Introduction to Robust

Estimation and HypothesisTesting3rd Edition

Rand Wilcox

AMSTERDAM • BOSTON • HEIDELBERG • LONDON

NEW YORK • OXFORD • PARIS • SAN DIEGO

SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

Academic Press is an imprint ofElsevier

Contents

Preface *'*

Chapter 1 Introduction 1

1.1 Problems with Assuming Normality 1

1.2 Transformations 5

1.3 The Influence Curve 7

1.4 The Central Limit Theorem 8

1.5 Is the ANOVA F Robust? •9

1.6 Regression 10

1.7 More Remarks 11

1.8 Using the Computer: R 11

1.9 Some Data Management Issues 13

1.9.1 Eliminating Missing Values 22

Chapter 2 A Foundation for Robust Methods 23

2.1 Basic Tools for Judging Robustness 23

2.1.1 Qualitative Robustness 24

2.1.2 Infinitesimal Robustness 27

2.1.3 Quantitative Robustness 28

2.2 Some Measures of Location and Their Influence Function 29

2.2.1 Quantiles 29

2.2.2 The Winsorized Mean 30

2.2.3 The Trimmed Mean 32

2.2.4 M-Measures of Location 32

2.2.5 R-Measures of Location 35

2.3 Measures of Scale 36

2.4 Scale Equivariant M-Measures of Location 38

2.5 Winsorized Expected Values 39

Chapter 3 Estimating Measures ofLocation and Scale 43

3.1 A Bootstrap Estimate of a Standard Error 43

3.1.1 R Function bootse 45

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Contents

3.2 Density Estimators 46

3.2.1 Normal Kernel 46

3.2.2 Rosenblatt's Shifted Histogram 47

3.2.3 The Expected Frequency Curve 47

3.2.4 An Adaptive Kernel Estimator 48

3.2.5 R Functions skerd, kerden, kdplot, rdplot, akerd, and splot 49

3.3 The Sample Trimmed Mean 54

3.3.1 R Functions mean, tmean, and Hoc 57

3.3.2 Estimating the Standard Error of the Trimmed Mean 57

3.3.3 Estimating the Standard Error of the Sample Winsorized Mean..

62

3.3.4 R Functions winmean, winvar, trimse, and winse 62

3.3.5 Estimating the Standard Error of the Sample Median, M 62

3.3.6 R Function msmedse 63

3.4 The Finite Sample Breakdown Point 63

3.5 Estimating Quantiles 64

3.5.1 Estimating the Standard Error of the Sample Quantile 65

3.5.2 R Function qse 66

3.5.3 The Maritz-Jarrett Estimate of the Standard Error of xq 67

3.5.4 R Function mjse 68

3.5.5 The Harrell-Davis Estimator 68

3.5.6 R Function hdv

69

3.5.7 A Bootstrap Estimate of the Standard Error of 0q 69

3.5.8 R Function hdseb 70

3.6 An M-Estimator of Location 70

3.6.1 R Function mad 75

3.6.2 Computing an M-estimator of Location 75

3.6.3 R Functions mest 77

3.6.4 Estimating the Standard Error of the M-estimator 78

3.6.5 R Function mestse 80

3.6.6 A Bootstrap Estimate of the Standard Error of j±m 80

3.6.7 R Function mestseb 81

3.7 One-Step M-estimator 82

3.7.1 R Function onestep 83

3.8 W-estimators 83

3.8.1 Tau Measure of Location 84

3.8.2 R Function tauloc 85

3.8.3 Zuo's Weighted Estimator 85

3.9 The Hodges-Lehmann Estimator 85

3.10 Skipped Estimators 85

3.10.1 R Functions mom and bmean 86

3.11 Some Comparisons of the Location Estimators 86

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Contents

3.12 More Measures of Scale 89

3.12.1 The Biweight Midvariance 90

3.12.2 R Function bivar 92

3.12.3 The Percentage Bend Midvariance and tau Measure of

Variation 92

3.12.4 R Functions pbvar, tauvar 94

3.12.5 The Interquartile Range 95

3.12.6 R Function idealf 96

3.13 Some Outlier Detection Methods 96

3.13.1 Rules Based on Means and Variances 96

3.13.2 A Method Based on the Interquartile Range 96

3.13.3 Carling's Modification 97

3.13.4 A MAD-Median Rule 97

3.13.5 R Functions outbox, out, and boxplot 98

3.13.6 Skewness and the Boxplot Rule 99

3.13.7 R Function adjboxout 100

3.14 Exercises 100

Chapter 4 Confidence Intervals in the One-Sample Case 103

4.1 Problems when Working with Means 103

4.2 The g-and-h Distribution,

107

4.2.1 R Functions ghdist and rmul ' 110

4.3 Inferences About the Trimmed and Winsorized Means Ill

4.3.1 R Functions trimci and winci 114

4.4 Basic Bootstrap Methods 115

4.4.1 The Percentile Bootstrap Method 115

4.4.2 R Function onesampb 116

4.4.3 Bootstrap-t Method : 117

4.4.4 Bootstrap Methods when Using a Trimmed Mean 118

4.4.5 Singh's Modification 123

4.4.6 R Functions trimpb and trimcibt 123

4.5 Inferences About M-Estimators 124

4.5.1 R Functions mestci and momci 126

4.6 Confidence Intervals for Quantiles 126

4.6.1 Beware of Tied Values when Using the Median 129

4.6.2 Alternative Method for the Median 130

4.6.3 R Functions qmjci, hdci, sint, sintv2, qci, and qint 131

4.7 Empirical Likelihood 132

4.7.1 Bartlett Corrected Empirical Likelihood 133

4.8 Concluding Remarks 135

4.9 Exercises 135

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Chapter 5 Comparing Two Croups 137

5.1 The Shift Function 138

5.1.1 The Kolmogorov-Smirnov Test 141

5.1.2 R Functions ks, kssig, kswsig, and kstiesig 144

5.1.3 The S Band and W Band for the Shift Function 146

5.1.4 R Functions sband and wband 147

5.1.5 Confidence Band for the Deciles Only 150

5.1.6 R Function shifthd 151

5.1.7 R Functions g2plot and splotg2 153

5.2 Student's Mest 153

5.3 Comparing Medians and Other Trimmed Means 157

5.3.1 R Function yuen 160

5.3.2 A Bootstrap-t Method for Comparing Trimmed Means 161

5.3.3 R Functions yuenbt and yhbt 163

5.3.4 Measuring Effect Size: Robust Analogs of Cohen's d 166

5.3.5 R Functions akp.effect, yuenv2, and ees.ci 169

5.3.6 Comments on Measuring Effect Size 170

5.4 Inferences Based on a Percentile Bootstrap Method 170

5.4.1 Comparing M-Estimators 171

5.4.2 Comparing TrimmedMeans and Medians 172

5.4.3 R Functions trimpb2, pb2gen, m2ci, and medpb2 173

5.5 Comparing Measures of Scale 174

5.5.1 Comparing Variances 174

5.5.2 R Function comvar2 175

5.5.3 Comparing Biweight Midvariances 176

5.5.4 R Function b2ci 176

5.6 Permutation Tests 176

5.6.1 R Function permg 177

5.7 Inferences About a Probabilistic Measure of Effect Size 177

5.7.1 R Function mee 179

5.7.2 The Cliff and Bruner-Munzel Methods: Handling Tied Values.. 180

5.7.3 R Functions cid, cidv2, bmp, and wmwloc 184

5.8 Comparing Two Independent Binomials 186

5.8.1 Storer-Kim Method 187

5.8.2 Beal's Method 188

5.8.3 KMS Method 189

5.8.4 R Functions twobinom, twobici, bi2KMS, bi2KMSv2, and

bi2CR 189

5.8.5 Comparing Discrete Distributions: R Functions binband

and disc2com 190

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5.9 Comparing Dependent Groups 190

5.9.1 A Shift Function for Dependent Groups 190

5.9.2 R Function lband 192

5.9.3 Comparing Deciles 192

5.9.4 R Function shiftdhd 193

5.9.5 Comparing Trimmed Means 195

5.9.6 R Functions yuend and yuendv2 197

5.9.7 A Bootstrap-t Method for Marginal Trimmed Means 198

5.9.8 R Function ydbt 198

5.9.9 Inferences about the Distribution of Difference Scores 199

5.9.10 R Functions loc2dif and 12drmci 200

5.9.11 Percentile Bootstrap: Comparing Medians, M-Estimators

and Other Measures of Location and Scale 201

5.9.12 R Function bootdpci 202

5.9.13 Handling Missing Values 203

5.9.14 R Functions rm2miss and rmmismcp 207

5.9.15 Comparing Variances 208

5.9.16 The Sign Test and Inferences about the Binomial Distribution... 208

5.9.17 R Functions binomci and acbinomci 211

5.10 Exercises 212

Chapter 6 Some Multivariate Methods 215

6.1 Generalized Variance 215

6.2 Depth 216

6.2.1 Mahalanobis Depth 216

6.2.2 Halfspace Depth 216

6.2.3 Computing Halfspace Depth 218

6.2.4 R Functions depth2, depth, fdepth, fdepthv2, and unidepth 221

6.2.5 Projection Depth 222

6.2.6 R functions pdis and pdisMC 223

6.2.7 Other Measures of Depth 223

6.2.8 R Function zdepth 224

6.3 Some Affine Equivariant Estimators 224

6.3.1 Minimum Volume Ellipsoid Estimator 225

6.3.2 The Minimum Covariance Determinant Estimator 226

6.3.3 S-Estimators and Constrained M-Estimators 227

6.3.4 R Function tbs 228

6.3.5 Donoho-Gasko Generalization of a Trimmed Mean 228

6.3.6 R Functions dmean and dcov 229

6.3.7 The Stahel-Donoho W-Estimator 230

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6.3.8 R Function sdwe 231

6.3.9 Median Ball Algorithm 231

6.3.10 R Function rmba 232

6.3.11 OGK Estimator 232

6.3.12 R Function ogk 233

6.3.13 An M-Estimator 234

6.3.14 R Function MARest 234

6.4 Multivariate Outlier Detection Methods 235

6.4.1 ARelplot 236

6.4.2 R Function relplot 23 8

6.4.3 The MVE Method 239

6.4.4 The MCD Method 239

6.4.5 R Functions covmve and covmcd 239

6.4.6 R function out 240

6.4.7 The MGV Method 241

6.4.8 R Function outmgv 243

6.4.9 A Projection Method 244

6.4.10 R functions outpro and out3d 246

6.4.11 Outlier Identification in High Dimensions 247

6.4.12 R Function outproad and outmgvad 247

6.4.13 Approaches Based on Geometric Quantiles 248

6.4.14 Comments on Choosing a Method 248

6.5 A Skipped Estimator of Location and Scatter 250

6.5.1 R Functions smean, wmcd, wmve, mgvmean, Llmedcen,

spat, mgvcov, skip, skipcov, and dcov 252

6.6 Robust Generalized Variance 254

6.6.1 R Function gvarg 255

6.7 Inference in the One-Sample Case 2556.7.1 Inferences Based on the OP Measure of Location 255

6.7.2 Extension of Hotelling's T2 to Trimmed Means 256

6.7.3 R Functions smeancrv2 and hotell.tr 257

6.7.4 Inferences Based on the MGV Estimator 259

6.7.5 R Function smgvcr 259

6.8 Two-Sample Case 2596.8.1 R Functions smean2, smean2v2, matsplit, and mat2grp 260

6.8.2 Comparing Robust Generalized Variances 262

6.8.3 R function gvar2g 262

6.9 Multivariate Density Estimators 262

6.10 A Two-Sample, Projection-Type Extension of the

Wilcoxon-Mann-Whitney Test 263

6.10.1 R functions mulwmw and mulwmwv2 265

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6.11 A Relative Depth Analog of the Wilcoxon-Mann-Whitney Test 267

6.11.1 R function mwmw 268

6.12 Comparisons Based on Depth 269

6.12.1 R Functions lsqs3 and depthg2 272

6.13 Comparing Dependent Groups Based on All Pairwise Differences 275

6.13.1 R Function dfried 277

6.14 Robust Principal Components Analysis 277

6.14.1 R Functions prcomp and regpca 279

6.14.2 Maronna's Method 279

6.14.3 The SPCA Method 280

6.14.4 Method HRVB 280

6.14.5 Method OP 281

6.14.6 Method PPCA 281

6.14.7 R Functions outpca, robpca, robpcaS, SPCA, Ppca, and

Ppca.summary 282

6.14.8 Comments on Choosing the Number of Components 283

6.15 Cluster Analysis 287

6.15.1 R Functions Kmeans, kmeans.grp, TKmeans, and

TKmeans.grp 288

6.16 Exercises 288

Chapter 7 One-Way and Higher Designs for Independent Croups 291

7.1 Trimmed Means and a One-Way Design 292

7.1.1 A Welch-Type Procedure and a Robust Measure of Effect Size.. 293

7.1.2 R Functions tlway, tlwayv2, esmcp, fac21ist, and tlwayF 295

7.1.3 A Generalization of Box's Method 298

7.1.4 R Function boxlway 299

7.1.5 Comparing Medians 300

7.1.6 R Function medlway 301

7.1.7 A Bootstrap-t method 301

7.1.8 R Functions tlwaybt and btrim 302

7.1.9 Percentile Bootstrap Methods 304

7.2 Two-Way Designs and Trimmed Means 304

7.2.1 R Functions t2way 308

7.2.2 Comparing Medians 310

7.2.3 R Function med2way 311

7.3 ThreesWay Designs and Trimmed Means 311

7.3.1 R Functions t3way and fac21ist 313

7.4 Multiple Comparisons Based on Medians and Other Trimmed Means ..316

7.4.1 An Extension of Yuen's Method to Trimmed Means 317

7.4.2 R Function lincon 319

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7.4.3 Multiple Comparisons for Two-way and Three-Way

Designs 322

7.4.4 R Functions mcp2atm, mcp2med, mcp3atm, mcp3med,

con2way, and con3way 323

7.4.5 A Bootstrap-t Procedure 325

7.4.6 R Functions linconb, bbtrim, and bbbtrim 327

7.4.7 Percentile Bootstrap Methods for Comparing Medians and

Other Trimmed Means 329

7.4.8 R Functions tmcppb, bbmcppb, bbbmcppb, medpb,

med2mcp, med3mcp, and mcppb20 331

7.4.9 Judging Sample Sizes 333

7.4.10 R Function hochberg 334

7.4.11 Explanatory Measure of Effect Size 335

7.4.12 R Functions ESmainMCP and eslmcp 335

7.5 A Random Effects Model for Trimmed Means 336

7.5.1 A Winsorized Intraclass Correlation 338

7.5.2 R Function rananova 339

7.6 Global Tests Based on M-Measures of Location 340

7.6.1 R Functions blway and pbadepth 343

7.6.2 M-estimators and Multiple Comparisons 344

7.6.3 R Functions linconm and pbmcp 347

7.6.4 M-Estimators and the Random Effects Model 348

7.6.5 Other Methods for One-Way Designs 348

7.7 M-Measures of Location and a Two-Way Design 348

7.7.1 R Functions pbad2way and mcp2a 351

7.8 Ranked-Based Methods for a One-Way Design 351

7.8.1 The Rust-Fligner Method 352

7.8.2 R Function rfanova 353

7.8.3 A Heteroscedastic Rank-Based Method that Allows

Tied Values 354

7.8.4 R Function bdm 354

7.8.5 Inferences about a Probabilistic Measure of Effect Size 356

7.8.6 R Functions cidmulv2, wmwaov and cidM 358

7.9 A Rank-Based Method for a Two-Way Design 359

7.9.1 R Function bdm2way 361

7.9.2 The Patel-Hoel Approach to Interactions 361

7.9.3 R Function rimul 362

7.10 MANOVA Based on Trimmed Means 363

7.10.1 R Functions MULtr.anova, MULAOVp, bw21ist, and

YYmanova 365

7.10.2 Linear Contrasts 367

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7.10.3 R Functions linconMpb, linconSpb, YYmcp, fac2Mlist,and fac2BBMlist 369

7.11 Nested Designs 371

7.11.1 R Functions anova.nestA, mcp.nestA, and anova.nestAP 374

7.12 Exercises 374

Chapter 8 Comparing Multiple Dependent Croups 379

8.1 Comparing Trimmed Means 379

8.1.1 Omnibus Test Based on the Trimmed Means of the

Marginal Distributions 380

8.1.2 R Function rmanova 380

8.1.3 Pairwise Comparisons and Linear Contrasts Based on

Trimmed Means 381

8.1.4 Linear Contrasts Based on the Marginal Random Variables 384

8.1.5 R Functionrmmcp and rmmismcp 385

8.1.6 Judging the Sample Size 386

8.1.7 R Functions stein1 .tr and stein2.tr 387

8.2 Bootstrap Methods Based on Marginal Distributions 387

8.2.1 Comparing Trimmed Means 387

8.2.2 R Function rmanovab 388

8.2.3 Multiple Comparisons Based on Trimmed Means 388

8.2.4 R Functions pairdepb and bptd 390

8.2.5 Percentile Bootstrap Methods 392

8.2.6 R Functions bdlway and ddep 394

8.2.7 Multiple Comparisons Using M-estimators or SkippedEstimators 395

8.2.8 R Functions lindm and mcpOV 397

8.3 Bootstrap Methods Based on Difference Scores 398

8.3.1 R Function rmdzero 400

8.3.2 Multiple Comparisons 400

8.3.3 R Functions rmmcppb, wmcppb, dmedpb, and lindepbt 402

8.4 Comments on which Method to Use 404

8.5 Some Rank-Based Methods 406

8.5.1 R Functions apanova and bprm 408

8.6 Between-by-Within and Within-by-Within Designs 408

8.6.1 Analyzing a Between-by-Within Design Based on

Trimmed Means 408

8.6.2 R Functions bwtrim and tsplit 410

8.6.3 Data Management: R Function bw21ist 412

8.6.4 Bootstrap-t Method for a Between-by-Within Design 413

8.6.5 R Functions bwtrimbt and tsplitbt 414

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8.6.6 Percentile Bootstrap Methods for a Between-by-WithinDesign 414

8.6.7 R Functions sppba, sppbb, and sppbi 417

8.6.8 Multiple Comparisons 418

8.6.9 R Functions bwmcp, bwamcp, bwbmcp, bwimcp, spmcpa,

spmcpb, and spmcpi 421

8.6.10 Within-by-Within Designs 422

8.6.11 R Functions wwtrim, wwtrimbt, wwmcppb, and wwmcpbt 423

8.6.12 A Rank-Based Approach 423

8.6.13 R Function bwrank 427

8.6.14 Rank-Based Multiple Comparisons 429

8.6.15 R Function bwrmcp 429

8.6.16 Multiple Comparisons when Using a Patel-Hoel Approachto Interactions 429

8.6.17 R Function sisplit 431

8.7 Some Rank-Based Multivariate Methods 431

8.7.1 The Munzel-Brunner Method 431

8.7.2 R Function mulrank 433

8.7.3 The Choi-Marden Multivariate Rank Test 434

8.7.4 R Function cmanova 435

8.8 Three-Way Designs 436

8.8.1 Global Tests Based on Trimmed Means 436

8.8.2 R Functions bbwtrim, bwwtrim, wwwtrim, bbwtrimbt,

bwwtrimbt, and wwwtrimbt 437

8.8.3 Data Management: R Functions bw21ist and bbw21ist 437

8.8.4 Multiple Comparisons 438

8.8.5 R Function rm3mcp 439

8.8.6 R Functions bbwmcp, bwwmcp, bbwmcppb, bwwmcppb,and wwwmcppb 439

8.9 Exercises 440

Chapter 9 Correlation and Tests ofIndependence 447

9.1 Problems with the Product Moment Correlation 441

9.1.1 Features of Data that Affect r and T 444

9.1.2 Heteroscedasticity and the Classic Test that p = 0 445

9.2 Two Types of Robust Correlations 446

9.3 Some Type M-Measures of Correlation 446

9.3.1 The Percentage Bend Correlation 446

9.3.2 A Test of Independence Based on Ppb 447

9.3.3 R Function pbcor 449

9.3.4 A Test of Zero Correlation among p Random Variables 449

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9.3.5 R Function pball 451

9.3.6 The Winsorized Correlation 452

9.3.7 R Functions wincor and winall 453

9.3.8 The Biweight Midcovariance 454

9.3.9 R Functions bicov and bicovm 455

9.3.10 Kendall's tau 456

9.3.11 Spearman's rho 457

9.3.12 R Functions tau, spear, cor, and taureg 457

9.3.13 Heteroscedastic Tests of Zero Correlation 458

9.3.14 R Functions corb, pcorb, and pcorhc4 459

9.4 Some Type O Correlations 460

9.4.1 MVE and MCD Correlations 460

9.4.2 Skipped Measures of Correlation 460

9.4.3 The OP Correlation 461

9.4.4 Inferences Based on Multiple Skipped Correlations 461

9.4.5 R Functions scor and mscor 463

9.5 A Test of Independence Sensitive to Curvature 464

9.5.1 R Functions indt, indtall, and medind 466

9.6 Comparing Correlations: Independent Case 467

9.6.1 Comparing Pearson Correlations 467

9.6.2 Comparing Robust Correlations 468

9.6.3 R Functions twopcor and twocor 468

9.7 Exercises 468

Chapter 10 Robust Regression 471

10.1 Problems with Ordinary Least Squares 472

10.1.1 Computing Confidence Intervals under Heteroscedasticity 475

10.1.2 An Omnibus Test 479

10.1.3 R Functions IsfitNci, lsfitci, olshc4, hc4test, and hc4wtest 480

10.1.4 Comments on Comparing Means via Dummy Coding 483

10.1.5 Comments on Trying to Salvage the HomoscedasticityAssumption 483

'

10.2 Theil-Sen Estimator 484

10.2.1 R Functions tsreg, correg, and regplot 486

10.3 Least Median of Squares 487

10.3.1 R Function lmsreg 487

10.4 Least Trimmed Squares Estimator 488

10.4.1 R Functions ltsreg and ltsgreg 488

10.5 Least Trimmed Absolute Value Estimator 488

10.5.1 R Function ltareg 489

10.6 M-Estimators ....: 489

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10.7 The Hat Matrix 490

10.8 Generalized M-Estimators 493

10.8.1 R Function bmreg 497

10.9 The Coakley-Hettmansperger and Yohai Estimators 498

10.9.1 MM-Estimator 499

10.9.2 R Functions chreg and MMreg 500

10.10 Skipped Estimators 500

10.10.1 R Functions mgvreg and opreg 501

10.11 Deepest Regression Line 502

10.11.1 R Function mdepreg 502

10.12 A Criticism of Methods with a High Breakdown Point 503

10.13 Some Additional Estimators 503

10.13.1 S-Estimators and t-Estimators 503

10.13.2 R Functions snmreg and stsreg 504

10.13.3 E-Type Skipped Estimators 505

10.13.4 R Functions mbmreg, tstsreg, and gyreg 506

10.13.5 Methods Based on Robust Covariances 507

10.13.6 R Functions bireg, winreg, and COVreg 509

10.13.7 L-Estimators 510

10.13.8 Lx and Quantile Regression 510

10.13.9 R Functions qreg and rqfit 511

10.13.10 Methods Based on Estimates of the Optimal Weights 511

10.13.11 Projection Estimators 512

10.13.12 Methods Based on Ranks 513

10.14 Comments About Various Estimators 514

10.14.1 Contamination Bias 515

10.15 Outlier Detection Based on a Robust Fit 520

10.15.1 Detecting Regression Outliers 520

10.15.2 R Function reglev 521

10.16 Logistic Regression and the General Linear Model 522

10.16.1 R Functions glm, logreg, wlogreg, and logreg.plot 523

10.16.2 The General Linear Model 524

10.16.3 R Function glmrob 525

10.17 Multivariate Regression 525

10.17.1 The RADA Estimator 526

10.17.2 The Least Distance Estimator 527

10.17.3 R Functions mlrreg and Mreglde 528

10.17.4 Multivariate Least Trimmed Squares Estimator 529

10.17.5 R Function MULTtsreg 530

10.17.6 Other Robust Estimators 530

10.18 Exercises 530

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Chapter 11 More Regression Methods 533

11.1 Inferences About Robust Regression Parameters 533

11.1.1 Omnibus Tests for Regression Parameters 534

11.1.2 R Function regtest 538

11.1.3 Inferences About Individual Parameters 539

11.1.4 R Functions regci and wlogregci 541

11.1.5 Methods Based on the Quantile Regression Estimator 543

11.1.6 R Functions rqtest, qregci, and qrchk 544

11.1.7 Inferences Based on the OP-Estimator 545

11.1.8 R Functions opregpb and opregpbMC 546

11.1.9 Hypothesis Testing when Using the Multivariate

Regression Estimator RADA 547

11.1.10 R Function mlrGtest 548

11.1.11 Robust ANOVA via Dummy Coding 549

11.2 Comparing the Parameters of Two Independent Groups 549

11.2.1 R Function reg2ci 551

11.3 Detecting Heteroscedasticity 553

11.3.1 A Quantile Regression Approach : 553

11.3.2 Koenker's Method 554

11.3.3 R Functions qhomt and khomreg 555

11.4 Curvature and Half-Slope Ratios 555

11.4.1 R Function hratio 556

11.5 Curvature and Nonparametric Regression 558

11.5.1 Smoothers 558

11.5.2 Kernel Estimators and Cleveland's LOWESS 559

11.5.3 R Functions lplot and kerreg 561

11.5.4 The Running Interval Smoother 561

11.5.5 R Functions runmean, rungen, runmbo, and runhat 566

11.5.6 Skipped Smoothers 569

11.5.7 Smoothers for Estimating Quantiles via Splines 569

11.5.8 R Function qsmcobs 570

11.5.9 Special Methods for Binary Outcomes 570

11.5.10 R Functions logrsm bkreg, logSM, and rplotbin 572

11.5.11 Smoothing with More than One Predictor 573

11.5.12 R Functions runm3d, run3hat, rung3d, run3bo, rung3hat,

rplot, rplotsm, and runpd 574

11.5.13 LOESS 578

11.5.14 Other Approaches 581

11.5.15 R Function adrun, adrunl, gamplot, and gamplotINT 583

11.6 Checking the Specification of a Regression Model 584

11.6.1 Testing the Hypothesis of a Linear Association 585

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11.6.2 R Function lintest 586

11.6.3 Testing the Hypothesis of a Generalized Additive Model 586

11.6.4 R Function adtest 587

11.6.5 Inferences About the Components of a GeneralizedAdditive Model 588

11.6.6 R Function adcom 588

11.7 Regression Interactions and Moderator Analysis 589

11.7.1 R Functions kercon, riplot, runsm2g, ols.plot.inter, and

reg.plot.inter 591

11.7.2 Mediation Analysis 594

11.7.3 R functions ZYmediate, regmed2, and regmediate 596

11.8 Comparing Parametric, Additive, and Nonparametric Fits 596

11.8.1 R Functions adpchk and pmodchk 597

11.9 Measuring the Strength of an Association Given a Fit to the Data 598

11.9.1 R Function RobRsq 600

11.9.2 Comparing Two Independent Groups via Explanatory Power.... 600

11.9.3 R Functions smcorcom and smstrcom 601

11.10 Comparing Predictors 601

11.10.1 Comparing Pearson Correlations 602

11.10.2 Methods Based on Estimating Prediction Error 603

11.10.3 R Functions TWOpov, regpre, and regpreCV 605

11.10.4 R Function larsR 607

11.10.5 Comparing Predictors via Explanatory Power and a

Robust Fit 607

11.10.6 R Functions ts2str and sm2strv7 608

11.11 ANCOVA 609

11.11.1 Methods Based on Specific Design Points 610

11.11.2 R Functions ancova, ancpb, runmean2g, lplot2g, ancboot,ancbbpb, and cobs2g 613

11.11.3 Multiple Covariates 618

11.11.4 R Functions ancdes, ancovamp, ancmppb, and ancmg 619

11.11.5 Some Global Tests 620

11.11.6 R Functions ancsm and Qancsm 624

11.12 Marginal Longitudinal Data Analysis: Comments on ComparingGroups 624

11.12.1 R Functions long2g, longreg, longreg.plot, and xyplot 626

11.13 Exercises 627

References 631

Index 687

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