Adaptive Tests of Significance Using Permutations of Residuals with R and SAS® -- CONTENTS -- Preface -- 1 Introduction -- 1.1 Why Use Adaptive Tests? -- 1.2 A Brief History of Adaptive Tests -- 1.2.1 Early Tests and Estimators -- 1.2.2 Rank Tests -- 1.2.3 The Weighted Least Squares Approach -- 1.2.4 Recent Rank-Based Tests -- 1.3 The Adaptive Test of Hogg, Fisher, and Randles -- 1.3.1 Level of Significance of the HFR Test -- 1.3.2 Comparison of Power of the HFR Test to the t Test -- 1.4 Limitations of Rank-Based Tests -- 1.5 The Adaptive Weighted Least Squares Approach -- 1.5.1 Level of Significance -- 1.5.2 Comparison of Power of the Adaptive WLS Test to the t Test and the HFR Test -- 1.6 Development of the Adaptive WLS Test -- 2 Smoothing Methods and Normalizing Transformations -- 2.1 Traditional Estimators of the Median and the Interquartile Range -- 2.2 Percentile Estimators that Use the Smooth Cumulative Distribution Function -- 2.2.1 Smoothing the Cumulative Distribution Function -- 2.2.2 Using the Smoothed c.d.f. to Compute Percentiles -- 2.2.3 R Code for Smoothing the c.d.f. -- 2.2.4 R Code for Finding Percentiles -- 2.3 Estimating the Bandwidth -- 2.3.1 An Estimator of Variability Based on Traditional Percentiles -- 2.3.2 R Code for Finding the Bandwidth -- 2.3.3 An Estimator of Variability Based on Percentiles from the Smoothed Distribution Function -- 2.4 Normalizing Transformations -- 2.4.1 Traditional Normalizing Methods -- 2.4.2 Normalizing Data by Weighting -- 2.5 The Weighting Algorithm -- 2.5.1 An Example of the Weighing Procedure -- 2.5.2 R Code for Weighting the Observations -- 2.6 Computing the Bandwidth -- 2.6.1 Error Distributions -- 2.6.2 Measuring Errors in Adaptive Weighting -- 2.6.3 Simulation Studies -- 2.7 Examples of Transformed Data -- Exercises -- 3 A Two-Sample Adaptive Test -- 3.1 A Two-Sample Model.

3.2 Computing the Adaptive Weights -- 3.2.1 R Code for Computing the Weights -- 3.3 The Test Statistics for Adaptive Tests -- 3.3.1 R Code to Compute the Test Statistic -- 3.4 Permutation Methods for Two-Sample Tests -- 3.4.1 Permutation of Observations -- 3.4.2 Permutation of Residuals -- 3.4.3 R Code for Permutations -- 3.5 An Example of a Two-Sample Test -- 3.6 R Code for the Two-Sample Test -- 3.6.1 R Code for Computing the Test Statistics -- 3.6.2 R Code to Compute the Traditional F Test Statistic and p-Value -- 3.6.3 An R Function that Computes the p-Value for the Adaptive Test -- 3.6.4 R Code to Perform the Adaptive Test -- 3.7 Level of Significance of the Adaptive Test -- 3.8 Power of the Adaptive Test -- 3.9 Sample Size Estimation -- 3.10 A SAS Macro for the Adaptive Test -- 3.11 Modifications for One-Tailed Tests -- 3.12 Justification of the Weighting Method -- 3.13 Comments on the Adaptive Two-sample Test -- Exercises -- 4 Permutation Tests with Linear Models -- 4.1 Introduction -- 4.2 Notation -- 4.3 Permutations with Blocking -- 4.4 Linear Models in Matrix Form -- 4.5 Permutation Methods -- 4.5.1 The Permute-Errors Method -- 4.5.2 The Permute-Residuals Method -- 4.5.3 The Permutation of Independent Variables Method -- 4.5.4 The Permutation of Dependent Variables Method -- 4.6 Permutation Test Statistics -- 4.7 An Important Rule of Test Construction -- 4.8 A Permutation Algorithm -- 4.9 A Performance Comparison of the Permutation Methods -- 4.10 Discussion -- Exercises -- 5 An Adaptive Test for a Subset of Coefficients -- 5.1 The General Adaptive Testing Method -- 5.1.1 Weighting Step -- 5.1.2 Permutation Step -- 5.2 Simple Linear Regression -- 5.2.1 The Significance of the Adaptive Test -- 5.2.2 The Power of the Adaptive Test -- 5.2.3 Justification of the Weighting Method -- 5.3 An Example of a Simple Linear Regression.

5.3.1 Using R Code to Perform the Adaptive Test for Slope -- 5.3.2 Using a SAS Macro to Perform the Adaptive Test -- 5.4 Multiple Linear Regression -- 5.4.1 Comments on the Weighting Method -- 5.4.2 Significance Level of the Adaptive Test -- 5.4.3 Power of the Adaptive Test -- 5.5 An Example of a Test in Multiple Regression -- 5.5.1 Example Using R Code -- 5.5.2 Example Using a SAS Macro -- 5.6 Conclusions -- Exercises -- 6 More Applications of Adaptive Tests -- 6.1 The Completely Randomized Design -- 6.1.1 Model Specification -- 6.1.2 Level of Significance and Power of the Adaptive Test -- 6.1.3 An Example of a Completely Randomized Design -- 6.1.4 Multiple Comparison Procedures -- 6.2 Tests for Randomized Complete Block Designs -- 6.2.1 The Significance Level and Power of the Adaptive RCB Test -- 6.2.2 An Example of the Analysis of RCB Design Data -- 6.3 Adaptive Tests for Two-way Designs -- 6.3.1 Tests for Interaction -- 6.3.2 An Example -- 6.3.3 Tests for Main Effects -- 6.4 Dealing with Unequal Variances -- 6.4.1 Tests with Stochastically Ordered Random Variables -- 6.4.2 Tests when the Random Variables Are Not Stochastically Ordered -- 6.5 Extensions to More Complex Designs -- 6.5.1 Analysis of Covariance -- 6.5.2 Multifactorial Designs -- 6.5.3 Other Designs -- Exercises -- 7 The Adaptive Analysis of Paired Data -- 7.1 Introduction -- 7.2 The Adaptive Test of Miao and Gastwirth -- 7.2.1 A Measure of Tail-Heaviness -- 7.2.2 Rank Score Functions -- 7.2.3 Selecting the Score Function -- 7.3 An Adaptive Weighted Least Squares Test -- 7.3.1 The Unweighted Test Statistic -- 7.3.2 Adaptive Weighting and Permutations -- 7.3.3 The Test Statistic for the Adaptive WLS Test -- 7.4 An Example Using Paired Data -- 7.4.1 Data from Twins -- 7.4.2 The Adaptive WLS Test Using R -- 7.4.3 The Adaptive WLS Test Using SAS -- 7.5 Simulation Study.

7.6 Sample Size Estimation -- 7.7 Discussion of Tests for Paired Data -- Exercises -- 8 Multicenter and Cross-Over Trials -- 8.1 Tests in Multicenter Clinical Trials -- 8.1.1 Level of Significance and Power -- 8.1.2 An Example of the Analysis of Data from a Multicenter Clinical Trial -- 8.2 Adaptive Analysis of Cross-over Trials -- 8.2.1 Tests for Two-Period Cross-Over Trials without Baseline Measurements -- 8.2.2 Tests for a Two-Period Cross-Over Design with Baseline Measurements -- 8.2.3 An Example -- 8.2.4 Recommendations for Cross-Over Trials -- Exercises -- 9 Adaptive Multivariate Tests -- 9.1 The Traditional Likelihood Ratio Test -- 9.2 An Adaptive Multivariate Test -- 9.2.1 The Projection Method -- 9.2.2 Adaptive Weighting -- 9.2.3 Permutation Method -- 9.2.4 Justification of the Projection Method -- 9.3 An Example with Two Dependent Variables -- 9.3.1 Using the SAS Macro -- 9.3.2 Using an R Function -- 9.4 Performance of the Adaptive Test -- 9.4.1 Significance Level of the Tests -- 9.4.2 Power of the Tests -- 9.5 Conclusions for Multivariate Tests -- Exercises -- 10 Analysis of Repeated Measures Data -- 10.1 Introduction -- 10.2 The Multivariate LR Test -- 10.3 The Adaptive Test -- 10.4 The Mixed Model Test -- 10.5 Two-Sample Tests -- 10.6 Two-Sample Tests for Parallelism -- 10.6.1 Traditional LR Test for Parallelism -- 10.6.2 Adaptive Test for Parallelism -- 10.6.3 Mixed Model Test for Parallelism -- 10.6.4 An Example -- 10.6.5 Comparison of the Tests for Parallelism -- 10.7 Two-Sample Tests for Group Effect -- 10.7.1 Simulation Results for Group Effects -- 10.8 An Example of Repeated Measures Data -- 10.8.1 Using the SAS Macro -- 10.8.2 Using R Code -- 10.9 Dealing with Missing Data -- 10.10 Conclusions and Recommendations -- Exercises -- 11 Rank-Based Tests of Significance -- 11.1 The Quest for Power -- 11.2 Two-Sample Rank Tests.

11.3 The HFR Test -- 11.4 Significance Level of Adaptive Tests -- 11.5 Büning's Adaptive Test for Location -- 11.6 An Adaptive Test for Location and Scale -- 11.7 Other Adaptive Rank Tests -- 11.8 Maximum Test -- 11.9 Discussion -- Exercises -- 12 Adaptive Confidence Intervals and Estimates -- 12.1 The Relationship Between Tests and Confidence Intervals -- 12.2 The Iterative Procedure of Garthwaite -- 12.3 Confidence Interval for a Difference -- 12.3.1 Comparison of Coverage Probabilities -- 12.3.2 Comparison of Average Width -- 12.4 A 95% Confidence Interval for Slope -- 12.5 A General Formula for Confidence Limits -- 12.6 Computing a Confidence Interval Using R -- 12.7 Computing a 95% Confidence Interval Using SAS -- 12.8 Adaptive Estimation -- 12.9 Adaptive Estimation of the Difference Between Two Population Means -- 12.10 Adaptive Estimation of a Slope in a Multiple Regression Model -- 12.11 Computing an Adaptive Estimate Using R -- 12.12 Computing an Adaptive Estimate Using SAS -- 12.13 Discussion -- Exercises -- Appendix A: R Code for Univariate Adaptive Tests -- Appendix B: SAS Macro for Adaptive Tests -- Appendix C: SAS Macro for Multiple Comparisons Procedures -- Appendix D: R Code for Adaptive Tests with Blocking Factors -- Appendix E: R Code for Adaptive Test with Paired Data -- Appendix F: SAS Macro for Adaptive Test with Paired Data -- Appendix G: R Code for Multivariate Adaptive Tests -- Appendix H: R Code for Confidence Intervals and Estimates -- Appendix I: SAS Macro for Confidence Intervals -- Appendix J: SAS Macro for Estimates -- References -- Index.