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

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 11.1 Why Use Adaptive Tests? 11.2 A Brief History of Adaptive Tests 21.3 The Adaptive Test of Hogg, Fisher, and Randies 51.4 Limitations of Rank-Based Tests 81.5 The Adaptive Weighted Least Squares Approach 91.6 Development of the Adaptive WLS Test 122 Smoothing Methods and Normalizing Transformations 152.1 Traditional Estimators of the Median and the Interquartile Range 152.2 Percentile Estimators that Use the Smooth Cumulative Distribution Function 162.3 Estimating the Bandwidth 212.4 Normalizing Transformations 232.5 The Weighting Algorithm 272.6 Computing the Bandwidth 302.7 Examples of Transformed Data 373 A Two-Sample Adaptive Test 433.1 A Two-Sample Model 443.2 Computing the Adaptive Weights 453.3 The Test Statistics for Adaptive Tests 473.4 Permutation Methods for Two-Sample Tests 503.5 An Example of a Two-Sample Test 543.6 R Code for the Two-Sample Test 563.7 Level of Significance of the Adaptive Test 613.8 Power of the Adaptive Test 633.9 Sample Size Estimation 653.10 A SAS Macro for the Adaptive Test 683.11 Modifications for One-Tailed Tests 703.12 Justification of the Weighting Method 703.13 Comments on the Adaptive Two-sample Test 714 Permutation Tests with Linear Models 754.1 Introduction 754.2 Notation 764.3 Permutations with Blocking 774.4 Linear Models in Matrix Form 774.5 Permutation Methods 784.6 Permutation Test Statistics 814.7 An Important Rule of Test Construction 824.8 A Permutation Algorithm 824.9 A Performance Comparison of the Permutation Methods 834.10 Discussion 845 An Adaptive Test for a Subset of Coefficients 875.1 The General Adaptive Testing Method 875.2 Simple Linear Regression 915.3 An Example of a Simple Linear Regression 935.4 Multiple Linear Regression 965.5 An Example of a Test in Multiple Regression 1005.6 Conclusions 1056 More Applications of Adaptive Tests 1116.1 The Completely Randomized Design 1116.2 Tests for Randomized Complete Block Designs 1206.3 Adaptive Tests for Two-way Designs 1276.4 Dealing with Unequal Variances 1346.5 Extensions to More Complex Designs 1407 The Adaptive Analysis of Paired Data 1497.1 Introduction 1497.2 The Adaptive Test of Miao and Gastwirth 1517.3 An Adaptive Weighted Least Squares Test 1537.4 An Example Using Paired Data 1607.5 Simulation Study 1617.6 Sample Size Estimation 1637.7 Discussion of Tests for Paired Data 1658 Multicenter and Cross-Over Trials 1698.1 Tests in Multicenter Clinical Trials 1708.2 Adaptive Analysis of Cross-over Trials 1769 Adaptive Multivariate Tests 1919.1 The Traditional Likelihood Ratio Test 1919.2 An Adaptive Multivariate Test 1929.3 An Example with Two Dependent Variables 1969.4 Performance of the Adaptive Test 1999.5 Conclusions for Multivariate Tests 20310 Analysis of Repeated Measures Data 20710.1 Introduction 20710.2 The Multivariate LR Test 20910.3 The Adaptive Test 20910.4 The Mixed Model Test 21010.5 Two-Sample Tests 21110.6 Two-Sample Tests for Parallelism 21210.7 Two-Sample Tests for Group Effect 21910.8 An Example of Repeated Measures Data 22310.9 Dealing with Missing Data 22710.10 Conclusions and Recommendations 22911 Rank-Based Tests of Significance 23511.1 The Quest for Power 23511.2 Two-Sample Rank Tests 23611.3 The HFR Test 24211.4 Significance Level of Adaptive Tests 24311.5 Biining's Adaptive Test for Location 24411.6 An Adaptive Test for Location and Scale 24511.7 Other Adaptive Rank Tests 24711.8 Maximum Test 24811.9 Discussion 24912 Adaptive Confidence Intervals and Estimates 25312.1 The Relationship Between Tests and Confidence Intervals 25312.2 The Iterative Procedure of Garthwaite 25412.3 Confidence Interval for a Difference 25912.4 A 95% Confidence Interval for Slope 26312.5 A General Formula for Confidence Limits 26412.6 Computing a Confidence Interval Using R 26612.7 Computing a 95% Confidence Interval Using SAS 26812.8 Adaptive Estimation 26812.9 Adaptive Estimation of the Difference Between Two Population Means 27112.10 Adaptive Estimation of a Slope in a Multiple Regression Model 27212.11 Computing an Adaptive Estimate Using R 27412.12 Computing an Adaptive Estimate Using SAS 27812.13 Discussion 278Exercises 279Appendix A: R Code for Univariate Adaptive Tests 283Appendix B: SAS Macro for Adaptive Tests 287Appendix C: SAS Macro for Multiple Comparisons Procedures 299Appendix D: R Code for Adaptive Tests with Blocking Factors 303Appendix E: R Code for Adaptive Test with Paired Data 305Appendix F: SAS Macro for Adaptive Test with Paired Data 309Appendix G: R Code for Multivariate Adaptive Tests 313Appendix H: R Code for Confidence Intervals and Estimates 317Appendix I: SAS Macro for Confidence Intervals 321Appendix J: SAS Macro for Estimates 329References 333Index 341

“Each chapter provides detailed information on R and SAS code, respectively. Moreover, each chapter closes with illustrating exercises (without solutions). This is ideal for researchers who wish to implement anadaptive test of significance for their specific problem.”  (Biometrical Journal, 1 May 2013)