Repeated Measurements and Cross-Over Designs

DAMARAJU RAGHAVARAO, PhD, was Laura H. Carnell Professor and Chairperson in the Department of Statistics at Temple University. With more than fifty years of research experience in all aspects of experimental design, sampling, and multivariate analysis, Dr. Raghavarao authored eight additional books and over 135 journal articles throughout his career. LAKSHMI PADGETT, PhD, is Senior Manager at Janssen R&D. Dr. Padgett has authored approximately twenty journal articles, and her research interests include phase I, II, and III trials.

Preface xi

1. Introduction 1

1.1 Introduction 1

1.2 One-Sample RMD 2

1.3 k-Sample RMD 4

1.4 Split-Plot Designs 7

1.5 Growth Curves 13

1.6 Cross-Over Designs 14

1.7 Two-Period Cross-Over Designs 18

1.8 Modifications in Cross-Over Designs 19

1.9 Nonparametric Methods 22

References 23

2. One-Sample Repeated Measurement Designs 25

2.1 Introduction 25

2.2 Testing for Sphericity Condition 26

2.3 Univariate ANOVA for One-Sample RMD 29

2.4 Multivariate Methods for One-Sample RMD 32

2.5 Univariate ANOVA Under Nonsphericity Condition 34

2.6 Numerical Example 35

2.7 Concordance Correlation Coefficient 41

2.8 Multiresponse Concordance Correlation Coefficient 44

2.9 Repeated Measurements with Binary Response 47

References 51

3. k-Sample Repeated Measurements Design 53

3.1 Introduction 53

3.2 Test for the Equality of Dispersion Matrices and Sphericity Condition of k-Dispersion Matrices 54

3.3 Univariate ANOVA for k-Sample RMD 57

3.4 Multivariate Methods for k-Sample RMD 60

3.5 Numerical Example 63

3.6 Multivariate Methods with Unequal Dispersion Matrices 67

3.7 Analysis with Ordered Categorical Response 72

References 75

4. Growth Curve Models 77

4.1 Introduction 77

4.2 Sigmoidal Curves 78

4.3 Analysis of Mixed Models 84

4.4 Simple Linear Growth Curve Model 90

4.5 Nonlinear Growth Curve Model 92

4.6 Numerical Example 93

4.7 Joint Action Models 100

References 103

5. Cross-Over Designs without Residual Effects 105

5.1 Introduction 105

5.2 Fixed Effects Analysis of CODWOR 107

5.3 Connectedness in CODWOR 113

5.4 Orthogonality in CODWOR 115

5.5 Latin Square Designs 116

5.6 Youden Square Design and Generalization 118

5.7 F-Squares 123

5.8 Lattice Square Designs 123

5.9 Analysis of CODWOR when the Units Effects Are Random 125

5.10 Numerical Example 127

5.11 Orthogonal Latin Squares 131

References 133

6. Cross-Over Designs with Residual Effects 135

6.1 Introduction 135

6.2 Analysis of CODWR 136

6.3 BRED 143

6.4 PBCOD(m) 148

6.5 Numerical Example 152

6.6 Analysis with Unit (or Subject) Effects Random 156

6.7 Concluding Remarks 159

References 160

7. Two-Period Cross-Over Designs with Residual Effects 163

7.1 Introduction 163

7.2 Two-Period Two-Treatment CODWR Analysis: Parametric Methods 164

7.2.1 Analysis of the design based on the model (7.2.9) 167

7.2.2 Decomposition of the model (7.2.9) into intra- and interunit components 169

7.2.3 Estimating direct effects contrast using cross-over nature of the treatments 170

7.2.4 Modified two-period two-treatment design 171

7.2.5 Cost analysis 171

7.3 Two-Period Two-Treatment CODWR Analysis: Nonparametric Methods 173

7.4 Two-Period t Treatment Cross-Over Design 174

7.5 Numerical Examples 177

References 186

8. Other Cross-Over Designs with Residual Effects 189

8.1 Introduction 189

8.2 Extra-Period Designs 191

8.2.1 Residual effect of a treatment effect on itself is the same as residual effect on other treatments 192

8.2.2 Residual effect of a treatment on itself is different from the residual effect on other treatments 193

8.3 Residual Effects Proportional to Direct Effects 194

8.4 Undiminished Residual Effects Designs 195

8.5 Treatment Balanced Residual Effects Designs 197

8.6 A General Linear Model for CODWR 199

8.7 Nested Design 201

8.8 Split-Plot Type CODWR 203

8.9 CODWR in Circular Arrangement 205

8.10 Numerical Examples 207

References 213

9. Some Constructions of Cross-Over Designs 215

9.1 Introduction 215

9.2 Galois Fields 215

9.3 Generalized Youden Designs 217

9.4 Williams’ Balanced Residual Effects Designs 221

9.5 Other Balanced Residual Effects Designs 226

9.6 Combinatorially Overall Balanced Residual Effects Designs 229

9.7 Construction of Treatment Balanced Residual Effects Designs 231

9.8 Some Construction of PBCOD (m) 232

9.9 Construction of Complete Set of MOLS and Patterson’s BRED 234

9.10 Balanced Circular Arrangements 235

9.11 Concluding Remarks 236

References 237

Index 245