Adaptive Tests of Significance Using Permutations of Residuals with R and SAS(R)

Thomas W. O'gorman , PhD, is Associate Professor in the Department of Mathematical Sciences at Northern Illinois University. Dr. O'Gorman's current research focuses on the analysis of adaptive methods for performing statistical tests and confidence intervals.

Preface xv 1 Introduction 1 1.1 Why Use Adaptive Tests? 1 1.2 A Brief History of Adaptive Tests 2 1.3 The Adaptive Test of Hogg, Fisher, and Randies 5 1.4 Limitations of Rank-Based Tests 8 1.5 The Adaptive Weighted Least Squares Approach 9 1.6 Development of the Adaptive WLS Test 12 2 Smoothing Methods and Normalizing Transformations 15 2.1 Traditional Estimators of the Median and the Interquartile Range 15 2.2 Percentile Estimators that Use the Smooth Cumulative Distribution Function 16 2.3 Estimating the Bandwidth 21 2.4 Normalizing Transformations 23 2.5 The Weighting Algorithm 27 2.6 Computing the Bandwidth 30 2.7 Examples of Transformed Data 37 3 A Two-Sample Adaptive Test 43 3.1 A Two-Sample Model 44 3.2 Computing the Adaptive Weights 45 3.3 The Test Statistics for Adaptive Tests 47 3.4 Permutation Methods for Two-Sample Tests 50 3.5 An Example of a Two-Sample Test 54 3.6 R Code for the Two-Sample Test 56 3.7 Level of Significance of the Adaptive Test 61 3.8 Power of the Adaptive Test 63 3.9 Sample Size Estimation 65 3.10 A SAS Macro for the Adaptive Test 68 3.11 Modifications for One-Tailed Tests 70 3.12 Justification of the Weighting Method 70 3.13 Comments on the Adaptive Two-sample Test 71 4 Permutation Tests with Linear Models 75 4.1 Introduction 75 4.2 Notation 76 4.3 Permutations with Blocking 77 4.4 Linear Models in Matrix Form 77 4.5 Permutation Methods 78 4.6 Permutation Test Statistics 81 4.7 An Important Rule of Test Construction 82 4.8 A Permutation Algorithm 82 4.9 A Performance Comparison of the Permutation Methods 83 4.10 Discussion 84 5 An Adaptive Test for a Subset of Coefficients 87 5.1 The General Adaptive Testing Method 87 5.2 Simple Linear Regression 91 5.3 An Example of a Simple Linear Regression 93 5.4 Multiple Linear Regression 96 5.5 An Example of a Test in Multiple Regression 100 5.6 Conclusions 105 6 More Applications of Adaptive Tests 111 6.1 The Completely Randomized Design 111 6.2 Tests for Randomized Complete Block Designs 120 6.3 Adaptive Tests for Two-way Designs 127 6.4 Dealing with Unequal Variances 134 6.5 Extensions to More Complex Designs 140 7 The Adaptive Analysis of Paired Data 149 7.1 Introduction 149 7.2 The Adaptive Test of Miao and Gastwirth 151 7.3 An Adaptive Weighted Least Squares Test 153 7.4 An Example Using Paired Data 160 7.5 Simulation Study 161 7.6 Sample Size Estimation 163 7.7 Discussion of Tests for Paired Data 165 8 Multicenter and Cross-Over Trials 169 8.1 Tests in Multicenter Clinical Trials 170 8.2 Adaptive Analysis of Cross-over Trials 176 9 Adaptive Multivariate Tests 191 9.1 The Traditional Likelihood Ratio Test 191 9.2 An Adaptive Multivariate Test 192 9.3 An Example with Two Dependent Variables 196 9.4 Performance of the Adaptive Test 199 9.5 Conclusions for Multivariate Tests 203 10 Analysis of Repeated Measures Data 207 10.1 Introduction 207 10.2 The Multivariate LR Test 209 10.3 The Adaptive Test 209 10.4 The Mixed Model Test 210 10.5 Two-Sample Tests 211 10.6 Two-Sample Tests for Parallelism 212 10.7 Two-Sample Tests for Group Effect 219 10.8 An Example of Repeated Measures Data 223 10.9 Dealing with Missing Data 227 10.10 Conclusions and Recommendations 229 11 Rank-Based Tests of Significance 235 11.1 The Quest for Power 235 11.2 Two-Sample Rank Tests 236 11.3 The HFR Test 242 11.4 Significance Level of Adaptive Tests 243 11.5 Biining's Adaptive Test for Location 244 11.6 An Adaptive Test for Location and Scale 245 11.7 Other Adaptive Rank Tests 247 11.8 Maximum Test 248 11.9 Discussion 249 12 Adaptive Confidence Intervals and Estimates 253 12.1 The Relationship Between Tests and Confidence Intervals 253 12.2 The Iterative Procedure of Garthwaite 254 12.3 Confidence Interval for a Difference 259 12.4 A 95% Confidence Interval for Slope 263 12.5 A General Formula for Confidence Limits 264 12.6 Computing a Confidence Interval Using R 266 12.7 Computing a 95% Confidence Interval Using SAS 268 12.8 Adaptive Estimation 268 12.9 Adaptive Estimation of the Difference Between Two Population Means 271 12.10 Adaptive Estimation of a Slope in a Multiple Regression Model 272 12.11 Computing an Adaptive Estimate Using R 274 12.12 Computing an Adaptive Estimate Using SAS 278 12.13 Discussion 278 Exercises 279 Appendix A: R Code for Univariate Adaptive Tests 283 Appendix B: SAS Macro for Adaptive Tests 287 Appendix C: SAS Macro for Multiple Comparisons Procedures 299 Appendix D: R Code for Adaptive Tests with Blocking Factors 303 Appendix E: R Code for Adaptive Test with Paired Data 305 Appendix F: SAS Macro for Adaptive Test with Paired Data 309 Appendix G: R Code for Multivariate Adaptive Tests 313 Appendix H: R Code for Confidence Intervals and Estimates 317 Appendix I: SAS Macro for Confidence Intervals 321 Appendix J: SAS Macro for Estimates 329 References 333 Index 341