An Introduction to Optimal Designs for Social and Biomedical Research

Preface xi

Acknowledgements xiii

1 Introduction to designs 1

1.1 Introduction 1

1.2 Stages of the research process 4

1.2.1 Choice of a ‘good’ design 5

1.3 Research design 6

1.3.1 Choice of independent variables and levels 6

1.3.2 Units of analysis 6

1.3.3 Variables 7

1.3.4 Replication 8

1.4 Types of research designs 8

1.5 Requirements for a ‘good’ design 9

1.5.1 Statistical conclusion validity 10

1.5.2 Internal validity 12

1.5.3 Control of (unwanted) variation 13

1.6 Ethical aspects of design choice 16

1.7 Exact versus approximate designs 17

1.8 Examples 19

1.8.1 Radiation dosage example 19

1.8.2 Designs for the Poggendorff and Ponzo illusion experiments 20

1.8.3 Uncertainty about best fitting regression models 22

1.8.4 Designs for a priori contrasts among composite faces 23

1.8.5 Designs for calibration of item parameters in item response theory models 24

1.9 Summary 26

2 Designs for simple linear regression 27

2.1 Design problem for a linear model 27

2.1.1 The design 28

2.1.2 The linear regression model 31

2.1.3 Estimation of parameters and efficiency 32

2.2 Designs for radiation-dosage example 35

2.3 Relative efficiency and sample size 36

2.4 Simultaneous inference 37

2.5 Optimality criteria 39

2.5.1 D-optimality criterion 40

2.5.2 A-optimality criterion 41

2.5.3 G-optimality criterion 41

2.5.4 E-optimality criterion 43

2.5.5 Number of distinct design points 43

2.6 Relative efficiency 44

2.7 Matrix formulation of designs for linear regression 44

2.8 Summary 49

3 Designs for multiple linear regression analysis 51

3.1 Design problem for multiple linear regression 51

3.1.1 The design 52

3.1.2 The multiple linear regression model 54

3.1.3 Estimation of parameters and efficiency 54

3.2 Designs for vocabulary-growth study 56

3.3 Relative efficiency and sample size 60

3.4 Simultaneous inference 61

3.5 Optimality criteria for a subset of parameters 62

3.6 Relative efficiency 64

3.7 Designs for polynomial regression model 65

3.7.1 Exact D-optimal designs for a quadratic regression model 69

3.7.2 Scale dependency of A- and E-optimality criteria 71

3.8 The Poggendorff and Ponzo illusion study 71

3.9 Uncertainty about best fitting regression models 76

3.10 Matrix notation of designs for multiple regression models 79

3.10.1 Design for regression models with two independent variables 80

3.10.2 Design for regression models with two non-additive independent variables 82

3.11 Summary 85

4 Designs for analysis of variance models 87

4.1 A typical design problem for an analysis of variance model 87

4.1.1 The design 89

4.1.2 The analysis of variance model 90

4.1.3 Formulation of an ANOVA model as a regression model 91

4.2 Estimation of parameters and efficiency 95

4.2.1 Measures of uncertainty 96

4.3 Simultaneous inference and optimality criteria 97

4.4 Designs for groups under stress study 98

4.4.1 A priori planned unequal sample sizes 99

4.4.2 Not planned unequal sample sizes 100

4.5 Specific hypotheses and contrasts 101

4.5.1 Loss of efficiency and power 103

4.6 Designs for the composite faces study 106

4.7 Balanced designs versus unbalanced designs 109

4.8 Matrix notation for Groups under Stress study 109

4.9 Summary 111

5 Designs for logistic regression models 113

5.1 Design problem for logistic regression 113

5.2 The design 114

5.3 The logistic regression model 115

5.3.1 Design for a single dichotomous independent variable 116

5.3.2 Design for multiple qualitative independent variables 122

5.3.3 Design for a single quantitative independent variable 125

5.3.4 Design for two independent quantitative variables 130

5.4 Approaches to deal with local optimality 133

5.5 Designs for calibration of item parameters in item response theory models 134

5.6 Matrix formulation of designs for logistic regression 137

5.6.1 Hours of practice experiment 138

5.6.2 Problem solving study 140

5.7 Summary 141

6 Designs for multilevel models 143

6.1 Design problem for multilevel models 143

6.1.1 The design 144

6.1.2 Validity considerations 146

6.2 The multilevel regression model 147

6.2.1 Cluster randomization of treatment 147

6.2.2 Subject randomization of treatment 149

6.3 Cluster versus subject randomization 151

6.4 Cost function 153

6.5 Example: Nursing home study 155

6.5.1 Cluster randomization 157

6.5.2 Subject randomization 159

6.6 Optimal design and power 160

6.6.1 Power for cluster randomized design 162

6.6.2 Power for multi-center design 164

6.6.3 Increase of efficiency and power by including covariates 165

6.6.4 Unequal sample sizes 165

6.7 Design effect in multilevel surveys 166

6.7.1 Values of intra-class correlation ρ 168

6.7.2 Cluster randomized sampling versus simple random sampling 168

6.8 Matrix formulation of the multilevel model 169

6.8.1 Cluster randomization of treatment 170

6.8.2 Subject randomization of treatment 172

6.9 Summary 174

7 Longitudinal designs for repeated measurement models 175

7.1 Design problem for repeated measurements 175

7.2 The design 179

7.3 Analysis techniques for repeated measures 180

7.4 The linear mixed effects model for repeated measurement data 181

7.4.1 Random intercept model 182

7.4.2 Random intercept and slope model 183

7.5 Variance–covariance structures 184

7.5.1 Compound symmetry structure 184

7.5.2 Auto-correlation structure 185

7.6 Estimation of parameters and efficiency 187

7.6.1 Small sample behaviour of estimators 188

7.7 Bone mineral density example 189

7.7.1 Improvement of the longitudinal design 194

7.8 Cost function 196

7.9 D-optimal designs for linear mixed effects models with autocorrelated errors 200

7.10 Miscellanea 207

7.10.1 Homoscedasticity 207

7.10.2 Uninformative dropout 208

7.11 Matrix formulation of the linear mixed effects model 208

7.12 Summary 211

8 Two-treatment crossover designs 213

8.1 Design problem for crossover studies 213

8.2 The design 216

8.3 Confounding treatment effects with nuisance effects 218

8.4 The linear model for crossover designs 221

8.5 Estimation of parameters and efficiency 223

8.6 Cost and efficiency of the crossover design 223

8.6.1 Cost function 226

8.7 Optimal crossover designs for two treatments 229

8.7.1 Some further observations 231

8.8 Matrix formulation of the mixed model for crossover designs 232

8.9 Summary 235

9 Alternative optimal designs for linear models 237

9.1 Introduction 237

9.2 Information matrix 238

9.3 DA- or Ds-optimal designs 239

9.4 Extrapolation optimal design 241

9.5 L-optimal designs 242

9.6 Bayesian optimal designs 244

9.7 Minimax optimal design 247

9.8 Multiple-objective optimal designs 250

9.8.1 Constrained optimal design 251

9.8.2 Compound optimal design 253

9.9 Summary 255

10 Optimal designs for nonlinear models 257

10.1 Introduction 257

10.2 Linear models versus nonlinear models 258

10.2.1 The Arrhenius equation 258

10.2.2 The compartmental model 259

10.2.3 The Michaelis–Menten model 260

10.2.4 The Emax model 261

10.3 Design issues for nonlinear models 261

10.3.1 Local optimality 262

10.4 Alternative optimal designs with examples 265

10.4.1 DA or Ds-optimal design 265

10.4.2 Extrapolation optimal design 266

10.4.3 Optimal design for estimating percentiles 266

10.5 Bayesian optimal designs 267

10.6 Minimax optimal design 269

10.7 Multiple-objective optimal designs 271

10.8 Optimal design for model discrimination 273

10.9 Summary 275

11 Resources for the construction of optimal designs 277

11.1 Introduction 277

11.2 Sequential construction of optimal designs 278

11.3 Exchange of design points 283

11.3.1 Exchange algorithms 283

11.4 Other algorithms 284

11.5 Optimal design software 285

11.6 A web site for finding optimal designs 286

11.6.1 Optimal designs for the Michaelis–Menten and Emax models 288

11.6.2 Optimal designs for discriminating among toxicological models 290

11.7 Summary 294

References 295

Author Index 313

Subject Index 319