the correlation matrix (R-1) for a SPD with four whole plots of size four.

-1 ⎡R01 0 0⎤ ⎢ ⎥ 0R00-1 R−1 = ⎢ 2 ⎥ (1.5) ⎢ -1 ⎥ 00R03 ⎢ -1 ⎥ ⎣⎢000R4 ⎦⎥

22 where ⎡⎤ηη(1++ ) ηη (1 ) ()1+−η − ⎢⎥11++wwη η ⎢⎥ ⎢⎥ηη(1+ ) − ⎢⎥1+ wη R−1 = ⎢⎥. i ⎢⎥ηη(1+ ) − ⎢⎥1+ wη ⎢⎥ ⎢⎥ηη(1++ ) ηη (1 ) ⎢⎥−− ()1+η ⎣⎦11++wwηη

where 1+η represents the inverse correlation effect of individual observations within the η(1+η ) whole plots and − represents the inverse correlation co-effect within the whole 1+ wη plots. In terms of design optimality interpretation, the impact of the correlation matrix on the information matrix implies designs which are optimal in a CRD structure might not be optimal for a SPD structure. According to Goos and Vandebroek (2001 and 2004) the difference in optimality properties should not be surprising because designing a completely randomized experiment consists of determining the design points, while designing a split plot experiment requires simultaneously choosing the number of whole plots and the number of subplots within each whole plot and the design points for each plot type. Several research efforts have involved characterizing the information matrix of experimental designs. Srivastava and Chopra (1971) obtained the characteristic roots of the information matrix for balanced 2k-m fractional factorial designs of resolution V. Similarly, Hoke (1975) derived the characteristic polynomial of the information matrix for second-order models. Both works resulted in the formulation of explicit expressions for the determinant of the information matrix and the trace of the inverse of the information matrix. These two expressions are associated with D- and A-optimality criteria. Giovannitti-Jensen and Myers (1989) and Myers et al. (1992) presented some measures of the prediction variance using the eigenvalue decomposition of the inverse of

23 the information matrix. These measures included the maximum and minimum prediction variance for a first order model. Borkowski (1995) considered the case for second-order models and developed closed-form expressions for the prediction variance function for CCD and BBD designs. These closed form expressions were developed by the determination of the analytical form of the information matrix, inverting this analytical form and then pre- and post-multiplying the result by the general form of the function describing the location and the nature of the design model. The similarity with all of these efforts is that they focused on situations where it is assumed that complete randomization of the experimental run order is possible. However, in industrial experimentation there are situations that will lead to restrictions on randomization and therefore affecting the structure of the design. The current research work considers the impact of this restriction on the information matrix and presents analytical expressions for the characterization of the inverse of the information matrix for SPD within a CCD and BBD structure. These expressions will then be used to analytically determine the determinant of the information matrix.

3.4 Structure of Second-Order SPD The general form of the model vector used for the structure of second-order SPD is given as ' f ()z,x = ⎡ 2222⎤ ⎣1,,z111…… zwp x ,, x sp zz21 ,, … z wp−− z wp zx11 ,, … z wp x sp xx 121 ,, … x sp x sp z1 ,, … z wp x 1 ,, … x sp ⎦ where z and x are the whole plot and subplot factors respectively while wp and sp are the number of whole plot and subplot factors respectively. According to the structure of SPD they are four distinct categories of whole plots.

These categories are denoted as a1 to a4 for factorial, whole plot axials, subplot axials and center runs respectively. Therefore each category would have whole plot sizes of w1 to w4 respectively. In general if there are two different whole plot sizes the resulting SPD is unbalanced. Table 1 and Table 2 show typical structures of the design matrix X for a

24 balanced SPD within a CCD and BBD structure respectively. Table 3 shows an example of the design matrix X for an unbalanced SPD within a BBD structure. Note that whole plot 4 in Table 3 is of size two (2) while the other whole plots are of size four (4). For details on the construction of second-order designs within a response surface structure, see Vining et al. (2005) and Parke r ( 20 05 a n d 2 00 6). C onsi deration is given to these designs becau se th ey pro vi de eq uiv al ent estim ation of ordinary least square and generali zed le ast sq uar e. Considera tio n i s also given to the standard second-order designs when they are conducted as SP D. H ow ev er, t he se d e sign s d o not s a tisfy the equivale nt esti mation property. The notation VKM is used to denote that the Vining method of construction is used while the notation MWP is used to denote that the Parker method of construction is used. For labeling designs the first numbe r in the e xtension portion, such as VKM D12, represents the numbe r o f w hol e plo ts while th e second represents the number of subplots in the design.

Table 1: The X matrix for a CCD with one whole plot and two subplot factors (VKM D12) 2 2 2 Whole Plot z1 x1 x2 z1x1 z1x2 x1x2 z1 x1 x2 Points 1 -1- 1 -1111111 1 1 -1 1 -1-11-1111 Factorial 1 -1 -111- 1-1111 1 -1 1 1-1-11111 1 1- 1- 1-1-11111 2 1 1 1- 11-1-1111 Factorial 1 1- 1 1-11-1111 1 1 1 1111111 1 - 0 00002 00 3 1 - 0 00002 00 Whole plot axials 1 - 00000 2 00 1 - 0 00002 00 1 0 00002 00 4 1 0 00002 00 Whole plot axials 1 0 00002 00 1 0 00002 00 1 0 -α 00000α2 0 5 1 0 α 00000α2 0 Subplot axials 1 0 0 -α 00000α2 1 0 0 α 00000α2 1 0 0 0000000 6 1 0 0 000 0 00 0 Centers 1 00 0000000 1 00 0000000

Table 2: The X matrix for a balanced BBD with one whole plot and two subplot factors (VKM D12) 2 2 2 Whole Plot z1 x1 x2 z1x1 z1x2 x1x2 z1 x1 x2 Points 1 -1 -1 0 100110 1 1 -1 1 0 -100110Edge centers 1 -1 0 -1 010101(Axial Structure) 1 -1 0 1 0-10101 1 0 -1 -1 001011 2 1 0 1 -1 00-1011Factorial 1 0 -1 1 00-1011 1 0 1 1 001011 1 1 -1 0 -100110 3 1 1 1 0 100110Edge centers 1 1 0 -1 0-10101(Axial Structure) 1 1 0 1 010101 1 0 0 0 000000 4 1 0 0 0 000000 Centers 1 0 0 0 000000 1 0 0 0 000000

Table 3: The X matrix for an unbalanced BBD with one whole plot and two subplot factors (VKM D12) 2 2 2 Whole Plot z1 x1 x2 z1x1 z1x2 x1x2 z1 x1 x2 Points 1 -1 -1 0 100110 1 1 -1 1 0 -100110Edge centers 1 -1 0 -1 010101(Axial Structure) 1 -1 0 1 0-10101 1 0 -1 -1 001011 2 1 0 1 -1 00-1011Factorial 1 0 -1 1 00-1011 1 0 1 1 001011 1 1 -1 0 -100110 3 1 1 1 0 100110Edge centers 1 1 0 -1 0-10101(Axial Structure) 1 1 0 1 010101 4 1 0 0 0 000000 Centers 1 0 0 0 000000

27 3.5 Critical Functions of the Variance Ratio and Whole Plot Sizes A thorough investigation of the impact of the variance ratio on the information matrix revealed that, for both “balanced” and “unbalanced” symmetric designs, there are several critical functions (Φ) of the variance ratio and whole plot sizes that alter the values of the information matrix for second-order SPD. These critical functions were derived by pre and post multiplying a symbolic representation of the inverse of the correlation matrix (R-1) by the design matrix X. Matlab was used to assist in the determination of the symbolic representation of the information matrix XR' −1 X. The process leads to the critical functions given in Equations (1.6) and (1.7) according to the nature of the design. In the case of unbalanced SPD these critical functions are also applic able but require some modification. This modification is due to the differences in whole plot sizes. However, these modifications are of the same form as Φ1 and Φ3 using the relevant whole sizes. If the unbalanced nature of the SPD is as a result of the different whole plots sizes for the whole plot or subplot axials, then Φ6 or Φ7 is used.

Further, in the case of the subplot axials, Φ3 is modified using the necessary whole plot size associated with the subplot axials. However, if the unbalanced nature of the SPD is as a result of different whole plot size for the whole plot centers then the modification to

Φ1 is reflected using Φ8 and is applicable to the center runs. Φ9 is utilized only when the center runs are grouped with the subplot axials. In the case of balanced SPD there is only one whole plot size (w1) to consider and therefore the critical functions are:

(1+η ) Φ=1 (1 + w1η )

Φ=2 1+η ⎛⎞ηη(1 + ) Φ=−3 2⎜⎟ (1.6) ⎝⎠1 + w1η

Φ=421 or Φ if whole plot axials are grouped with factorial whole plot

Φ=Φ52 or Φ 1 if subplot axials are in sparate whole plots while Φ3 goes to zero

Φ6 = ΦΦ=ΦΦ=781 = and 9 0.

In the case of unbalanced SPD there is more than a single whole plot size to consider and therefore, the critical functions are modified as follows:

(1+η ) Φ=1 (1 + w1η )

Φ=2 1+η ⎛⎞ηη(1 + )⎛⎞ ηη (1 + ) Φ=−3 2⎜⎟ or −2⎜⎟ if unbalanced subplot axials ⎝⎠1 + ww13ηη⎝⎠ 1 +

Φ=421 or Φ if whole plot axials are grouped with factorial whole plots (1.7)

Φ=Φ52 or Φ 1 if subplot axials are in sparate whole plots while Φ3 goes to zero (1+η ) ΦΦ67, , and Φ= 8 for i =2,3,4 (1 + wiη ) ⎛⎞(1 + η )2 Φ=9 0 or 2⎜⎟ if the centers are grouped with subplot axials . ⎝⎠1 + w3η

For both the balanced and unbalanced cases the whole plot factor linear (zi) and two- factor (zizj) interaction terms are affected by critical functions of the form Φ1 while the subplot factor linear (xi) terms and its interactions (zixi and xixj) are affected by Φ2. This is an indication that for a first order model only critical functions of the form Φ1 and Φ2 are applicable.

3.6 Information Matrix for CCD and BBD Structures This section outlines the symbolic representation of the information matrix for the split plot CCD and BBD designs. The symbolic representations are done using the design parameters for the characterization of the designs. Table 4 gives the list of notations used throughout the derivation of the analytical expression for the information matrix for a CCD structure.

29 Table 4: Table of notations for the split plot CCD Notations Meanings

kk−m f # of factorial runs () 2 or 2V fw # of whole plot factor runs th wii whole plot size awii# of whole plots with size rw # of repeated whole plot axials rs # of repeated subplot axials β Whole plot axial setting α Subplot axial setting ζ Factor level setting

The information matrix for the split plot CCD can be partitioned as follows:

XR' −1 X= ⎛ 22'' 22 ⎞ ⎜ ΠΦ0J( 16frwwwζβ+ΦΦ22) p( 17 fr ζα+Φ ss) Jp ⎟ ⎜ ⎟ ⎜ 00Diag() di 0⎟ ⎜ ⎟ ⎜ ' ⎟ Φ+frζβ22Φ22J0 Φ r β 4 I +ΦΦ f ζ 4 JJJ'' f ζ 4⎡⎤J ⎜ 16w w) wp 6 w wp 1 w wp wp 1⎢⎥ sp wp ⎟ ⎜( ⎣⎦ ⎟ ⎜ 22 4'' 4 44⎟ ⎜ Φ+frζαΦ22J0 Φ f ζ JJIJ Φ+ r αΦ +ΦΦ+ fr ζαΦ2J⎟ ⎝ ( 17s ) sp 1 sp wp( 5 s 9) sp( 413 s) sp sp ⎠ (1.8)

where 0’s are zero matrices of appropriate sizes, Jwp and Jsp are unit vectors of wp×1 and sp×1 respectively, I wp and Isp are wp-dimensional and sp-dimensional identity matrices and Diag (di) are diagonal elements which are given as follows:

22 dfiw=Φ16ζβ +Φ2 rf w ori 1 ≤ ≤wp 22 =Φ2 ()frζα +2s for wpi +≤≤ 1 k , where k =+wpsp wp 4 ⎛⎞ =Φ1 ffwζ or k + 1 ≤i ≤k +⎜⎟ ⎝⎠2

4 ⎛⎞wp ⎛⎞ wp ⎛⎞ sp =Φ 2 ffζ or ki +⎜⎟ +≤≤+ 1k ⎜⎟ +×+wpsp ⎜⎟ . ⎝⎠22 ⎝⎠ ⎝⎠2

30 The first entry, Π, in Equation (1.8) is a scalar quantity. For a balanced SPD Π=N Φ1 and for an unbalanced SPD Π=aw11 Φ 1 + aw 2 2 Φ 6 + aw 33 Φ 7 + aw 4 4 Φ 8. The total number

4 of runs is Na= w. ∑i=1 ii Equation (1.8) represents a useful form of the information matrix for the split plot CCD and shows the location of the critical functions. The partitioning of the matrix is done to differentiate between the whole plot and subplot design moments. Because the information matrix is symmetric the components on either side of the main block diagonal are identical and represent the quadratic design moments of order two. The upper entries on the main diagonal represent design moments of order two for the intercept, linear and two-factor interactions. The entries of the lower block diagonal represent the quadratic design moments of order four. The following outlines a similar form for the split plot BBD. Table 5 gives the list of notations used throughout the derivation of the analytical expression for the information matrix for the split plot BBD.

Table 5: Table of notations for the split plot BBD Notations Descriptions f # of factorial runs per block () 2t

fc # of factorial runs at edge centers th wii whole plot size awii# of whole plots with size rw # of blocks within which a whole plot factor appears rs # of blocks within which a subplot factor appears t# of active subplot factors per block (t≥ 1)

λc # of edge centers

λλw # of times a pair of whole plot factors appears in the same block (w ≥ 1)

λs # of times a pair of subplot factors appears in the same block

λint # of times a whole plot factor appears with a subplot factor in the same block β whole plot factor level setting α subplot factor level setting

31 The information matrix for a symmetric SPD, based on a BBD structure, can be partitioned as shown in Equation (1.9). The partitioning of the inf ormation matrix for the split plot BBD is similar to the split plot CCD however the computations for the design moments are different. Note that some of the critical functions are not applicable because of the nature of the split plot BBD.

X'RX−1 =

⎛ ΠΦ0Jfr βα2 '' Φfr 2 J ⎞ ⎜ 1 ()wwp 1 ()ssp ⎟ ⎜ 00Diag d 0⎟ ⎜ ()i ⎟ ⎜ ⎟ ΦΦfrβλ24J0 f r−β I+ f λβ4 JJJ'' Φf λα 4J' ⎜ 11()w wp()() w w wp w wp wp ()1int sp wp ⎟ ⎜ ⎟ ⎜ 244' 444' ⎟ ⎜ ΦΦ11frsαλJ0sp fintαλ JspJIJ wpΦ5 f() r s− sα+Φ1Φ4 fλ sα+Φ3 f () r s−λ sα+ f cλ cα spJ sp ⎟ ⎝ () ( ) sp ()()⎠ (1.9) where

2 dfiw=Φ1 rfβ ori 1 ≤ ≤wp 2 =Φ2 frsα for wp + 1 ≤ i ≤ k

4 ⎛⎞wp =Φ1 ffλβw or k + 1 ≤i ≤k +⎜⎟ ⎝⎠2

4 ⎛⎞wp ⎛⎞ wp =Φ 2intffλα or ki ++≤≤++×⎜⎟ 1 k ⎜⎟wpsp ⎝⎠22 ⎝⎠

4 ⎛⎞wp ⎛⎞ wp ⎛⎞ sp =Φ2 ffλαs or k+⎜⎟ +wp ×+≤≤+ sp1. i k ⎜⎟ + wp ×+ sp ⎜⎟ ⎝⎠22 ⎝⎠ ⎝⎠2

The symbolic representation of the information matrix for the split plot CCD and BBD, given in Equations (1.8) and (1.9) respectively, illustrates how the critical functions and design points are mapped into the information matrix. The last entry of the main block diagonal for both design structures involves several critical functions. Therefore, the assi gnment of subplot axial points plays a crucial role in the application of these critical functions. The next section details the analysis of the impact of design structure on the application of the critical functions.

32 3.7 Impact of Design Structure on the Application of the Critical Functions

In general the information matrix, XR' −1 X, may be partitioned into four (4) sub- matrices according to the parameters needing to be estimated as follows:

' -1 ⎛⎞AA11 12 XR X== A ⎜⎟. ⎝⎠AA21 22

The details of the partitioned matrix A, as shown in Equation (1.8), for the split plot CCD is given as:

⎛⎞Π 0 A11 = ⎜⎟ ⎝⎠0 Diag() di ⎛⎞Φ+frζβ22Φ2222JJ' Φ+ fr ζαΦ ' A = ⎜⎟()16wwp() 17 ssp 12 ⎜⎟ ⎝⎠00 (1.10) ⎛⎞Φ+frζβ2 Φ2 2 J0 ⎜⎟()16wwp A21 = ⎜⎟Φ+frζα22Φ2 J0 ⎝⎠()17ssp ⎛ ' ⎞ Φ+2rfβζ44IJΦΦJJ' f ζ 4⎡⎤J' ⎜ 16wwp wpwp 1⎣⎦spwp ⎟ A22 = ⎜ ⎟. ⎜ ΦΦfrfζαζ444JJ' 22+Φ I+ΦΦ+Φrα4 JJ' ⎟ ⎝ 15sp wp() s9 sp()413 s spsp⎠

Further, it can be observed that A22 is partitioned into four (4) sub-matrices representing the quadratics of the whole plot and subplot terms in the model. As a result, A22 can be partitioned as

⎛⎞CC11 12 A22 = ⎜⎟ ⎝⎠CC21 22 where the components of the C matrix is given as:

33 CI=Φ2rfβζ44 + JJ' 11 1 ()w wp w wp wp CJ=Φ f ζ 4 J'' 12 1 sp wp (1.11) ' CC21= 12

444' CI22=()Φ 52rfsαζ + Φ 9sp +ΦΦ() 4 1 +Φ 3 2r sαJspJsp .

Note that both C11 and C22 repr es ent the whole plot and s ubp lo t qu ad rati c t erm s of order four respectively while C12 and C21 represent the interactio n of w ho le plo t a nd subplot quadratic terms of order four respectively. In situations where the design structure is such that t he wh ole p lot ax ials are placed in separate whole plots and the subplot axials are also placed in s epa rat e whole plots, as shown in Ta bl e 6 , t hen th e imp act o f the varian c e ra tio is th e sa m e fo r both C11 and C22. Therefore, Φ3 = 0, Φ4 = 1 , Φ5 = Φ1 and Φ9 = 0. In other words, the desig n momen ts of o rder fo ur for bo th th e subplot and whole plot axials are affected in a similar manner by Φ1. The r es ulti ng fo rm of C22 is given in Equation (1.12). Observe that Φ1 is the only critical f unction active which is also the case for C11. No te tha t in T ab le 6 the whole plot axials are placed in whole plots 5 through 8 while the subplot axials are placed in whole plots 9 through 11. Further, observe that the subplot axials are repeated in order to preserve the balanced nature of the SPD. Consequently, C22 is given as follows:

CI=Φ2rfαζ44 + JJ' . (1.12) 22 1 ( s sp sp sp )

Table 6: A resolution V CCD with = α = 1, two whole plots and three subplot factors (VKM)

Whole Plot z1 z2 x1 x2 x3 Whole Plot z1 z2 x1 x2 x3 1 -1 -1 -1 -1 1 7 0 - 00 0 -1 -1 1 -1 -1 0 - 00 0 -1 -1 -1 1 -1 0 - 00 0 -1 -1 1 1 1 0 - 00 0 2 1 -1 -1 -1 -1 8 0 00 0 1 -1 1 -1 1 0 00 0 1 -1 -1 1 1 0 00 0 1 -1 1 1 -1 0 00 0 3 -1 1 -1 -1 -1 9 0 0 - α 0 0 -1 1 1 -1 1 0 0 α 0 0 -1 1 -1 1 1 0 0 - α 0 0 -1 1 1 1 -1 0 0 α 0 0 4 1 1 -1 -1 1 10 0 0 0 - α 0 1 1 1 -1 -1 0 0 0 α 0 1 1 -1 1 -1 0 0 0 - α 0 1 1 1 1 1 0 0 0 α 0 5 - 0 0 00 11 0000 - α - 00 0 0 0000 α - 00 00 00 00 - α - 00 00 00 00 α 6 00 00 12 00 00 0 00 00 00 00 0 0 0 00 00 00 0 0 0 00 00 00 0

However, if all th e subplot axials ar e placed in the same whole plot, as shown in Table 7, then the sub plo t ax ia ls w o uld b e impacted differently by the variance ratio.

Therefore the critic al fu nc tio ns are ad justed accordingly with Φ4 = 1, Φ5 = Φ 2 and Φ9 = 0.

The resulting form of C22 is given in Equation (1.13). Observe that along with Φ1 there are tw o addition al a cti ve critical f u nc tion s given as Φ2 and Φ 3. Φ2 represent the inverse correlation effect on the individual ob servation of subplot axials while Φ3 represents the inverse correlation co-eff ect within th e whole plot. The overall effect of these critical functions is such that the design moments of order four for the subplot axials are correc ted for th e in flue nce of th e in ver se correlation co-effect within the whole plot. As a result C22 is affected as follows:

35 444' CI22=Φ 222rfrsαζαsp +( Φ 1 +Φ 3 s) J spJ sp . (1.13)

In contrast, if the whole plot axials are placed in the same whole plots as the factorial runs for t hei r respecti ve se ttings (±) then Φ4 = Φ2 and Φ9 = 0 if whole plot centers are used. T her efore C22 will now be affected based on w het her or not the subplot factors are in the same or sep ar ate w hole plots. Consequently, having the subplot axials in the same whole plot, as shown in Table 8, will result in C22 being modified as shown in Equation (1.14). Note that the inverse correlation effect is now affecting the contribution of the factorial points to the design momen ts of o rder four for the subplot as follows:

444' C22=Φ() 2 22rfsα +Φ9IJsp +ΦΦ( 2 1 ζ+ Φ 3 rsα) spJsp . (1.14)

Table 7: A CCD with = α = 5 , one whole plot factor and three subplot factors (MWP)

Whole Plot z1 x 1 x 2 x 3Whole Plot z 1 x 1 x 2 x 3 1 1 -1 -1 -1 4 00 0 1 -1 -1 1 00 0 1 -1 1 -1 00 0 1 -1 1 1 00 0 1 1 -1 -1 00 0 1 1 -1 1 00 0 1 1 1 -1 00 0 1 1 1 1 00 0 2 -1 -1 -1 -1 5 0 - α 0 0 -1 -1 -1 1 0 α 0 0 -1 -1 1 -1 0 0 - α 0 -1 -1 1 1 0 0 α 0 -1 1 -1 -1 0 0 0 - α -1 1 -1 1 0 0 0 α -1 1 1 -1 0 0 0 0 -1 1 1 1 0 0 0 0 3 - 0 0 0 - 0 0 0 - 0 0 0 - 0 0 0 - 0 0 0 - 0 0 0 - 0 0 0 - 0 0 0

Table 8: A CCD with = α = 1, one whole plot factor and three subplot factors (MWP)

Whole Plot z1 x1 x2 x3 Whole Plot z1 x1 x2 x3 1 -1 -1 -1 -1 3 1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 -1 -1 -1 1 1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 1 1 -1 1 1 1 -1 -1 1 1 1 1111 - 0 0 0 000 2 0 - α 0 0 0 α 0 0 0 0 - α 0 0 0 α 0 0 0 0 - α 0 0 0 α 0 0 0 0 0 0 0 0 0 0 0 0

In situations where we have separate whole plots for the subplot axials C22 is affected as shown in Equation (1.15). Note that Φ3 = 0 which means the inv erse correlation co-effect has no influence on the subplot axials. However, the inverse correlation effect is still affecting the contribution of the factorial points to the design moments of order four for the subplot as follows:

44' CI22=Φ 12rfsαζsp +( Φ 2 Φ 1 ) J spJ sp . (1.15)

The observations above imply that the structure of the design plays a significant role in deciding the impact of the variance ratio on the design moments of order four for the subplot factors. A similar effect exists for the split plot BBD. This type of behavior is not expected for the first order design and model because only linear and interaction terms are involved. The inverse of the information matrix plays a critical role in the determination of optimal design properties pertinent to second-order models such as the

37 maximum and integrated prediction variance. Further analytical characterization of the inverse of the information matrix would provide more insights into the effect of these critical functions on design optimality. The following section outlines the analytical characterization scheme used to determine the i nverse o f the information matrix.

3.8 Outline of Analytical Characterization Scheme This section outlines the procedure followed to derive the analytical expressions for the characterization of the inverse of the information matrix for second-order SPD. In

−1 general the inverse of information matrix (XR'1− X) may be partitioned into four (4) sub-matrices as follows:

−1 ⎛⎞BB11 12 AB= =⎜⎟. ⎝⎠BB21 22

Assuming a symmetric structure of the SPD, then both A11 and A22 are nonsingular. Consequently, it is possible to apply theorem 8.5.11 of Harville (1997) to construct the inverse of the information matrix A. The theorem makes use of a function call the Schur

−1 Complement. It turns out that the Schur Complement of A11 and A22 are equal to B11

−1 and B22 respectively. For the purposes of this work we use the Schur Complement of

A11, which is given as

−−11 BAAAA11=− 11 12 22 21 . (1.16)

In addition, Harville (1997) showed that

AA−−11=− BB 12 22 11 12 (1.17) −−11 ABBBB22=− 22 21 11 12.

38 The two equations given in (1.17) are used to solve for B12 and B22 respectively. Because of the symmetric nature of the information ma trix once B12 is known the components of

B22 are also known. The derived expressions for each component of the information matrix were computed using

1 −1 − BA11 =( 11−A 12AA 22 21 ) BB=− AA−1 12 11 12 22 (1.18) ' BB21= 12 −−11 BABBB22=+ 22 21 11 12.

Using the re sults of the E quation (1.18) the inverse of the information matrix is constructed in terms of the design parameters and variance ratio. The analytical characterization of the inverse of the information matrix was arrived at using the following procedures:

−1 1. Determin e A22 . 2. Form and invert the Schur Complement. 3. Use Equation (1.18) to compute the components of the inverse of the information matrix. Figure 2 summarizes the analytical computation scheme used for the determination of the inverse of the information matrix. Information on the principle of partitioned matrix inversion and the Schur Complement is detailed in Harville (1997) and Searle (1982).

' −1 ⎛⎞AA11 12 XR X= ⎜⎟ Partition information matrix ⎝⎠AA21 22

⇓

⎛⎞AA11 12 ⎜⎟ ⎛⎞CC Partition A ⎜⎟A 11 12 22 ⎜⎟21 ⎜⎟ ⎝⎠⎝⎠CC21 22

⇓

⎛⎞AA11 12 ⎜⎟ ⎛⎞DD Determine A-1 ⎜⎟A 11 12 22 ⎜⎟21 ⎜⎟ ⎝⎠⎝⎠DD21 22

⇓ −−11 BAAAA11=− 11 12 22 21 Formulate and invert the Schur Complem ent

⇓

−1 −1 ''−1 ⎛⎞BB11 12 −1 ()XR X = ⎜⎟ Determine the ()XR X ⎝⎠BB21 22 Figure 2: Analytical Characterization Scheme

−1 3.9 Characterization of (XR' −1 X) for Split Plot CCD

−1 In order to begin the process, the determination of the components of A22 is required. The components of A22 are given in Equation (1.11) and reflect the nature of SPD in terms of the two distinct design spaces. Because of this partitioning the principles

−1 outlined in Figure 2 above must also be applied to obtain the components of A22 .

−1 Consequently, A22 is denoted as

−1 ⎛⎞DD11 12 AD22 ==⎜⎟. ⎝⎠DD21 22

First, the formulation of the expression of the Schur Complement for C11 is given as

−11− D11 =−CCCC11 12 22 21 (1.19) where the components of the C matrix is given in Equation (1.11). Then it follows that the components of D are given as:

−1 −1 DD11= ( 11 ) DD=− CC−1 12 11 12 22 ' DD21= 12 −−11 DCDDD22=+ 22 21 11 12.

−1 In order to solve the Schur Complement in Equation (1.19), the matrix C22 must be determined. Since C22 is symmetric its inverse is of the form

−1 CIJ−1 =+abJ' 22 ( sp sp sp ) (1.20) 1 ⎛⎞b ' =−⎜⎟IJspspJsp aa⎝+ spb⎠

444 where ar=Φ592ssα +Φ and b =Φ 4 Φ13 fζα +Φ 2 r. Therefore,

1 ⎛⎞ΦΦfrζα44 +Φ2 CI−1 =−⎜⎟41 3 s JJ' . 22 Φ+2r α 4 Φ⎜⎟spsΦ+22rsαζ44Φ+ΦpΦ+frΦα4psp 59s ⎝⎠594ss()13

For details of the principle involved in finding the inverse of matrices of the form given in Equation (1.20) see Graybill (1969). This method is used extensively throughout this derivation process. See Appendices A1 and A2 for details. The result of

41 the computations after some tedious matrix manipulation and linear algebra, the

−1 components of D, and by extension the components of A22 where determined as

1 ⎛⎞c DI=−1 ⎡ JJ' ⎤ 11 44⎜⎟wp⎢ wp wp ⎥ ΦΦ66122rrwwββ⎝⎠+wpc⎣ ⎦ Φ f ζ 4 1 '' (1.21) DJ12 =− spJ wp c2 ' DD21= 12

2 ⎛⎞ΦΦfRrcζα44 +22 − Φ r α 4 +Φ Φ fw ζ 4p 1 ⎜⎟()41 3ss 2(( 5 9)( 1 )) DI=− JJ' . 22 Φ+2r α 4 Φ⎜⎟sp 444sp sp 59s ⎜⎟()Φ+59422rsssαζΦ+Φp()Φ+132frΦαc ⎝⎠ where

2 4 (Φ1 fsζ ) p cf=Φζ 4 − 11w 444 Φ+59422rsssαζΦ+Φp()Φ+13frΦα

4444 cr25=Φ()22ssαζ +Φ+Φ9sp() 4 Φ 1 f +Φ 3 rα() Φ+ 12 rwwβpc 1.

Returning to the information matrix, A, and applying the results obtained in

−1 Equation (1.21) for A22 the Schur Complement for A11 can now be formed and inverted. This then leads to the analytical characterization of the inverse of the information matrix for a CCD structure. Recall that the Schur Complement for A11 is given in Equation (1.16) and that the components of matrix A are defined in Equation (1.10). The components of B can now be determined using Equation (1.18). The results are

42 ⎛⎞γ1 0 ⎜⎟ B11 = ⎛⎞ ⎜⎟0 Diag ⎜⎟1 ⎝⎠⎝⎠di ⎛⎞γγJJ'' B = 23wp sp 12 ⎜⎟ ⎝⎠00 (1.22) ⎛⎞γ 2J0wp B21 = ⎜⎟ ⎝⎠γ 3J0sp ⎛⎞1 ' IJ−γγJJ''⎡⎤J ⎜⎟Φ 2r β 4 ()wp45 wp wp⎣⎦ sp wp B = ⎜⎟6 w . 22 ⎜⎟1 γγJJ'' I− JJ ⎜⎟56sp wp 4 ()sp sp sp ⎝⎠Φ+592rsα Φ

By combining the results of B11, B12, B21 and B22, the components of the inverse of the

−1 information matrix can be obtained. Therefore (XR' −1 X) for CCD is given as

⎛ γγγ0J'' J⎞ ⎜ 123wp sp ⎟ ⎜ ⎛⎞ ⎟ 000Diag ⎜⎟1 ⎜ ⎝⎠di ⎟ ⎜ ⎟ 1 ' . ⎜γγJ0 IJ− JJ''γ⎡⎤J ⎟ ⎜ 24wp Φ 2r β 4 ()wp wp wp5⎣⎦ sp wp ⎟ ⎜ 6 w ⎟ 1 ⎜ γγJ0 JJI''−γJJ⎟ ⎜ 35sp sp wp 4 ()sp6 sp sp ⎟ ⎝ Φ+592rsα Φ ⎠

Table 9 summarizes the scalar quantities represented by γ i in the inverse of the information matrix while Table 10 provides the details of the pertinent constants (c) that

−1 were derived. The inverse of the information, (XR' −1 X) , represents a useful form that will facilitate an understanding of the contribution of each of the distinct design spaces to the optimality of the design.

43 Table 9: Table of scalar quantities for the split plot CCD Sca lars Equations 1 γ1 c11

c5 γ 2 − c11 c γ − 9 3 c11 4 2 c1 Φ652rcwβ () γ 4 4 − Φ+6112rwwβ pc c1 4 Φ1 f ζ cc59 γ 5 −+ cc211 4 2 ()Φ+592rcsα Φ()9 γ 67c − c11

Table 10: Table of constant quantities for the split plot CCD Constants Equations 2 4 ()Φ1 fsζ p cfΦ−ζ 4 11w 444 Φ+59422rsssαζΦ+Φpfr()Φ+13Φα

4444 cr259()Φ+22ssαζΦ+Φsp()4Φ+1fΦ36rα() Φ2rwβ +wpc1 22 Φ+16frwwζβΦ2 c3 4 Φ+612rwwβ pc 42 2 ΦΦ+11ffζζ()Φ 72 rss αp c4 − c2 cc53+ c4 42 2 ΦΦ+11ffζζ()wwΦ 62 rw βp c6 − c2 2 ΦΦf ζ 4 +Φ22rcααζ444 − Φ r +Φ Φ f wp ()41 325ss(( 91)()) c 7 444 ()Φ+5942rsssαζ Φ+Φp()Φ+132frΦ2αc 22 ()Φ+17frcζαΦ21s () − 7sp c 8 4 Φ+592rsα Φ cc96+ c8

2222 cc10 5()Φ+ 1frwwζβΦ 622wp +Φ+c 9() 1fr ζαΦ 7 ssp cc11Π− 10

44 −1 3.10 Characterization of (XR' −1 X) for Split Plot BBD The same analytical computational scheme was followed for the BBD structure.

−1 Therefore, the ()XR' −1 X for BBD is given as

⎛ γγ0J'' γ J⎞ ⎜ 12wp 3sp ⎟ ⎜ ⎛⎞ ⎟ 00Diag ⎜⎟1 0 ⎜ ⎝⎠di ⎟ ⎜ ⎟ . 1 ' ⎜γγJ0 IJ− JJ''γ⎡⎤J ⎟ ⎜ 24wp Φ−fr()λβ4 ()wp wp wp5⎣⎦ sp wp ⎟ ⎜ 1 ww ⎟ 1 ⎜ γγJ0 JJ'' IJ−γJ⎟ ⎜ 35sp sp wp 4 ()sp6 sp sp ⎟ ⎝ Φ−5 fr()ssλα ⎠

Table 11 summarizes the scalar quantities represented by γ i in the inverse of the information matrix while Table 12 provides the details of the pertinent constants (c) that were derived.

Table 11: Table of scalar quantities for the split plot BBD Scalars Equations 1 γ1 c11

c5 γ 2 − c11

c9 γ 3 − c11 4 2 c1 Φ−15f ()rcwwλβ() γ 4 4 − Φ−11fr()wwλβ + wpc c11 4 Φ1intfcλα 59c γ 5 −+ cc211 4 2 Φ−59fr()ssλα( c) γ 67c − c11

45 Table 12: Table of constant quantities for the split plot BBD Constants Equations 2 4 ()Φ1intfsλα p cfΦ−λβ4 11w 44 44 Φ−51fr()ssλα+Φ sp()Φ4 f λα s +Φ3() fr () ss − λα + f cc λα cfΦ−rλα44 +ΦspΦff λα +Φ−+rff λα 4 λα4 Φ−+rw λ β 4pc 25()()ss() 143 s() () ss cc() 1() ww 1 2 Φ1 ()frwβ c3 4 Φ−11fr()wwλβ + wpc

2 24 ()Φ1ifrsαλαnt sp c4 − c2 c53c+ c4 2 2 4 ()Φ1 frwβ λαint wp c6 − c2 2 ΦΦfλα44 +Φ fr− λα + fλα4c− Φ fr − λ α 4Φ fw λ α4p ()14ssc 3()()sc2 ( 5() s s(1 int )) c7 ΦΦfr−λα4+ spΦfλ α4 +Φ fr − λα 4 +fλ α4c ()51()ss (4s 3() ()ss cc) 2 Φ−rα 2 1 csp 17()s () c8 4 Φ−5 ()rsλsα cc96+ c8 22 cc10 5ΦΦ 1frwβwpf+c9 1 rsαsp cc11Π− 10

3.11 Generaliz ed Va ria nc e of Pa ra met er Est ima tes To demonstrate th e u se of t he se ana lytic al expressions we obtain t he generalize d varianc e of para me ter estim ates which is measured by the D-optim a lity cr iteri on . Th e D- optima lity crite rion is the de term inant of t he info rmation m a trix. In o rde r to d eter m ine the determinant of the infor mation m atrix (A), theorem 13.3.8 of Harvil le (1 9 97) was applied. See App end ix B1 fo r details. According to the theo rem , if a m atrix A is partitioned such that A11 is nonsingular and the inverse of A is given as B, then the determinant of A is com p uted a s fo llo ws:

A A = 11 . (1.23) B22

3.11.1 Determinant for split plot CCD Using Equation (1.23) the analytical expressions for the determinant of second- order split plot CCD is given as follows:

wp sp cw c cs 22 224 4+ ΠΦ()16frζβ +Φ22ws( Φ 2( frf ζ +αζζ)) () Φ 1() Φ 2 f XR' −1X = . wp−1 ⎛⎞4 2 sp−1 ⎛⎞1−γ 4wp ⎛11⎞1−γ 6sp Φ652rwwβγ() psp⎛ ⎞ ⎜⎟44⎜⎟⎜⎟ 4− ⎜4 ⎟ Φ22rββ Φr⎜⎟ Φ+ 2 r αΦ 1 − γ wp Φ+ 2 r αΦ ⎝⎠61w⎝w⎠⎝⎠ 59 s 4⎝ 59s ⎠

3.11.2 Determinant for split plot BBD Using Equation (1.23) the analytical expressions for the determinant of second- order split plot BBD is given as follows:

wp sp cw c cs ΠΦfrβαλ22 Φ fr Φ fβλ 4 Φ fαλ 44 Φ f α ()1212( ) ()ws()int()2 XR' −1 X = ws . wp−1 2 sp−1 ⎛⎞11−γ wp ⎛⎞⎛⎞1−γ sp Φ−fr()λβγ4 () wpsp ⎛⎞1 ⎜⎟4 ⎜⎟⎜⎟6 − 15ww ⎜⎟ ⎜⎟Φ−frλβ44⎜⎟ Φ− fr λβ⎜⎟ Φ− fr λα 41 − γ wp ⎜⎟ Φ− fr λα4 ⎝⎠11()ww⎝⎠() ww⎝⎠ 5() ss 45 ⎝⎠()ss

⎛⎞wp where cw = ⎜⎟ is the number of cross whole plots, c = wp * sp is the number of cross ⎝⎠2 ⎛⎞sp products and cs = ⎜⎟ is the number of cross subplots. ⎝⎠2

See Appendices B2 and B3 for details on the derivation of these equations. The actual

1/ p computation used for the D-criterion is XR' −1 X where p is the number of parameters to be estimated.

47 3.12 Application To demonstrate the application of these equa tions, the D-op timality criterion was computed for the BBD g iven in T ab le 2 and the CCD given in Table 6. In addition, a modified design of th e CCD i n Table 6 was also studied . This m odified design is given in Table 13. The modified de s ign ha s two who le plots l ess tha n the VKM D23 due to the fact that the all the subplot axials are pl aced in o ne wh ol e plot . T his m eans the modified design also requires few er run s.

Table 13: M odified VK M D2 3 with = α = 1, tw o who l e plots a nd thre e subplot factors – Unbalanced

Whole Plot z1 z2 x1 x2 x3 Whol e Plot z1 z2 x1 x2 x3 1 -1 -1 -1 -1 1 7 0 - 0 0 0 -1 -1 1 -1 -1 0 - 0 0 0 -1 -1 -1 1 -1 0 - 0 0 0 -1 -1 1 1 1 0 - 0 0 0 2 1 -1 -1 -1 -1 8 0 0 0 0 1 -1 1 -1 1 0 0 0 0 1 -1 -1 1 1 0 0 0 0 1 -1 1 1 -1 0 0 0 0 3 -1 1 -1 -1 -1 9 0 0 0 0 0 -1 1 1 -1 1 0 0 0 0 0 -1 1 -1 11 00 0 0 0 -1 1 1 1 -1 0 0 0 0 0 4 1 1 -1 -1 1 10 0 0 - α 0 0 1 1 1 -1 -1 0 0 α 0 0 1 1 -1 1 -1 0 0 0 - α 0 1 1 1 1 1 0 0 0 α 0 5 - 0 0 0 0 0 0 0 0 - α - 0 0 0 0 0 0 0 0 α - 0 0 0 0 - 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

The parameters for the two CCD designs are given in Table 14. The results are presented in Table 15 and are computed for three (3) commonly used variance ratios of η = 0.5, 1 and 10. Both the standardized and scaled D-criterion values are given.

48 Table 14: Design parameters for the split plot CCD Parameters VKM D23 Modified VKM D23 f 16 16

a1 12 9

a3 0 1

w1 4 4

w3 0 6 wp 2 2 sp 3 3 N 48 42

rw 4 4

rs 2 1 1 1 α 1 1 1 1 p 21 21

An examination of the critical functions conveys the influence of the design structures on the application of these functions. Note that for the VKM D23 design given in Table 6 the subplot axial points are in separate whole plots. Therefore Φ3 = 0 indicates that the design moments for the subplot axial runs are not influenced by the inverse correlation co-effect within the whole plot. The resulting impact is that both the whole plot axial points and the subplot axial points are affected by the same critical function Φ1.

In contrast, the critical functions Φ3 is active for the modified design and therefore influence the subplot design moments. Note that value of Φ3 as shown in Table 15 for the modified design is negative across the variance ratios. Therefore, the design moments for the subplot axial runs are corrected for the influence of the inverse correlation co-effect within the whole plot. For both designs Φ4 = 1 which indicates that the factorial points are not affected by the inverse correlation effect within the whole plot.

On the other hand, Φ5 = Φ1 for the VKM D23 design which also indicates that the subplot axial points are not affected by the inverse correlation effect within the whole plot.

However, for the modified design Φ5 = Φ2 meaning that the subplot ax ial points are affected by the invers e corr elation e ffect within the whole plot and is a direct consequence of placing al l the subp lot axials in the sam e wh ole p lot.

49 Table 15: Constants, functions and D-value results for CCD across the different variance ratios VKM D23 Modified VKM D23 Variance Ratio (η) 0.5 1 10 0.5 1 10 Constants

c1 0.615 0.492 0.330 0.242 0.185 0.117

c2 136.000 87.040 39.158 111.000 70.583 31.511

c3 2.294 2.294 2.294 2.676 2.689 2.705

c4 -1.765 -1.765 -1.765 -1.892 -1.896 -1.902

c5 0.52 9 0.529 0.529 0.784 0.793 0.803

c6 -1.412 -1.412 -1.412 -1.730 -1.741 -1.754

c7 0.23 5 0.2 35 0.235 0.153 0.031 -2.164

c8 1.471 1.4 71 1.471 1.577 1.580 1.585

c9 0.05 9 0.0 59 0.059 -0.153 -0.161 -0.170

c10 14.4 71 11.5 76 7.765 14.791 11.861 7.980

c11 9.52 9 7.62 4 5.113 5.460 4.253 2.760 Scalars

1 0.10 5 0.1 31 0.196 0.183 0.235 0.362

2 -0.05 6 -0.0 69 -0.104 -0.144 -0.186 -0.291

3 -0.00 6 -0.0 08 -0.012 0.028 0.038 0.061

4 0.000 0.000 0.000 -0.396 -0.421 -0.453

5 -0.056 -0.069 -0.104 -0.094 -0.121 -0.186

6 0.235 0.235 0.235 0.140 0.007 -2.393 Critical Functions

Ф1 0.500 0.400 0.268 0.500 0.400 0.268

Ф2 1.500 2.000 11.000 1.500 2.000 11.000

Ф3 0.000 0.000 0.000 -0.375 -0.571 -3.607

Ф4 1.000 1.000 1.000 1.000 1.000 1.000

Ф5 0.500 0.400 0.268 1.500 2.000 11.000

Ф6 0.500 0.400 0.268 0.500 0.400 0.268

Ф7 0.500 0.400 0.268 0.375 0.286 0.180

Ф8 0.500 0.400 0.268 0.500 0.400 0.268 Standardized D value 14.252 15.266 34.077 14.073 15.801 43.015 Scaled by Design Size 0.297 0.318 0.710 0.335 0.376 1.024

Further, it can be observed that for the VKM D23 design Φ6 = Φ7 = Φ8 = Φ1 which is reflecting the balanced nature of the design. However, for the modified design

Φ6 = Φ8 = Φ1 while Φ7 is assigned the appropriate value to reflect the unbalanced nature of the design due to the subplot axials. Note also that Φ6 and Φ8 are the same for both designs because whole plot axials and whole plot centers do not make either design

50 unbalanced. It is important to note that the design structure does not affect the application of the critical functions Φ1 and Φ2 as their values remain unchanged. Therefore, for a first or der model the impact of the critical functions will not influence the information mat rix beyon d Φ1 and Φ2. In fact, the linear and interaction terms of the first-order model are only influenced by Φ1 and Φ2. However, the first entry in the information matrix for the fir st o rder desig ns will be changed in a similar manner as explained for second-order des ign s. A discussion on the co mparison of the D-optimality criterion is presented next. The standardized D-value shows that for η = 1 and η = 10 the modified VKM D23 is more desirable while for η = 0.5 the V KM D23 appears to be better. However, scaling the D-value by the desig n size in dicate that the modified VKM D23 is more desirable across t he variance ratios. The relative efficiencies are given in Table 16 for the standardized and scaled D-valu es. The relati ve efficiencies for the standardized D-values are very similar except for η = 10 wher e th e VKM D23 design is 79%. In contrast, for the scaled D-values ther e are c lea r diffe renc es. Fo r η = 10 the relative efficiency is 69% for the VKM D23 design while for η = 0.5 an d η = 1 the relative efficiencies are 89% and 85% respectively. As a note o f c aution , sc aling by N is not necessarily the best way to scale the designs since it does not take into account the cost associated with number of whole plots . Liang et a l. (2006) prese nt so me su ggestions on incorporating cost in the decision ma king process . How eve r, for these two designs if a cost function is considered the modified VKM D23 design would show an even higher level of desirability. This is because the design is m ore e ffic ient i n terms of the number of whole plots and the number of runs required. Therefore, for this example scaling by the design size (N) represents a conservative measure of the D-optimality criterion.

Table 16: Relative efficiencies Standardized Scaled by Design Size Variance Ratio (η) 0.5 1 10 0.5 1 10 VKM D23 100% 97% 79% 89% 85% 69% Modified VKM D23 99% 100% 100% 100% 100% 100%

51 To complete the demonstration of these derived expressions the BBD design given in Table 2 is presented. The design is balanced and the design parameters are given in Table 17. The results are presented in Table 18.

Table 17: Design parameters for the BBD Parameters Settings Parameters Settings

f 2 rw 4

fc 4 rs 4

t 1 λw 1

a1 4 λs 2

a2 0 λint 2

w1 4 λc 0

w2 0 1 wp 1 α 1 sp 2 p 10 N 16

An important observation to make is that the application of the critical function is restriction to Φ1 through Φ5. The main reason for the restriction is the nature of the BBD design structure which dictates that the factorial or edge center whole plots have similar structure and the same whole plot size.

52 Table 18: Constants, functions and D-value results for BBD across the different variance ratios VKM D12-BBD Variance Ratio (η) 0.5 1 10 Constants

c1 -0.333 -0.267 -0.179

c2 16.000 10.240 4.607

c3 1.500 1.500 1.500

c4 -1.000 -1.000 -1.000

c5 0.500 0.500 0.500

c6 -0.500 -0.500 -0.500

c7 -0.250 -0.750 -9.750

c8 1.000 1.000 1.000

c9 0.500 0.500 0.500

c10 6.000 4.800 3.220

c11 2.000 1.600 1.073 Scalars

1 0.500 0.625 0.932

2 -0.250 -0.313 -0.466

3 -0.250 -0.313 -0.466

4 -0.500 -0.500 -0.500

5 0.000 0.000 0.000

6 -1.000 -2.000 -20.000 Critical Functions

Ф1 0.500 0.400 0.268

Ф2 1.500 2.000 11.000

Ф3 -0.500 -0.800 -5.366

Ф4 1.000 1.000 1.000

Ф5 1.500 2.000 11.000 Standardized D value 5.468 5.943 14.088 Scaled by Design Size 0.342 0.371 0.881

53 3.13 Conclusion Our investigation on the impact of restricted randomization on the information matrix has led to the identification of several critical functions of the variance ratio and whole plot size. The study reveals some important and useful insights for the construction of second-order SPD and provides an understanding of the relationships among the critical functions identified and the design structure. The structure of the design does have an influence on how the critical functions are applied to the information matrix. This influence is particularly evident with the design moments of order four for the su bplot facto rs. In situations where the design structure is such that the whole plot axials are placed in separate whole plots and the subplot axials are also placed in separate whole plots then the effect of the appropriate critical function is the same for both the whole plot and subplot design moments of order four. However, if all the subplot axials are placed in the same whole plot then the subplot axials would be affected by the inverse correlation effect and co-effect within the whole plot. The overall effect is such that the design moments of order four for the subplot are corrected for the influence of the correlation co-effect within the whole plot. In situations where the whole plot axials are placed in the same whole plots as the factorial runs and all the subplot axials in the same whole plot then the inverse correlation effect is now affecting the contribution of the factorial points to the design moments of order four for the subplot. In contrast, if we have separate whole plots for the subplot axials then the inverse correlation co-effect has no influence on the subplot axials. However, the inverse correlation effect is still affecting the contribution of the factorial points to the design moments of order four for the subplot. In addition, the analytical characterization of the inverse of the information matrix was achieved for both balanced and unbalanced second-order SPD. The critical functions identified are used throughout the analytical characterization process. The entire characterization process was done using the design parameters therefore making it possible for practitioners to assess the impact of changing any of the design parameters on the optimality of the design. The results of this analytical characterization process were then used to derive explicit functions for the D-optimality criterion for second-order models. The application of these functions was demonstrated by the evaluation of the

54 generalized variance of parameter estimates for several second-order designs. The use of the D-optimality criterion was strictly for demonstration purposes and to show how design properties can be obtained and designs can be compared. It is clearly recognized that prediction variance properties are more appropriate for the evaluation of second- order designs. Currently, work is underway to apply the results of the characterization of the inverse of information matrix to analytically determine the other design optimality criteria such as maximum and integrated prediction variance for second-order SPD.

55 CHAPTER 4

4.0 Prediction Variance Properties and G-Criterion Location for Second-Order Split Plot Designs

4.1 Abstract Prediction variance properties for completely randomized designs (CRD) are fairly well covered in the response surface literature for both spherical and cuboidal designs. However, for split plot designs (SPD) there are still many unanswered questions about the predication capabilities of SPD for second-order response surface models. This paper evaluates the impact of changes in the variance ratio on the prediction properties of second-order SPD. It is shown that the variance ratio not only influences the value of the G-criterion but also its location, in contrast with the G-criterion tendencies in CRD. An analytical method, rather than a heuristic optimization algorithm is used to compute the prediction variance properties, which include the maximum, minimum, and integrated variances for second-order SPD. The analytical equations are functions of the design parameters, radius and variance ratio. As a result, the exact values for these quantities are reported along with the location of the maximum prediction variance used in the G- criterion. For various designs, these measures are graphically represented using variance dispersion graphs. The results indicate that the G-criterion has a tendency to be located at the center of the whole plot factor and at the extreme of the subplot factor(s). The two distinct design spaces of the whole plot and the subplot are separated and studied. As a result, relative efficiency values for these distinct spaces are suggested.

56 4.2 Introduction In recent years there is a growing interest in conducting and analyzing industrial experiments in which there are restrictions on randomization. In statistical designs, experimentation with a restriction on randomization is referred to as split plot designs (SPD). SPD are used when there are factors which are difficult or costly to adjus t during an experiment. The split plot appr oa ch is to separate the experiment into two design spaces known as the whole plot, for the hard to change factors and the subplot, for the easy to change factors. Wooding (1973) gives a complete outline of the nature and use of split plot designs. The issue of the location of the maximum prediction variance within the design space might be intuitive for CRD but for SPD it is not as clear, because the design optimality criteria are jointly affected by a function of the variance ratio and whole plot sizes. It is therefore important that the maximum prediction variance be reported together with its location in the design space. The variance ratio can be thought of as applying relative weights to each of the factors within the model to emphasize the relative importance of each factor. This emphasis results in changes in the magnitude and location of the maximum prediction variance. As a result, the traditional location of the maximum prediction variance in the design space, which is farthest from the design center, might not necessarily be the same location for SPD. Graphical techniques have been developed to facilitate an understanding of prediction variance dispersion within a particular region of interest for a given design. The two graphical techniques currently available are the variance dispersion graph (VDG) developed by Giovannitti-Jensen and Myers (1989) and fraction of design space (FDS) plots developed by Zahran et al. (2003). Both techniques plot the prediction variance values for regions within the design space and do not necessarily give the exact location of the maximum predication variance. These graphical techniques have been used for comparing and evaluating competing designs as demonstrated by Myers et al. (1992) and Park et al. (2005). Liang et al. (2006) demonstrated the application of these graphical techniques to SPD. This research investigates the impact of changes in the variance ratios on the location of the maximum prediction variance as well as the performance of the prediction variance in the whole plot and subplot spaces relative to the design. For each of the selected designs, the

57 maximum and integrated prediction variances are reported and analyzed. Specifically, the designs considered are second-order SPD for both spherical and cuboidal regions.

4.3 The Split Plot Model The general form for split plot models in matrix form is given as

⎡⎤Σ1 00...... ⎢⎥0 00... ⎢⎥Σ2 Σ = ⎢⎥0 (1.24) ⎢⎥ ⎢⎥0 ⎢⎥ ⎣⎦00 0Σa

2' 2 where a is for the number of whole plots, Σiw=σδ11×11××ww+σεIw and w represents the th number of subplot runs (whole plot size). In matrix form the structure of Σ i for the i whole plot is represented as

222 2 ⎡⎤σσσδεδ+ σ δ ⎢⎥ σ 2 Σ = ⎢⎥δ (1.25) i ⎢⎥2 σδ ⎢⎥2222 ⎣⎦⎢⎥σδ σσσδδε+

58 2 2 2 where σ δ + σ ε represents the variance of individual observations, σ δ is the covariance

2 and whole plot error variance while σ ε is the subplot error variance. Given the structure of the two error terms, estimation of the model parameters is accomplished using the

−1 generalized least squares method of estimation given by Θˆ = (X''Σ−−11XX) Σ y , where

−1 ˆ ' −1 2 2 Var ()Θ = ()X Σ X . The two variance components σ δ and σ ε are needed to obtain parameter estimates and standard errors. Letsinger et al. (1996) point out that design optimality criteria, which are functions of the Var ()Θˆ , depend on the structure of the design, particularly the information matrix (A). In the case of CRD the information matrix is given as X' X . However, for SPD alphabetic optimality is now dependent on an information matrix of the form X'Σ−1X , where Σ is the variance-covariance matrix described in Equation (1.1). This situation complicates the process because here the optimal design depends on the

2 2 variance ratio or degree of correlation η = σ δ σ ε , given by the correlation matrix (R). The correlation matrix (R) of the observations is derived by dividing the variance

22 covariance matrix, Σ, by σ δ +σ ε . In the actual computation of the information matrix the correlation matrix is used because it provides a unitless measure of the information matrix. An information matrix without units is desirable especially for design comparison purposes. As a result the information matrix would be without units and is given as

⎛⎞−1 ''⎛⎞Σ −1 AX==⎜⎟⎜⎟ X XRX ⎜⎟σσ22+ () ⎝⎠⎝⎠δε where, R-1 denotes the inverse of the correlation matrix and it is of the form

59 −1 ⎡⎤R01 ...... 0 ⎢⎥−1 ⎢⎥0R2 0... 0 −1 R = ⎢⎥0 . (1.26) ⎢⎥ ⎢⎥0 ⎢⎥−1 ⎣⎦00 0Ra

4.4 Prediction Optimality Criteria The G-optimality criterion is associated with the prediction variance v(z, x) and seeks to minimize the maximum prediction variance over the region (Ξ). In response surface methods, the two most commonly encountered region shapes are the spherical and cuboidal regions. G-optimality emphasizes the use of design points for which the maximum v(z, x) in the region of the design is not too large. Thus a G-optimal design (ξ ) is one in which

⎡ ⎤ Min Max v()zx, ξ ⎣⎢ x∈Ξ ⎦⎥ and v(z, x) is given as

Var⎡⎤yˆ (zx , ) −1 vfzx,,==⎣⎦ zx' XR' −1 Xf zx, () 22 ()()() σσδε+ where f ()zx, is the general form of the model vector for second-order SPD and is given as

f z,x ' = () ⎡ 2222⎤ ⎣1,,z111…… zwp x ,, x sp zz21 ,, … z wp− z wp zx11 ,, … z wp x sp xx 121,,………xsp− x sp z1 ,, z wp x 1 ,, x sp ⎦ (1.27) where, z and x are the whole plot and subplot factors respectively while wp and sp are the number of whole plot and subplot factors respectively. The integrated prediction variance (IV) criterion is used to generate a single average measure of prediction performance throughout the region of interest.

60 Computationally, this is done by integrating the prediction variance v(z, x) over Ξ. The standardized IV criterion for a split plot design can be expressed as follows:

1 IV= vd()zx , zxd Ω ∫ Ξ 1 −1 = ∫ f ()zx,,' () XR' −1 Xfd() zx zxd Ω Ξ where, Ω= ddzx is the volume of the region Ξ. ∫Ξ

4.5 Second-Order Split Plot Designs Draper and John (1998) introduce the concept of constructing response surface designs within a split plot structure. The paper focuses on cube and star designs that require less experimental levels of the hard to change factors. The authors illustrate how the designs can be augmented to fit the required quadratic model. The rotatability criterion was used to determine acceptable or optimal designs. Vining et al. (2005) also address the issue of applying second-order response surface models to split plot designs. They have demonstrated how second-order designs such as Central Composite Design (CCD) and Box-Behnken Designs (BBD) can be developed within a split plot structure. Their work includes a detailed proof of the conditions that guarantee the equivalence of ordinary least squares (OLS) and generalized least squares (GLS) coefficient estimates. The conditions outlined are that each whole plot contains the same number of subplots, the subplot runs must be orthogonal and the axial runs for the subplot factors are contained in a single whole plot. Parker et al. (2005) generalize the conditions given by Vining et al. (2005) for the equivalence of OLS and GLS. In a subsequent paper, Parker et al. (2006) used the results of their generalization to illustrate the construction of balanced and unbalanced equivalent estimation designs for second-order split plot models and focus on the minimum number of whole plots as an additional criterion. A balance SPD refers to designs with equal whole plot sizes while unbalance SPD refers to designs with several whole plot sizes. Due to the nature and structure of split plot designs it is widely accepted that second-order response surface designs which require the least

61 amount of factor level changes at the whole plot level are most desirable. This requirement therefore makes the CCD with α = 1 and BBD designs very attractive for second-order SPD. These designs require only three factor level settings, which is the minimum number required for second-order response surface designs. For the purposes of this paper designs such as the CCD and BBD will be evaluated for their prediction variance properties when they are being executed within a split plot structure. Parker (2005) has developed a catalog of all such split plot designs accounting for different split plot scenarios with respect to whole plot sizes, whole plot factors, subplot factors and design augmentation. All the designs in the catalog satisfy the equivalent estimation property.

4.6 Critical functions of the variance ratio and whole plot sizes In chapter 3, the study of the information matrix for SPD reveals that there are several critical functions (Φ) of the variance ratio and whole plot sizes. The locations o f these critical functions in the info rmation matrix are given in Equations (1.28) and (1.30) for split plot CCD and split plot BBD. According to the structure of SPD there are four distinct categories of whole plots, denoted a1 through a4. These categories have whole plot sizes of w1 through w4 for factorial, whole plot axials, subplot axials and center runs respectively. In the case of balanced SPD there is only one whole plot size (w1) to consider and therefore the critical functions are:

(1+η ) Φ=1 (1 + w1η )

Φ=2 1+η ⎛⎞ηη(1 + ) Φ=−3 2⎜⎟ ⎝⎠1 + w1η

Φ=421 or Φ if whole plot axials are grouped with factorial whole plot

Φ=Φ52 or Φ 1 if subplot axials are in sparate whole plots while Φ3 goes to zero

Φ6 = ΦΦ=ΦΦ=781 = and 9 0.

62 In the case of unbalanced SPD there are several whole plot sizes to consider and therefore, the critical functions are modified as follows:

(1+η ) Φ=1 (1 + w1η)

Φ=2 1+η ⎛⎞ηη(1 + )⎛⎞ ηη (1 + ) Φ=−3 2⎜⎟ or −2⎜⎟ if unbalanced subplot axials ⎝⎠1 + ww13ηη⎝⎠ 1 +

Φ=421 or Φ if whole plot axials are grouped with factorial whole plots

Chapter 3 gives a full discussion on the impact of these functions on the information

−1 matrix and the analytical derivation of (XR' −1 X) . An outline of the results is presented in this chapter.

4.7 Information Matrix for the Split Plot CCD Using the parameters characterizing the split plot CCD as given in Table 19, the information matrix for the split plot CCD can be partitioned as in Equation (1.28) and its inverse given in Equation (1.29). These equations are applicable to both balanced and unbalanced SPD and for both spherical and cuboidal regions.

Table 19: Table of notations for the split plot CCD Notations Meanings

kk−m f # of factorial runs () 2 or 2V

fw # of whole plot factor runs th wii whole plot size awii# of whole plots with size rw # of repeated whole plot axials rs # of repeated subplot axials β Whole plot axial setting α Subplot axial setting ζ Factor level setting

The information matrix for the split plot CCD is given as

XR'1− X= ⎛ 22'' 22 ⎞ ⎜ ΠΦ0J( 16frwwwζβ+ΦΦ22) p( 17 fr ζα+Φ ss) Jp ⎟ ⎜ ⎟ ⎜ 00Diag() di 0⎟ ⎜ ⎟ . ⎜ ' ⎟ Φ+frζβ22Φ22J0 Φ r β 4 I +Φ f ζ 4 JJ'Φfζ4⎡⎤JJ' ⎜ 16w w) wp 6 w wp 1 w wp wp 1⎢⎥sp wp ⎟ ⎜( ⎣⎦ ⎟ ⎜ 22 4'' 4 44⎟ ⎜ Φfζα+Φ22rJ0 Φ f ζ JJIJ Φ+ r αΦ +ΦΦ+ fr ζαΦ2J⎟ ⎝ ( 17s ) sp 1 sp wp( 5 s 9) sp( 413 s) sp sp ⎠ (1.28)

where 0’s are zero matrices of appropriate sizes, Jwp and Jsp are unit column vectors of wp×1 and sp×1 respectively, I wp and I sp are wp-dimensional and sp-dimensional identity matrices and Diag (di) are diagonal elements which are given as follows:

64 22 dfiw=Φ16ζβ +Φ2 rf w ori 1 ≤ ≤wp 22 =Φ2 ()frζα +2s for wpi +≤≤ 1 k , where k =+wp sp

4 ⎛⎞wp =Φ1 ffwζ or k + 1 ≤i ≤k +⎜⎟ ⎝⎠2

4 ⎛⎞wp ⎛⎞ wp ⎛ sp⎞ = Φ2 ffζ or k ++⎜⎟ 1≤i≤k++× ⎜⎟wpsp+ ⎜⎟. ⎝⎠22 ⎝⎠ ⎝2⎠

The first entry, Π, in Equation (1.28) is a scalar quantity. For a balanced SPD Π=N Φ1 and for an unbalanced SPD Π=aw11 Φ 1 + aw 2 2 Φ 6 + aw 33 Φ 7 + aw 4 4 Φ 8. The total number

4 of runs is Na= w. ∑i=1 ii

−1 Therefore ()XR' −1 X for both balanced and unbalanced split plot CCD is given as

⎛ γγ0J'γJ'⎞ ⎜ 123wp sp ⎟ ⎜ ⎛⎞ ⎟ 000Diag ⎜⎟1 ⎜ ⎝⎠di ⎟ ⎜ ⎟ . (1.29) 1 ' ⎜γγJ0 IJ− JJ''γ⎡⎤J ⎟ ⎜ 24wp Φ 2r β 4 ()wp wp wp5⎣⎦ sp wp ⎟ ⎜ 6 w ⎟ 1 ⎜ γγJ0 JJI''− γJJ⎟ ⎜ 35sp sp wp 4 ()sp6 sp sp ⎟ ⎝ Φ+592rsα Φ ⎠

Table 20 summarizes the scalar quantities represented by γ i in the inverse of the information matrix while Table 21 provides the details of the pertinent constants (c) that were derived.

Table 20: Table of scalar quantities for the split plot CCD Scalars Equations 1 γ1 c11

c5 γ 2 − c11 c γ − 9 3 c11 4 2 c1 Φ652rcwβ () γ 4 4 − Φ+6112rwwβ pc c1 4 Φ1 f ζ cc59 γ 5 −+ cc211 4 2 ()Φ+592rcsα Φ()9 γ 67c − c11

Table 21: Table of constant quantities for the split plot CCD Constants Equations 2 4 ()Φ1 fsζ p cΦ−fζ 4 11w 444 Φ+59422rsssαζΦ+Φpfr()Φ+13Φα

4444 cr259()Φ+22ssαζΦ+Φsp()4Φ+1fΦ361rα() Φ2rwβ +wpc Φ+frζβ22Φ2 c 16ww 3 4 Φ62rwwβ +pc1 42 2 ΦΦ+11ffζζ()Φ 72 rss αp c4 − c2

cc53+ c4 42 2 ΦΦ+11ffζζ()wwΦ 62 rw βp c6 − c2 2 ΦΦf ζ 4 +Φ22rcααζ444 − Φ r +Φ Φ f wp ()41 32591ss(()()) c 7 444 ()Φ+5942rsssαζ Φ+Φpfr()Φ+132Φ2αc 22 ()Φ+17frcζαΦ21s () − 7sp c8 4 Φ+592rsα Φ

cc96+ c8

2222 cc10 5()Φ+ 1frwwζβΦ 622wp +Φ+c 9() 1fr ζαΦ 7 ssp

cc11Π− 10

4.8 Information Matrix for the Split Plot BBD The parameters characterizing the split plot BBD are given in Table 22. Using these parameters the information matrix for the split plot BBD can be partitioned as in Equation (1.30) and its inverse given in Equation (1.31).

Table 22: Table of notations for the split plot BBD Notations Descriptions f # of factorial runs per block () 2t

fc # of factorial runs at edge centers th wii whole plot size

awii# of whole plots with size

rw # of blocks within which a whole plot factor appears

rs # of blocks within which a subplot factor appears t# of active subplot factors per block (t≥ 1)

λc # of edge centers

λλw # of times a pair of whole plot factors appears in the same block (w ≥ 1)

λs # of times a pair of subplot factors appears in the same block

λint # of times a whole plot factor appears with a subplot factor in the same block β whole plot factor level setting α subplot factor level setting

The information matrix for the split plot BBD is given as

XR' −1 X=

⎛ ΠΦ0Jfr βα2 '' Φfr 2 J ⎞ ⎜ 1 ()wwp 1 ()ssp ⎟ ⎜ 00Diag d 0⎟ ⎜ ()i ⎟ . ⎜ ' ⎟ 244'' 4⎡⎤ ⎜ΦΦ11frwβλJ0 wp f() r w− wβ I wp+ f λ wβ J wpJJ wp Φ1intf λαspJ wp ⎟ ⎜ () ()()⎣⎦⎢⎥ ⎟ ⎜ 244' 444' ⎟ ⎜ ΦΦ11frsαλJ0sp fintαλ JspJIJ wpΦ5 f() r s− sα+Φ1Φ4 fλ sα+Φ3 f () r s−λ sα+ f cλ cα spJ sp ⎟ ⎝ () ( ) sp ()()⎠ (1.30)

67 where

2 dfiw=Φ1 rβ fori 1≤ ≤wp 2 =Φ2 frsα for wp + 1 ≤ i ≤ k

4 ⎛⎞wp =Φ1 ffλβw or k + 1 ≤i ≤k +⎜⎟ ⎝⎠2

4 ⎛⎞wp ⎛⎞ wp = Φ2intffλα or k ++⎜⎟ 1≤ i≤k++× ⎜⎟ wpsp ⎝⎠22 ⎝⎠

4 ⎛⎞wp⎛⎞ wp ⎛⎞sp =Φ2 ffλαs or k +⎜⎟ +×+≤≤+ wpsp1ik⎜⎟+×+wp sp ⎜⎟. ⎝⎠2 ⎝⎠22 ⎝⎠

−1 Therefore, the (XR' −1 X) for both balanced and unbalanced split plot BBD is given as

⎛ γγ0J'γJ'⎞ ⎜ 12wp 3sp ⎟ ⎜ ⎛⎞ ⎟ 000Diag ⎜⎟1 ⎜ ⎝⎠di ⎟ . (1.31) ⎜ ⎟ 1 ' ⎜γγJ0 IJ− JJ''γ⎡⎤J ⎟ ⎜ 24wp Φ−fr()λβ4 ()wp wp wp5⎣⎦ sp wp ⎟ ⎜ 1 ww ⎟ 1 ⎜ γγJ0 JJ'' IJ− γJ⎟ ⎜ 35sp sp wp 4 ()sp 6 sp sp ⎟ ⎝ Φ−5 fr()ssλα ⎠

Table 23 summarizes the scalar quantities represented by γ i in the inverse of the information matrix while Table 24 provides the details of the pertinent constants (c) that were derived.

68 Table 23: Table of scalar quantities for the split plot BBD Scalars Equations 1 γ1 c11

c5 γ 2 − c11

c9 γ 3 − c11 4 2 c1 Φ−15f ()rcwwλβ() γ 4 4 − Φ−11fr()wwλβ + wpc c11 4 Φ1intfcλα 59c γ 5 −+ cc211 4 2 Φ−59fr()ssλα( c) γ 67c − c11

Table 24: Table of constant quantities for the split plot BBD Constants Equations 2 4 ()Φ1intfsλα p cfΦ−λβ4 11w 44 44 Φ−51fr()ssλα +Φ sp()Φ4 f λα s +Φ3() fr () ss − λα + f cc λα cfΦ−rsλα44 +ΦpΦff λα +Φ−+rff λα 4 λα4 Φ−+r λ β 4wpc 25()()ss() 143 s() () ss cc() 1() ww 1 2 Φ1 ()frwβ c3 4 Φ−11fr()wwλβ + wpc

2 24 ()Φ1ifrsαλαnt sp c4 − c2

cc53+ c4 2 2 4 ()Φ1 frwβ λαint wp c6 − c2 2 ΦΦffλα44 +Φrfc − λα + λα4 − Φfr − λα 4 Φ fw λα4p ()14ss 3()()scc2( 5() ss() 1 int ) c7 Φ−frλα44 +Φ spΦ f λα +Φ fr − λα 4 + f λα4 c ()51()ss()4 s3() () ss cc 2 2 Φ−17()rcsα ()1 sp c8 4 Φ−5 ()rssλα

cc96+ c8 22 cc10 5Φ+ 1frwsβαwpc 9Φ 1 frsp

cc11Π− 10

69 A similar characterization of the inverse of the information matrix for CRD was presented by Borkowki (1995). However, setting η = 0 and the whole plot size to 1 gives the same result for a CRD. The split plot design can be considered the more general structure while the CRD actually represents a specific instance of the SPD.

4.9 Minimum and Maximum Prediction Variance Borkowski (1995) developed closed-form expressions for the determination of the exact minimum, maximum and spherical (or average) prediction variances on the surface of a sphere of any radius ρ. The principles he outlined are adopted here for SPD. However in extending the method to SPD, provision is made to account for the impact of the variance ratio and the number of whole plots. To analytically characterize the prediction variance v(z, x) for SPD, the inverse of the information matrix (as in Equations (1.29) and (1.31)) was pre- and post-multiplied by the model vector in Equation (1.27). The result of this matrix operation is presented in Equation (1.32) for the split plot CCD and Equation (1.33) for the split plot BBD. For details of this matrix operation see Appendices C1 and C2.

v()z,x =

wp sp wp sp ⎛⎞⎛⎞1122 ⎛⎞ 1 22 γγ12++⎜⎟⎜⎟222∑∑zxii ++ γ 3 ++ ⎜⎟ γ 5 ∑∑z i xj ⎝⎠⎝⎠ddiiii==11 ⎝⎠ d i i = 1j=1 wp 2 wp . (1.32) ⎛⎞11γ 4 ⎛⎞24⎛⎞1 +−⎜44⎟⎜⎟∑∑zzii+⎜⎟ − ⎝2diwΦ662rββ⎠⎝⎠ii==11⎝⎠Φ 22 rdwi 2 ⎛⎞11γ ⎛⎞sp ⎛⎞1 sp 6 x2 x4 ⎜⎟−+44⎜⎟∑ i ⎜⎟− ∑ i ⎝⎠22drisΦ+59ααΦ⎝⎠i=1 ⎝⎠ Φ+ 5 2 r sΦ9 2di i=1

70 v()z,x =

wp sp wp sp ⎛⎞⎛⎞1122⎛⎞122 γγ12++⎜⎟⎜⎟22∑∑zxii ++ γ 3 +2 ⎜⎟γ 5+∑∑z ixj ⎝⎠⎝⎠ddiiii==11 ⎝⎠ d i i = 1j=1 2 ⎛⎞11γ ⎛⎞wp ⎛⎞1wp . (1.33) +−4 zz24 + − ⎜⎟⎜⎟∑ ii⎜⎟∑ ⎝⎠22dfiwΦ−11()rλλw⎝⎠ii==11⎝⎠ Φ− f()r wwi d 2 ⎛⎞11γ ⎛⎞sp ⎛⎞1 sp −+6 x2 − x4 ⎜⎟⎜⎟∑ i ⎜⎟∑ i ⎝⎠2dfisΦ−55()rλs⎝⎠i=1 ⎝⎠Φfr()ss− λ 2 d ii=1

Therefore the general form for prediction variance v(z, x) may be expressed as

2 2 wp sp⎛⎞ wp wp⎛⎞ sp sp wp sp 22 2 4 2 4 22 vA()z,x =+++B∑∑∑zijCxD⎜⎟zE i ++ ∑∑z iFxG⎜⎟ j ++ ∑∑x jHz∑ixj ij===11⎝⎠ i 1 i = 1⎝⎠ j = 1 j == 1 i 1j=1 (1.34) where the values of A, B, C, D, E, F and H are given in Table 25 for both the split plot CCD and the split plot BBD.

The optimization problem therefore is to

wp sp 222 max vz()z,x subject to ∑∑ij+ x= ρ . (1.35) ij==11

22 2 Without loss of generality the constraint can be expressed as ρ zx+ ρρ= where

wp 22 ∑ ziz= ρ corresponds to the whole plot design space i=1 sp 22 ∑ x jx= ρ corresponds to the subplot design space. j=1

22 Substituting ρz and ρxinto Equation (1.34) the general form of v(z, x) may be simplified to

wp sp 224 44 4 22. (1.36) vA()z,x =++++BCDEρz ρρxz∑∑z i ++FG ρ xx j +H ρ zρx ij==11

Table 25: Table of coefficients of the variance function v(z, x) for CCD and BBD Coefficients CCD BBD

A γγ11 11 B2γγ++2 2222 Φ fr Φ+1 ()frζβ2 w 1 w 11 C2γγ++ 2 3322 Φ fr Φ+2 ()frζα2 s 2 s

11γγ44 D 44−− 222ΦΦ16frζ wwβλ ΦΦ− 11 ff()rw λw 11 1 1 E 44−− ΦΦΦ61122rfwwβ ζλ f()r−Φw 2 1 fλw

11γγ66 F 44−− 222ΦΦ25frζ sαλλ+ΦΦΦ− 925 ffss()rs 1111 G 4 − 4 − Φ+52rsα Φ922ΦΦ25f ζ fr()s −Φλλss2 f

⎛⎞11⎛⎞ H2⎜γγ5++4 ⎟2⎜⎟5 ⎝Φ2ffζλ⎠⎝⎠Φ2int

4.10 Vminρ and Vmaxρ for Spherical Regions The principle of Langrage multipliers is used to find the critical points to analytically determine the maximum (V max ρ ) and minimum (V min ρ ) value for the predication variance, v(z, x) at radius ρ. See Appendix D1. Table 26 summarizes the three (3) critical points.

Table 26: Critical Points for the evaluation of v(z, x)

Location Critical Points (zxij ,) Conditions ⎛⎞ρ 2 1 Whole plot space ⎜⎟±= , 0 ρ 2 0 ⎜⎟wp− n x ⎝⎠wp ⎛⎞ρ 2 2 Subplot space ⎜⎟ 0, ±= ρ 2 0 ⎜⎟sp− n z ⎝⎠sp ⎛⎞ρ 2 ρ 2 3 Design space ⎜⎟±±z , x ρ 22 + ρρ=2 ⎜⎟wp−− n sp n zx ⎝⎠()wp() sp

Evaluation of Equation (1.36) according to the critical points given in Table 26 results in the exact computation of V min ρ and V max ρ for whole plot, subplot and combined design spaces. Appendix D2 details the derivation of the evaluation of Equation (1.36) according to the critical points given in Table 26 for spherical regions. The results are now presented for the respective design spaces.

4.11 Vmin and Vmax for Whole Plot Design Space for Spherical Regions ρz ρz The exact computation of V min and V max values for the whole plot design ρz ρz space are given in Equation (1.37). Therefore depending on whether E > 0 or E ≤ 0, the locations of V min and V max would occur at zw= ρρp,, wp and any ρz ρz ( 1 … wp ) permutation of z = ()ρ12,0 ,… ,0wp respectively. Depending on whether E > 0 or E ≤ 0, the minimum and maximum prediction variances for the subplot are computed as follows:

73 ⎧⎫24⎛⎞E ⎪AB++ρρ⎜⎟D+ for E > 0 ⎪ V min = wp ρz ⎨⎝⎠ ⎬ ⎪⎪AB+++ρρ24() DE for E ≤ 0 ⎩⎭. (1.37) ⎧⎫AB+++ρρ24() DE for E > 0 ⎪⎪ V max = ρz ⎨⎬24⎛⎞E ⎪⎪AB+++ρρ⎜⎟ D for E ≤ 0 ⎩⎭⎝⎠wp

4.12 Vmin and Vmax for Subplot Design Space for Spherical Regions ρx ρx Similar to the whole plot design space the exact computation of V min and ρx V max values for the subplot design space are given in Equation (1.38) and their ρx locations occur at x = ρρsp,, sp and any permutation of x = ρ ,0 , ,0 ()1 … sp ()12… sp respectively. Depending on whether G > 0 or G ≤ 0, the minimum and maximum prediction variances for the subplot are computed as follows:

⎧⎫24⎛⎞G ⎪⎪AC+++ρρ⎜⎟ F for G > 0 V min = sp ρx ⎨⎬⎝⎠ ⎪⎪AC+ρ2++()FGρ4for G ≤ 0 ⎩⎭. (1.38) ⎧⎫AC+++ρρ24() FG for G > 0 ⎪⎪ V max = ρx ⎨⎬24⎛⎞G ⎪⎪AC+++ρρ⎜⎟ F for G ≤ 0 ⎩⎭⎝⎠sp

4.13 Vminρ and Vmaxρ for Combined Design Space for Spherical Regions Combining both the whole plot and subplot design spaces will give the spherical region or the region of operability for the design. Therefore, the exact computation for

V min ρ and V max ρ for the region of operability (combined design spaces) can be determined a s given in Equation (1.39). Therefore depending on the combination of whether E > 0 or E ≤ 0 and G > 0 or G ≤ 0, their respective locations would occur at the respective combination of zw= ρρp,, wp or z = ρ ,0 , ,0 for the ()zz1 … wp ( zw2 … p)

74 whole plot terms and x = ρρsp,, sp or x = ρ ,0 , ,0 for the subplot ()xx1 … sp ( xs2 … p) terms. Depending on the combination of whether E > 0 or E ≤ 0 and G > 0 or G ≤ 0, the minimum and maximum prediction variances are computed as follows:

⎧⎫22⎛⎞⎛⎞EG 4 4 22 ⎪⎪AB++++ρρzx C⎜⎟⎜⎟ D ρ z ++ F ρ x + H ρ zρxfor E and G > 0 ⎪⎪⎝⎠⎝⎠wp sp ⎪⎪224422 AB+++++++ρzx Cρρρ()() DE z FG x Hρ zρx for E and G ≤ 0 ⎪⎪ V min ρ = ⎨⎬22⎛⎞E 4 4 22 AB++++ρρzx C⎜⎟ D ρ z +++() FG ρ x H ρ zρxfor E > 0 and G ≤ 0 ⎪⎪wp ⎪⎪⎝⎠ ⎪⎪ 22 4⎛⎞G 422 ⎪⎪AB++++++ρρzx C() DE ρ z⎜⎟ F ρρxz+≤H ρxfor E 0 and G > 0 ⎩⎭⎪⎪⎝⎠sp

22 4 4 22 ⎧⎫AB+++++++ρzx Cρρρ()() DE z FG x Hρ zρx for E and G > 0 ⎪⎪ ⎛⎞⎛⎞EG ⎪⎪AB++++ρρ22 C D ρ 4 ++ F ρ 4 + H ρ 2ρ2for E and G ≤ 0 ⎪⎪zx⎜⎟⎜⎟ z x zx ⎝⎠⎝⎠wp sp ⎪⎪ V max ρ = ⎨⎬22 44⎛⎞G 22 ⎪⎪AB++++ρρzx C() DEρρzx++⎜⎟FH +ρzρxfor E > 0 and G ≤ 0 ⎪⎪⎝⎠sp ⎪⎪ 22⎛⎞E 4 4 22 ⎪⎪AB++++ρρzx C⎜⎟ D ρ z +++() FG ρ x H ρ zρxfor E ≤ 0 and G > 0 .(1.39) ⎩⎭⎪⎪⎝⎠wp

4.14 Vminρ and Vmaxρ for Cuboidal Regions

The evaluation of Vminρ and Vmaxρ for cuboidal regions requires the consideration

of two conditions where 0 ≤ ρ ≤ 1 and ρ > 1. For the case 0 ≤ ρ ≤ 1 both Vminρ and

Vmaxρ are the same for the spherical and cuboidal regions because for ρ ≤ 1 the hyper- sphere is a subset of the hypercube. However, for ρ > 1 the hypercube is restricted to the corners of the cuboidal region. Appendix D3 details the derivation of Vminρ and Vmaxρ for the cuboidal region. The results are now presented.

75 4.15 Vmin and Vmax for Whole Plot Design Space for Cuboidal Regions ρz ρz The exact computation of V min and V max values for the whole plot design ρz ρz space are given in Equation (1.40). Therefore, depending on whether E > 0 or E ≤ 0, the locations of V min and V max occur at zw= ρρp,, wp and ρz ρz ( 1 … wp )

22 z =±1,…… ± 1, ±ρρ −⎣⎦⎢⎥ , 0, 0 respectively. The number of occurrences at ±1 is ( )

2 2 given by ⎣⎦⎢⎥ρ which is the greatest integer ≤ ρ . Depending on whether E > 0 or E ≤ 0 the minimum and maximum prediction variances for the whole plot are computed as follows:

⎧⎫24⎛⎞E ⎪⎪AB+++ρρ⎜⎟ D for E > 0 ⎪wp ⎪ V min = ⎝⎠ ρz ⎨⎬ 2 ⎪⎪24⎛⎞ 222 AB+++ρρ D E⎜⎟⎢⎥ ρρ +−() ⎢⎥ρ for E ≤ 0 ⎩⎭⎪⎪⎝⎠⎣⎦ ⎣⎦ . (1.40) 2 ⎧⎫24⎛⎞ 222 ⎪A+BDEρρ++⎜⎟⎣⎦⎢⎥ ρρρ +−() ⎣⎦⎢⎥ for E > 0 ⎪ ⎪⎝⎠⎪ V max = ρz ⎨⎬ 24⎛⎞E ⎪⎪AB++ρρ⎜⎟D+ for E ≤ 0 ⎩⎭⎪⎪⎝⎠wp

4.16 Vmin and Vmax for Subplot Design Space for Cuboidal Regions ρx ρx Similar to the whole plot design space the exact computation of V min and ρx V max values for the subplot design space are given in Equation (1.41). Thus ρx depending on whether G > 0 or G 0, the locations of V min and V max occur at ≤ ρx ρx

x = ρρsp,, sp and x =±1, ± 1, ±ρρ22 −⎢⎥ , 0, 0 respectively. ()xx1 … sp ……⎣⎦ ( ) Depending on whether G > 0 or G ≤ 0, the minimum and maximum prediction variances for the subplot are computed as follows:

76 ⎧⎫24⎛⎞G ⎪⎪AC+++ρρ⎜⎟ F for G > 0 ⎪⎪sp V min = ⎝⎠ ρx ⎨⎬ 2 ⎪⎪24⎛⎞ 2 22 AC+++ρρ F G⎜⎟⎢⎥ ρρρ +−() ⎢⎥ for G ≤ 0 ⎪⎪⎩⎭⎝⎠⎣⎦ ⎣⎦ . (1.41) 2 ⎧⎫24⎛⎞ 2 22 ⎪⎪AC+++ρρ F G⎜⎟⎣⎦⎢⎥ ρρρ +−() ⎣⎦⎢⎥ for G > 0 ⎪⎪⎝⎠ V max = ρx ⎨⎬ 24⎛⎞G ⎪⎪AC+++ρρ⎜⎟ F for G ≤ 0 ⎩⎭⎪⎪⎝⎠sp

4.17 Vminρ and Vmaxρ for the Combined Design Space for Cuboidal Regions Combining b oth th e whole p lot and s ubplot design spaces will give the cuboidal region or the region of operability for the design. Therefore, the exact computation for

V min ρ and V max ρ for the region of operability (combined design spaces) can be determined as give n in E q u ati on ( 1.42). Therefore depending on the combination of whether E > 0 or E ≤ 0 and G > 0 or G ≤ 0, their respective locations would occur at the respective combination of zw= ρρp,, wp or ( z1 … zwp ) 22⎢⎥ z =±1,……± 1,±ρρzz −⎣⎦, 0 , 0 for the whole plot terms and ( )

x = ρsp ,,ρsp or x =±1, ±1,±ρ2 −⎢⎥ρ2, 0 , 0 for the subplot ()x1 … xsp …x⎣⎦x… ( ) terms. Depending on the combination of whether E > 0 or E ≤ 0 and G > 0 or G ≤ 0, the minimum and maximum prediction variances are computed as follows:

77 ⎧ 22⎛⎞⎛⎞EG 4 4 22 ⎫ ⎪AB++++ρρzx C⎜⎟⎜⎟ D ρ z ++ F ρ x + H ρρ zx for E and G > 0 ⎪ ⎪ ⎝⎠⎝⎠wp sp ⎪ ⎪ 2 ⎪ 224⎛⎞ 2 22 4 ⎪AB++++ρρρ C D E⎜⎟⎢⎥ ρ +− ρρ ⎢⎥ ++ F ρ… ⎪ zxz⎝⎠⎣⎦ z() zz ⎣⎦ x ⎪ for E and G ≤ 0 ⎪ 2 ⎪ ⎛⎞222 22 ⎪ V min = GH⎜⎟⎢⎥ρρρxxx+− ⎢⎥ + ρ zρx ρ ⎨ ⎝⎠⎣⎦() ⎣⎦ ⎬ ⎪ ⎪ 2 ⎪ 22⎛⎞E 442222⎛⎞2 ⎪ AB++++ρρzx C⎜⎟ D ρ z + Fρρρρρxxxxz++GH⎜⎟⎢⎥−+ ⎢⎥ ρxfor E > 0 and G ≤ 0 ⎪ wp ⎝⎠⎣⎦() ⎣⎦ ⎪ ⎪ ⎝⎠ ⎪ ⎪ 2 ⎛⎞G ⎪ 224⎛⎞⎢⎥ 2 22 ⎢⎥ 4 22 ⎪AB++++ρρρzxz C D E⎜⎟⎣⎦ ρ z +−() ρρ zz ⎣⎦ ++⎜⎟ F ρ x + H ρρ zxfor E ≤ 0 and G > 0⎪ ⎪⎩ ⎝⎠⎝⎠sp ⎪⎭

2 ⎧ 2242224⎛⎞ ⎫ AB+++ρρρ C D ++EF⎜⎟⎢⎥ρρρ−++ ⎢⎥ ρ… ⎪ zxz⎝⎠⎣⎦zzz() ⎣⎦ x ⎪ ⎪ for E and G > 0 ⎪ 2 ⎪ ⎛⎞222 22 ⎪ GH⎜⎟⎢⎥ρρρ+− ⎢⎥ + ρρ ⎪ ⎝⎠⎣⎦xxx() ⎣⎦ zx ⎪ ⎪ ⎪ ⎛⎞⎛⎞EG ⎪AB++++ρρ22 C D ρ 4 ++ F ρ 4 + H ρρ 22 for E and G ≤ 0 ⎪ V max = zx⎜⎟⎜⎟ z x zx ρ ⎨ ⎝⎠⎝⎠wp sp ⎬ ⎪ ⎪ .(1.42) 2 ⎪ 224⎛⎞ 2 22 ⎛⎞G 422 ⎪ AB++++ρρρzxz C D E⎜⎟⎢⎥ ρ z +− ρρ zz ⎢⎥ ++⎜⎟ F ρρxz+≤H ρxfor E > 0 and G 0 ⎪ ⎝⎠⎣⎦() ⎣⎦ sp ⎪ ⎪ ⎝⎠ ⎪ ⎪ ⎛⎞E 2 ⎪ 22 44⎛⎞⎢⎥2 22 ⎢⎥ 22 ⎪AB++++ρρzx C⎜⎟ D ρρ zx ++ F G⎜⎟⎣⎦ρx +−() ρρ xx ⎣⎦ + H ρρ zxfor E ≤ 0 and G > 0⎪ ⎪⎩ ⎝⎠wp ⎝⎠ ⎪⎭

78 4.18 Delta G-Criterion Measures for Whole Plot and Subplot Design Spaces A consequence of the analytical characterization process is the successful separation of the prediction variance according to the contributions from the two distinct design spaces, namely the whole plot and subplot spaces. This separation allows for the investigation of the performance of the design optimality criteria in the combined and separate design spaces of the SPD. As a result, a measure of the performance of the design optimality criterion in the whole plot and subplot spaces relative to the combined design space can be determined. The following Equations are used to compute the relative measures. Since Vmaxρ (combined) is either equal to or greater than Vmax (subplot) or Vmax (whole plot) then the delt a G-criterion ( ∆G ) measure of the ρx ρz separate design spaces can be determined as follow s:

Subplot space VVmax− max ρ ρx ∆=Gx Vmax ρ Whole plot space VVmax− max ρ ρz ∆=Gz . Vmax ρ

In terms of comparing designs, Vmaxρ for the best design identified should be used.

4.19 Prediction Variance Assessment of Second-Order SPD 4.19.1 CCD with one whole plot factor and two subplot factors In this example a standard CCD with one whole plot factor and two subplot factors as shown in Table 27 will be utilized. Table 28 shows the prediction variance results and the location of the maximum prediction variance for the different variance ratios. It can be observed that the location of the maximum prediction variance for η = 0.5 and η = 1 remain the same but differs for η = 10. The location for η =0.5 and 1 is at the extreme of the whole p lot and a t the center of the subp lot. Howev er for a η =10 the location is at the center of the whole plot and subplot. Therefore such moveme nt of the loca tio n is a n i ndicat ion that c hanges in the variance ratio is affecting the location in the

79 desi gn spac e w here the max im um p redi ction variance w ill occ u r. Figure 3 shows the variance dispersion graph for the design which presents further insights in understanding the nature of the prediction variance throughout the design space. As expected, the value of the prediction variances increases as η increases. The results also indicate that the maximum prediction variances in the subplot space are less than or equal to the maximum prediction variances in both the whole plot and combined design spaces.

Table 27: Standard CCD with = α = k and N=17

Factors WP w # of runs per z1 x1 x2 whole plot 1 4 -1 ±1 ±1 4 2 4 1 ±1 ±1 4 3 1 - 0 0 1 4 1 0 0 1 0 ± 0 2 5 4 α 0 0 ±α 2 6 4 0 0 0 3

Table 28: Prediction variance properties and G-criterion location. Standardized Scaled Location of Max Vmax Vmax Vmax Vmax η ρx ρz Vmaxρ IV ρx ρz Vmaxρ IV z1 x1 x2 0.5 0.620 0.747 0.747 0.420 10.537 12.705 12.705 7.139 1.732 0.000 0.000 1 0.667 0.783 0.783 0.431 11.333 13.312 13.312 7.331 1.732 0.000 0.000 10 0.939 0.939 0.939 0.454 15.970 15.970 15.970 7.716 0.000 0.000 0.000

80 η = 0.5 η = 1 η = 10

15 15 15 10 SPV 10 SPV 10

SPV 5

0 5 1 5 0 1 0 1 0 0.2 0.8 0.2 0.8 0.2 0.6 0.6 0.4 0.4 0.5 0.4 0.6 0.4 0.6 0.6 0.4 0.8 0.8 0.2 0.8 0.2 0 ρ 0 ρ ρ ρ 1 sp ρ 1 sp ρ 1 0 sp wp wp wp

Figure 3: 3D VDGs for the standard CCD with one whole plot factor and two subplot factors

To further understand the performance of each of the design spaces the predication variance for both spaces along with the maximum prediction variance of the design is plotted as shown in Figure 4. It can be observed that the subplot space is performing better than the whole plot space for the variance ratios presented.

η = 0.5 η = 1 η = 10

16 16 16

14 14 14

12 12 12

10 10 10

8 8 8 SPV SPV SPV

6 6 6

4 4 4

2 2 2

0 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Radius Radius Radius

Whole plot space Subplot space G max Design space Figure 4: Comparison of whole plot and subplot spaces for the standard CCD with one whole plot factor and two subplot factors

81 Table 29 presents the relative measures of the whole and subplot design spaces to the combined design spaces. The subplot design space is 17% and 15% better than the whole plot space when compared to the combined design space for η = 0.5 and η =1. However, for η = 10, the relative measures of whole plot and subplot spaces are 0% because their maximum prediction variances are the same as that of the combined design space. Even though both design spaces recorded 0% efficiency the scaled prediction variance of the subplot space is consistently less than that of the whole plot space.

Table 29: Relative measures of whole plot and subplot design spaces

η ∆Gx ∆Gz 0.5 17% 0% 1 15% 0% 10 0% 0%

To improve the performance of the standard CCD the design was modified as shown in Table 30. The subplot axials are placed in separate whole plots which also include one additional center run. This modification res ults in the number of whole plots remaining same b ut the nu m ber of runs reduces from 17 to 16. Since the only difference between the two designs is the number of runs, the comparison of the designs can be done by scaling the designs by N. Both designs would incur the same cost for the number of who le p lo ts sinc e the nu mbe r o f whole p lo ts r em ained the same.

Table 30: Modified CCD with = α = k and N=16 Facto rs WP w # of runs per z1 x1x 2 whole plot 1 4 -1 ±1 ± 1 4 2 4 1 ±1 ±1 4 3 1 - 0 0 1 4 1 0 0 1 0 ± 0 2 5 3 α 0 0 0 1 0 0 ± 2 6 3 α 0 0 0 1

82 Table 31 gives the prediction variance results for the modified CCD. The modification resulted in the stabilization of the location for the maximum prediction variance which now appears at the extreme of the whole plot and the center of the subplot regardless of the variance ratio. Further, the relative efficiency of the maximum scaled prediction variance is 7% better for = 0.5 and = 1 and 13% better for = 10 when compared to the values given in Table 28 for the standard CCD. There is also improvement in the integrated prediction variances and the maximum prediction variance for the whole plot and subplot design spaces. Figure 5 gives the 3D VDGs which shows improvement in the stability of the prediction variance for the modified design.

Table 31: Prediction variance properties and G-criterion location for the modified CCD Standardized Scaled Location of Max Vmax V max VVmax max z x x η ρx ρz Vmaxρ IV ρx ρz Vmaxρ IV 1 1 2 0.5 0.645 0.739 0.739 0.422 10.322 11.820 11.820 6.749 1.732 0.000 0.000 1 0.615 0.774 0.774 0.409 9.844 12.386 12 .386 6.547 1.732 0.000 0.000 10 0.423 0.830 0.830 0.358 6.765 13.286 13 .286 5.729 1.732 0.000 0.000

η = 0.5 η = 1 η = 10

15 14 14

12 12 10

10 10

SPV 8 SPV

SPV 8 5 6 6 1 4 0 1 1 0 4 0 0 0.5 0.5 0.5 0.5 0.5 0.5

1 0 ρ 1 0 ρsp 1 0 ρ ρ sp ρ ρ sp wp wp wp

Figure 5: 3D VDGs for the modified CCD with one whole plot factor and two subplot factors

83 Figure 6 shows the plot of the scaled prediction variance for each of the design spaces. The plots indicate that the whole plot space is performing better than the subplot space up to a radius of 0.6 for η = 0.5 and η =1 and up to a radius of 0.5 for η = 10. However, the subplot space begins to dominate for radii greater than 0.6 and 0.5 η = 0.5, η = 1 and η = 10 respectively. The advantage of the two design spaces becomes more apparent as the variance ratio increases.

η = 0.5 η = 1 η = 10 15 15 15

10 10 10 SPV SPV SPV

5 5 5

0 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Radius Radius Radius

Whole plot space Subplot space G max Design space Figure 6: Comparison of whole plot and subplot spaces for modified CCD with one whole plot factor and two subplot factors

The relative measures of the two distinct design spaces are given in Table 32. The results indicate that for the modified design the relative measure of the subplot space is from 13% to 49% better than the whole plot and combined design spaces depending on the variance ratio used.

Table 32: Relative measures of whole plot and subplot design spaces for modified CCD

η ∆Gx ∆Gz 0.5 13% 0% 1 21% 0% 10 49% 0%

4.19.2 BBD wi t h tw o whole plot factors and two subplot factors The BBD given in Table 33 was developed by Parker (2006) using minimum whole plot (MWP ) criteria while satisfying the equivalent estimation property. Table 34 summaries the prediction v ari an ce results for this design.

Table 33: MWP_BBD with two whole plot factors and two subplot factors

Whole Plot z1 z2 x1 x 2 Whole Pl ot z1 z2 x1 x2 1 -1 -1 0 0 6 0 1 -1 0 -1 -1 0 0 0110 -1 -1 0 0 0 1 0 -1 -1 -1 0 0 0101 -1 -1 0 0 0100 2 1 -1 0 0 7 0 -1 -1 0 1 -1 0 0 0 -1 1 0 1 -1 0 0 0 -1 0 -1 1 -1 0 0 0 -1 0 1 1 -1 0 0 0 -1 0 0 3 -1 1 0 0 8 1 0 0 -1 -1 1 0 0 1001 -1 1 0 0 1 0 -1 0 -1 1 0 0 1010 -1 1 0 0 1000 4 1 1 0 0 9 0 0 -1 -1 1 1 0 0 0 0 -1 1 1 1 0 0 0 0 1 -1 1 1 0 0 0011 1 1 0 0 0000 5 -1 0 0 -1 -1 0 0 1 -1 0 -1 0 -1 0 1 0 -1 0 0 0

85 An examination of the results in Table 34 reveals that the variance ratio is affecting the location of the maximum prediction variance. The maximum prediction variance is located at the center of the whole plot and at the extreme of the subplot for = 0.5 and η = 1. It is important to note that the location of the maximum prediction variance for these variance ratios are not among those points that make up the design. In contrast, for η = 10 the location of the maximum prediction variance is now repositioned at the extreme of the whole plot factor level setting and at the center of the subplot. This can be observed in the 3D VDG given in Figure 7 for the scaled prediction variance throughout the design region. Table 34 also indicates that for η = 0.5 and η = 1 the maximum prediction variance in the whole plot is less than the maximum prediction variance in the subplot. However for η = 10 the relationship is reverse.

Table 34: Prediction variance properties and G-criterion location for the MWP BBD with two whole plot factors and two subplot factors Standardized Scaled Location of Max Vmax Vmax Vmax Vmax z z x x η ρx ρz Vmaxρ IV ρx ρz Vmaxρ IV 1 2 1 2 0.5 0.831 0.726 0.831 0.387 37.417 32.667 37.417 17.396 0 0 ±1.414 ±1.414 1 0.971 0.933 0.971 0.413 43.688 42.000 43.688 18.594 0 0 ±1.414 ±1.414 10 1.313 1.442 1.442 0.479 59.080 64.909 64.909 21.534 ±1 ±1 0 0

η = 0.5 η = 1 η = 10

100

90 100 100 80 80 70 80

60 60 60 50 SPV

SPV 40

40 SPV 40 20 30 20 0 20 1 0 1 10 0 1 0 0 0.5 0.5 0.5 0.5 0.5 0.5 1 0 1 0 ρ 1 0 ρ ρ sp ρ sp ρ sp ρ wp wp wp

Figure 7: 3D VDGs the MWP BBD with two whole plot factors and two subplot factors.

86 A comparison of the two design subspaces within the design region, shown graphically in Figure 8, reveals that better prediction is expected for the whole plot space when compared to the subplot space. For η = 0.5 and η = 1 the whole is better across all radii but for η = 10 the subplot space only perform better for radius greater than approximately 0.9. Therefore, while the subplot space is 9% better (see Table 35) than the whole space for η = 10, in terms of maximum prediction variance, the overall performance of the whole space appears more appealing.

η = 0.5 η = 1 η = 10 70 70 70

60 60 60

50 50 50

40 40 40

SPV 30 SPV 30 SPV 30

20 20 20

10 10 10

0 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Radius Radius Radius

Whole plot space Subplot space G max Design space Figure 8: Comparison of whole plot and subplot spaces for the MWP BBD with two whole plot factors and two subplot factors.

The change in the performance of the two design spaces is supported by the relative measures in Table 35. When compared to the entire design space the whole plot space is 13% and 4% better than the subplot space for η = 0.5 and η = 1. This relationship is reversed for η = 10 with the subplot space being 9% better than the whole plot space.

87 Table 35: Relative measures of whole plot and subplot design spaces MWP BBD with two whole plot factors and two subplot factors

η ∆Gx ∆Gz 0.5 0% 13% 1 0% 4% 10 9% 0%

4.19.3 FCC with two whole plot factors and three subplots factors Table 36 gives the detail of a cuboidal design which is developed according to the method presented by Vining et al. (2005). The method, referred to as VKM, focuses on developing equivalent estimation designs. The design is balanced with all the whole plots having equal sizes.

Table 36: A resolution V VKM FCC with = α = 1, two whole plots and three subplot factors Factors # of runs per WP w whole plot z1 z2 x1 x2 x3 1 4 -1 -1 ±1 ±1 ±1 4 2 4 1 -1 ±1 ±1 ±1 4 3 4 -1 1 ±1 ±1 ±1 4 4 4 1 1 ±1 ±1 ±1 4 5 4 - 0 0 0 0 4 6 4 0 0 0 0 4 7 4 0 - 0 0 0 4 8 4 0 0 0 0 4 9 4 0 0 ±α 0 0 4 10 4 0 0 0 ±α 0 4 11 4 0 0 0 0 ±α 4 12 4 0 0 0 0 0 4

Table 37 gives the prediction variance results for the VKM FCC design. Again it is shown that the location of the maximum prediction variance differs according to the variance ratio. For this cuboidal design the location for η = 0.5 and η = 1 is at the extreme of the whole plot and the edge center of the subplot plot factors while for = 10 the location remain at the extreme of the whole plot factor but changes to the face center

88 of the subplot factors. This behavior is displayed in Figure 9 which shows the 3D VDGs for the cuboidal region. Notice that at a radius of approximately 0.5 the prediction variance appears stable for both the whole plot and subplot but becomes very unstable for radii greater than one. This appears to be the case across all the variance ratios.

Table 37: Prediction variance properties and G-criterion location for the VKM FCC with two whole plot factors and three subplot factors Standardized Scaled Location of Max Vmax Vmax Vmax Vmax z z x x x η ρx ρz Vmaxρ IV ρx ρz Vmaxρ IV 1 2 1 2 3 0.5 0.737 0.674 1.011 0.340 35.393 32.370 48.533 16.336 ±1 ±1 ±1 ±1 0 1 0.845 0.843 1.126 0.388 40.567 40.463 54.067 18.620 ±1 ±1 ±1 ±1 0 10 1.154 1.257 1.549 0.505 55.376 60.327 74.343 24.226 ±1 ±1 ±1 0 0

η = 0.5 η = 1 η = 10

80 70 80

60 60 60 50

40 40 40 SPV

SPV 30 SPV

20 20 20

10 0 0 0 1.5 1.5 1.5 2 2 1 1 1 1 2 1 0.5 0.5 1.5 0.5 1 0 0 0.5 0 ρ 0 ρ ρ 0 ρ 0 ρ wp sp wp ρ wp sp sp Figure 9: 3D VDG the VKM FCC with two whole plot factors and three subplot factors.

The apparent difference in the plots of Figure 9 and Figure 10 when compared the other plots of Figure 3 through Figure 8 is because of the design region. Figure 9 and Figure 10 are for a cuboidal design region while the others are for spherical design regions. The difference is that for a cuboidal region the design variables are confined according to −≤11zi ≤ and −≤11x j ≤ such that all points are either positioned on or inside a hypercube while for a spherical region the design variables are confined

89 wp sp according to zx22+≤1 such that all points are either positioned on or inside a ∑∑ij==11ij hyper-sphere of radius one. Notice that the all the plots are similar for a radius of 1. In terms of an overall pe rformance Figure 10 shows that the whole plot space is evidently performing better than the subplot space up to a radius of approximately 1.4 for η = 0.5 and η = 1 and up to a radius of approximately 1.3 for η = 10. The subplot space appears to dominates for radii larger than 1.4 and 1.3 for η = 0.5, η =1 and η = 10 respectively.

η = 0.5 η = 1 η = 10 80 80 80

70 70 70

60 60 60

50 50 50

40 40 40 SPV SPV SPV

30 30 30

20 20 20

10 10 10

0 0 0 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 Radius Radius Radius

Whole plot space Subplot space G max Design space Figure 10: Comparison of whole plot and subplot spaces for the MWP BBD with two whole plot factors and two subplot factors.

Table 38 shows that the relative measure of the whole plot is 6% better than the subplot and 33% better than the combined design space for η = 0.5. The relative efficiency are the same for both the whole plot and subplot design spaces which is 25% better than the combined design space. However, η = 10 the relative measure of the subplot space is 7% better than the whole plot space and 26% better than the combined space.

Table 38: Relative measures of whole plot and subplot design spaces for the VKM FCC

η ∆Gx ∆Gz 0.5 27% 33% 1 25% 25% 10 26% 19%

4.20 Conclusion The research work reveals that the location of the maximum prediction variance can change based on the variance ratio used. Therefore, it is recommended that when reporting the value of the maximum prediction variance, its location should also be reported. Reporting the location is especially important for designs where the location is not stable and moves around depending on the magnitude of the variance ratio. In terms of design development and selection, the practitioner should select a design where the location of the maximum prediction variance is robust to changes in the variance ratio. The practitioner should at least be aware of the possible locations where the maximum prediction variance may occur based on the magnitude of the variance ratio. Knowing the location of the maximum prediction variance will alert the practitioner prior to data collection as to where in the design region he should expect the worse prediction and where to collect more data if necessary. The analytical method presented also results in the successful separation of the SPD into the two distinct design spaces that characterizes the design. The separation of the design spaces facilitates an understanding of the relative efficiencies of each distinct design space with respect to the combined design space. A graphical approach and the use of a delta G-criterion measure for these two distinct design spaces are suggested as evaluation methods for assessing the prediction capability of a given design. Several examples covering both spherical and cuboidal regions of interest were studied. The results shows various ways a practitioner can use the information presented to make a decision in selecting the best design for experimentation. The versatility of the derived analytical expressions for the determination of the maximum predication variance was demonstrated. The equations presented are easily coded and represent an efficient and effective way of studying the prediction capabilities of second-order SPD. One of the

91 attractive features of this method is that the Equations presented are functions of the design parameters, variance ratio and radius.

92 CHAPTER 5

5.0 Integrated Prediction Variance for Response Surface Designs

5.1 Abstract Over the years, design optimality evaluation of response surface designs focused mainly on D-optimality and G-optimality criteria. However, integrated prediction variance (IV) is sometimes more important to practitioners than either D-optimality or G- optimality criterion. Attempts have been made by several researchers to use the IV- optimality criterion but in most cases its value is approximated. In situations where the exact value is used it is often for designs that are small in nature. The apparent limited use of the IV-optimality criterion appears to be influenced by the computational challenges associated with the criterion. Computing IV-optimality criterion numerically is time consuming for several reasons, the main one is the integration required over the specified design region. In this chapter, an efficient and exact method is presented for computing the IV-optimality criterion for selected response surface designs. The investigation examines both spherical and cuboidal regions of interest. In addition, an analytical approach is outlined for computing the IV-optimality criterion for second-order split plot designs. This analytical method is also capable of computing the IV-optimality criterion for completely randomized designs. A particular feature of the analytical expressions is that they are derived using the design parameters. Several comparisons of second-order response surface designs are illustrated for split plots and completely randomized designs. The evaluations include a variety of second-order designs such as equiradial, Hoke, hybrid and Box-Draper.

93 5.2 Introduction In response surface literature, noticeably absent are studies involving the use of the exact integrated prediction variance (IV) for response surface designs, particularly for spherical designs were the computational time required increases exponentially. Meyers et al. (1992) present some integrated prediction variance values for selected designs ( k ≤ 5) while evaluating the variance dispersion properties of second-order response surface designs. No detail is given on the computational procedure used to generate the IV values. Ozol-Godfrey et al. (2005) illustrate the use of fraction of design space plots for examining model robustness using G-optimality and IV-optimality criteria. However, the IV values presented are approximated values which they argue are unbiased estimates of the true IV values. Hussey (1997) utilizes the IV-optimality criterion in a study on correlated simulation experiments but focuses only on first order models which are essentially cuboidal designs. Borkowski and Valerose (2001) study design optimality criteria for reduced models but focus only on cuboidal designs and presented IV values for designs with35≤≤k . An analytical approach is presented which computes the exact IV values for the central composite design (CCD). However, for other designs the exact IV value is computed by evaluating the appropriate integrals, which is time consuming. Haines (1987) and Borkowski (2003) utilize the IV-optimality criterion in generating exact optimal designs but its application is restricted to second-order models with 2 or 3 factors for the cuboidal region. Hardin and Sloane (1993) present an algorithm capable of generating IV-optimal designs for a number of “classical” situations, such as linear, quadratic or cubic response-surface designs with 1 ≤ k ≤ 12 continuous variables in a cube or a sphere. Borkowski (2003) demonstrated the inaccuracies of computer packages in generating average prediction variance (APV) as an approximation of the IV-criterion for cuboidal designs. The author gave a cautionary note that the practitioner should be careful in using APV values given by these computer packages to compare designs. A Monte Carlo approach was suggested for computing the IV-criterion which is not necessarily exact but provides better approximation values than what is currently available in computer packages. However this suggestion can still be time consuming depending on the number of sampled points. The most obvious reason for the seemingly limited use of the exact

94 value of the IV-optimality criterion is the computational time required to generate IV values. Even though published research work gives IV values for selected designs there is no published work on how to efficiently compute the exact IV values without the need for numerical computation involving integration. It is the objective of this chapter to present a method to efficiently compute the exact value of the IV-optimality criterion for second-order response surface designs.

5.3 The Split Plot Model The split plot model is developed here as the general case and the completely randomized design model is considered a specific instance of the split plot model. The general form for split plot models in matrix form is given as

yX= Θ++ where y is an N x 1 vector of expected responses, N is the total number of observations, Θ is a p x 1 vector of unknown model p a rameters, p is the number of model parameters, X is the N x p matrix of the levels of the independent variables, and and are N x 1 vectors of random variables for whole plot and subplot errors respectively. N and It is assumed that + ~ N (0, Σ). The structure of the variance-covariance matrix Σ is

⎡⎤Σ1 00...... ⎢⎥0 00... ⎢⎥Σ2 Σ = ⎢⎥0 (1.43) ⎢⎥ ⎢⎥0 ⎢⎥ ⎣⎦00 0Σa

22' where a is for the number of whole plots, Σiwww=+σσδε11×11×× Iw and w represents the th number of subplot runs (whole plot size). In matrix form the structure of Σ i for the i whole plot is represented as

95 222 2 ⎡⎤σσσδεδ+ σ δ ⎢⎥ σ 2 Σ = ⎢⎥δ (1.44) i ⎢2 ⎥ σδ ⎢2222⎥ ⎣⎦⎢⎥σδ σσσδδε+

2 2 2 where σ δ + σ ε represents the variance of individual observations, σ δ is the covariance

−1 Θˆ = (X''Σ−−11XX) Σ y

−1 ˆ ' −1 2 2 where Var ()Θ = ()X Σ X . The two variance components σ δ and σ ε are needed to obtain parameter estimates and standard errors. Letsinger et al. (1996) point out that design optimality criteria, which are functions of the Var ()Θˆ , depend on the structure of the design, particularly the information matrix (A). In the case of CRD the information matrix is given as X' X . However, for SPD alphabetic optimality is now dependent on an information matrix of the form X'Σ−1X , where Σ is the variance-covariance matrix described in Equation (1.1). This situation complicates the process because here the optimal design will depend on the

2 2 variance ratio or degree of correlation η = σ δ σ ε , through the correlation matrix (R). The correlatio n matrix (R) of the observations is derived by dividing the variance

96 ⎛⎞−1 ''⎛⎞Σ −1 AX==⎜⎟⎜⎟ X XRX ⎜⎟σσ22+ () ⎝⎠⎝⎠δε where, R-1 denotes the inverse of the correlation matrix and is of the form

−1 ⎡⎤R01 ...... 0 ⎢⎥−1 ⎢⎥0R2 0... 0 −1 R = ⎢⎥0 (1.45) ⎢⎥ ⎢⎥0 ⎢⎥−1 ⎣⎦00 0Ra where ⎡⎤ηη(1++ ) ηη (1 ) ()1+−η − ⎢1+wη 1+wη ⎥ ⎢⎥ ⎢⎥ηη(1+ ) − ⎢⎥1+ wη R−1 = ⎢⎥. i ⎢⎥ηη(1+ ) − ⎢⎥1+ wη ⎢⎥ ⎢⎥ηη(1++ ) ηη (1 ) ⎢⎥−− ()1+η ⎣⎦11++wwηη

5.4 IV – Optimality Criterion The pioneers of the IV-optimality criterion are Box and Draper (1959 and 1963). They present the mean squared deviation from the true response as the averaged over the region of interest, normalized by the design points and variance. The average mean square error criterion was comprised of two main components. The first component is the “variance error” resulting from sampling errors and the second component is the “bias error” which reflects the inadequacy of the fitted model. The first component is the IV- optimality criterion which generates a single measure of prediction performance throughout the entire region of interest, Ξ. Computationally, this is done by integrating

97 the prediction variance v(z, x) over Ξ. The standardized IV function for a split plot design can be expressed as follows:

1 IV= ∫ vd()zx , zxd Ω Ξ 1 ' −1 = ∫ ff()zx,,() XR' −1 X() zxd zxd Ω Ξ

−1 ⎡⎤' −1 1 ' = trace⎢⎥()XR X∫ f()() z,, x f z x d z d x ⎣⎦Ω Ξ

where, Ω= ddzx is the volume of the region Ξ and the general form of the model ∫Ξ vector for second-order SPD is given as

f z,x ' = () ⎡ 2222⎤ ⎣1,,z111…… zwp x ,, x sp zz21 ,, … z wp−− z wp zx11 ,, … z wp x sp xx 121 ,, … x sp x sp z1 ,, … z wp x 1 ,, … x sp ⎦ where, z and x are the whole plot and subplot factors respectively while wp and sp are the number of whole plot and subplot factors respectively. The regions of interest, Ξ, in response surface methodology are normally either cuboidal or spherical. The difference is that for a cuboidal region the design variables are confined according to −≤11xi ≤ for ik=1, 2,… , such that all points are either positioned on or inside a hypercube while for a spherical region the design variables are

k 2 confined according to ∑ xi ≤ k such that all points are either positioned on or inside a i=1 hyper-sphere of radius k. In addition, the volume of the respective region is given as follows:

⎧ 2k for cuboidal region ⎪ k ⎪ Ω= ()π . (1.46) ⎨ for spherial region ⎪ ⎛⎞k + 2 ⎪ Γ⎜⎟ ⎩ ⎝⎠2

Computing IV-optimality numerically is time consuming for several reasons, the main one is that integration is required over the specified design region. The computation is less extensive for cuboidal regions than it is for spherical regions. However, the computational time increases exponentially for both regions as the dimensionality of the problem increases. That is, as the number of factors increases both the dimension of information matrix and the number of variables to integrate expand rapidly. In the case of the spherical region, the integration process is even more time consuming because the process involves converting from rectangular to hyper-spherical coordinates, formulation of the Jacobean due to change of variables, and integration of trigonomical functions. In addition, numerical computations are prone to numerical errors and error propagation. These issues are normally overcome when an analytical approach to the problem can be obtained. In general analytical computation is much more efficient than numerical computation and often gives a better relationship among the parameters governing the system and the system optimality performance measure. These features of the analytical computational approach can then be exploited to design optimal systems or study the optimality of a system. To develop the analytical approach to compute IV-optimality of a design, the IV function was separated into two components. The first component is the inverse of the

−1 information matrix, BXRX= ()' −1 , and the second component is 1 Mz= ∫ f ()(),,x' fdzxzdx which is known as the matrix of region moments. The Ω Ξ methodology use is to obtain analytical expressions for these components and then use those expressions for the analytical determination of IV-optimality. The trace element in the IV function suggests that its computation is a linear function and therefore the determination of the IV-optimality criterion will simply be the sum of these expressions. In the next 2 sections the analytical characterization of the 2 components are presented.

99 −1 5.5 Analytical Characterization of (XR' −1 X)

−1 Chapter 3 gives a full discussion of the analytical characterization of (XR' −1 X) for both the central composite design (CCD) and the Box-Behnken Design (BBD) within a split plot structure. For additional details see Appendices A1 and A2. The following discussion briefly outlines the important elements of the analytical characterization process for both CCD and BBD split plot designs.

5.5.1 Critical functions of the variance ratio and whole plot sizes In Chapter 3, the study of the information matrix for SPD reveals that there are several critical functions (Φ) of the variance ratio and whole plot sizes. The locations of these critical functions in the information matrix are given in Equations (1.28) and (1.30) for split plot CCD and split plot BBD. According to the structure of SPD there are four distinct categories of whole plots, denoted a1 through a4. These categories have whole plot sizes of w1 through w4 for factorial, whole plot axials, subplot axials and center runs respectively. In the case of balanced SPD there is only one whole plot size (w1) to consider and therefore the critical functions are:

(1+η ) Φ=1 (1 + w1η )

Φ=2 1+η ⎛⎞ηη(1 + ) Φ=−3 2⎜⎟ ⎝⎠1 + w1η

Φ=421 or Φ if whole plot axials are grouped with factorial whole plot

Φ=Φ52 or Φ 1 if subplot axials are in sparate whole plots while Φ3 goes to zero

Φ6 = ΦΦ=ΦΦ=781 = and 9 0.

In the case of unbalanced SPD there are several whole plot sizes to consider and therefore, the critical functions are modified as follows:

100

(1+η ) Φ=1 (1 + w1η )

Φ=2 1+η ⎛⎞ηη(1 + )⎛⎞ ηη (1 + ) Φ=−3 2⎜⎟ or −2⎜⎟ if unbalanced subplot axials ⎝⎠1 + ww13ηη⎝⎠ 1 +

Φ=421 or Φ if whole plot axials are grouped with factorial whole plots

Chapter 3 gives a full discussion of the impact of these functions on the information

−1 matrix and the analytical derivation of (XR' −1 X) . An outline of the results is presented in this paper.

5.5.2 Information Matrix for the Split Plot CCD Using the parameters characterizing the split plot CCD given in Table 39, the information matrix for the split plot CCD can be partitioned as in Equation (1.28) and its inverse given in Equation (1.29). These equations are applicable to both balanced and unbalanced SPD and for both spherical and cuboidal regions.

101

Table 39: Table of notations for the split plot CCD Notations Meanings

kk−m f # of factorial runs () 2 or 2V

fw # of whole plot factor runs th wii whole plot size

awii# of whole plots with size

rw # of repeated whole plot axials

rs # of repeated subplot axials β Whole plot axial setting α Subplot axial setting ζ Factor level setting

The information matrix for the split plot CCD is given as

XR'1− X= ⎛ 22'' 22 ⎞ ⎜ ΠΦ0J( 16frwwwζβ+ΦΦ22) p( 17 fr ζα+Φ ss) Jp ⎟ ⎜ ⎟ ⎜ 00Diag() di 0⎟ ⎜ ⎟ ⎜ ' ⎟ Φ+frζβ22Φ22J0 Φ r β 4 I +ΦΦ f ζ 4 JJJ'' f ζ 4⎡⎤J ⎜ 16w w) wp 6 w wp 1 w wp wp 1⎢⎥ sp wp ⎟ ⎜( ⎣⎦ ⎟ ⎜ 22 4'' 4 44⎟ ⎜ Φ+frζαΦ22J0 Φ f ζ JJIJ Φ+ r αΦ +ΦΦ+ fr ζαΦ2J⎟ ⎝ ( 17s ) sp 1 sp wp( 5 s 9) sp( 413 s) sp sp ⎠ (1.47)

where 0’s are zero matrices of appropriate sizes, Jwp and Jsp are unit vectors of wp×1 and

sp×1 respectively, I wp and Isp are wp-dimensional and sp-dimensional identity matrices and Diag (di) are diagonal elements which are given as follows:

102 22 dfiw=Φ16ζβ +Φ2 rf w ori 1 ≤ ≤wp 22 =Φ2 ()frζα +2s for wpi +≤≤ 1 k , where k =+wpsp wp 4 ⎛⎞ =Φ1 ffwζ or k + 1 ≤i ≤k +⎜⎟ ⎝⎠2

4 ⎛⎞wp ⎛⎞ wp ⎛⎞ sp =Φ 2 ffζ or k +⎜⎟ +≤≤+ 1 ik ⎜⎟ +×+wpsp ⎜⎟. ⎝⎠22 ⎝⎠ ⎝2⎠

The first entry, Π, in Equation (1.28) is a scalar quantity. For a balanced SPD Π=N Φ1 and for an unbalanced SPD Π=aw11 Φ 1 + aw 2 2 Φ 6 + aw 33 Φ 7 + aw 4 4 Φ 8. The total number

4 of runs is Na= w. ∑i=1 ii

−1 After performing extensive matrix manipulation the (XR' −1 X) for the split plot CCD is given as

⎛ γγγ0J'' J⎞ ⎜ 123wp sp ⎟ ⎜ ⎛⎞1 ⎟ ⎜ 000Diag ⎜⎟ ⎟ ⎝⎠di ⎜ ⎟ ' . (1.48) ⎜ 1 ''⎡⎤ ⎟ γγ24J0wp IJ wp− wpJJ wpγ5 spJ wp ⎜ Φ 2r β 4 ()⎣⎦ ⎟ ⎜ 6 w ⎟ ⎜ 1 ⎟ γγJ0 JJI''−γJJ ⎜ 35sp sp wp4 () sp6 sp sp ⎟ ⎝ Φ+592rsα Φ ⎠

Table 40 summarizes the scalar quantities represented by γ i in the inverse of the information matrix while Table 41 provides the details of the pertinent constants (c) that were derived.

103 Table 40: Table of scalar quantities for the split plot CCD Scalars Equations 1 γ1 c11

c5 γ 2 − c11

c9 γ 3 − c11 4 2 c1 Φ652rcwβ () γ 4 4 − Φ+6112rwwβ pc c1 4 Φ1 f ζ cc59 γ 5 −+ cc211 4 2 ()Φ+592rcsα Φ()9 γ 67c − c11

Table 41: Table of constant quantities for CCD Constants Equations

4 2 ()Φ1 fsζ p cfΦ−ζ 4 11w 444 Φ+59422rsssαζΦ+Φpfr()Φ+13Φα

4444 cr259()Φ+22ssαζΦ+Φsp()4Φ+1fΦ36rα() Φ2rwβ +wpc1 22 Φ+16frwwζβΦ2 c3 4 Φ+612rwwβ pc 42 2 ΦΦ+11ffζζ()Φ 72 rss αp c4 − c2

cc53+ c4 42 2 ΦΦ+11ffζζ()wwΦ 62 rw βp c6 − c2 2 ΦΦf ζ 4 +Φ22rcααζ444 − Φ r +Φ Φ f wp ()41 325ss(( 91)()) c 7 444 ()Φ+5942rsssαζ Φ+Φp()Φ+132frΦ2αc 22 ()Φ+17frcζαΦ21s () − 7sp c8 4 Φ+592rsα Φ

cc96+ c8

2222 cc10 5()Φ+ 1frwwζβΦ 622wp +Φ+c 9() 1fr ζαΦ 7 ssp

cc11Π− 10

104

5.5.3 Information Matrix for the Split Plot BBD The parameters characterizing the split plot BBD are given in Table 42. Using these parameters the information matrix for the split plot BBD can be partitioned as in Equation (1.30) and its inverse given in Equation (1.31).

Table 42: Table of notations for the split plot BBD Notations Descriptions f # of factorial runs per block () 2t

fc # of factorial runs at edge centers th wii whole plot size

awii# of whole plots with size

rw # of blocks within which a whole plot factor appears

rs # of blocks within which a subplot factor appears tt# of active subplot factors per block (≥ 1)

λc # of edge centers

λλw # of times a pair of whole plot factors appears in the same block (w ≥ 1)

λs # of times a pair of subplot factors appears in the same block

λint # of times a whole plot factor appears with a subplot factor in the same block β whole plot factor level setting α subplot factor level setting

The information matrix for the split plot BBD is given as

XR−1 X =

⎛ ΠΦ0Jfr βα2 '' Φfr 2 J ⎞ ⎜ 1 ()wwp 1 ()ssp ⎟ ⎜ 00Diag d 0⎟ ⎜ ()i ⎟ ⎜ ' ⎟ 244'' 4⎡⎤ ⎜ΦΦ11frwβλJ0 wp f() r w− wβ I wp+ f λ wβ J wpJJ wp Φ1intf λαspJ wp ⎟ ⎜ () ()()⎣⎦⎢⎥ ⎟ ⎜ 244' 444' ⎟ ⎜ ΦΦ11frsαλJ0sp fintαλ JspJIJ wpΦ5 f() r s− sα+Φ1Φ4 fλ sα+Φ3 f () r s−λ sα+ f cλ cα spJ sp ⎟ ⎝ () ( ) sp ()()⎠ (1.49)

105 where

2 diw=Φ1 f rwβ for 1 ≤ i ≤ wp 2 =Φ2 frsα for wp + 1 ≤ i ≤ k

4 ⎛⎞wp =Φ1 ffwwλβ or k + 1 ≤i ≤k +⎜⎟ ⎝⎠2

4 ⎛⎞wp ⎛⎞ wp = Φ2intffλα or ki ++⎜⎟ 1≤≤k++× ⎜⎟wpsp ⎝⎠22 ⎝⎠

4 ⎛⎞wp ⎛⎞ wp ⎛⎞ sp =Φ2 ffλαs or kw++×+⎜⎟psp1.≤i≤kw++×+ ⎜⎟psp ⎜⎟ ⎝⎠22 ⎝⎠ ⎝⎠2

−1 Therefore, the ()XR' −1 X for the split plot BBD is given as

⎛ γγ0J'' γ J⎞ ⎜ 12wp 3sp ⎟ ⎜ ⎛⎞1 ⎟ ⎜ 000Diag ⎜⎟ ⎟ ⎝⎠di ⎜ ⎟ . (1.50) 1 ' ⎜γγJ0 IJ− JJ''γ⎡⎤J ⎟ 24wp 4 ()wp wpwp 5 sp wp ⎜ Φ−fr λβ ⎣⎦ ⎟ ⎜ 1 ()ww ⎟ ⎜ 1 ⎟ γγJ0 JJ'' IJ− γJ ⎜ 35sp sp wp 4 ()sp 6 sp sp ⎟ ⎝ Φ−5 fr()ssλα ⎠

γ Table 43 summarizes the scalar quantities represented by i in the inverse of the information matrix while Table 44 provides the details of the pertinent constants (c) that were derived.

106 Table 43: Table of scalar quantities for the split plot BBD Scalars Equations 1 γ1 c11

c5 γ 2 − c11

c9 γ 3 − c11 4 2 c1 Φ−15f ()rcwwλβ() γ 4 4 − Φ−11fr()wwλβ + wpc c11 4 Φ1intfcλα 59c γ 5 −+ cc211 4 2 Φ−59fr()ssλα( c) γ 67c − c11

Table 44: Table of constant quantities for the split plot BBD Constants Equations 2 4 ()Φ1intfsλα p cfΦ−λβ4 11w 44 44 Φ−51fr()ssλα +Φ sp()Φ4 f λα s +Φ3() fr () ss − λα + f cc λα cfΦ−rsλα44 +ΦpΦff λα +Φ−+rff λα 4 λα4 Φ−+r λ β 4wpc 25()()ss() 143 s() () ss cc() 1() ww 1 2 Φ1 ()frwβ c3 4 Φ−11fr()wwλβ + wpc

2 24 ()Φ1ifrsαλαnt sp c4 − c2

cc96+ c8 22 cc10 5Φ+ 1frwsβαwpc 9Φ 1 frsp

cc11Π− 10

107 A similar characterization of the inverse of the information matrix for CRD was presented by Borkowki (1995). However, setting η = 0 and the whole plot size to 1 gives the same result for a CRD. The split plot design can be considered the more general structure while the CRD actually represents a specific instance of the SPD. In the following section we examine the second component M, the matrix of region moments.

5.6 Matrix of Region Moments In response surface methodology region moments of order two and four are required for second-order designs. Giovannitti-Jensen and Myers (1989) present some expressions for these region moments when the region of interest is the spherical surface area. They illustrated the matrix of region moments for the case of k = 3 for both the first and second-order models. Haines (1987) also presents the matrix of region moments for k = 2 for a cuboidal region. Hussey et al. (1989) show a similar moment matrix but consider only first order models. In general the matrix ∫ f ()()zx,,' fd zx zxd is a (p × p) matrix of region moments Ξ (M) of the form given in Equation (1.51). The variable p represents the number of model parameters.

⎛⎞ΩΩ0Jφ ' ⎜⎟1 k M0=Ω⎜⎟Diag() mi 0 . (1.51) ⎜⎟ ⎜⎟ΩΩφφJ0 IJ+ΩφJ' ⎝⎠12kk3kk

where Ω is as described in Equation (1.46), φ1 represents region moment of order two

φ2 and φ3 represent region moments of order four. Therefore, dividing the matrix of region moments (M) by the volume of the region will reduce the matrix of region moments to

108 ⎛⎞1 0Jφ ' M ⎜⎟1 k = ⎜⎟00Diag() mi (1.52) Ω ⎜⎟ ⎜⎟φφJ0 IJ+φJ' ⎝⎠12kk3kk where

mfi = φ1 ori 1≤≤k ⎛⎞k =+φ3 for k 1 ≤i ≤⎜⎟ . ⎝⎠2

It is important to point out that the matrix of region moments is not affected by restriction s on ra ndomizatio n and thus the variance ratio will not have any impact on its structure. Therefore, the matrix of region moments for a completely randomized design and a SPD are exactly the same. The hard to change factor is just seen as another variable in the design. After a careful study of the region moments for cuboidal designs it became apparent that t he second and fourth order moments are constant values regardless of the number of factors. Therefore, only the dimensionality of the matrix M changes to reflec t the number of parameters in the model. Thus for a cuboidal region the second and fourth order moments are given as

14 1 φφ= , == and φ. (1.53) 12345 39

In the case of the spherical region a unit sphere was considered. The investigation revealed that unlike the cuboidal region the second and fourth order moments are functions of k. Table 45 gives the values of these quantities for 26≤ k ≤ .

109

Table 45: Spherical second and forth order region moments for 26≤ k ≤ Region Mo m ents

k ø1 ø2 ø3 2 1/4 1/12 1/24 3 1/5 2/3 5 1/35 4 1/6 1/ 2 4 1/48 5 1/7 2/63 1/63 6 1/8 1/4 0 1/80

The values in Table 45 agree with the expressions given in Equation (1.54) as presented by Box and Draper (1959 and 1963)

1 φ = 1 k + 2 2 φ = or 2φ . (1.54) 23()kk++24() 1 φ = 3 ()kk++24()

To reflect the nature of th e SPD, the m atrix of region moments can be partitioned as

⎛⎞1 0Jφφ'' J ⎜⎟11wp sp ⎜⎟00Diag() mi 0 M = ⎜⎟' . (1.55) ''⎡ ⎤ ⎜⎟φφ12J0IJwp wp+φ3 wpJJ wpφ3 spJ wp ⎜⎟⎣ ⎦ ⎜⎟'' ⎝⎠φφ13J0sps JpJwpφ2 IJk+φ3kJk

5.7 Analytical Determination of Integrated Prediction Variance The analytical determination of the integrated prediction variance can now be accomplished by using Equations (1.29), (1.31) and (1.55).

110 Therefore,

−1 ⎡⎤' −1 1 IV= trace⎢⎥()XR X∫ f()() z , x ' f z , x d z d x ⎣⎦ω Ξ = trace[]BM

⎡⎛B11BMM 12⎞⎛ 11 12 ⎞⎤ =×trace ⎢⎜⎟⎜ ⎟⎥ ⎣⎦⎝B21BMM 22⎠⎝ 21 22 ⎠

⎡⎛B11M 11++B1 2 M 21 B11M 12 BM 12 22 ⎞⎤ = trace ⎢⎜⎟⎥ ⎣⎦⎝B21M 11++B2 2 M 21 B21M 12 BM 22 22 ⎠

⎛Q11Q 12 ⎞ = trace⎜⎟. ⎝Q21Q 22 ⎠

However, since we need o nly the tra ce of th e Q matrix we need only to compute the diagonal elements which are the com p onents o f Q11 and Q22. Therefore,

IV =+trace(QQ11) trace( 22 ).

The fol lowin g sectio n s now pr es ent the exp licit func tions that can be used to compute the integrated prediction variance for the split plot CCD and the split plot BBD.

5.8 Integrated Variance for the Split Plot CCD Performing the necessary matrix multiplication and addition Equations (1.56) and (1.57) are obtained. For details on the derivation of Equations (1.56) and (1.57) see Appendix E1. Therefore, the IV-optimality criterion for a CCD can be determined by adding the results of Equations (1.56) and (1.57) as shown in Equation (1.58)

⎛⎞k k +⎜⎟ ⎝⎠2 m traceQ =+γφ wp γ + sp γ + Diag ⎛⎞i (1.56) ()11 1 1 ( 2 3 )∑ ⎜⎟d i=1 ⎝⎠i

111 ⎛⎞φγ34()11−+−wp φγ24( ) trace()Q22 = ⎜⎟4 ++φγ3 sp 5 φγ 1 2 wp + ⎝⎠Φ6 2rwβ (1.57) ⎛⎞φγ36()11−+sp φ2()− γ6 ⎜⎟4 +φ3513wpγφγ+sp. ⎝⎠Φ5 2rsα

IV =+trace(Q11 ) tra ce(Q22 ) ⎛⎞k k +⎜⎟ ⎝⎠2 m ⎛⎞φγ34()11−+−wp φγ24() =+++γφwp γ sp γ Diag⎛⎞i + +++φγφγ sp wp 11() 2 3 ∑ ⎜⎟d ⎜⎟4 3512 i=1 ⎝⎠i ⎝⎠Φ6 2rwβ

⎛φγφγ3626()11−+−sp () ⎞ ⎜⎟4 ++φγφγ3513wp sp. ⎝⎠Φ5 2rsα (1.58)

5.9 Integrated Variance for the Split Plot BBD Following the same procedure as for the CCD, Equations (1.59) and (1.60) are obtained. Therefore, the IV-optimality criterion for a CCD can be determined by adding the results of Equations (1.59) and (1.60) as shown in Equation (1.61). For details on the derivation of Equations (1.59) and (1.60) see Appendix E2

⎛⎞k k +⎜⎟ ⎝⎠2 m traceQ =+γφ wp γ + sp γ + Diag ⎛⎞i (1.59) ()11 1 1 ( 2 3 )∑ ⎜⎟d i=1 ⎝⎠i

⎛φγ()11−+−wp φγ( ) ⎞ traceQ = 3424+φ spγ+ φγwp + ()22 ⎜⎜⎟4 3 5 1 2 ⎟ ⎝⎠Φ−1 fr()wwλβ (1.60) ⎛⎞φγφγ()11−+−sp () 3626++φγφγwp sp. ⎜⎟4 3513 ⎝⎠Φ−5 fr()ssλα

112 Therefore,

IV =+trace(QQ11) trace( 22 ) ⎛⎞k k +⎜⎟ ⎝⎠2 m ⎛⎞φγ34()11−+−wp φγ24() =+++γφwp γ sp γ Diag⎛⎞i + +++φγφγ sp wp 11() 2 3 ∑ ⎜⎟d ⎜⎟4 3512 i=1 ⎝⎠i ⎝⎠Φ−1 fr()wwλβ ⎛⎞φγφγ()11−+−sp () 3626++φγφγwp sp. ⎜⎟4 3513 ⎝⎠Φ−5 fr()ssλα (1.61)

In terms o f imp lem ent atio n of the se a nalytica l IV expre ssions, for a cuboidal region the applicable second and fourth order region moments are given in Equation (1.53). For a spherical region, the applicable second and fourth order region moments are given i n Equati on (1.54 ). Furth er, it is importa nt to po int out that the computation of IV values for second-order designs whether CRD or SPD c an also be easily computed by

−1 −1 simply taking th e trac e of th e prod uc t of (XX' ) or (XR' −1X) respectively and Equation (1.52). The fo llowi ng secti on shows the a pp lication of these functions in computing integrated prediction variance for several types of designs for both spherical and cubo idal re gio ns of int erest.

5.10 Integrated Prediction Variance for Split Plot Designs In this section we illustrate the utilization of the analytical method for computing integrated prediction variance values for second-order SPD. The response surface designs considered are the central composite design (CCD) and the Box-Behnken Design (BBD) for both spherical and cuboidal regions. These designs were developed within a split plot structure by Vining et al. (2005) and Parker et al. (2005 and 2006). The notation VKM is used to denote that the Vining method of construction is used while the notation MWP is used to denote that the Parker method of construction is used. These designs were selected because they possess the equivalent estimation property which is very attractive to practitioners. Several combinations of whole plot and subplot factors are considered. Further these designs were selected to demonstrate the flexibility of the

113 methods presented and how easy it is to compute the IV values for large designs. Table 46 list the split plot designs selected for evaluated in this section. For details on these designs and others see the catalog split plot designs developed by Parker et al. (2005 and 2006). The spherical designs are scaled such that the design points lie on or inside a sphere of radius 1. For each of the designs p res ent ed b oth the standardized and scaled IV values are presented. Essentially, the standard ized IV value is obtained using Equations (1.58) and (1.61). The sta ndard ize d IV val ue i s th en m ulti plied by th e design size (N) to obtain the scaled IV value. Since all the design s com p ared have the same amount of whole plots then scaling by the design size is ap pro priat e.

Table 46: Split plot de si gn s sele cted f or evaluat ion Factor Combination Numb er o f Design whole plots wp sp Spherical Region k = 4 VKM BBD 1 3 5 MWP BBD 1 3 5 VKM BBD 2 2 11 MWP BBD 2 2 11

k = 6 VKM CCD 2 4 11 MWP CCD 2 4 11 VKM BBD 2 4 11

Cuboidal Region k = 7 VKM CCD 3 4 17 MWP CCD 3 4 17

5.10.1 SPD with one whole plot factor and three subplot factors The first design to consider is the BBD. Table 47 and Table 48 show the design matrices representing both balanced and unbalanced VKM BBD and MWP BBD respectively. The main difference between the balanced and unbalanced designs is that the size of whole plots 4 and 5 would be 9 for the balanced case and 2 for the unbalanced

114 situation. Table 49 gives the details of the design parameters used to compute the IV values.

Table 47: Design matrix for a balan ced VKM BBD with one whole plot factor and three subplot factors

Whole Plot z1 x1 x2 x3 Whol e Plot Siz e -1 ±1 ±1 0 4 1 -1 ±1 0 ±1 4 -1 0 ±1 ±1 4 1 ±1 ±1 0 4 2 1 ±1 0 ±1 4 1 0 ±1 ±1 4 0 ±1 ±1 0 4 3 0 ±1 0 ±1 4 0 0 ±1 ±1 4 4 0 0 0 0 12 5 0 0 0 0 12

Table 48: Design ma trix for a ba lanced M W P BBD with one whole plot factor and three subplot factors

Whole Plot z1 x1 x2 x3 Whole Plot Size -1 ±1 0 0 2 -1 0 ±1 0 2 1 -1 0 0 ±1 2 -1 0 0 0 7 1 ±1 0 0 2 1 0 ±1 0 2 2 1 0 0 ±1 2 1 0 0 0 7 0 ±1 ±1 0 4 0 ±1 0 ±1 4 3 0 0 ±1 ±1 4 0 0 0 0 1 4 0 0 0 0 13 5 0 0 0 0 13

115

Table 49: Design parameters for VKM BBD and MWP BBD with one whole plot factor and three subplot factors VKM M WP Parameters Balanced U nbalanced B alanced Unb alanced f 4 4 2 2

fc 4 4 2 2 t 2 2 1 1

a1 3 3 3 3

a4 2 2 2 2

w1 12 12 13 13

w4 12 2 13 2 wp 1 1 1 1 sp 3 3 3 3 N 60 40 65 43

rw 6 6 13 13

rs 6 6 6 6

λw 1 1 1 1

λs 3 3 2 2

λint 4 4 2 2

λc 3 3 2 2 0.57735 0.57735 0.70711 0.70711 α 0.57735 0.57735 0.70711 0.70711 p 15 15 15 15

Equation (1.61) is evaluated using the parameters given in Table 49. The results were obtained for variance ratios 0.5, 1 and 10. Table 50 summarizes the results obtained. It can be observed that the unbalanced MWP BBD design has the best scaled integrated prediction variance when compared to the other designs. The balanced MWP BBD design is the next best design while the balanced VKM BBD design performs the worst. Since all the designs have the same number of whole plots the unbalance MWP design would be more cost effective and therefore the best design.

116

Table 50: Integrated prediction variance for VKM BBD and MWP BBD with one whole plot factor and three subplot factors Standardized IV IV S caled by Design size Varian ce Ra tio (η) Bala nced U nb ala nced Balanced Unbalanced VKM MW P VK M MWP VK M MWP VKM MWP 0.5 0.658 0 .37 3 0. 68 6 0 .397 39 .453 24.270 27.431 17.075 1 0.827 0.40 4 0. 848 0 .425 49 .629 26.246 33.932 18.261 10 1.243 0 .47 2 1.24 7 0 .477 74.606 30.661 49.891 20.490

An in ves tig at ion w as al so pe r form ed on the similarity between VKM CCD and VKM BBD with 4 factors. It is an accepted fact that for a completely randomized design the BB D is considered to be a rotation of the CDD for a 4 factor design. Therefore both designs will give the same predictio n var iance r esu lts. The reason for this investigation is to verify whether or not the VKM CCD and VKM BBD designs with 4 factors are equivalent. It was observed that for SPD, the condition of equivalency is dependent on the number of whole plot and subplot factors. The details are presented only for the designs that are equivalent. The scaled integrated prediction variances for the balanced VKM CCD and VK M BBD with 2 w hole plot factors and 2 subplot factors that are equivalent are e qual to 1 4.68, 15 .98 , and 19.0 3 for varia nce ratios 0.5, 1 and 10 respective ly. Both desig ns ha ve wh ole p lot sizes of 4 obse rvations, 11 whole plots 2 of which are dedicated to center runs. The designs will also give the same result if the sizes of the 2 whole plots of center runs are increased or reduced. The results indicate that only SPD involving 2 whole plot factors and 2 subplot factors will give the same prediction variance properties. Any other combination of whole plot and subplot factors will result in different prediction variance properties.

5.10.2 SPD with two whole plot factors and four subplot factors We now consider the VKM and the MWP CCD with 2 whole plot factors and 4 subplot factors as shown in Table 51 and

117 Table 52. In both cases the unbalanced nature of the designs are as a result of reducing the size of the whole plot centers from 8 to 2 for the VKM and 9 to 2 for the MWP. The design parameters are given in Table 53.

Table 51: Design matrix fo r a b alan ced VK M C CD wit h = α = 2.4495, two whole plot factors and four su bplot fac tors

Whole P lot z1 z2 x1 x2 x3 x4 Whole Pl ot Size 1 -1 -1 ±1 ±1 ±1 ±1 8 2 1 -1 ±1 ±1 ±1 ±1 8 3 -1 1 ±1 ±1 ±1 ±1 8 4 1 1 ±1 ±1 ±1 ±1 8 5 -β 0 0 0 0 0 8 6 β 0 0 0 0 0 8 7 0 -β 0 0 0 0 8 8 0 β 0 0 0 0 8 0 0 -α 0 0 0 0 0 α 0 0 0 0 0 0 -α 0 0 0 0 0 0 0 9 α 8 0 0 0 0 -α 0 0 0 0 0 α 0 0 0 0 0 0 -α 0 0 0 0 0 α 10 0 0 0 0 0 0 8 11 0 0 0 0 0 0 8

118

Table 52: Design matrix for a balanced MWP CCD with = α = 2.4495, two whole plot factors and four subplot factors

Whole P lot z1 z2 x1 x2 x3 x4 Whole Pl ot Size -1 -1 ±1 ±1 ±1 ±1 8 1 -1 -1 0 0 0 0 1 -1 1 ±1 ±1 ±1 ±1 8 2 -1 1 0 0 0 0 1 1 1 ±1 ±1 ±1 ±1 8 3 1 1 0 0 0 0 1 -1 1 ±1 ±1 ±1 ±1 8 4 -1 1 0 0 0 0 1 5 -β 0 0 0 0 0 9 6 β 0 0 0 0 0 9 7 0 -β 0 0 0 0 9 8 0 β 0 0 0 0 9 0 0 -α 0 0 0 0 0 α 0 0 0 0 0 0 -α 0 0 0 0 0 α 0 0 9 0 0 0 0 -α 0 9 0 0 0 0 α 0 0 0 0 0 0 -α 0 0 0 0 0 α 0 0 0 0 0 0 10 0 0 0 0 0 0 9 11 0 0 0 0 0 0 9

119

Table 53: Design parameters for VKM CCD and MWP CCD with = α = 2.4495, two whole plot factors and four subplot factors VKM M WP Parameters Balanced Unbalanced Balanced Unbalanced f 32 32 32 32

fw 32 32 36 36

a1 4 4 4 4

a2 4 4 4 4

a3 1 1 1 1

a4 2 2 2 2

w1 8 8 9 9

w2 8 8 9 9

w3 8 8 9 9

w4 8 2 9 2 wp 2 2 2 2 sp 4 4 4 4 N 88 76 99 85

rw 8 8 9 9

rs 1 1 1 1 1 1 1 1 α 1 1 1 1 0.4 0825 0.40825 0.4 0825 0.40 825 p 28 28 28 28

An examination of the results in Table 54 reveals that the unbalanced VKM CCD performs the best among the designs considered. The unbalanced VKM CCD has the lowest scaled integrated prediction variance across the variance ratios considered. The next best design is the unbalanced MWP CCD. The worst design is the balanced MWP.

120

Table 54: Integrated prediction variance for VKM CCD and MWP CCD with = α = 2.4495, two whole plot factors and four subplot factors Standardized IV IV Scaled by Design size Variance Ratio (η) Balanced Unbalanced Balanced Unbalanced VKM MWP VKM MWP VKM MWP VKM MWP 0.5 0.365 0.356 0.378 0.365 32.152 35.247 28.718 30.986 1 0.372 0.358 0.382 0.365 32.756 35.450 29.001 31.057 10 0.389 0.360 0.391 0.361 34.236 35.601 29.697 30.714

The VKM BBD given in Table 55 with 2 whole plot factors and 4 subplot factors was also considered. A close examination of the design structure shows that subplot factor x2 has a total of 48 settings at the ±1 levels while the remaining subplot factors have 24 settings at the ±1 levels. Therefore, the design moments among the subplot factors are different and as a result the analytical method presented is not capable of computing the IV values for this design. However, the matrix manipulation method presented is used to compute the IV value. This method still represents an efficient and exact way of computing the IV values. The unbalanced structure of the design is as a result of the size of the whole plot centers reducing form 8 to 2. Table 56 gives the results which indicate that the unbalanced design performs better than the balanced design. However, when compared with IV results given in Table 54 the unbalanced VKM CCD performs better than the VKM BBD. This comparison is possible because both designs have the same number of whole plots and where scaled to be on a sphere of radius 1.

121

Table 55: Design matrix for a balanced VKM BBD with two whole plot factors and four subplot factors

Whole Plot z1 z2 x1 x2 x3 x4 Whole Plot Size 1 -1 -1 0 ±1 0 0 8 -1 0 0 ±1 ±1 0 4 2 -1 0 ±1 0 0 ±1 4 3 -1 1 0 ±1 0 0 8 0 -1 ±1 0 ±1 0 4 4 0 -1 0 0 ±1 ±1 4 5 0 0 ±1 ±1 0 ±1 8 0 1 ±1 0 ±1 0 4 6 0 1 0 0 ±1 ±1 4 7 1 -1 0 ±1 0 0 8 1 0 0 ±1 ±1 0 4 8 1 0 ±1 0 0 ±1 4 9 1 1 0 ±1 0 0 8 10 0 0 0 0 0 0 8 11 0 0 0 0 0 0 8

Table 56: Integrated prediction variance for VKM BBD with two whole plot factors and four subplot factors Standard ized IV IV Sca led by D esign size Variance Ratio (η) Balanced Unbalanced Balanced Unba lanced 0.5 0.441 0.453 38.780 34.44 1 1 0.490 0.499 43.110 37.944 10 0.611 0.612 53.738 46.540

5.10.3 SPD with three whole plot factors and four subplot factors To this point we have been dealing with designs with spherical regions. However, the region of interest can sometimes be cuboidal. Table 57 and Table 58 give the details of the design matrix for VKM and MWP CCD which are cuboidal designs. Note that both designs have the same number of whole plots. Therefore, scaling these designs by the design size for comparison is a reasonable way of selecting the most efficient design. Details of the design parameters are given in Table 59.

122

Table 57: Design matrix for a balanced VKM CCD with = α = 1, three whole plot factors and four subplot factors

Whole Plot z1 z2 z3 x1 x2 x3 x4 Whole Plot Size 1 -1 -1 -1 ±1 ±1 ±1 ±1 8 2 1 -1 -1 ±1 ±1 ±1 ±1 8 3 -1 1 -1 ±1 ±1 ±1 ±1 8 4 1 1 -1 ±1 ±1 ±1 ±1 8 5 -1 -1 1 ±1 ±1 ±1 ±1 8 6 1 -1 1 ±1 ±1 ±1 ±1 8 7 -1 1 1 ±1 ±1 ±1 ±1 8 8 1 1 1 ±1 ±1 ±1 ±1 8 9 -β 0 0 0 0 0 0 8 10 β 0 0 0 0 0 0 8 11 0 -β 0 0 0 0 0 8 12 0 β 0 0 0 0 0 8 13 0 0 -β 0 0 0 0 8 14 0 0 β 0 0 0 0 8 0 0 0 -α 0 0 0 0 0 0 α 0 0 0 0 0 0 0 -α 0 0 0 0 0 0 0 0 15 α 8 0 0 0 0 0 -α 0 0 0 0 0 0 α 0 0 0 0 0 0 0 -α 0 0 0 0 0 0 α 16 0 0 0 0 0 0 0 8 17 0 0 0 0 0 0 0 8

123

Table 58: Design matrix for a balanced MWP CCD with = α = 1, three whole plot factors and four subplot factors

Whole Plot z1 z2 z3 x1 x2 x3 x4 Whole Plot Size -1 -1 -1 ±1 ±1 ±1 ±1 8 1 -1 -1 -1 0 0 0 0 1 1 -1 -1 ±1 ±1 ±1 ±1 8 2 1 -1 -1 0 0 0 0 1 -1 1 -1 ±1 ±1 ±1 ±1 8 3 -1 1 -1 0 0 0 0 1 1 1 -1 ±1 ±1 ±1 ±1 8 4 1 1 -1 0 0 0 0 1 1 1 1 ±1 ±1 ±1 ±1 8 5 1 1 1 0 0 0 0 1 1 -1 1 ±1 ±1 ±1 ±1 8 6 1 -1 1 0 0 0 0 1 -1 1 1 ±1 ±1 ±1 ±1 8 7 -1 1 1 0 0 0 0 1 -1 -1 1 ±1 ±1 ±1 ±1 8 8 -1 -1 1 0 0 0 0 1 9 -β 0 0 0 0 0 0 9 10 β 0 0 0 0 0 0 9 11 0 -β 0 0 0 0 0 9 12 0 β 0 0 0 0 0 9 13 0 0 -β 0 0 0 0 9 14 0 0 β 0 0 0 0 9 0 0 0 -α 0 0 0 0 0 0 α 0 0 0 0 0 0 0 -α 0 0 0 0 0 0 α 0 0 15 0 0 0 0 0 -α 0 9 0 0 0 0 0 α 0 0 0 0 0 0 0 -α 0 0 0 0 0 0 α 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 9 17 0 0 0 0 0 0 0 9

124

Table 59: Design parameters for VKM CCD and MWP CCD with = α = 1, three whole plot factors and four subplot factors. VKM MWP Parameters Balanced Unbalanced Balanced Unbalanced f 64 64 64 64

fw 64 64 72 72

a1 8 8 8 8

a2 6 6 6 6

a3 1 1 1 1

a4 2 2 2 2

w1 8 8 9 9

w2 8 8 9 9

w3 8 8 9 9

w4 8 2 9 2 wp 3 3 3 3 sp 4 4 4 4 N 136 124 153 139

rw 8 8 9 9

rs 1 1 1 1 1 1 1 1 α 1 1 1 1 1 1 1 1 p 36 36 36 36

Note that the parameters presented in Table 59 for both designs indicate that = α = 1 which means that the designs are Face Centered Cube (FCC) and thus have cuboidal regions. A comparison of the designs according to the IV values given in Table 60 indicates that the unbalanced MWP design performs the best among the designs for each of the variance ratio considered. The balanced MWP design is next best design. The worst design is the balanced VKM design. Therefore, in this case the MWP designs should be the design of choice rather than the VKM design. The VKM and MWP designs are not the only cuboidal designs available. Draper and John (1998) presented some split plot cuboidal designs but they do not satisfy the equivalent estimation criteria. Similarly, Goos and Vandebroek (2001) presented some D-optimal designs but they do not satisfy the equivalent estimation criteria.

125

Table 60: Integrated prediction variance for VKM CCD and MWP CCD with = α = 1, 3 whole plot factors and 4 subplot factors. Standardized I V IV Scaled by Design size Variance Ratio (η) Balanced Unbala nce d Ba lanced Unbalanced VKM MWP VKM MWP VKM MWP VKM MW P 0.5 0.323 0.262 0.339 0.267 43.956 40.01 0 41.98 6 37.16 0 1 0.362 0.277 0.375 0.282 49.288 42.406 46.536 39.18 9 10 0.459 0.315 0.461 0.316 62.376 48.195 57.205 43.92 0

5.11 Integrated Prediction Variance for Completely Randomized Designs Lucas (1974 and 1976) examined design optimality properties for several second- order designs, however the examination did not include integrated prediction variance. Park et al. (2005) studied the prediction variance properties of second-order designs for cuboidal regions. Their investigation focused on maximum prediction variance (G- optimal) and presented the average prediction variance for the outer edge of the design space. This approach will not give the exact value for the integrated prediction variance even though it can provide some insight into the behavior of the average prediction variance over the selected set of points. Also, several of the saturated designs developed for second-order models focus on the generalized variance of the parameter estimates (D- optimality). Designs such as those developed by Hoke (1974), Box and Draper (1974) and Notz (1982) all use the generalized variance as the criterion for optimal designs. Since, the designs are generated based on the variance of the parameter estimates and second-order designs should be primarily concerned with prediction variance then these designs may not be the best for model prediction. According to the equivalence theorem of Kiefer-Wolfowitz (1959 and 1961) D- optimal designs are also G-optimal, in the limit of an infinite number of trials. However, G-optimality does not necessarily provide the best average prediction variance over the design region. Table 61 presents the integrated prediction variance for the CCD and

BBD designs fo r 2 ≤ k ≤ 9. The number of centers ( nc) use is also present ed. The results indicate that for k = 3, 5 and 6 the CCD performs better than the BBD. However, for k = 7 the BBD is a be tt er design. Note t hat for k = 4 both the BBD and CCD give the same average scaled prediction variance, because in this case the BBD is simply a rotation of

126 the CCD design. Therefore, for k = 4 either design would be produce the same level of confidence in terms of predicting new observations. The data given for k = 2, 8 and 9 is presented for completeness. Also note that for k = 2 the CCD is equivalent to the equiradial design, thus giving the same prediction variance results.

Table 61: Integrated prediction variance for standard CCD and BBD spherical designs. IV k Design N nc Standardized Scaled 2 CCD 11 3 0.3611 3.9722

3 BBD 15 3 0.4619 6.9286 CCD 17 2 0.3890 6.6121

4 BBD 28 4 0.3611 10.1110 CCD 28 4 0.3611 10.1110

5 BBD 44 4 0.3360 14.7830 CCD 29 3 0.4130 11.9780

6 BBD 52 4 0.4242 22.0590 CCD 48 4 0.3887 18.6590

7 BBD 59 3 0.4423 26.0980 CCD 81 3 0.3784 30.6480

8 CCD 84 4 0.3967 33.3200

9 BBD 128 0 0.3587 45.9080

5.11.1 Hybrid and Small Composite Designs Giovannitti-Jensen and Myers (1989) and Meyers et al. (1992) compare the hybrid and SCD designs for three factors using variance dispersion graphs but do not present any information on the integrated prediction variance properties for these designs. The information presented here complements the work done by these authors and confirms their findings. For details on the construction of these designs see Roquemore (1976) and Hartley (1959). In computing and ranking the IV values for these designs the number of center runs and the ranking system used by Meyers et al. (1992) were utilized as shown

127 in Table 62. The ranking is done such that 1 and 4 represent the best and worst design respectively. The design matrices have been scaled such that the points in the design either lie on or inside a sphere of radius 1. The scaling factor use d for each design is also given in T able 62. In te rm s of integrated prediction variance properties the design with the smallest scaled IV value is desira ble. The results in Table 62 show that for k = 3 the hybrid 311B with 3 center runs performs the best followed by 311A. The SCD performs the worst as a result of havin g the l arg est scal ed IV value. It is im portant to point out that scaled optimali ty values are used to compare designs of different number of observations (N) that have the sam e number of factors. H oweve r, the standardized optimality values are ap pro priate for com par ing designs across different number of factors. A similar ranking is obtained for k = 4 with t he hybrid 4 16C w ith 5 center runs performing the best followed by 416B. Again the SCD performs the worst . Thes e results agree with the rankings given by Meyers et al. (1992). However, as pointed out by Meyers et al. (1992), while the hybrid design outperforms the S CD in terms of prediction variance properties, the use of the SCD is sti ll encou raged. This is part icularly so in situations where seq uential ex per imen tation an d augm entation is desirab le. The SCD is very appealing in this regard.

Table 62: Comparison of IV values for near-saturated designs over spherical regions for k = 3 an d 4 . IV Ranked Scaling Design N nc Standardized Scaled Performance Factor k = 3 SCD α = √k 13 3 0. 798 1 0.369 4 0.57735 Hybrid 310 12 2 0.67 0 8.0 34 3 0.64445 311A 12 2 0. 576 6.911 2 0.44721 311B 13 3 0. 519 6.745 1 0.40825 k = 4 SCD α = √k 19 3 0. 816 1 5.503 4 0.50000 Hybri d 41 6A 20 4 0.57 1 11.4 30 3 0.52244 416B 20 4 0. 569 1 1.387 2 0.54189 416C 20 5 0.568 1 1.362 1 0.54865

128 Table 63 presents the comparison of the integrated prediction variance for hybrid and CCD for k = 6 and 7. According to the variance dispersion results presented by Meyers et al. (1992) for k = 6 the hybrid design 628A performs much better than the half fraction CCD over a considerable portion of the design region. However, it is clear from the scaled IV values given in Table 63 that the half fraction CCD performs better. Therefore depending on the interest of the practitioner, the most suitable design can be chosen. If the average variance is more of a concern then the ½ fraction CCD provides the better option. However, if the cost constraint is more critical then the hybrid 628A design is the better option. According to Roquemore (1976) the hybrid 746 design for k = 7 is not a near-saturated design but represents an economical alternative to the half fraction CCD with α = 2.828. The comparison of both design in terms of their scaled IV values given in Table 63 indicate that the half fraction CCD outperforms the hybrid 746. Therefore, the only time the hybrid 746 should be used is when very limited resources are available.

Table 63: Comparison of IV values for Hybrid and CCD designs over spherical regions for k = 6 and 7. IV Ranked Scaling Design N nc Standardized Scaled Performance Factor k = 6

½ fraction CCD α 48 4 0.389 18.659 1 0.42052 = 2.378 Hybrid 628A 30 2 0.683 19.817 2 0.43301 628B 31 2 0.723 21.687 3 0.41547 k = 7

½ fraction CCD α 81 3 0.378 30.648 1 0.35355 = 2.828 Hybrid 746 47 2 1.311 61.601 2 0.30995

5.11.2 Equiradial Designs Myers and Montgomery (2002) discuss the construction and properties of equiradial designs and point out that these designs represent a reasonably alternative to the central composite design involving two variables. The classification of these designs

129 includes pentagon, hexagon, heptagon, octagon and nonagon. A major feature of these designs is that the points are equally spaced on a circle. Incidentally, the octagon is the same as the CCD for two factors. Myers and Montgomery (2002) stress that the octagon design should be used whenever possible and should only be overlooked if a cost constraint prohibits its use. These designs considered are rotatable. For more details see Myers and Montgomery (2002). Equation (1.62) was presented by Myers and

Montgomery (2002) to generate equiradial designs. Designs were constructed for n1 = 5,

6, 7 8 and 9 where n1 represents the number of points on a sphere of radius ρ. These design were generated using = 0 and ρ = 1. This was done to facilitate the computation of the integrated prediction variance and the proper comparison of the designs.

xx12 , un= 0,1,2,… ,1 − 1. (1.62) ⎣⎦⎡⎤ρθπcos()++ 2un11 ρθπ sin() 2 un

The designs were augmented with center runs to stabilize the prediction variance. It was suggested that two to four center runs are sufficient to achieve stabilization therefore the designs were assessed for all three cases with nc = 2, 3 and 4. Table 64 gives the results for both the standardized and scaled integrated variance properties for the selected equiradial designs. The scaled results reveal that for the case of two center runs the pentagon design performs marginally better than the hexagon design while the nonagon performs the worst. However, the pentagon is a saturated design and should only be used if there are limited resources available. In situations where three center runs are used, the heptagon performs marginally better than the octagon (CCD) with the pentagon performing the worst. For the case of four center runs, the nonagon design performs marginally better than the octagon (CCD) design with the pentagon design performing the worst.

130

Table 64: Integrated variances of selected equiradial designs

Equiradial nc= 2 nc = 3 nc = 4 Designs Standardized Scaled Standardized Scaled Standardized Scaled Pentagon 0.5667 3.9667 0.5111 4.0 889 0.4833 4.3500 Hexagon 0.5000 4.0000 0.4444 4.00 00 0.4167 4.1667 Heptagon 0.4524 4.0714 0.3968 3.9683 0.3691 4.0595 Octagon (CCD) 0.4167 4.1667 0.3611 3.9722 0.3333 4.0000 Nonagon 0.3889 4.2778 0.3333 4.0000 0.3056 3.9722

Surprisingly, the octagon (CCD) design did not perform the best for any of the three cases considered but it has other properties such as orthogonal blocking and sequential experimentation that still make it the design of choice. However, it is good to know that if resource constraints prevent the use of the octagon (CCD) then one might experience a relatively good average prediction variance. Observe that for two or three centers the hexagon and heptagon are attractive alternatives to the octagon (CCD). Note that the octagon (CCD) with three centers and the nonagon with four centers both have the same average scaled prediction variance. The G-efficiency value for each of the designs is also computed to further understand the predication capability of these designs. The results are presented in Table 65 for the case where 3 center runs are used. Table 65 shows that the nonagon designs recorded a 90% efficiency which outperforms the other designs in terms of G-efficiency followed by the octagon design with 87% efficiency. The pentagon design performs the worst with 75% efficiency.

Table 65: Maximum prediction variance properties of selected equiradial designs for k = 2 factors Equiradial Maximum Prediction Variance G- N nc Designs Standardized Scaled Efficiency Pentagon 8 3 1.0000 8.0000 75% Hexagon 9 3 0.8333 7.5000 80% Heptagon 10 3 0.7143 7.1429 84% Octagon (CCD) 11 3 0.6250 6.8750 87% Nonagon 12 3 0.5556 6.6667 90%

131 5.12 Comparison of Selected Cuboidal Designs The IV-criterion for selected cuboidal designs was computed using 0 through 5 center points. The designs with the lowest scaled integrated prediction variance are presented in Table 66. The standardized values are also presented. It is clear from the data that for 2 ≤ k ≤ 6 the FCC design is superior to the other cuboidal designs. For FCC designs with k equal to 4, 5 and 6, there are no degrees of freedom for lack of fit. Therefore, to facilitate the test for lack of fit 2 or 3 center runs can be used. The resulting scaled IV values are 8.45 and 8.53 for k = 4, 11.17 and 11.41 for k = 5 and 17.47 and 17.67 for k = 6. Even with these additional runs, the FCC design maintains the advantage over the other designs except for k = 6 when 3 center runs are used. For k = 6 the Hoke design offers a slightly better scaled IV value of 11.58. The Hoke design performs the best for 7 ≤ k ≤ 11 even with the addition of 2 to 3 center runs. In general, the best saturated designs are the Hoke designs because they consistently outperform the Notz, Box and Draper and the SCD designs. .

132 Table 66: Integrated prediction properties of selected cuboidal designs IV k Design N nc Standardized Scaled 2 FCC 11 3 0.3260 3.5863 Notz 9 3 0.4635 4.1716 Box-Draper 7 1 0.6060 4.2417

3 FCC 16 2 0.3405 5.4483 Notz 12 2 0.7073 8.4875 Hoke 16 3 0.3524 5.6382 SCD 10 0 1.0333 10.3330 Box-Draper 12 2 0.4944 5.9327

4 FCC 25 1 0.3376 8.4397 Notz 16 1 0.7430 11.8870 Hoke 21 2 0.4918 10.3280 SCD 16 0 1.0269 16.4300 Box-Draper 18 3 0.5708 10.2740

5 FCC 26 0 0.4155 10.8030 Notz 24 3 0.6213 14.9100 Hoke 28 2 0.4382 12.2700 SCD 21 0 1.9375 40.688 Box-Draper 24 3 0.8201 19.6820

6 FCC 44 0 0.3902 17.1670 Notz 32 4 0.6788 21.7200 Hoke 36 2 0.4883 17.5800 Box-Draper 31 3 1.2146 37.6540

7 FCC 78 0 0.3840 29.9490 Hoke 44 1 0.5768 25.3810 Box-Draper 39 3 1.7784 69.3590

8 Hoke 53 0 0.6790 35.9870 Box-Draper 48 3 2.5403 121.9400

9 Hoke 64 0 0.7814 50.0120 Box-Draper 57 2 3.5967 205.0100

10 Hoke 76 0 0.9625 73.1530 Box-Draper 68 2 4.8529 330.0000

11 Hoke 89 0 1.0813 96.2360 Box-Draper 80 2 6.4077 512.6100

133 5.13 Conclusion In most cases the computational time required to compute the IV-optimality criterion is a deterrent to most practitioners interested in using the IV-optimality criterion. Our investigation has led to the development of a computationally efficient method for computing the exact value of the IV-optimality criterion for second-order response surface designs. The functions developed are capable of dealing with completely randomized and split plot designs for both spherical and cuboidal regions. In particular, these functions are derived for the analytical determination of the IV-criterion for standard response surface designs such as the CCD and the BBD. Further, an attractive feature of these derived expressions is that they are functions of the design parameters. Therefore, the effect of changes in any of the design parameters can be easily evaluated without the need to generate the actual design. The application of these functions was demonstrated by the evaluation of several response surface second-order designs. The results reveal that the unbalanced split plot designs have a tendency to give better IV values when compared with the balanced split plot designs. It was also shown that for 4 factors the CCD and BBD are only equivalent for situations involving 2 whole plot factors and 2 subplot factors. Any other combination of whole plot and subplot factors will result in two different designs thus giving different optimality results. With respect to the complete randomized designs on spherical region, the results indicate that for k = 3, 5 and 6 the CCD performs better than the BBD. However, for k = 7 the BBD is a better design. For saturated and near saturated designs on spherical region the hybrid designs for k = 3 and 4 perform the best. However, for k = 6 and 7 the ½ fraction CCD is perform better than the hybrid designs. For equiradial designs best performer is pentagon using 2 center runs. However, the use of this design should only be considered when there are limited resources available. For a cuboidal region of interest the FCC designs for 2 ≤ k ≤ 6 are the best designs while for 7 ≤ k ≤ 11 the Hoke designs are the best. The resulting equations and matrices presented provide and efficient and exact way of computing integrated prediction variance for second-order designs. Therefore, it is now possible for some of the cheaper commercially available software packages to present accurate results for integrated prediction variance without having do extensive

134 numerical computations. Currently work is underway to utilize the results presented in this paper to generate optimal second-order split plot designs. The method also has potential for dealing with higher order models such as cubic models.

135 CHAPTER 6

6.0 CONCLUSION

The intention of this dissertation was to facilitate a deeper understanding of the impact of restricted randomization on response surface designs. The focus of the dissertation was governed by three research objectives dealing with the impact of restricted randomization on the information matrix, prediction variance properties and G- criterion location for split plot designs and integrated prediction properties of split plot designs and complete randomized designs. These issues were thoroughly investigated and a full report presented in Chapters 3 through 5. The methodology utilized was to analytically characterize the structure of the information matrix and its inverse for split plot designs. The analytical characterization of the inverse of the information matrix was achieved for both balanced and unb al anced second-order SPD and is applicable to both spherical and cuboidal regions of interest. The entire characterization process was done using the design parameters therefo r e making it possible for practitioners to assess the impact of changing any of the design parameters on the optimality of the design. The results of the analytical characterization process were then used to develop a computationally efficient method to compute the exact values of design optimality properties such as the generalized variance of parameter estimates, maximum and minimum prediction variance and integrated prediction variance. In addition, the methods presented are also capable of dealing with completely randomized designs. The following section presents a synopsis of the conclusions associated with each research objectives

136 6.1 Research Objective 1 Chapter 3 presents a thorough investigation on the impact of restricted randomization on the information matrix. Several important insights and implications are presented for the construction of second-order split plot designs within a response surface structure. The investigation reveals several critical functions of the variance ratio and whole plot size(s). It was observed that, depending on the assignment of subplot axial points, the pure quadratic terms of the subplot factor will have similar correlation influence as the whole plot pure quadratic terms. This correlation influence has a tendency to negatively affect the optimality of the design. The structure of the design does have an influence on how the critical functions are applied to the information matrix. This influence is particularly evident with the design moments of order four for the subplot factors. In situations where the design structure is such that the whole plot axials are placed in separate whole plots and the subplot axials are also placed in separate whole plots then the effect of the appropriate critical function is the same for both the whole plot and subplot design moments of order four. However, if all the subplot axials are placed in the same whole plot then the subplot axials would be affected by the inverse correlation effect and co-effect within the whole plot. The overall effect is such that the design moments of order four for the subplot is corrected for the influence of the correlation co-effect within the whole plot. In situations where the whole plot axials are placed in the same whole plots as the factorial runs and all the subplot axials in the same whole plot then the inverse correlation effect is now affecting the contribution of the factorial points to the design moments of order four for the subplot. In contrast, if separate whole plots for the subplot axials then the inverse correlation co-effect has no influence on the subplot axials. However, the inverse correlation effect is still affecting the contribution of the factorial points to the design moments of order four for the subplot.

6.2 Research Objective 2 Chapter 4 evaluates the impact of changes in the variance ratio on the prediction properties of second-order SPDs. An analytical method, rather than a heuristic

137 optimization algorithm is used to compute the prediction variance properties, which include the maximum, minimum, and integrated variances for second-order SPD. For various designs, these measures are graphically represented using variance dispersion graphs. The research work reveals that the location of the maximum prediction variance can change based on the variance ratio used. The results indicate that the G-criterion location has a tendency to be positioned at the center of the whole plot factor and at the extreme of the subplot factor(s). Therefore, it is recommended that when reporting the value of the maximum p redic tio n variance, its location should also be reported. Reporting the location is especially important for designs where the location is not stable and changes position depending on the magnitude of the variance ratio. In terms of design development and selection, the practitioner should select a design where the location of the maximum prediction variance is robust to changes in the variance ratio. The practitioner should at least be aware of the possible locations where the maximum prediction variance may occur based on the magnitude of the variance ratio. Knowing the location of the maximum prediction variance will alert the practitioner prior to data collection as to where in the design region to expect the worse prediction and where to collect more data if necessary. The analytical method presented also results in the successful separation of the SPD into the two distinct design spaces that characterizes the design. The separation of the design spaces facilitates an understanding of the relative efficiencies of each distinct design space with respect to the combined design space. A graphical approach and the use of relative efficiencies for these two distinct design spaces are suggested as evaluation methods for assessing the prediction capability of a given design. The versatility of the methods presented provides various ways a practitioner can use the information to make a decision in selecting the best design for experimentation.

6.3 Research Objective 3 Chapter 5 develops an efficient method to compute the exact IV-optimality criterion for response surface designs. In most cases the computational time required to compute the IV-optimality criterion is a deterrent to most practitioners interested in using

138 the IV-optimality criterion. Our investigation has led to the development of a computationally efficient method for computing the exact value of the IV-optimality criterion for second-order response surface designs. The application of these functions was demonstrated by the evaluation of several response surface second-order designs. The results reveal that the unbalanced split plot designs have a tendency to give better IV-values than the balanced split plot designs. It was also shown that for 4 factors the CCD and BBD are only equivalent for situations involving 2 whole plot factors and 2 subplot factors. Any other combination of whole plot and subplot factors will result in two different designs thus giving different optimality results. With respect to the complete randomized designs on a spherical region, the results indicate that for k = 3, 5 and 6 factors the CCD performs better than the BBD. However, for k = 7 the BBD is a better design. For saturated and near saturated designs on a spherical region the hybrid designs for k = 3 and 4 factors perform the best. However, for k = 6 and 7 the half fraction CCD performs better than the hybrid designs. For equiradial designs the best performer is the pentagon using 2 center runs. However, the use of this design should only be considered when there are limited resources available. For a cuboidal region of interest the FCC designs for 2 ≤ k ≤ 6 are the best designs while for 7 ≤ k ≤ 11 the Hoke designs are the best. The resulting equations and matrices presented provide an efficient and exact way of computing integrated prediction variance for second-order designs. Therefore, it is now possible for some of the economical commercially available software packages to present accurate results for integrated prediction variance without having do extensive numerical computations.

6.4 Future Work Currently work is underway to utilize the results presented in this dissertation develop algorithms for generating G-optimal and IV-optimal second-order split plot designs. The matrix manipulation method presented for the computation of the integrated prediction variance also has potential for application to higher-order models such as cubic models.

139 APPENDIX A

A1: Analytical Characterization of (X'R-1X)-1 for Split Plot CCD

The details of the partitioned matrix A for the split plot CCD is given as:

⎛⎞Π 0 A11 = ⎜⎟ ⎝⎠0 Diag() di ⎛⎞Φ+frζβ22Φ2222JJ'' Φ+ fr ζαΦ A = ⎜⎟()16wwp() 17 ssp 12 ⎜⎟ ⎝⎠00 ⎛⎞Φ+frζβ2 Φ2 2 J0 ⎜⎟()16wwp A21 = ⎜⎟Φ+frζα22Φ2 J0 ⎝⎠()17ssp ⎛ ' ⎞ 44' 4⎡⎤' ⎜Φ+162rfwwβζIJpΦΦ wpwJJp 1 f ζspwJp ⎟ A = ⎣⎦ 22 ⎜ ⎟ ⎜ ΦΦfrfζαζ444JJ' 22+Φ I+ΦΦ+Φrα4 JJ' ⎟ ⎝ 15sp wp() s9 sp()413 s spsp⎠

It can be observed that A22 is further partitioned into four (4) sub-matrices to reflect the nature of split plot designs in terms of the two distinct design spaces. Because of this partitioning the principles outlined in Figure 2 must also be applied to obtain the

−1 −1 components of A22 . Consequently, A22 and A22 are denoted as

⎛⎞CC11 12-1 ⎛ DD 11 12 ⎞ AA22 ==⎜⎟ and 22 D= ⎜⎟ ⎝⎠CC21 22 ⎝ DD 21 22 ⎠ where

44' CIJ11=Φ() 62rfwβζ wp +Φ 1 w wpJ wp CJ=Φ f ζ 4 J' ' 12 1 sp wp ' CC21= 12 444' CI22=Φ 522rfsαζsp +() Φ 4 Φ 1 +Φ 3 r sα J spJ sp

140 • The components of D11

The expression of the Schur Complement for C11 is

−−11 DCCCC11=− 11 12 22 21

−1 (i) In order to solve this expression we need to first findC22 . Since C22 is symmetric its inverse is of the form

−1 −1' CIJ22 =+(absp spJ sp )

1 ⎛⎞b ' =−⎜⎟IJspspJsp a⎝⎠ a+ spb

444 where ar=Φ592ssα +Φ and b =Φ 4 Φ13 fζα +Φ 2 r. Therefore

1 ⎛⎞ΦΦfrζα44 +Φ2 CI−1 =−⎜⎟41 3 s JJ' 22 Φ+2r α 4 Φ⎜⎟spsΦ+22rsαζ44Φ+ΦpΦ+frΦα4psp 59s ⎝⎠594ss()13

−1 (ii) Pre-multiplying C22 by C12

4 44 Φ f ζ ' ⎛⎞ΦΦfrζα +Φ2 CC−1'=−1 ⎡⎤ JJ⎜⎟ I 41 3 s JJ' 12 22 Φ 2r α 4 ⎣⎦spwp⎜⎟ sp Φ+22rsαζ44Φ+ΦpΦ+frΦα4spsp 5 s ⎝⎠594ss()13

4 ⎛⎞44 Φ f ζ ' sp()ΦΦ41 fζα +Φ 32 rs =−1 ⎜⎟⎡⎤JJ' JJ' Φ 2r α 4 ⎜⎟⎣⎦sp wp Φ+22rsαζ44Φ+ΦpΦ+frΦα4sp wp 5 s ⎝⎠594ss()13 4 Φ f ζ ' = 1 ⎡⎤JJ' 4 44⎣⎦sp wp Φ+592rssα Φ+p()ΦΦ41frζα +Φ 32 s

(iii) Post-multiplying by C21

4 Φ f ζ ' CC−1Cf=Φ1 ⎡JJJJ'4⎤⎡ζ '⎤ 12 22 21 444⎣ spwp⎦⎣1 spwp⎦ Φ+59422rsssαζΦ+Φp()Φ+13frΦα 2 4 ()Φ1 fsζ p = ⎡⎤JJ' 444⎣⎦wp wp Φ+59422rsssαζΦ+Φp()Φ+13frΦα

141 Consequently,

−−11 DCCCC11=− 11 12 22 21 2 4 ()Φ1 fsζ p =Φ2rfβζ44IJ +Φ⎡⎤JJ' − ⎡⎤J' ()11wwp w⎣⎦ wpwp 444 ⎣⎦wp wp Φ+59422rsssαζΦ+Φpfr()Φ+13Φα 2 4 ()Φ1 fsζ p =Φ2rfβζ44IJ +Φ⎡⎤JJ' − ⎡J' ⎤ 11w wp w⎣⎦ wp wp 444 ⎣wp wp ⎦ Φ+59422rsssαζΦ+Φpfr()Φ+13Φα 2 ⎛⎞4 ()Φ1 f ζ sp =Φ2rfβζ44I +⎜⎟ Φ − ⎡⎤JJ' 11wwp⎜⎟ w Φ+22rsαζ44Φ+ΦpfrΦ+Φα4⎣⎦wp wp ⎝⎠594ss()13 thus

−1 −1 DD11= () 11 −1 2 ⎛⎞⎛⎞4 ()Φ1 fsζ p =Φ⎜⎟2rfβζ44IJ +Φ⎜⎟ − ⎡⎤J' 61wwp w 444⎣⎦wp wp ⎜⎟⎜⎟Φ+22rsαζΦ+ΦpfrΦ+Φα ⎝⎠⎝⎠594ss()13

2 ⎛ 4 ⎞ ⎜ ()Φ1 fsζ p ⎟ Φ−f ζ 4 ⎜ 1 w 444⎟ 1 Φ+59422rsssαζΦ+Φpfr()Φ+13Φα =−⎜I ⎡⎤JJ' ⎟ 4 wp 2 wp wp ⎜ 4 ⎣⎦⎟ Φ6 2rwβ ⎛⎞Φ fsζ p ⎜ 44 ()1 ⎟ Φ+Φ−2rwβζpf⎜⎟ ⎜ 61ww444 ⎟ ⎜ ⎜⎟Φ+2rsαζΦ+ΦpfΦ+Φ2r α ⎟ ⎝ ⎝⎠594s ()13 s ⎠ 1 ⎛⎞c =−IJ1 ⎡⎤J' 44⎜⎟wp⎣⎦ wp wp ΦΦ66122rrwwββ⎝⎠+wpc where

2 4 (Φ1 fsζ ) p cf=Φζ 4 − 11w 444 Φ+59422rsssα Φ+Φp()Φ+13frζαΦ

142 • Components of D12

−1 DD12=− 11C 12C 22 4 ⎛⎞ ' 1 ⎛⎞cf11' Φ ζ ' =−⎜⎟⎜⎟IJwp − ⎡ wpJ wp ⎤⎡JspJ wp ⎤ ⎜⎟ΦΦ22rrββ44+wpc⎣ ⎦⎣444⎦ ⎝⎠661ww⎝⎠Φ+59422rsssαζΦ+Φpfr()Φ+13Φα

4 ⎛⎞' Φ1 fcζ ''⎛⎞1 =− ⎜⎟⎡⎤JJsp wp −⎜⎟ ⎡⎤ JJsp wp 4444⎜⎟⎣⎦Φ+2rwβ 4 pc ⎣⎦ ()Φ+59422rsssαζΦ+Φpfrr()Φ+136ΦΦα2wβ⎝⎠⎝⎠61w

4 ' Φ11fwζ ⎛⎞pc' =− ⎜⎟1− ⎡⎤JJsp wp 4444Φ+2rwβ 4 pc⎣⎦ ()Φ+59422rsssαζΦ+Φpfrr()Φ+136ΦΦα2wβ⎝⎠61w

4 4 ' Φ1 f ζ ⎛⎞Φ6 2rwβ ' =− ⎜⎟⎡⎤JJsp wp 4444Φ+2rwβ 4 pc⎣⎦ ()Φ+59422rsssαζΦ+Φpfrr()Φ+136ΦΦα2wβ⎝⎠61w 4 Φ f ζ ' 1 ⎡⎤' =− ⎣⎦JJsp wp c2 4444 where, cr25=Φ()22ssαζ +Φ+ 9 spfr() ΦΦ 41 +Φ 3α( Φ 12 rwwβ + pc 1)

• The components of D21

' DD21= 12 Φ f ζ 4 1 ⎡ ' ⎤ =− ⎣JJspwp⎦ c2

• The components of D22

−−11 DCDDD22=+ 22 21 11 12

−1 (i) Pre-multiplying D11 by D21

Φ f ζ 4 −1'1 ⎡⎤4 ⎡' ⎤ DD21 11 =− ⎣⎦ JJsp wp()Φ62rc wβ I wp + 1 ⎣ J wp J wp ⎦ c2 ΦΦffζζ44 114'⎡ ⎤⎡ '⎤ =−() Φ612rcwβ ⎣JJ spwp⎦⎣ − ()wp JJ spwp⎦ cc22 Φ f ζ 4 1 4'⎡⎤ =−() Φ612rcwsβ + wp⎣⎦JJpwp c2

−1 (ii) Post-multiplying DD21 11 by D12

44 ΦΦffζζ⎛⎞' −1411⎡ '⎤⎡'⎤ DDD21 11 12=−() Φ 62rcwβ + 1wp⎣ JJ spwp⎦⎣⎜⎟ − JJ spwp⎦ cc22⎝⎠ 2 ⎛⎞Φ f ζ 4 4'1 ⎡⎤ =Φ()612rwsβ + c wp⎜⎟ − wp ⎣⎦JJpsp ⎝⎠c2

143 (iii) Consequently

−−11 DCDDD22=+ 22 21 11 12 1 ΦΦfrζα44 +Φ2 =−IJ41 3 s J''+∆⎡⎤JJ Φ 2r α 4 sp 44 4 4sp sp⎣⎦ sp sp 5 s ΦΦ+5522rrsssαα()pfr()Φ 4Φ+13 ζΦ 2 s α ⎛⎞ 1 ΦΦfrζα44 +Φ2 =−IJ⎜⎟41 3 s −∆⎡⎤J' 4 sp 44 4 4⎣⎦sp sp Φ5 2rsα ⎜⎟ΦΦ+22rrsααpfrΦΦ+ ζΦ 2 α ⎝⎠55ss()() 413 s 2 ⎛⎞ΦΦf ζα44+Φ22rc − Φ r αζ 44 Φ f wp 1 ⎜⎟()41 3251ss( ()) =−I ⎡⎤JJ' Φ 2r α 4 ⎜⎟sp 444⎣⎦sp sp 5 s ⎜⎟()Φ+Φ5422rsssαζp()Φ+1 fΦ3 rcα2 ⎝⎠

2 4 4 ⎛⎞Φ1 f ζ where, ∆=() Φ612rcwβ +wp⎜⎟ − wp ⎝⎠c2

−1 By combining the results of D11, D12, D21 and D22, the components of A22 matrix can be obtained.

1 ⎛⎞c DI=−1 ⎡⎤ JJ' 11 44⎜⎟wp⎣⎦ wp wp ΦΦ66122rrwwββ⎝⎠+wpc 4 Φ f ζ ' 1 ⎡⎤' DJ12 =− ⎣⎦spJ wp c2 ' DD21= 12 2 ⎛⎞ΦΦfrζα44 +Φ22cr − Φ α 4 +Φ Φ f ζ 4wp 1 ⎜⎟()41 3ss 2(( 5 9)( 1 )) DI=− ⎡JJ' ⎤ 22 Φ+2r α 4 Φ⎜⎟sp 444⎣ sp sp ⎦ 59s ⎜⎟()Φ+59422rsssαζΦ+Φp()Φ+132frΦαc ⎝⎠

Returning to the information matrix represented by the matrix A and applying the results

−1 obtained for A22 the Schur Complement can now be formed and inverted. This then leads to the analytical characterization of the information matrix for split plot CCD design. Recall that the Schur Complement is given as

−−11 ABAAAA11==− 11 11 12 22 21

144 • The components of B11

−−11 −−11 BB11==−() 11 ( AA 11 12AA 22 21 )

−1 (i) Pre-multiplying A22 by A12

⎛⎞Φ+frζβ2'2'Φ2222JJ Φ+ fr ζαΦ ⎛⎞DD AA−1 = ⎜⎟()16wwwp( 17 s) sp11 12 12 22 ⎜⎟⎜⎟ 00⎝⎠DD21 22 ⎝⎠

⎛⎞XYXYDDDD11++ 21 12 22 = ⎜⎟ ⎝⎠00

−1 (ii) Post-multiplying AA12 22 by A21

⎛⎞2 2 XYXYDDDD++(Φ+16frwwζβΦ2 )J0wp −1 ⎛⎞11 21 12 22 ⎜⎟ AAA12 22 21 = ⎜⎟ ⎝⎠00⎜⎟Φ+frζα2 Φ2 2 J0 ⎝⎠()17ssp

⎛⎞()XYXXYYDD11+++ 21() DD 12 22 0 = ⎜⎟ ⎝⎠00 ⎛⎞M 0 = ⎜⎟ ⎝⎠00

where, M =+()XYXXYDD11 21 ++( DD 12 22 )Y

⎡ 1 ⎛⎞c ⎤ X2DJ=Φfrζβ2' +Φ2 I − 1 ⎡⎤JJ' 11() 1w 6 w wp⎢ 44⎜⎟ wp⎣⎦ wp wp ⎥ ⎣⎢ΦΦ66122rrwwββ⎝⎠+wpc⎦⎥ Φ+frζβ2 Φ2 2 ()16ww⎡⎤''wpc1 =−44⎢⎥JJwp wp ΦΦ6622rrwwββ⎣⎦+wpc1 Φ+frζβ2 Φ2 2 ()16ww⎡⎤wpc1 ' =−44⎢⎥1 Jwp ΦΦ6622rrwwββ⎣⎦+wpc1 Φ+frζβ2 Φ2 2 ()16ww' = 4 J wp Φ+6 2rwβ wpc1 ' = c3J wp

2 2 Φ+16frwwζ Φ2 β where, c3 = 4 Φ+612rwwβ pc

145 ⎛⎞Φ f ζ 4 2'2 1 ⎡ '⎤ Y2DJ21 =Φ()17frζα +Φssp⎜⎟ − ⎣J spJwp⎦ ⎝⎠c2 ' = c4J wp

42 2 ΦΦ+11f ζζ()frΦ 72 s αsp where, c4 =− c2 therefore,

'' XYDD11+=+ 21 cc34 Jwp J wp ' =+()cc34Jwp ' = c5J wp

'22 ()XYXDD11+ 21 =Φ+()cf51JJwp( wζβΦ 62 r w) wp

2 2 =Φcf51()wwζβ +Φ 62 rwp

4 ⎛⎞Φ f ζ ' 2'2 1 ⎡ '⎤ X2DJ12 =Φ()16frwζβ +Φ w wp⎜⎟ − ⎣J spJ wp ⎦ ⎝⎠c2 ' = c6J sp

42 2 ΦΦ+11f ζζ()frwwΦ 62 βwp where, c6 =− c2

⎡ 1 ⎤ Y2DJ=Φfrζα2' +Φ2 I − c⎡JJ'⎤ 22() 1 7 ssp⎢ 4 () sp7 ⎣ spsp⎦ ⎥ ⎣Φ+592rsα Φ ⎦

2 2 Φ+17frζαΦ2 s '' =−4 ()JJspcsp7 sp Φ+592rsα Φ

2 2 Φ+17frζαΦ2 s ' =−4 ()1 csp7 Jsp Φ+592rsα Φ ' = c8J sp where

2 ΦΦf ζα44 +Φ22rc − Φ r α 4 +Φ Φ f ζ 4 wp ()41 3ss 2(( 5 9)( 1 )) c = 7 444 ()Φ+59422rsssαζΦ+Φp()Φ+132frΦαc

2 2 Φ+17frζαΦ2 s cc87=−4 ()1 sp Φ+592rsα Φ

146 '' XYDD122268+=+cc JJspsp ' =+()cc68J sp ' = c9Jsp

'2 2 ()XYYDD1222+=Φ(cf 9 Jsps)( 1ζα+Φ 7 2 r) Jsp

2 2 =Φcf91()ζα +Φ 72 rss p Consequently

M =+()XYXXYYDD11 21 ++( DD 12 22 )

2222 =cf51() Φζβ +Φ 622 rwwspcf + 91( Φ ζα +Φ 7 rs)p

= c10 Therefore

−1 −1 BAA11=−() 11 12AA 22 21 −1 ⎡⎤⎛⎞Π 0 ⎛⎞c10 0 =−⎢⎥⎜⎟ 0 Diag d ⎜⎟ ⎣⎦⎢⎥⎝⎠()i ⎝⎠00 −1 ⎛⎞Π−c10 0 = ⎜⎟ ⎝⎠0 Diag() di ⎛⎞1 0 ⎜⎟Π−c = ⎜⎟10 ⎜⎟⎛⎞ ⎜⎟0 Diag ⎜⎟1 ⎝⎠⎝⎠di

⎛⎞γ1 0 ⎜⎟ = ⎛⎞ ⎜⎟0 Diag ⎜⎟1 ⎝⎠⎝⎠di

where 1 γ1 = c11

cc11=Π− 10

147 • The components of B12

−1 BB12=− 11A 12A 22 ⎛⎞1 ⎜⎟0 c11 ⎛⎞XYXYDDDD11++ 21 12 22 =−⎜⎟⎜⎟ ⎜⎟⎛⎞⎝⎠00 ⎜⎟0 Diag ⎜⎟1 ⎝⎠⎝⎠di ⎛⎞1 0 ⎜⎟c ⎛⎞ccJJ'' =−⎜⎟11 ⎜⎟59wp sp ⎜⎟⎛⎞⎝⎠00 ⎜⎟0 Diag ⎜⎟1 ⎝⎠⎝⎠di ⎛⎞cc 59JJ'' =−⎜⎟ccwp sp ⎜⎟11 11 ⎜⎟ ⎝⎠00 ⎛⎞γγJJ'' = ⎜⎟23wp sp ⎝⎠00

• The components of B21

' BB21= () 12

⎛⎞γ 2J0wp = ⎜⎟ ⎝⎠γ 3J0sp where c γ =− 5 2 c 11 c9 γ 3 =− c11

148 • The components of B22

−−11 BABBB22=+ 22 21 11 12

−1 (i) Pre-multiplying B11 by B21

⎛⎞c 5 J0 ⎜⎟c wp ⎛⎞c 0 −1 ⎜⎟11 11 BB21 11 =− ⎜⎟ 0 Diag d ⎜⎟c9 ⎝⎠()i ⎜⎟J0sp ⎝⎠c11

⎛⎞c5J0wp =−⎜⎟ ⎝⎠c9J0sp

−1 (ii) Post-multiplying BB21 11 by B12

⎛⎞cc ⎛⎞c J0−−59JJ'' BBB−1 =− 5 wp ⎜⎟ccwp sp 21 11 12 ⎜⎟⎜⎟11 11 ⎝⎠c9J0sp ⎜⎟ ⎝⎠00 2 ⎛⎞()c cc ' ⎜⎟5 ''59⎡⎤ JJwp wp⎣⎦ JJ sp wp ⎜⎟cc = 11 11 ⎜⎟2 cc ()c ⎜⎟59JJ''9 JJ ⎜⎟sp wp sp sp ⎝⎠cc11 11

⎛⎞EE11 12 = ⎜⎟ ⎝⎠EE21 22

Consequently

⎛⎞DD11 12⎛ EE 11 12 ⎞ B22 =+⎜⎟⎜⎟ DD EE ⎝⎠21 22⎝ 21 22 ⎠

⎛⎞DE11++ 11 DE 12 12 = ⎜⎟ ⎝⎠DEDE21++ 21 22 22

149 2 1 ⎛⎞c (c ) DE+= I −1 ⎡⎤ JJ'' +5 JJ 11 11 44⎜⎟wp⎣⎦ wp wp wp wp ΦΦ661122rrwwββ⎝⎠+wpcc1 2 1 c ()c =−IJ1 ⎡⎤J'' +5 JJ Φ 2rcβ 4 wp 44⎣⎦wp wp wp wp 6 w ΦΦ+6622rrwwwββ()pc 1 11

2 1 ⎛⎞c ()c =−IJ⎜⎟1 −5 J' Φ 2rcβ 4 wp ⎜⎟ΦΦ+22rrwββ44pc wp wp 61w ⎝⎠66ww() 11 2 1 ⎛⎞⎛⎞c Φ 2rcβ 4 () =−⎜⎟I ⎜⎟1 − 65w JJ' ΦΦ22rrβ 4 ⎜⎟wp ⎜⎟β 4 + wpc c wp wp 66ww⎝⎠⎝⎠111

1 ' =−4 ()IJwpγ 4 wpJ wp Φ6 2rwβ where

4 2 c1 Φ652rcwβ () γ 4 =−4 Φ+6112rwwβ pc c1

4 Φ f ζ ''cc 1 ⎡ ''⎤⎡59 ⎤ DE12+= 12 −⎣ JJsp wp⎦⎣ + JJ sp wp ⎦ cc211 4 ⎛⎞Φ f ζ cc ' 1 59 ⎡⎤' =−⎜⎟ + ⎣⎦JJsp wp ⎝⎠cc211 ' ⎡⎤' = γ 5 ⎣⎦JJsp wp

4 Φ1 f ζ ''cc59 D21+= E 21 − JJspwp + JJ spwp cc211 4 ⎛⎞Φ1 f ζ cc59 ' =−⎜⎟ + JJsp wp ⎝⎠cc211 ' = γ 5JJsp wp where

4 Φ1 f ζ cc59 γ 5 =− + cc211

150 2 1 ()c DE+= I −c ⎡⎤ JJ'' +9 JJ 22 224 ()sp 7 ⎣⎦ sp sp sp sp Φ512rcsα 1 2 1 ⎛⎞⎛⎞Φ 2rcα 4 () =−⎜⎟IJ⎜⎟c −59s J' Φ 2rcα 4 ⎜⎟sps⎜⎟7 psp 51s ⎝⎠⎝⎠1

1 ' =−4 ()IJspγ 6 spJ sp Φ5 2rsα

4 2 Φ592rcsα () where ,γ 67=−c c11

By combining the results of B11, B12, B21 and B22, the components of the inverse of the information matrix A−1 can be obtained.

⎛⎞γ1 0 ⎜⎟ B11 = ⎛⎞ ⎜⎟0 Diag ⎜⎟1 ⎝⎠⎝⎠di '' ⎛γ23Jwp γJsp ⎞ B12 = ⎜⎟ ⎝⎠00

⎛⎞γ 2J0wp B21 = ⎜⎟ ⎝⎠γ 3J0sp ⎛⎞1 ' ''⎡⎤ 4 ()IJwp−γγ45 wpJJ wp spJ wp ⎜⎟Φ 2r β ⎣⎦ ⎜⎟6 w B22 = ⎜⎟''1 ⎜⎟γγ56JJsp wp 4 () Isp− JJ sp sp ⎝⎠Φ+592rsα Φ

−1 Therefore (XR'1− X) for SPD within a CCD Structure is given as

⎛ γγγ0J'' J⎞ ⎜ 123wp sp ⎟ ⎛⎞ ⎜ 000Diag ⎜⎟1 ⎟ ⎜ ⎝⎠di ⎟ ⎜ ⎟ 1 ' ⎜γγJ0 IJ− JJ''γ⎡⎤J ⎟ 24wp 4 ()wp wp wp5⎣⎦ sp wp ⎜ Φ6 2rwβ ⎟ ⎜ ⎟ ⎜ ''1 ⎟ γγ35J0sp JspJI wp 4 ()sp−γ6J spJ sp ⎜ Φ+2r α Φ ⎟ ⎝ 59s ⎠

151 A2: Analytical Characterization of (X'R-1X)-1 for Split Plot BBD

The components of the sub-matrices for the information matrix, A, for the split plot BBD structure is given as follows:

⎛⎞Π 0 A11 = ⎜⎟ ⎝⎠0 Diag() di

⎛⎞ΦΦfrβα2'JJ fr 2' A = ⎜⎟11()wwp () ssp 12 ⎜⎟ ⎝⎠00

⎛⎞Φ fr β 2 J0 ⎜⎟1 ()wwp A21 = ⎜⎟Φ frα 2 J0 ⎝⎠1 ()ssp

⎛ 44' 4'' ⎞ Φ−1 ()fr()wwλβIJ wpw + f λβ wpwpJJ( Φ1intf λα)spwpJ A = ⎜ ⎟ 22 ⎜ 4' 4 4 4 4 ' ⎟ ⎜ ()ΦΦ1intffλαJJsp wp 5()r s− λα s I +()Φ 14Φf λα s+Φ 3()f ()r s− λαλα s+f c cJJ sp sp ⎟ ⎝ sp ⎠

−1 Again lets denote A22 and A22 as

⎛⎞CC11 12-1 ⎛ DD 11 12 ⎞ AA22 ==⎜⎟ and 22 D= ⎜⎟ ⎝⎠CC21 22 ⎝ DD 21 22 ⎠ where

44' CI11=Φ 1 ()fr()ww −λβ wp + f λβ wJ wpwJp 2'' CJ12=Φ() 1f λα int spJ wp ' CC21= 12 44 44' CI22=Φ 5fr()s −λαs sp +ΦΦ() 1 4 f λα s +Φ 3 () fr () s − λα s + f c λα c J spJ sp

• The components of D11

The expression of the Schur Complement for C11 is

−−11 DCCCC11=− 11 12 22 21

152 where

−1 C22 = ⎛⎞444 1 ()ΦΦ14ffλαss +Φ 3()()rf − λs α + c λαc ⎜⎟IJ− J' 4 sp 44 44sp sp Φ−5 fr()ssλα⎜⎟Φ−frλα +Φ spΦ f λα +Φ fr − λα + f λα ⎝⎠51()ss()4 s3() () ss cc

−1 (i) Pre-multiplying C22 by C12

444 4 ⎛⎞ Φ f λα ' (ΦΦ14ffλαss +Φ 3( ()rf − λs α + c λαc)) CC−1'=−1int ⎡⎤ JJ⎜⎟ I JJ' 12 22 4 ⎣⎦spwp sp 44 44sp sp Φ5 fr()ss−λα ⎜⎟Φ−frλα +Φ spΦ f λα +Φ fr − λα + f λα ⎝⎠51()ss()4 s3() () ss cc

444 4 ⎛ ⎞ Φ f λα ' ()ΦΦ14ffλαss +Φ 3()()rf − λs α + c λαc =1int ⎜ ⎡⎤JJ' − JJ' ⎟ 4 ⎣⎦sp wp 4 444sp wp Φ−5 fr()ssλα⎜ Φ−frλα +Φ sp Φ+ffλαΦ−+rf λ α λα ⎟ ⎝ 5 ()ss ()14ss 3()()scc⎠ 4 Φ f λα ' = 1int ⎡⎤JJ' 4 ⎣⎦sp wp Φ−5 fr()ssλα + aa

444 where, aa=Φ sp()14Φ fλαss +Φ 3( f() r − λs α + fc λαc)

(ii) Post-multiplying by C21

4 Φ f λα ' CC−1'Cf=Φ1int ⎡ JJ⎤⎡λα4 JJ'⎤ 12 22 21 4 ⎣ spwp⎦⎣1 int spwp⎦ Φ−5 fr()ssλα + aa 2 4 ()Φ1intfsλα p = ⎡⎤JJ' 4 ⎣⎦wp wp Φ−5 fr()ssλα + aa

Consequently,

−−11 DCCCC11=− 11 12 22 21 2 Φ fsλα4 p 44'()1int ' =Φ1 ()fr()ww −λβIJ wp + f λβ w wpwJJp − 4 wpwJp Φ−5 fr()ssλα + aa 2 Φ fsλα4 p 44'()1int ' =Φ11fr()w −λβ wIJ wp +Φ f λβ w wpwpJJ − 4 wpwpJ Φ−5 fr()ssλα + aa 2 ⎛⎞4 ()Φ1intfsλα p =Φfr() −λβ44IJ +⎜⎟ Φ f λβ − J' 11ww wp⎜⎟ w Φ−fr()λα4 + aa wp wp ⎝⎠5 ss

153 Thus

−1 −1 DD11= () 11 −1 2 ⎛⎞⎛⎞4 ()Φ1intfsλα p =Φ⎜⎟fr() −λβ44IJ +Φ⎜⎟ f λβ − J' ⎜⎟11ww wp⎜⎟ w 4 wpwp ⎜⎟Φ−5 fr()ssλα + aa ⎝⎠⎝⎠ 1 ⎛⎞c =−IJ1 J' 44⎜⎟wp wp wp Φ−111fr()wwλβ⎝⎠ Φ− fr() ww λβ + wpc

2 4 (Φ1intfsλα) p where, cf=Φλβ4 − 11w 44 44 Φ−51fr()ssλα +Φ sp()Φ4 f λα s +Φ3() fr () ss − λα + f cc λα

The components of D12

−1 DD12=− 11C 12C 22

4 ⎛⎞1 ⎛⎞c Φ f λα ' =−⎜⎟⎜⎟IJ − 1 J''1int ⎡JJ⎤ ⎜⎟Φ−frλβ44⎜⎟wp Φ− fr λβ + wpcwp wp Φ− fr λα 4 + aa⎣ sp wp ⎦ ⎝⎠111()ww⎝⎠() ww 5()ss

4 Φ f λα ⎛⎞' ⎛⎞c =− 1int ⎜⎟⎡⎤JJ''−⎜⎟1 ⎡⎤ JJ Φ−+Φ−frλλ aa fr β4 ⎜⎟⎣⎦sp wp ⎜⎟Φ−frλβ4 + wpc ⎣⎦sp wp ()51()ss() ww ⎝⎠⎝⎠11()ww

Φ f λ ⎛⎞wpc ' =− 1 int 1− 1 ⎡⎤JJ' ⎜⎟4 ⎣⎦sp wp ()Φ−+Φ−51fr()ssλλ aa fr() ww⎝⎠Φ−11fr()wwλβ + wpc 4 4 Φ f λα ⎛⎞Φ−fr()λβ ' =− 1int ⎜⎟1 ww ⎡⎤JJ' 44⎜⎟Φ−frλβ4 + wpc ⎣⎦sp wp ()Φ−51fr()ssλα +Φ− aa fr() ww λ β⎝⎠11()ww 4 Φ f λα ' 1int ⎡⎤' =− ⎣⎦JJsp wp c2

Where c2 = Φ−f rsλα44 +ΦpΦf λα +Φ−+ fr λα 4 f λα4 Φ−+ fr λ β 4 wpc ()51()ss()4 s3() () ss cc() 1() ww 1

154 • The components of D21

' DD21= 12 Φ f λα4 1int ⎡ ' ⎤ =− ⎣JJspwp⎦ c2

• The components of D22

−−11 DCDDD22=+ 22 21 11 12

−1 (i) Pre-multiplying D11 by D21

Φ f λα4 −1'1int ⎡⎤ ⎡ ' ⎤ DD21 11 =−⎣⎦ JJsp wp() Φ1fr() w −λ w I wp + c 1 ⎣ J wp J wp ⎦ c2 ΦΦffλα44 λα 1int 4'⎡ ⎤⎡1int '⎤ =−() Φ11fr()w −λβ w⎣JJ sp wp ⎦⎣ − () cwp JJsp wp ⎦ cc22 Φ f λα4 1int 4'⎡⎤ =−() Φ11fr()ww −λβ + cwp⎣⎦JJ spwp c2

−1 (ii) Post-multiplying DD21 11 by D12

4 4 ΦΦffλα ⎛⎞λα ' −141int ⎡ '⎤⎡1int '⎤ DDD21 11 12=−() Φ 1fr()w −λβ w + cwp 1 ⎣JJJJ sp wp⎦⎣⎜⎟ − sp wp ⎦ cc2 ⎝⎠2 2 ⎛⎞Φ f λα4 4'1int ⎡⎤ =Φ()11fr()w −λβ w + cwp⎜⎟ − wp⎣⎦JJsp sp ⎝⎠c2 (iii) Consequently

−−11 DCDDD22=+ 22 21 11 12 ⎛⎞444 1 ()ΦΦ14ffλαss +Φ 3()()rf − λs α + c λαc =−⎜⎟IJJ'' +∆⎡⎤JJ Φ−fr()λα44⎜⎟sp Φ− fr() λα + aa sp sp⎣⎦ sp sp 55ss ⎝⎠ss 444 1 ΦΦ14ffλαss +Φ 3()()rf − λs α + c λαc =−IJJ'' +∆⎡⎤JJ Φ−fr λα4 sp 44sp sp⎣⎦ sp sp 5 ()ss Φ−55fr()ssλα() Φ− fr () ss λα + aa ⎛⎞444 1 ΦΦ14ffλαss +Φ 3()()rf − λs α + c λαc = IJ−−⎜⎟∆⎡⎤J' Φ−fr λ α 4 sp ⎜⎟Φ−frλα44 Φ− fr λα + aa ⎣⎦sp sp 5 ()s s ⎝⎠55()ss() () ss 2 ⎛ ΦΦffλα44 +Φrf − λαλα +4c − Φfr − λα 4 Φ fw λα4p⎞ 1 ⎜ ()14ss 3()()scc2( 5() ss() 1 int ) ⎟ =−I ⎡JJ' ⎤ Φ−fr λα4 ⎜ sp 4 ⎣ sp sp ⎦ ⎟ 5 ()ss ⎜ ()Φ−52fr()ssλα + aac ⎟ ⎝ ⎠

4 2 4 ⎛⎞Φ1intf λα where, ∆=() Φ11f ()rww −λβ + c wp⎜⎟ − wp ⎝⎠c2

155 −1 By combining the results of D11, D12, D21 and D22, the components of A22 matrix can be obtained.

1 ⎛⎞c DI=−1 JJ' 11 44⎜⎟wp wp wp Φ−11fr()wwλβ⎝⎠ Φ− fr() ww λβ + wpc1 4 Φ f λα ' 1int ⎡⎤' DJ12 =− ⎣⎦spJ wp c2 Φ f λα4 1int ⎡⎤' DJ21 =− ⎣⎦spJ wp c2

2 ⎛ ΦΦffλα44 +Φrfc − λα + λα4 − Φfr − λα 4 Φ fw λα4p⎞ 1 ⎜ ( 14ss 3()()scc) 2( 5() ss() 1 int ) ⎟ D=I− ⎡⎤JJ' 22 Φfr−λα4 ⎜ sp 4 ⎣⎦sp sp ⎟ 5 ()ss ⎜ ()Φ−52fr()ssλα + aac ⎟ ⎝ ⎠

• The components of B11

−−11 −−11 BB11==−() 11 ( AA 11 12AA 22 21 )

−1 (i) Pre-multiplying A22 by A12

⎛⎞2' 2' −1 ΦΦ11()frwwβαJJp( fr ss) p⎛⎞DD11 12 AA12 22 = ⎜⎟⎜⎟ ⎜⎟'' φφ⎝⎠DD21 22 ⎝⎠23

⎛⎞XYXYDDDD11++ 21 12 22 = ⎜⎟'' ⎝⎠φφ23

−1 (ii) Post-multiplying AA12 22 by A21

⎛⎞2 ⎛⎞XYXYDDDD++Φ12( frwwβ )J pφ AAA−1 = 11 21 12 22 ⎜⎟ 12 22 21 ⎜⎟'' 2 φφ23⎜⎟Φ fr α J φ ⎝⎠⎝⎠13()ssp

⎛⎞()XYXXYYDD11+++ 21() DD 12 22 φ23 = ⎜⎟'' ⎝⎠φφ23 23

⎛⎞M φ23 = ⎜⎟'' ⎝⎠φφ23 23

Where, M =+(XYXXYDD11 21) ++( DD 12 22 )Y

156 ⎡⎤1 ⎛⎞c XDJ=Φ()fr ' ⎢⎥⎜⎟ I − 1 JJ' 11 1 wwpΦ−frλβ44⎜⎟ wp Φ− fr λβ + wpc wpwp ⎣⎦⎢⎥111()ww⎝⎠() ww () Φ fr β 2 ⎡⎤ 1 ()w ''wpc1 =−44⎢⎥JJwp wp Φ−111fr()wwλβ⎣⎦⎢⎥ Φ− fr() ww λβ + wpc Φ fr β 2 ⎡⎤ 1 ()w wpc1 ' =−44⎢⎥1 J wp Φ−111fr()wwλβ⎣⎦⎢⎥ Φ− fr() ww λβ + wpc

Φ1 ()frw ' = 4 Jwp Φ−11fr()wwλβ + wpc ' = c3J wp

2 Φ1 ()frwβ where, c3 = 4 Φ−11f ()rwwwλβ +pc

⎛⎞Φ f λα4 2'1int ⎡ '⎤ YDJ21 =Φ1 ()frssα p⎜⎟ − ⎣ J spJwp⎦ ⎝⎠c2 ' = c4J wp

2 24 ()Φ1if rssαλαntp where, c4 =− c2 Therefore,

'' XYDD11+=+ 21 cc34 Jwp J wp ' =+()cc34Jwp ' = c5J wp

'2 ()XYXDD11+=Φ 21 (cf51 Jwp) ( r wβ ) J wp

2 =Φcf51rwwβ p

4 ⎛⎞Φ f λα ' 2'1int ⎡ '⎤ XDJ12 =Φ1 ()frwwβ p⎜⎟ − ⎣ J spJwp⎦ ⎝⎠c2 ' = c6J sp

2 24 ()Φ1if rwwβλαntp where, c6 =− c2

157 ⎡ 1 ⎤ YDJ=Φfrα 2' I − c ⎡JJ'⎤ 22 1 ()ssp⎢ 4 () sp7 ⎣ spsp⎦ ⎥ ⎣⎢Φ−5 fr()ssλα ⎦⎥ Φ frα 2 1 ()s '' =−4 ()JJspcsp7 sp Φ−5 fr()ssλα Φ frα 2 1 ()s ' =−4 ()1 csp7 J sp Φ−5 fr()ssλα ' = c8J sp where

2 ΦΦf λα44 +Φfr − λα + f λα4 c − Φ fr − λα 4 Φ f λα4 wp ()14ss 3()()scc2( 5() ss() 1 int ) c7 = Φ−frλα44 +Φ spΦ f λα +Φ fr − λα 4 + f λα4 c ()51()ss()4 s3() () ss cc 2 2 Φ−17()frsα ()1 c sp c8 = 4 Φ−5 fr()ssλα

'' XYDD122268+=+cc JJspsp ' =+()cc68J sp ' = c9Jsp

'2 ()XYYDD1222+=Φ(cf 91 Jsps) ( rα ) Jsp

2 =Φcf91rssα p

Consequently

M =+()XYXXYYDD11 21 ++( DD 12 22 ) 22 =Φcf51rwwsβαpcf +Φ 91rsp

= c10

158 Therefore

−1 −1 BAA11=−() 11 12AA 22 21 −1 ⎡⎤' ⎛⎞Π φ1 ⎛⎞c10φ 23 =−⎢⎥⎜⎟⎜⎟'' ⎜⎟φ Diag d φφ ⎣⎦⎢⎥⎝⎠1 ()i ⎝⎠23 23

' −1 ⎛⎞Π−c10φ 1 = ⎜⎟ ⎝⎠φ1 Diag() di ⎛⎞1 φ ' ⎜⎟Π−c 1 = ⎜10 ⎟ ⎜⎟⎛⎞1 ⎜⎟φ1 Diag ⎜⎟ ⎝⎠⎝⎠di ' ⎛⎞γφ11 = ⎜⎟ ⎜⎟⎛⎞1 ⎜⎟φ1 Diag ⎜⎟ ⎝⎠⎝⎠di where 1 γ1 = c11

cc11=Π− 10

• The components of B12

−1 BB12=− 11A 12A 22 ⎛⎞1 φ ' ⎜⎟c 1 ⎛⎞XYXYDDDD++ ⎜⎟11 11 21 12 22 =− ⎜⎟'' φφ ⎜⎟⎛⎞1 ⎝⎠23 ⎜⎟φ1 Diag ⎜⎟ ⎝⎠⎝⎠di ⎛⎞1 φ ' ⎜⎟c 1 ⎛⎞ccJJ'' ⎜⎟11 59wp sp =− ⎜⎟'' ⎜⎟⎛⎞1 ⎝⎠φφ23 ⎜⎟φ1 Diag ⎜⎟ ⎝⎠⎝⎠di ⎛⎞cc 59JJ'' =−⎜⎟ccwp sp ⎜⎟11 11 ⎜⎟'' ⎝⎠φφ23 '' ⎛γγ23JJwp sp ⎞ = ⎜ ''⎟ ⎝ φφ23⎠

159 • The components of B21

' BB21= () 12

⎛⎞γ 22J wp φ = ⎜⎟ ⎝⎠γ 33J sp φ where c γ =− 5 2 c 11 c9 γ 3 =− c11

• The components of B22

−−11 BABBB22=+ 22 21 11 12

−1 (i) Pre-multiplying B11 by B21

⎛⎞c 5 J φ ⎜⎟wp 2 ' c11 ⎛⎞c φ BB−1 =−⎜⎟⎜⎟11 1 21 11 ⎜⎟⎜⎟ c9 ⎝⎠φ1 Diag() di ⎜⎟J sp φ3 ⎝⎠c11

⎛⎞c52Jwp φ =−⎜⎟ ⎝⎠c93J sp φ

−1 (ii) Post-multiplying BB21 11 by B12

⎛⎞cc c J φ −−59JJ'' −1 ⎛⎞52wp ⎜⎟wp sp BBB21 11 12 =−⎜⎟cc11 11 c J φ ⎜⎟ ⎝⎠93sp ⎜⎟'' ⎝⎠φφ23 2 ⎛⎞()c cc ' ⎜⎟5 ''59⎡⎤ JJwp wp⎣⎦ JJ sp wp ⎜⎟cc = 11 11 ⎜⎟2 cc ()c ⎜⎟59JJ''9 JJ ⎜⎟sp wp sp sp ⎝⎠cc11 11

⎛⎞EE11 12 = ⎜⎟ ⎝⎠EE21 22

160 Consequently

⎛⎞DD11 12⎛⎞ EE 11 12 B22 =+⎜⎟⎜⎟ DD EE ⎝⎠21 22⎝⎠ 21 22

⎛DE11++ 11DE 12 12 ⎞ = ⎜⎟ ⎝⎠D21++EDE 21 22 22

2 1 ⎛⎞c (c ) DE+= I −1 JJ'' +5 JJ 11 11 44⎜⎟wp wp wp wp wp Φ−111fr()wwλβ⎝⎠ Φ− fr() ww λβ + wpc c11 2 1 c ()c =−IJ1 ⎡⎤J'' +5 JJ Φ−fr λβ4 wp 44⎣⎦wp wpc wp wp 1 ()ww Φ−11fr()wwλβ() Φ− fr () ww λβ + wpc 111

2 1 ⎛⎞c ()c =−IJ⎜⎟1 −5 J' Φ−fr λβ4 wp ⎜⎟Φ−frλβ44 Φ− fr λβ + wpc c wp wp 1 ()ww ⎝⎠11()ww() () ww 111 2 1 ⎛⎞⎛⎞c Φ−fr()λβ4 ( c) =−⎜⎟IJ⎜⎟1 −15ww J' Φ−frλβ44⎜⎟wp ⎜⎟ Φ− fr λβ + wpc c wp wp 11()ww⎝⎠⎝⎠() ww 111

1 4' =−βγ()IJwp4 wpJ wp Φ−1 fr()wwλ 4 2 c1 Φ−15f ()rcwwλβ() where , γ 4 =−4 Φ−11fr()wwλβ + wpc c11

4 Φ fcλα ''c 1int⎡ ''⎤⎡ 59 ⎤ DE12+= 12 −⎣ JJspwp⎦⎣ + JJ spwp⎦ cc211 4 ⎛⎞Φ fcλα c ' 1int 59⎡⎤' =−⎜⎟ + ⎣⎦JJsp wp ⎝⎠cc211 ' ⎡⎤' = γ 5 ⎣⎦JJsp wp

4 Φ1intfcλα '' 59c D21+= E 21 − JJspwp + JJ spwp cc211 4 ⎛⎞Φ1intfcλα 59c' =−⎜⎟ + JJsp wp ⎝⎠cc211 ' = γ 5JJsp wp

4 Φ1intf λα cc 59 Where, γ 5 =− + cc211

161 2 1 ()c DE+= I −c ⎡⎤ JJ'' +9 JJ 22 22 4 ()sp7 ⎣⎦ sp sp sp sp Φ−51fr()ssλα c1

2 1 ⎛⎞⎛⎞Φ−fr()λα4 ( c) =−⎜⎟IJ⎜⎟c −59ss J' Φ−frλα4 ⎜⎟sps⎜⎟7 c psp 51()ss ⎝⎠⎝⎠1

1 ' =−4 ()IJspγ 6 spJ sp Φ−5 fr()ssλα

4 2 Φ−59f ()rcssλα() where, γ 67=−c c11

By combining the results of B11, B12, B21 and B22, the components of the inverse of the information matrix A−1 can be obtained.

' ⎛⎞γφ11 B = ⎜⎟ 11 ⎜⎟⎛⎞1 ⎜⎟φ1 Diag ⎜⎟ ⎝⎠⎝⎠di '' ⎛⎞γγ23JJwp sp B12 = ⎜⎟'' ⎝⎠φφ23

⎛⎞γφ22J wp B21 = ⎜⎟ ⎝⎠γφ33J sp ⎛⎞1 ' ''⎡⎤ 4 ()IJwp−γγ45 wpJJ wp spJ wp ⎜⎟Φ−fr λβ ⎣⎦ ⎜⎟1 ()ww B22 = ⎜⎟''1 ⎜⎟γγ56JJsp wp 4 () Isp− JJ sp sp ⎝⎠Φ−5 fr()ssλα

−1 Therefore ()XR'1− X for split plot BBD is given as

⎛ γφ'' γJJ γ '⎞ ⎜ 11 2wp 3sp ⎟ ⎜ φφφDiag ⎛⎞1 ''⎟ 12⎜⎟d 3 ⎜ ⎝⎠i ⎟ −1 ⎜ ⎟ XR'1− X = 1 ' ()⎜γφJI− γJJ'' γ⎡⎤JJ⎟ 22wp 4 ()wp 4 wp wp 5⎣⎦ sp wp ⎜ Φ−1 fr()wwλβ ⎟ ⎜ ⎟ 1 ⎜ γφJJ γJ''I− γJJ⎟ ⎜ 33sp 5sp wp 4 ()sp6 sp sp ⎟ ⎝ Φ−5 fr()ssλα ⎠

162 APPENDIX B

B1: Determinant of the Information Matrix In order to analytically determine the determinant of the information matrix, A, theorem 13.3.8 of Harville (1997) was applied. According to the theorem if a matrix A is partitioned such that A11 is nonsingular and the inverse of A is given as B, then the determinant of A is computed as follows:

−1 AAA=⋅−11 22 AAA 21 11 12

−1 =AB11⋅ 22 (1.63) A = 11 B22

The form of A11 is a diagonal matrix and therefore its determinant is simply the product of its diagonal elements. Further, the principle of diagonal expansion by Searle (1982) was used to facilitate computation of the determinant of matrix B22. The diagonal expansion principle states that if a matrix is of the form as shown in Equation (1.64), then its determinant is given as shown it Equation (1.65).

' TI=+abkk JJk (1.64) T =+aakk−1 ( kb) (1.65)

It turns out that the diagonal block matrices of B22 are of the form of Equation (1.64) and therefore their determinants are computed according to Equation (1.65).

Equation (1.63) dictates that, in order to analytically compute the determinant, the inverse of the information matrix is required. If the model being considered is a first order model then the determinant would be given as A11 which is simply the product of the diagonal elements of A11 . Therefore the inverse of the information matrix would not be required for a first order model.

163 B2: Derivation of the Determinant for Split Plot CCD

The B22 matrix for a CCD can be partitioned as follows

⎛⎞1 ' ''⎡⎤ 4 ()IJwp−γγ45 wpJJ wp spJ wp TU ⎜⎟Φ 2r β ⎣⎦ ⎛⎞⎜⎟6 w B22 ==⎜⎟ ⎝⎠VW ⎜⎟''1 ⎜⎟γγ56JJsp wp 4 () Isp− JJ sp sp ⎝⎠Φ+592rsα Φ

-1 Therefore it determinant is given as BT22 = W-VTU Where,

−1 -1 ⎛⎞1 γ 4 ' TIJ=−⎜⎟wp wpJ wp ΦΦ22rrββ44 ⎝⎠66ww

4'⎛⎞γ 4 =Φ6 2rwwβ ⎜⎟IJp + wpJwp ⎝⎠1−γ 4wp

-1 4'⎛⎞γ 4 ' VT=Φγβ562rw J spwpwp J⎜⎟ I + J wpwp J ⎝⎠1−γ 4wp

4 ⎛⎞γβ56Φ 2rw ' = ⎜⎟JJsp wp ⎝⎠1−γ 4wp

⎛⎞γβΦ 2r 4 ' VT-1 U= 56 w J J' *γ ⎡ J J' ⎤ ⎜⎟sp wp5 ⎢ sp wp ⎥ ⎝⎠1−γ 4wp ⎣ ⎦

⎛⎞Φ 2rwβγ4 ()2 p = ⎜⎟65w JJ' ⎜⎟1−γ wp sp sp ⎝⎠4

2 1 ⎛⎞γ ⎛⎞Φ 2rwβγ4 () p W-VT-1 U=−+ I ⎜⎟6 ⎜⎟65w JJ' Φ+221rrααγ44Φsps⎜⎟ Φ+Φ⎜⎟ − wppsp 5959ss⎝⎠⎝⎠ 4 ⎛⎞4 2 sp−1 1−γ sp Φ652*rwwβγ() psp⎛⎞1 =−⎜⎟6 ⎜⎟ ⎜⎟Φ+21rwαγ44Φ −p Φ+ 2r αΦ ⎝⎠59ss 4⎝⎠ 59

1 ' T =−4 ()IJwpγ 4 wpJ wp Φ6 2rwβ wp−1 ⎛⎞11−γ 4wp ⎛⎞ = ⎜⎟44⎜⎟ ⎝⎠ΦΦ6622rrwwββ⎝⎠

164

The determinant of A11 is given as:

wp sp cw c cs 22 22 4 4+ A11=Π() Φ 1frζβ +Φ 622ws( Φ 2( frf ζ +α)) () Φ 1 ζζ() Φ 2 f This would also be the determinant for a first order model.

Therefore the determinant of A for CCD is A A = 11 B22 wp sp cw c+ cs 22 22 4 4 ΠΦ()16frζβ +Φ22ws() Φ 2( frf ζ +αζζ)() Φ 1() Φ 2 f = wp−1 ⎛⎞4 2 sp−1 ⎛⎞1−γ wp ⎛⎞111−γ sp Φ652rwwβγ() psp ⎛ ⎞ ⎜⎟4 ⎜⎟⎜⎟6 − ⎜ ⎟ ΦΦ22rrββ44⎜⎟ Φ+ 2 r α 4Φ 1 − γ wp Φ+ 2 r α4Φ ⎝⎠61ww⎝⎠⎝⎠ 59 s 4 ⎝ 59s ⎠

165 B3: Derivation of the Determinant for Split Plot BBD

The B22 matrix for a BBD can be partitioned as follows

⎛⎞1 ' ''⎡⎤ 4 ()IJwp−γγ45 wpJJ wp spJ wp ⎜⎟Φ−fr λβ ⎣⎦ ⎛⎞TU ⎜⎟1 ()ww B22 ==⎜⎟ ⎝⎠VW ⎜⎟''1 ⎜⎟γγ56JJsp wp 4 () Isp− JJ sp sp ⎝⎠Φ−5 fr()ssλα

-1 Therefore it determinant is given as BT22 = W-VTU Where,

−1 ⎛⎞1 γ TI-1 =−⎜⎟4 JJ' ⎜⎟Φ−fr()λβ44wp Φ− fr() λβ wp wp ⎝⎠11ww ww

4'⎛⎞γ 4 =Φ1 fr()w −λβ w⎜⎟IJ wp + wpJ wp ⎝⎠1−γ 4wp

-1 4'⎛⎞γ 4 ' VT=Φγλ51fr()w − wβ J sp J wp⎜⎟ I wp + J wp J wp ⎝⎠1−γ 4wp

⎛⎞γλΦ−fr()β4 = 51 ww JJ' ⎜⎟sp wp ⎝⎠1−γ 4wp ⎛⎞4 ' -1 γλ51Φ−fr()wwβ' ⎡ ' ⎤ VT U= ⎜⎟ J J*γ J J ⎜⎟sp wp5 ⎣⎢ sp wp ⎦⎥ ⎝⎠1−γ 4wp

⎛⎞Φ−fr()λβγ4 ()2 wp = ⎜⎟15ww JJ' ⎜⎟1−γ wp sp sp ⎝⎠4

2 1 ⎛⎞γ Φ−fr()λβγ4 () wp W-VT-1 U=−+ I ⎜⎟6 15ww JJ' Φ−frλα44sp ⎜⎟ Φ− fr λα1 + γ wp sp sp 5()ss⎝⎠5() ss 4 2 sp−1 ⎛⎞1− γ sp Φ−fr()λβγ4 () wpsp* ⎛⎞1 =−⎜⎟6 15ww ⎜⎟ ⎜⎟Φ−frλα4 1 − γ wp⎜⎟ Φ− fr λα4 ⎝⎠54()ss ⎝⎠5()ss

166 1 ' T =−4 ()IJwpγ 4 wpJ wp Φ−1 fr()wwλβ wp−1 ⎛⎞1−γ wp ⎛⎞1 = 4 ⎜⎟44⎜⎟ ⎝⎠Φ−11fr()wwλβ⎝⎠ Φ− fr() ww λβ

The determinant of A11 is given as:

wp sp cw c cs A =Π Φfrβαλ22 Φ fr Φ fβλ 4 Φ fαλ 4 Φ f α4 11() 1ws() 2() 1ws() 2 int( 2 )

Therefore the determinant of A for a BBD is

A A = 11 B 22 wp sp cw c cs ΠΦfrβαλ22 Φ fr Φ fβλ 4 Φ fαλ 44 Φ f α ()1212()()ws()int()2 = ws wp−1 2 sp−1 ⎛⎞11−γ wp ⎛⎞⎛⎞1−γ sp Φ−fr()λβγ4 () wpsp ⎛1⎞ ⎜⎟4 ⎜⎟⎜⎟6 − 15ww ⎜⎟ ⎜⎟Φ−frλβ44⎜⎟ Φ− fr λβ⎜⎟ Φ− fr λα 41 − γ wp ⎜ Φ− fr λα4 ⎟ ⎝⎠11()ww⎝⎠() ww⎝⎠ 5() ss 45 ⎝()ss ⎠

167 APPENDIX C

C1: Analytical Determination of v(z, x) for Split Plot CCD

The analytical determination was done by pre- and post-multiplying the information matrix by the model vector. The steps are detailed below.

1. Pre-multiplying by the model vector

−1 f ''()z,x() X R−1 X =

wp sp wp sp wp−−11 wp wp sp sp sp ⎡ 2211 1 1 1 ⎢γγ12++∑∑∑∑zii γ 3 x z i x i ∑∑ zz ij ∑∑ zx ij ∑∑ xx ij… ⎣ iiii====1111ddii i = 1j=i+1 d i i = 1j=1 d i i = 1j=i+1 d i ⎛⎞1 γγwp2 wp sp γγ+−44zz22 − +x2 2 ⎜⎟44∑ ij 4∑∑5 i… ⎝⎠22rrwwββΦΦ11ij==11 2 r w β Φ 1i+i=1 ⎛⎞1 γγsp2 sp wp ⎤ γγ+−66xx22 − +z2 3 ⎜⎟44∑∑ij 4 5 ∑i⎥ ⎝⎠22rrssααΦ+Φ59 Φ+Φ 59ij==11 2 r s α Φ+Φ 59i+i=1⎦⎥

2. Post-multiplying by the model matrix

−1 ff''()z,x( X R−1 X) () z,x = wp sp wp sp wp−−11 wp wp sp sp sp 2211 2 2 1 22 1 22 122 γγ12+++∑∑∑∑zxi γ 3 i z i + x i + ∑∑ z ijz +2 ∑∑ z ijx + ∑∑ x ijx iiii====1111ddii i = 1j=i+1 d i i = 1j=1 d i i = 1j=i+1 d i

wp ⎛⎞1 γγwp2 wp−1 wp wp sp ++γ zz24 −44 − z2z22+ γ zx2 2 ∑ ii⎜⎟44∑∑ 4∑iji5 ∑∑j i=1 ⎝⎠22rrwwββΦΦ11ii==11 2 r w β Φ 1j=i+11i=ij= sp ⎛⎞1 γγsp2 sp−1 sp wp sp x2 66xx4 22xz 22x ++γγ3 ∑∑ii⎜⎟44 − − 4∑∑ij +5 ∑∑ij ii==11⎝⎠22rrssααΦ+Φ59 Φ+Φ 59 2 r s α Φ+Φ 59i=1j=i+1i=ij=1

168 v(z,x) =

wp sp wp−−11 wp sp sp ⎛⎞⎛⎞1122112222 γγ12++⎜⎟⎜⎟22∑∑zxii ++ γ 3 +∑∑zizj +∑∑xixj ⎝⎠⎝⎠ddiiii==11ddiii=1j=i+1i=1j=i+1 ⎛⎞wp wp2 wp sp (1.66) 1142⎛⎞⎛⎞ 22 +−+4 ⎜⎟zziiγγ45⎜⎟2⎜⎟+ z ixj 2rdβ Φ ⎜⎟∑∑ ∑∑ wi1 ⎝⎠ii==11⎝⎠⎝⎠ i = 1j=1 ⎛⎞sp sp 2 1 4 ⎛⎞2 + 4 ⎜⎟xi −γ 6 ⎜⎟xi 2r α Φ+Φ⎜⎟∑∑ s 59⎝⎠i=1 ⎝⎠i=1

wp−1 wp⎛⎞ wp2 wp 221 ⎛⎞ 2 4 zzij=−⎜⎟⎜⎟ z i z i ∑∑2 ⎜⎟ ∑ ∑ iji==+11⎝⎠⎝⎠ i = 1 i = 1 (1.67) sp−1 sp⎛⎞ sp2 sp 221 ⎛⎞ 2 4 xijxx=−⎜⎟⎜⎟ ix i ∑∑2 ⎜⎟ ∑ ∑ iji==+11⎝⎠⎝⎠ i = 1 i = 1

Substituting the identities in Equation (1.67) into Equation (1.66) the prediction variance, v(z, x), is simplified to:

v()z,x =

wp sp wp sp ⎛⎞⎛⎞1122 ⎛⎞ 122 γγ12++⎜⎟⎜⎟222∑∑∑zxii ++ γ 3 ++ ⎜⎟ γ 5 ∑zixj ⎝⎠⎝⎠ddiiii==11 ⎝⎠ d i i = 1j=1 wp 2 wp (1.68) ⎛⎞11γ 4 ⎛⎞24⎛⎞1 +−⎜⎟44⎜⎟∑∑zzii +⎜⎟ − ⎝⎠22driwββΦΦ11⎝⎠ii==11⎝⎠ 2 r w 2 d i 2 ⎛⎞11γ ⎛⎞sp ⎛⎞1 sp 6 x2 x4 ⎜⎟−+44⎜⎟∑ i ⎜⎟− ∑ i ⎝⎠22drisααΦ+Φ59⎝⎠i=1 ⎝⎠ 2 r s Φ+ 5Φ9 2di i=1

169 C2: Analytical Determination of v(z, x) for Split Plot BBD

The procedure utilized for the split plot CCD case is also applied here for the BBD. The results are as follows:

1. Pre-multiplying by the model vector

−1 f ''()z,x( X R−1 X) =

wp sp wp sp wp−−11 wp wp sp sp sp ⎡ 2211 1 1 1 ⎢γγ12++∑∑∑∑zii γ 3 x z i x i ∑∑ zz ij ∑∑ zx ij ∑∑ xx ij… ⎣ iiii====1111ddii i = 1j=i+1 d i i = 1j=1 d i i = 1j=i+1 d i ⎛⎞1 γγwp 2 wp sp γγ+−44zz22 − +x2 2 ⎜⎟∑ ij∑∑5 i… ⎝⎠Φ−11fr()wwλλ Φ− fr() wwi=1 Φ− 1 fr() ww λji=+11 i = ⎛⎞1 γγsp2 sp wp ⎤ γγ+−⎜⎟66xx22 − +z2⎥ 3 ⎜⎟Φ−Φ−frλλ fr∑∑ij Φ− fr λ5 ∑i ⎝⎠55()ss() ssij==11 5() ssi+i=1⎦⎥

2. Post-multiplying by the model matrix

−1 ff''()z,x*( X R−1 X) () z,x = wp sp wp sp wp−−11 wp wp sp sp sp 2 211 2 2 1 22 1 22 1 22 γγ12+++∑∑∑∑zxii γ 3 z i + x i + ∑∑ z izj + ∑∑ z ixj + ∑∑ x ixj iiii====1111ddii i = 1j=i+1 d i i = 1j=1 d i i = 1j=i+1 d i

wp ⎛⎞1 γγwp2 wp−1 wp wp sp ++γ zz24 −44 − zz22+ γ zx 22 2 ∑ ii⎜⎟∑∑∑ij5 ∑∑ ij i=1 ⎝⎠Φ−11fr()wwλλ Φ− fr() wwii==11 Φ− 1 fr() w λw j=i+1i=ij=1 sp ⎛⎞1 γγsp2 sp−1 sp wp sp ++γγx2 −66xx4 − 22x + z 22x 3 ∑∑i ⎜⎟i∑∑ ij5 ∑∑ ij ii==11⎝⎠Φ−Φ−55fr()ssλλ fr() ss Φ− 5 fr() ss λi=1j=i+1i=ij=1

170 Therefore v(z,x) =

wp sp wp−−11 wp sp sp ⎛⎞⎛⎞1122112222 γγ12++⎜⎟⎜⎟22∑∑zxii ++ γ 3 +∑∑zizj +∑∑xixj ⎝⎠⎝⎠ddiiii==11ddiii=1j=i+1i=1j=i+1 ⎛⎞wp wp2 wp sp (1.69) 1142⎛⎞⎛⎞ 22 +−⎜⎟zziiγγ45⎜⎟+⎜⎟2 + z ixj Φ−frλ ⎜⎟∑∑ d ∑∑ 1 ()ww⎝⎠ii==11⎝⎠⎝⎠i i = 1j=1 ⎛⎞spsp2 1 4 ⎛⎞2 + ⎜⎟xi −γ 6 ⎜⎟xi Φ−fr λ ⎜⎟∑∑ 5 ()ss⎝⎠i=1 ⎝⎠i=1

wp−1 wp⎛⎞ wp2 wp 221 ⎛⎞ 2 4 zzij=−⎜⎟⎜⎟ z i z i ∑∑2 ⎜⎟ ∑ ∑ iji==+11⎝⎠⎝⎠ i = 1 i = 1 (1.70) sp−1 sp⎛⎞ sp2 sp 221 ⎛⎞ 2 4 xijxx=−⎜⎟⎜⎟ ix i ∑∑2 ⎜⎟ ∑ ∑ iji==+11⎝⎠⎝⎠ i = 1 i = 1

Substituting the identities in Equations (1.67) into equation (1.66) v(z, x) is simplified to:

v()z,x =

wp sp wp sp ⎛⎞⎛⎞1122 ⎛⎞ 122 γγ12++⎜⎟⎜⎟222∑∑∑zxii ++ γ 3 ++ ⎜⎟ γ 5 ∑zixj ⎝⎠⎝⎠ddiiii==11 ⎝⎠ d i i = 1j=1 2 ⎛⎞11γ ⎛⎞wp ⎛⎞1wp (1.71) +−4 zz24 + − ⎜⎟⎜⎟∑ ii⎜⎟∑ ⎝⎠22dfiwΦ−11()rλλw⎝⎠ii==11⎝⎠ Φ− f()r wwi d 2 ⎛⎞11γ ⎛⎞sp ⎛⎞1 sp −+6 x2 − x4 ⎜⎟⎜⎟∑ i ⎜⎟∑ i ⎝⎠2dfisΦ−55()rλs⎝⎠i=1 ⎝⎠ Φfr()ss− λ 2 d ii=1

171 APPENDIX D

D1: Determination of Critical Points The principle of Langrage multipliers is used to find the critical points to analytically determine the maximum and minimum value for the predication variance, v (z, x). As show in Equations (1.35) and (1.36) the predication variance is a function of two variables subject to a single constraint. If is the Lagrange multiplier, the Lagrange function L is defined as wp sp⎛⎞ wp sp 224 44 4 22 2 22 LA()z,x,=++++BρzxzCρρDE∑∑z i ++F ρ xGx j +H ρ zρx +⎜⎟ ∑ z i +−∑x j ρ ij==11 ⎝⎠ ij The system of partial derivatives is

∂L(z,x,) 3 =+4Ezii 2λ z for 1 ≤ i≤ wp ∂zi ∂L z,x, () 3 =+4Gxjj 2λ x for 1 ≤ j≤ sp ∂x j wp sp ∂L()z,x, 222 =+−∑∑zxiiρ ∂ ii Equating these partial derivatives to zero yields the following critical points for the analytical determination of the exact maximum and minimum value:

22 zii==0 or − 2 Ez for any i ; x j == 0 or − 2 Gx j for any j wp sp sp wp wp sp 22 2 22 2 2 2 ∑∑zxij=−ρρ ; ∑∑∑xzzjii =− and or ∑xj = 0 ijjiij

Let nwp and nsp represent the number of occurrences where zxijand = 0 for 1 ≤≤inwp

and 1 ≤≤jnsp respectively. Further let wp− nwp and sp− nsp represent the number of

2 2 occurrences where =−2Ezi and =−2Gx j for niwp +1 ≤≤wp and njsp +≤1 ≤sp respectively. These conditions give rise to the following three (3) critical points:

wp 2 (1) when ∑ zi = 0 then zi = 0 and xi is given as follows i=1

172 sp wp 22 2 ∑∑xzji=−ρ ji==11 sp 22 2 ∑ xsjs==ρ ()p −npxsp (1.72) j=1

2 x js=±ρ sp − nps np + 1 ≤ j ≤ sp

sp 2 (2) when ∑ xi = 0 then xi = 0 and zi is given as follows i=1

wp sp 22 2 ∑∑zxii=−ρ ii==11 wp 22 2 ∑ zwiw==ρ ()p −npzwp (1.73) i=1

2 zwiw=±ρ p −np niwp + 1 ≤ ≤wp

wp sp 22 22 (3) when ∑ ziz= ρ and ∑ x jx= ρ then zi and xj are given as follows i=1 j=1

wp 22 2 ∑ zwi==ρ z()p −n wpz wp i=1 (1.74) 222 zwiz=±ρρ()p −n wp = ±−ρ x() wp −nn wp wp +≤≤ 1 iwp

sp 22 2 ∑ xsj==ρ x()p −n spx sp j=1 (1.75) 222 x jx=±ρρ()sp − n sp = ±−ρ z() sp − n sps np +≤≤ 1 j sp

22 22 Note that ρ − ρz or ρ − ρx is to ensure that the constraint is not violated.

173 D2: Evaluation of Vminρ and Vmaxρ for Spherical Regions

wp sp 4 4 Substitution of the critical points 1 and 2 into ∑ zi and ∑ x j give the following i=1 j=1

wp sp 44 44 ∑ zwiw=−ρ pnpand ∑ x js=−ρ sp n pwhich are both increasing functions of nwp i=1 j=1 and nsp respectively. Therefore setting nwwp = p−1 and nssp = p−1 will minimize while

wp sp n = 0 and n = 0 will maximize z 4 and x4 for ∀∈Ξzx and ∀∈Ξ wp sp ∑ i ∑ j ( SSρρ) i=1 j=1 respectively. Where zx, and ΞS are the whole and subplot locations within the spherical ρ region Ξ governed by the radius, . Therefore, the V min and V max for whole plot Sρ ρ ρ ρ and subplot design spaces and combined design spaces and be determined as follows:

Vmin and Vmax for whole plot design space ρz ρz

wp ρ 4 min z4 = occurs at zw= ρρp,, wp ∑ i ( 1 … wp ) z∈Ξ S ρ i=1 wp

wp 44 max ∑ zi = ρ occurs at z = (ρ12,0 ,… ,0wp ) z∈Ξ S ρ i=1 Therefore, depending on whether E > 0 or E ≤ 0, the minimum and maximum prediction variances for the whole plot are computed as follows:

⎧⎫24⎛⎞E ⎪⎪AB+++ρρ⎜⎟ D for E > 0 V min = wp ρz ⎨⎬⎝⎠ ⎪⎪AB+++ρρ24() DE for E ≤ 0 ⎩⎭ ⎧⎫AB+++ρρ24() DE for E > 0 ⎪⎪ V max = ρz ⎨⎬24⎛⎞E ⎪⎪AB+++ρρ⎜⎟ D for E ≤ 0 ⎩⎭⎝⎠wp

Vmin andVmax for subplot design space ρ x ρx sp ρ 4 min x4 = occurs at x = ρρsp,, sp ∑ j ( 1 … sp ) x∈Ξ S ρ j=1 sp

174 sp 44 max ∑ x j = ρ occurs at x = (ρ12,0 ,… ,0sp ) x∈Ξ S ρ j=1 Therefore, depending on whether G > 0 or G ≤ 0, the minimum and maximum prediction variances for the subplot are computed as follows:

⎧⎫24⎛⎞G ⎪⎪AC+++ρρ⎜⎟ F for G > 0 V min = sp ρx ⎨⎬⎝⎠ ⎪⎪AC+++ρρ24() FG for G ≤ 0 ⎩⎭ ⎧⎫AC+++ρρ24() FG for G > 0 ⎪⎪ V max = ρx ⎨⎬24⎛⎞G ⎪⎪AC+++ρρ⎜⎟ F for G ≤ 0 ⎩⎭⎝⎠sp

Vminρ and Vmaxρ for combined design space

wp sp 4 4 Substitution of these critical points 3(i) and 3(ii) into ∑ zi and ∑ x j give the following i=1 j=1

wp sp 2 422 4222 ∑ zwix=−()ρρ ()p −n wpand ∑ x jz=−()ρρ ()sp − n spwhich are both i=1 j=1 increasing functions of nwp and nsp respectively. Therefore, setting nwwp =−p1 and

wp 4 nssp =−p1 will minimize while nwp = 0 and nsp = 0 will maximize ∑ zi and i=1

sp x4 for ∀∈Ξzx and ∀∈Ξ respectively. Where zx, and Ξ are the whole and ∑ j ()SSρρ Sρ j=1 subplot locations within the spherical region Ξ governed by the radius, . Therefore, Sρ ρ

V min ρ and V max ρ for the combined design spaces can be determined as follows:

wp ρ 4 min z4 = z occurs at zw= ρρp,, wp ∑ i ( zz1 … wp ) z∈Ξ S ρ i=1 wp

wp 44 max ∑ ziz= ρ occurs at z = (ρzw,02 ,… ,0 p) z∈Ξ S ρ i=1 sp ρ 4 min x4 = x occurs at x = ρρsp,, sp ∑ j ( xx1 … sp ) x∈Ξ S ρ j=1 sp

175 sp 44 max ∑ x jx= ρ occurs at x = (ρxs,02 ,… ,0 p) x∈Ξ S ρ j=1

Therefore depending on the combination of whether E > 0 or E ≤ 0 and G > 0 or G ≤ 0, the minimum and maximum prediction variances for the combined design space are computed as follows:

⎧⎫22⎛⎞⎛⎞EG 4 4 22 ⎪⎪AB++++ρρzx C⎜⎟⎜⎟ D ρ z ++ F ρ x + H ρ zρxfor E and G > 0 ⎪⎪⎝⎠⎝⎠wp sp ⎪⎪22 4 4 22 AB+++++++ρρzx C()() DE ρ z FG ρ x H ρ zρx for E and G ≤ 0 ⎪⎪⎪⎪ V min ρ = ⎨⎬22⎛⎞E 4 4 22 AB++++ρρzx C⎜⎟ D ρ z +++() FG ρ x H ρ zρxfor E > 0 and G ≤ 0 ⎪⎪wp ⎪⎪⎝⎠ ⎪⎪ 22 4⎛⎞G 422 ⎪⎪AB++++++ρρzx C() DE ρ z⎜⎟ F ρρxz+≤H ρxfor E 0 and G > 0 ⎩⎭⎪⎪⎝⎠sp

224422 ⎧⎫AB+++++++ρzx Cρρρ()() DE z FG x Hρ zρx for E and G > 0 ⎪⎪ ⎛⎞⎛⎞EG ⎪⎪AB++++ρρ22 C D ρ 4 ++ F ρ 4 + H ρ 2ρ2for E and G ≤ 0 ⎪⎪zx⎜⎟⎜⎟ z x zx ⎝⎠⎝⎠wp sp ⎪⎪ V max ρ = ⎨⎬22 44⎛⎞G 22 ⎪⎪AB++++ρρzx C() DEρρzx++⎜⎟FH +ρzρxfor E > 0 and G ≤ 0 ⎪⎪⎝⎠sp ⎪⎪ 22⎛⎞E 4 4 22 ⎪AB++++ρρzx C⎜⎟ D ρ z +++() FG ρ x H ρ zρxfor E ≤ 0 and G > 0⎪ wp ⎩⎭⎪⎝⎠ ⎪

176 D3: Evaluation of Vminρ and Vmaxρ for Cuboidal Regions

Let Ξ=−≤≤∈Hi{zziw:1 1 ()1,,……p ; xxj:1 −≤≤∈ j 1 (1,, sp)} be the k-dimensional wp sp wp sp 22 4 4 hypercube where kwpsp=+. Further let ω =+∑ zxtztxij∑∑; zixj = and = ∑ ij i j

For the trivial case where ρ = 0 then VVmaxΞΞ= min therefore it will be assumed HHρρ that ρ > 0 . Given the fact that −≤11zii ≤ ∈(1,…, wp) and −≤11x j ≤ js ∈()1,…,p then it follows that 0 <≤ω wp + sp . In addition, it can be assumed that there exist integers iz ∈()0,1,… , wp and jsx ∈()0,1,… , p along with incremental adjustments δ z ∈[0,1) and

wp sp 22 22 δ x ∈[0,1) such thatω =+ijzzδδ + xx + . Since ω =+∑ zxij∑ is both sign and ij permutation invariant with respec t to the locations z and x it can be postulated without lost of generality that 10≥≥≥≥zz12… zwp ≥ and 10≥≥≥≥xx12… xsp ≥. Let uz be the largest integer such that u > 0 and z = 0 while ux be the largest integer such that z uz +1 u > 0 and x = 0 . By fixing the values of ω;,,zzxx ;,, both z and t can x ux +1 21……uuzx− 2−1uz z be considered as functions of z while x and t be considered as functions of x . 1 ux x 1 Therefore,

uuzz−−11sp zz2=−−−ω 22zxt 2 and =++z 444zz uiz ∑∑1 jz ∑iu1z ij==21 i = 2

uuxx−−11wp x22=−ω xx −2 − z2 and t=xxx2 +4 +4 ujix ∑∑1x∑j1ux ji==21 j = 2

The systems of partial derivatives with respect to z1 and x1 are

(i) Functions of z1

∂t ∂z ⎛⎞∂z z =+44zz33uz 22zzuz =− ∂zz11∂ uz ⎜⎟ 1 11 ⎝⎠∂z1 and =−44zz32z ∂z 11uz uz z1 =− =−4zz22 z ∂zz 11()uz 1 uz

177 (ii) Functions of x1

∂t ∂x ⎛⎞∂x x =+44xx33ux 22x ux =− x ∂x 11∂x ux ⎜⎟ 1 11 ⎝⎠∂x1 and =−44xx32x ∂x 11ux ux x1 =− =−4xx22 x ∂x x 11()ux 1 ux

Since zz≥>0and xx≥> 0 then increasing z and x will simultaneous 1 uz 1 ux 1 1 decease z and x and increase t and t while preserving the ordering uz ux z x 1≥zz≥≥≥> z0 and10≥≥≥≥xx x > respectively. Optimally, z and 12… u 12… u uz x are decreased to 0 by in creasin g z and x respectively. The process is iterat ed until ux 1 1 tz andtx are maximized. The maximization of tz and tx should be accomplished under the cases outlined by the three (3) critical points given in Appendix A. Using these cases

2 2 it follows thattz and tx are maximized by fixing the value of ω = izz+ δ and ω =+jxxδ respectively, whiles the combined maxim ization o f tz and tx is maximized by fixing the

22 value ofω =+ijzzδδ + xx + .

Vmin and Vmax for whole plot design space ρz ρz

2 Maximizatio n oftz for a fixed value of ω = izz+ δ

⎧zz12====… zi 1 ⎪ zzmax==⎨ iz+ 1 δ ⎪ ⎩zzii++23====… z wp0

wp 44 2 Therefore when tz is evaluated at zmax then tziizi= ∑ =+δω z =+() − i. Let the floor i function ⎢⎥ω represent the greatest integer ω and Ξ =Ξ ∩Ξ be the set of points ⎣⎦ ≤ H ρ HSρ

2 2 radius ρ inside the hypercube. If tz is restricted toΞ , then ω = ρ andi = ⎢⎥ρ . Hρ ⎣⎦

wp 2 22⎢⎥ 42⎢ ⎥⎢22⎥ Alsoδ z =−ρρ⎣⎦. Therefore, max∑ ztiz==+− max ⎣ρρρ⎦⎣()⎦. zz∈ΞHH ∈Ξ ρρi=1

178 Therefore, depending on whether E > 0 or E 0, V min andV max can be determined ≤ ρz ρz as follows:

⎧⎫24⎛⎞E ⎪⎪AB++ρρ⎜⎟ D+ for E > 0 ⎪wp ⎪ V min = ⎝⎠ ρz ⎨⎬ 2 ⎪⎪24⎛⎞ 222 AB+++ρρ D E⎜⎟⎢⎥ ρρρ +−() ⎢⎥ for E ≤ 0 ⎩⎭⎪⎪⎝⎠⎣⎦ ⎣⎦

2 ⎧24⎛ 222⎞⎫ ⎪⎪AB+++ρρ D E⎜⎣⎦⎢⎥ ρρ +(−⎢⎥ ⎣⎦ρ)⎟for E > 0 ⎪⎪⎝⎠ V max = ρz ⎨⎬ 2⎛⎞E 4 ⎪AB++ρ⎜ D+⎟ρfor E ≤ 0⎪ ⎪⎪⎩⎭⎝⎠wp

Vmin and Vmax for subplot design space ρ x ρx

2 Maximization oftx for a fixed value of ω = jxx+ δ

⎧xx12====… xj 1 ⎪ xxmax==⎨ jx+ 1 δ ⎪ ⎩xxjj++23====… x sp0

sp 44 2 Therefore when tx is evaluated at xmax then txjjxj==∑ +=δω j+−() j. Using j=1 similar arguments as were presented for tz then

sp 2 42⎢⎥2 ⎢⎥2 max ∑ xtjx==+−max ⎣⎦ρρρ() ⎣⎦. Thus depending on whether G > 0 or G ≤ 0, x∈Ξ HHx∈Ξ ρρj=1 V min and V max can be determined as follows: ρx ρx

⎧⎫24⎛⎞G ⎪⎪AC++ρρ⎜⎟ F+ for G > 0 ⎪⎪sp V min = ⎝⎠ ρx ⎨⎬ 2 ⎪⎪24⎛ 2 22⎞ AC+++ρρ F G⎜⎟⎢⎥ ρρρ +−() ⎢⎥ for G ≤ 0 ⎪⎪⎩⎭⎝⎠⎣⎦ ⎣⎦

2 ⎧⎫24⎛⎞ 2 22 ⎪⎪AC+++ρρ F G⎜⎟⎣⎦⎢⎥ ρρρ +−() ⎣⎦⎢⎥ for G > 0 ⎪⎪⎝⎠ V max = ρx ⎨⎬ 24⎛⎞G ⎪⎪AC+++ρρ⎜⎟ F for G ≤ 0 ⎩⎪⎪⎝sp ⎠ ⎭

179 Vminρ and Vmaxρ for the combined design space

22 Maximization oftz andtx for a fixed value of ω = ijzz+++δδ xx then

wp sp 44 4 4 22 tz+tx=∑∑ z i + x j =+++ iδδ z j x =+ j() ω − j ++− i () ω i. Let the floor function ij=1

ω represent the greatest integer ω and Ξ =Ξ ∩Ξ be the set of points of radius ⎣⎦⎢⎥ ≤ H ρ HSρ

2 2 2 ρ inside the hypercube. If tz + t x is restricted toΞ , thenω = ρ , i = ⎢ρ ⎥ and j = ⎢ρ ⎥ . Hρ ⎣ z ⎦ ⎣ x ⎦

wp 2 42⎢ ⎥⎢22⎥ Therefore, max∑ ztizzz==+− max ⎣ρρρ⎦⎣()z⎦and zz∈ΞHH ∈Ξ ρρi=1

sp 2 42⎢⎥2 ⎢⎥2 max∑ xtjxxxx==+− max ⎣⎦ρρρ() ⎣⎦. Therefore, depending on the combination of xx∈ΞHH ∈Ξ ρρj=1 whether E > 0 or E 0 and G > 0 or G 0, V min and V max can be determined as ≤ ≤ ρx ρx follows:

⎧ 22⎛⎞⎛⎞EG 4 4 22 ⎫ ⎪AB++++ρρzx C⎜⎟⎜⎟ D ρ z ++ F ρ x + H ρρ zx for E and G > 0 ⎪ ⎪ ⎝⎠⎝⎠wp sp ⎪ ⎪ 2 ⎪ 224⎛⎞ 2 22 4 ⎪AB++++ρρρ C D E⎜⎟⎢⎥ ρ +− ρρ ⎢⎥ ++ F ρ… ⎪ zxz⎝⎠⎣⎦ z() zz ⎣⎦ x ⎪ for E and G ≤ 0 ⎪ 2 ⎪ ⎛⎞222 22 ⎪ V min = GH⎜⎟⎢⎥ρρρxxx+− ⎢⎥ + ρ zρx ρ ⎨ ⎝⎠⎣⎦() ⎣⎦ ⎬ ⎪ ⎪ 2 ⎪ 22⎛⎞E 442222⎛⎞2 ⎪ AB++++ρρzx C⎜⎟ D ρ z + Fρρρρρxxxxz++GH⎜⎟⎢⎥−+ ⎢⎥ ρxfor E > 0 and G ≤ 0 ⎪ wp ⎝⎠⎣⎦() ⎣⎦ ⎪ ⎪ ⎝⎠ ⎪ ⎪ 2 ⎛⎞G ⎪ 224⎛⎞⎢⎥ 2 22 ⎢⎥ 4 22 ⎪AB++++ρρρzxz C D E⎜⎟⎣⎦ ρ z +−() ρρ zz ⎣⎦ ++⎜⎟ F ρ x + H ρρ zxfor E ≤ 0 and G > 0⎪ ⎩⎪ ⎝⎠⎝⎠sp ⎭⎪

2 ⎧ 22 42224⎛⎞ ⎫ AB+++ρρρ C D ++EF⎜⎟⎢⎥ρρρ−++ ⎢⎥ ρ… ⎪ zxz⎝⎠⎣⎦zzz() ⎣⎦ x ⎪ ⎪ for E and G > 0 ⎪ 2 ⎪ ⎛⎞222 22 ⎪ GH⎜⎟⎢⎥ρρρ+− ⎢⎥ + ρρ ⎪ ⎝⎠⎣⎦xxx() ⎣⎦ zx ⎪ ⎪ ⎪ ⎛⎞⎛⎞EG ⎪AB++++ρρ22 C D ρ 4 ++ F ρ 4 + H ρρ 22 for E and G ≤ 0 ⎪ V max = zx⎜⎟⎜⎟ z x zx ρ ⎨ ⎝⎠⎝⎠wp sp ⎬ ⎪ ⎪ 2 ⎪ 22 4⎛⎞ 2 22 ⎛⎞G 422 ⎪ AB++++ρρρzxz C D E⎜⎟⎢⎥ ρ z +− ρρ zz ⎢⎥ ++⎜⎟ F ρρxz+≤H ρxfor E > 0 and G 0 ⎪ ⎝⎠⎣⎦() ⎣⎦ sp ⎪ ⎪ ⎝⎠ ⎪ ⎪ ⎛⎞E 2 ⎪ 22 44⎛⎞⎢⎥ 2 22 ⎢⎥ 22 ⎪AB++++ρρzx C⎜⎟ D ρρ zx ++ F G⎜⎟⎣⎦ ρ x +−() ρρ xx ⎣⎦ + H ρρ zxfor E ≤ 0 and G > 0⎪ ⎪⎩ ⎝⎠wp ⎝⎠ ⎪⎭

180 APPENDIX E

E1: Derivation of trace (Q11) and trace (Q22) for Split Plot CCD

In gene ra l the IV criterion is given as:

1 ⎡⎤' −1 − 1 IV = trace ⎢⎥()XR X ∫ f()()z,', x f z x d z d x ⎣⎦ω Ξ

⎡⎤⎛⎞BB11 12⎛ MM 11 12 ⎞ =×trace ⎢⎥⎜⎟⎜ ⎟ ⎣⎦⎝⎠BB21 22⎝ MM 21 22 ⎠

⎡⎤⎛⎞B11M11++BM 12 21 BM 11 12 BM 12 22 = trac e ⎢⎥⎜⎟ ⎣⎦⎝⎠B21M11+ BM 22 21 BM 21 12 +BM22 22

⎛⎞QQ11 12 = trace⎜⎟ ⎝⎠QQ21 22

The components of the matrix M is given as:

⎛⎞1 0 M11 = ⎜⎟ ⎝⎠0 Diag() mi ⎛φφJ'J'⎞ M = ⎜11wp sp ⎟ 12 ⎜⎟ ⎝00⎠

⎛⎞φ1Jwp 0 M21 = ⎜⎟ ⎝⎠φ1J0sp ' ⎛⎞φI+φJJJ'' φ⎡ J⎤ ⎜⎟2wp 3wp wp 3⎣ sp wp ⎦ M22 = ⎜⎟'' ⎝φ3JspwpJφ2 I k+φ3 JJ kk⎠

181 For the CCD the matrix B is given as:

⎛⎞γ1 0 B = ⎜⎟ 11 ⎛⎞1 ⎜0 Diag ⎜⎟⎟ ⎝⎝⎠di ⎠ '' ⎛γ2Jwp γ3Jsp ⎞ B12 = ⎜⎟ ⎝00⎠

⎛γ 2Jwp 0⎞ B21 = ⎜⎟ ⎝γ 3J0sp ⎠ ⎛⎞1 ' ''⎡⎤ 4 ()IJwp−γγ45 wpJ wp JspJ wp ⎜⎟Φ 2r β ⎣⎦ ⎜⎟1 w B22 = ⎜⎟'1 ' ⎜⎟γγ56JJsp wp 4 ( Isp − JJsp sp ) ⎝⎠Φ52rsα +Φ9

Diagonal elements of Q11

QBMBM11=+ 11 11 12 21

⎛γ 0 ⎞ 1 ⎛⎞1 0 BM = ⎜⎟ 11 11 ⎜⎛⎞1 ⎟⎜⎟ ⎜0 Diag ⎜⎟⎟⎝⎠0 Diag() mi ⎝⎝⎠di ⎠

⎛⎞γ1 0 ⎜⎟ = ⎛⎞m ⎜⎟⎜0 Diag ⎜⎟i ⎟ ⎝⎝⎠di ⎠ ''φ J0 ⎛γγ23JJwp sp ⎞⎛⎞1 wp BM12 21 = ⎜⎟⎜⎟ ⎝00⎠⎝⎠φ1J0sp

⎛⎞φγφγ1213wp+ sp 0 = ⎜⎟ ⎝⎠00

⎛⎞γ1 0 ⎜⎟⎛⎞φγφγ1213wp+ sp 0 BBM11M 11 +=12 21 ⎛⎞m +⎜⎟ ⎜⎟0 Diag ⎜⎟i ⎝00⎠ ⎝⎠⎝⎠di

⎛γφ11++wp γφγ 213 sp 0 ⎞ ⎜⎟ = ⎛⎞m ⎜⎟0 Diag ⎜⎟i ⎝⎠⎝⎠di

182 Diagonal elements of Q22

QBM22=+ 21 12BM 22 22 '' ⎛⎞γ 2J0wp ⎛⎞ φφ11JJwp sp BM21 12 = ⎜⎟⎜⎟ γ J0⎜⎟ ⎝⎠3 sp ⎝⎠00

⎛φγ JJ'φγ JJ'⎞ = ⎜⎟12 wp wp 12 wp sp ⎜⎟'' ⎝⎠φγ13JJsp wp φγ 13 JJ sp sp

⎛⎞1 ' ''⎡⎤ 4 ()Iwp−γγ45J wpJJ wp spJ wp ' ⎜⎟Φ 2r β ⎣⎦ ⎛⎞φφIJ+ JJ'' φ⎡⎤J ⎜⎟6 w ⎜⎟23wp wp wp 3⎣⎦ sp wp BM22 22 = '' ⎜⎟''1 ⎜⎟ φφ32JJspwp I k+φ3 JJ kk ⎜⎟γγ56JJsp wp 4 () Isp− JJ sp sp ⎝⎠ ⎝⎠Φ+592rsα Φ

The diagonal elements of BM22 22 matrix is determined in two parts as follows:

Part 1

⎡⎤1 ' I−+γφ J J''⎡⎤ Iφγφ J J+⎡ JJ'⎤ JJ' ⎢⎥4 ()wp42 wp wp⎣⎦ wp353 wp wp⎣ sp wp⎦ sp wp ⎣⎦Φ6 2rwβ ⎡⎤ ⎡⎤φφ22⎡γ4' ⎤ ⎢⎥⎢⎥44IJwp−+⎢ wpJ wp ⎥ ΦΦ22rrββ ⎢⎥⎣⎦66ww⎣ ⎦ ' =+⎢⎥φγ35sp J wpJ wp ⎡⎤⎛⎞⎡⎤ ⎢⎥φφ33''wpγ4 ⎢⎥⎜⎟44JJwp wp− ⎢⎥ JJ wp wp ⎢⎥ΦΦ22rrββ ⎣⎦⎣⎦⎢⎥⎝⎠66ww⎣⎦

φ2 ⎛⎞φγ34()1−−wp φ2γ4 ' =+44IJwp ⎜⎟ +φγ35sp wpJ wp ΦΦ6622rrwwββ⎝⎠

Part 2

' ⎡⎤1 γφJJ''⎡⎤ JJ+−+ I γ JJ' ⎡ φφ I JJ'⎤ 53spwp⎣⎦ spwp⎢⎥4 () sp6 spsp ⎣ 23 sp spsp⎦ ⎣⎦Φ+592rsα Φ ⎡ ⎤ ⎡⎤φ2 ⎡φγ26 ' ⎤ ⎢⎥⎢⎥44IJsp−+⎢ spJ sp ⎥ Φ+22rrααΦ Φ+Φ ' ⎢⎥⎣⎦5959ss⎣ ⎦ =+φγwp JJ 35sp sp ⎢⎥ ⎡⎤φφ⎡⎤spγ ⎢⎥33JJ''− 6 JJ ⎢⎥⎢⎥44sp sp⎢⎥ sp sp ⎣⎦⎣⎦Φ+5922rrssααΦ⎣⎦ Φ+ 59Φ

φ2 ⎛⎞φγφ362()1−−sp γ6 ' =+44IJsp ⎜⎟ +φγ35wp spJ sp Φ+5922rrssααΦ⎝⎠ Φ+ 59Φ

183 Consequently,

⎛⎞k k+⎜⎟ ⎝⎠2 m traceQ =+γφ wp γ + sp γ + Diag ⎛⎞i ()11 1 1 ( 2 3 )∑ ⎜⎟d i=1 ⎝⎠i

⎛⎞φγ34()11−+−wp φγ24( ) trace()Q22 = ⎜⎟4 ++φγ3 sp 5 φγ 1 2 wp + ⎝⎠Φ6 2rwβ

⎛⎞φγφγ36()11−+−sp 26() ⎜⎟4 ++φγφγ3513wp sp ⎝⎠Φ+592rsα Φ

184 E2: Derivation of trace (Q11B )B and trace (Q22B )B for Split Plot BBD

For the BBD the matrix B is given as

⎛⎞γ1 0 ⎜⎟ B11 = ⎛⎞ ⎜⎟0 Diag ⎜⎟1 ⎝⎠⎝⎠di '' ⎛⎞γγ23JJwp sp B12 = ⎜⎟ ⎝⎠00

⎛⎞γ 2J0wp B21 = ⎜⎟ ⎝⎠γ 3J0sp ⎛⎞1 ' ''⎡⎤ 4 ()IJwp−γγ45 wpJJ wp spJ wp ⎜⎟Φ−fr λβ ⎣⎦ ⎜⎟1 ()ww B22 = ⎜⎟''1 ⎜⎟γγ56JJsp wp 4 () Isp− JJ sp sp ⎝⎠Φ−5 fr()ssλα

Diagonal elements of Q11B

QBMBM11=+ 11 11 12 21

⎛⎞γ 0 1 ⎛⎞1 0 BM = ⎜⎟ 11 11 ⎜⎟⎛⎞1 ⎜⎟ ⎜⎟0 Diag ⎜⎟⎝⎠0 Diag() mi ⎝⎠⎝⎠di

⎛⎞γ1 0 ⎜⎟ = ⎛⎞m ⎜⎟0 Diag ⎜⎟i ⎝⎠⎝⎠di ''φ J0 ⎛⎞γγ23JJwp sp ⎛⎞1 wp BM12 21 = ⎜⎟⎜⎟ ⎝⎠00⎝⎠φ1J0sp

⎛⎞φγφγ1213wp+ sp 0 = ⎜⎟ ⎝⎠00

185 ⎛⎞γ1 0 ⎜⎟⎛⎞φγφγ1213wp+ sp 0 BM11 11+= BM 12 21 ⎛⎞m +⎜⎟ ⎜⎟0 Diag ⎜⎟i ⎝⎠00 ⎝⎠⎝⎠di

⎛⎞γφ11++wp γφγ 213 sp 0 ⎜⎟ = ⎛⎞m ⎜⎟0 Diag ⎜⎟i ⎝⎠⎝⎠di

Diagonal elements of Q22B

QBMBM22=+ 21 12 22 22 '' ⎛⎞γ 2J0wp ⎛⎞ φφ11JJwp sp BM21 12 = ⎜⎟⎜⎟ γ J0⎜⎟ ⎝⎠3 sp ⎝⎠00

⎛⎞φγJJ'' φγ JJ = ⎜⎟12wp wp 12 wp sp ⎜⎟'' ⎝⎠φγ13JJsp wp φγ 13 JJ sp sp

⎛⎞1 ' IJJ−γγ''⎡⎤ JJ ⎜⎟4 ()wp45 wp wp⎣⎦ sp wp ''' Φ−fr()λβ ⎛⎞φφIJJJ+ φ⎡⎤J ⎜⎟1 ww ⎜⎟23wp wp wp 3⎣⎦ sp wp BM22 22 = '' ⎜⎟''1 ⎜⎟ φφ32JJspwp I k+φ3 JJ kk ⎜⎟γγ56JJsp wp 4 () Isp− JJ sp sp ⎝⎠ ⎝⎠Φ−5 fr()ssλα

The diagonal elements of BM22 22 matrix is determined in two parts as follows: Part 1

⎡⎤1 ' I−+γφ J J''⎡⎤ Iφγφ J J+⎡ JJ'⎤ JJ' ⎢⎥4 ()wp42 wp wp⎣⎦ wp353 wp wp⎣ sp wp⎦ sp wp ⎣⎦⎢⎥Φ−1 fr()wwλβ ⎡⎤⎡⎤⎡ ⎤ φφ22γ4' ⎢⎥⎢⎥44IJwp−+⎢ wpJ wp ⎥ Φ−11fr()wwλβ Φ− fr() ww λβ ⎢⎥⎣⎦⎢⎥⎣⎢ ⎦⎥' = ⎢⎥+φγ35sp JJwp wp ⎢⎥⎡⎤⎛⎞φφ⎡⎤wpγ ⎢⎥⎜⎟33JJ''− ⎢ 4 JJ⎥ ⎢⎥⎜⎟Φ−frλβ44wp wp Φ− fr λβ wp wp ⎣⎦⎢⎥⎣⎦⎝⎠11()ww⎣⎢ () ww ⎦⎥ φ ⎛⎞φγφ()1−−wp γ =++2 IJ3424φγsp J' 44wp ⎜⎟35wp wp Φ−11fr()wwλβ⎝⎠ Φ− fr() ww λβ

186 Part 2

' ⎡⎤1 γφJJ''⎡⎤ JJ+− I γ JJ'' ⎡ φφ I+ JJ ⎤ 53spwp⎣⎦ spwp⎢⎥4 () sp6 spsp ⎣ 23 sp spsp ⎦ ⎣⎦⎢⎥Φ−5 fr()ssλα ⎡⎤⎡⎤φ ⎡φγ ⎤ ⎢⎥⎢⎥2 IJ−+⎢26 J' ⎥ Φ−frλα44sp Φ− fr λα sp sp ' ⎢⎥⎣⎦⎢⎥55()ss⎣⎢() ss ⎦⎥ =+φγ35wp JJsp sp ⎢⎥ ⎢⎥⎡⎤φφ' ⎡⎤spγ' ⎢⎥33JJ− ⎢⎥6 JJ ⎢⎥Φ−frλα44sp sp Φ− fr λα sp sp ⎣⎦⎣⎦⎢⎥55()ss⎣⎦⎢⎥() ss φ ⎛⎞φγφ()1−−sp γ =++2 IJ3626φγwp J' 44sp ⎜⎟35sp sp Φ−55fr()ssλα⎝⎠ Φ− fr() ss λα

Consequently,

⎛⎞k k+⎜⎟ ⎝⎠2 m traceQ =+γφ wp γ + sp γ + Diag ⎛⎞i ()11 1 1 ( 2 3 )∑ ⎜⎟d i=1 ⎝⎠i ⎛⎞φγ()11−+−wp φγ( ) traceQ = 3424++φγ sp φγ wp + ()22 ⎜⎟4 3 5 1 2 ⎝⎠Φ−1 fr()wwλβ

⎛⎞φγφγ()11−+−sp () 3626++φγφγwp sp ⎜⎟4 3513 ⎝⎠Φ−5 fr()ssλα

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192 BIOGRAPHICAL SKETCH

EDUCATION Doctor of Philosophy in Industrial Engineering Emphasis: Quality Engineering/Applied Statistics Florida State University Summer 2006

Dissertation Advisor: Dr. James Simpson Dissertation Title: Design and Analysis of Response Surface Designs with Restricted Randomization

Master of Science in Manufacturing Systems Southern Illinois University at Carbondale December 1999

Thesis Advisor: Dr. Julie K. McBride Thesis Title: An Expert System Approach to Automation Decisions in Manufacturing Systems

Bachelor of Education in Industrial Technology – First Class Honors University of Technology, Jamaica. January 1996

Diploma in Mechanical Technology University of Technology, Jamaica. March 1992

WORKSHOPS (Attended) Certificate of Participation in: Alternative Teaching/Training Methods University of Technology, Jamaica and Westviking College. June 1996

Certificate of Participation in: Designs for Adult Learning, a Program Development workshop College of Arts, Science and Technology (CAST). September 1995

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EXPERIENCE Lecturer School of Engineering and School of Technical and Vocational Education University of Technology, Jamaica. January 1993 – present (Currently on leave)

Graduate Research Assistant - Quality Engineering and Simulation Department of Industrial Engineering Florida State University Summer 2004 - Summer 2006

Graduate Teaching Assistant - Integrated Production Systems (IPS) Department of Industrial Engineering Florida State University Spring 2004

- Computer Aided Manufacturing 1 College of Engineering - Department of Technology Southern Illinois University at Carbondale Summer 1998 – Fall 1999

- Computer Aided Drawing (CAD) College of Engineering - Department of Technology Southern Illinois University at Carbondale Summer 1998

CONSULTANCY Project to improve the quality of Technical Vocational Education and Training (TVET) Provision at the St Vincent Technical College ● Program Development for Technology ● Conduct Building and Equipment Audit Ministry of Education St. Vincent and the Grenadines June 2001 – April 2002

Institutional Audit of Excelsior Community College Ministry of Education and Culture, Jamaica Tertiary Unit February 29 – March 1, 2000

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Commonwealth of Learning TVET Teacher Training ● Revision of learning materials for units of the TVET Teacher Training Curriculum. ● Module Advisor Commonwealth of Learning (CoL) April 2000 – August 2003

External Exam Moderator National Council on Technical and Vocational Education and Training (NCTVET) September 1996 – August 2003

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