## Table of contents

Table of contents :

Preface

1 Introduction

1.1 Problems with Assuming Normality

1.2 Transformations

1.3 The Inﬂuence Curve

1.4 The Central Limit Theorem

1.5 Is the ANOVA F Robust?

1.6 Regression

1.7 More Remarks

1.8 R Software

1.9 Some Data Management Issues

1.9.1 Eliminating Missing Values

1.10 Data Sets

2 A Foundation for Robust Methods

2.1 Basic Tools for Judging Robustness

2.1.1 Qualitative Robustness

2.1.2 Inﬁnitesimal Robustness

2.1.3 Quantitative Robustness

2.2 Some Measures of Location and Their Inﬂuence Function

2.2.1 Quantiles

2.2.2 The Winsorized Mean

2.2.3 The Trimmed Mean

2.2.4 M-Measures of Location

2.2.5 R-Measures of Location

2.3 Measures of Scale

2.4 Scale Equivariant M-Measures of Location

2.5 Winsorized Expected Values

3 Estimating Measures of Locationand Scale

3.1 A Bootstrap Estimate of a Standard Error

3.1.1 R Function bootse

3.2 Density Estimators

3.2.1 Silverman’s Rule of Thumb

3.2.2 Rosenblatt’s Shifted Histogram

3.2.3 The Expected Frequency Curve

3.2.4 An Adaptive Kernel Estimator

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

3.3 The Sample Trimmed Mean

3.3.1 R Functions mean, tmean and lloc

3.3.2 Estimating the Standard Error of the Trimmed Mean

3.3.3 Estimating the Standard Error of the Sample Winsorized Mean

3.3.4 R Functions winmean, winvar, trimse and winse

3.3.5 Estimating the Standard Error of the Sample Median

3.3.6 R Function msmedse

3.4 The Finite Sample Breakdown Point

3.5 Estimating Quantiles

3.5.1 Estimating the Standard Error of the Sample Quantile

3.5.2 R Function qse

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

3.5.4 R Function mjse

3.5.5 The Harrell-Davis Estimator

3.5.6 R Functions qest and hd

3.5.7 A Bootstrap Estimate of the Standard Error of thetaq

3.5.8 R Function hdseb

3.6 An M-Estimator of Location

3.6.1 R Function mad

3.6.2 Computing an M-Estimator of Location

3.6.3 R Functions mest

3.6.4 Estimating the Standard Error of the M-Estimator

3.6.5 R Function mestse

3.6.6 A Bootstrap Estimate of the Standard Error of µm

3.6.7 R Function mestseb

3.7 One-Step M-Estimator

3.7.1 R Function onestep

3.8 W-Estimators

3.8.1 Tau Measure of Location

3.8.2 R Function tauloc

3.8.3 Zuo’s Weighted Estimator

3.9 The Hodges-Lehmann Estimator

3.10 Skipped Estimators

3.10.1 R Functions mom and bmean

3.11 Some Comparisons of the Location Estimators

3.12 More Measures of Scale

3.12.1 The Biweight Midvariance

3.12.2 R Function bivar

3.12.3 The Percentage Bend Midvariance and Tau Measure of Variation

3.12.4 R Functions pbvar, tauvar

3.12.5 The Interquartile Range

3.12.6 R Functions idealf and idrange

3.13 Some Outlier Detection Methods

3.13.1 Rules Based on Means and Variances

3.13.2 A Method Based on the Interquartile Range

3.13.3 Carling’s Modiﬁcation

3.13.4 A MAD-Median Rule

3.13.5 R Functions outbox, out and boxplot

3.13.6 R Functions adjboxout and adjbox

3.14 Exercises

4 Conﬁdence Intervals in theOne-Sample Case

4.1 Problems when Working with Means

4.2 The g-and-h Distribution

Multivariate g-and-h Distributions

4.2.1 R Functions ghdist, rmul, rngh and ghtrim

4.3 Inferences About the Trimmed and Winsorized Means

4.3.1 R Functions trimci, winci and D.akp.effect

4.4 Basic Bootstrap Methods

4.4.1 The Percentile Bootstrap Method

4.4.2 R Functions onesampb and hdpb

4.4.3 Bootstrap-t Method

4.4.4 Bootstrap Methods when Using a Trimmed Mean

4.4.5 Singh’s Modiﬁcation

4.4.6 R Functions trimpb and trimcibt

4.5 Inferences About M-Estimators

4.5.1 R Functions mestci and momci

4.6 Conﬁdence Intervals for Quantiles

4.6.1 Beware of Tied Values when Making Inferences About Quantiles

4.6.2 A Modiﬁcation of the Distribution-Free Method for the Median

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

4.7 Empirical Likelihood

4.7.1 Bartlett Corrected Empirical Likelihood

4.8 Concluding Remarks

4.9 Exercises

5 Comparing Two Groups

5.1 The Shift Function

5.1.1 The Kolmogorov-Smirnov Test

5.1.2 R Functions ks, kssig, kswsig, and kstiesig

5.1.3 The B and W Band for the Shift Function

5.1.4 R Functions sband and wband

5.1.5 Conﬁdence Band for Speciﬁed Quantiles

Method Q1

Method Q2

5.1.6 R Functions shifthd, qcomhd, qcomhdMC and q2gci

5.1.7 R Functions g2plot and g5plot

5.2 Student’s t Test

5.3 Comparing Medians and Other Trimmed Means

Yuen’s Method

Comparing Medians

5.3.1 R Functions yuen and msmed

5.3.2 A Bootstrap-t Method for Comparing Trimmed Means

5.3.3 R Functions yuenbt and yhbt

5.3.4 Measuring Effect Size

A Standardized Difference

Explanatory Power

A Classiﬁcation Perspective

A Probabilistic Measure of Effect Size

5.3.5 R Functions akp.effect, yuenv2, ees.ci, med.effect and qhat

5.4 Inferences Based on a Percentile Bootstrap Method

5.4.1 Comparing M-Estimators

5.4.2 Comparing Trimmed Means and Medians

5.4.3 R Functions trimpb2, pb2gen, m2ci, medpb2 and M2gbt

5.5 Comparing Measures of Scale

5.5.1 Comparing Variances

5.5.2 R Function comvar2

5.5.3 Comparing Biweight Midvariances

5.5.4 R Function b2ci

5.6 Permutation Tests

5.6.1 R Function permg

5.7 Rank-Based Methods and a Probabilistic Measure of Effect Size

5.7.1 The Cliff and Brunner-Munzel Methods

Cliff’s Method

Brunner-Munzel Method

5.7.2 R Functions cid, cidv2, bmp, wmwloc, wmwpb and loc2plot

5.8 Comparing Two Independent Binomial and Multinomial Distributions

5.8.1 Storer-Kim Method

5.8.2 Beal’s Method

5.8.3 KMS Method

5.8.4 R Functions twobinom, twobici, bi2KMS, bi2KMSv2 and bi2CR

5.8.5 Comparing Discrete (Multinomial) Distributions

5.8.6 R Functions binband, splotg2, cumrelf

5.9 Comparing Dependent Groups

5.9.1 A Shift Function for Dependent Groups

5.9.2 R Function lband

5.9.3 Comparing Speciﬁed Quantiles

Method D1

Method D2

Method D3

5.9.4 R Functions shiftdhd, Dqcomhd, qdec2ci, Dqdif and difQpci

5.9.5 Comparing Trimmed Means

5.9.6 R Functions yuend, yuendv2 and D.akp.effect

5.9.7 A Bootstrap-t Method for Marginal Trimmed Means

5.9.8 R Function ydbt

5.9.9 Inferences About the Distribution of Difference Scores

5.9.10 R Functions loc2dif and l2drmci

5.9.11 Percentile Bootstrap: Comparing Medians, M-Estimators and Other Measures of Location and Scale

5.9.12 R Function bootdpci

5.9.13 Handling Missing Values

Method M1

Method M2

Method M3

Comments on Choosing a Method

5.9.14 R Functions rm2miss and rmmismcp

5.9.15 Comparing Variances

5.9.16 R Function comdvar

5.9.17 The Sign Test and Inferences About the Binomial Distribution

5.9.18 R Functions binomci, acbinomci and binomLCO

5.10 Exercises

6 Some Multivariate Methods

6.1 Generalized Variance

6.2 Depth

6.2.1 Mahalanobis Depth

6.2.2 Halfspace Depth

6.2.3 Computing Halfspace Depth

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

6.2.5 Projection Depth

6.2.6 R Functions pdis, pdisMC, and pdepth

6.2.7 Other Measures of Depth

6.2.8 R Functions zdist, zoudepth and prodepth

6.3 Some Afﬁne Equivariant Estimators

6.3.1 Minimum Volume Ellipsoid Estimator

6.3.2 The Minimum Covariance Determinant Estimator

6.3.3 S-Estimators and Constrained M-Estimators

6.3.4 R Function tbs

6.3.5 Donoho-Gasko Generalization of a Trimmed Mean

6.3.6 R Functions dmean and dcov

6.3.7 The Stahel-Donoho W-Estimator

6.3.8 R Function sdwe

6.3.9 Median Ball Algorithm

6.3.10 R Function rmba

6.3.11 OGK Estimator

6.3.12 R Function ogk

6.3.13 An M-Estimator

6.3.14 R Functions MARest and dmedian

6.4 Multivariate Outlier Detection Methods

6.4.1 A Relplot

6.4.2 R Function relplot

6.4.3 The MVE Method

6.4.4 The MCD Method

6.4.5 R Functions covmve and covmcd

6.4.6 R Function out

6.4.7 The MGV Method

6.4.8 R Function outmgv

6.4.9 A Projection Method

6.4.10 R Functions outpro and out3d

6.4.11 Outlier Identiﬁcation in High Dimensions

6.4.12 R Functions outproad and outmgvad

6.4.13 Methods Designed for Functional Data

6.4.14 R Functions FBplot, Flplot, medcurve, func.out, spag.plot, funloc and funlocpb

6.4.15 Comments on Choosing a Method

6.5 A Skipped Estimator of Location and Scatter

6.5.1 R Functions smean, wmcd, wmve, mgvmean, L1medcen, spat, mgvcov, skip, skipcov

6.6 Robust Generalized Variance

6.6.1 R Function gvarg

6.7 Multivariate Location: Inference in the One-Sample Case

6.7.1 Inferences Based on the OP Measure of Location

6.7.2 Extension of Hotelling’s T2 to Trimmed Means

6.7.3 R Functions smeancrv2 and hotel1.tr

6.7.4 Inferences Based on the MGV Estimator

6.7.5 R Function smgvcr

6.8 Comparing OP Measures of Location

6.8.1 R Functions smean2, matsplit and mat2grp

Data Management

6.8.2 Comparing Robust Generalized Variances

6.8.3 R Function gvar2g

6.9 Multivariate Density Estimators

6.10 A Two-Sample, Projection-Type Extension of the Wilcoxon-Mann-Whitney Test

6.10.1 R Functions mulwmw and mulwmwv2

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

6.11.1 R Function mwmw

6.12 Comparisons Based on Depth

6.12.1 R Functions lsqs3 and depthg2

6.13 Comparing Dependent Groups Based on All Pairwise Differences

6.13.1 R Function dfried

6.14 Robust Principal Components Analysis

6.14.1 R Functions prcomp and regpca

6.14.2 Maronna’s Method

6.14.3 The SPCA Method

6.14.4 Method HRVB

6.14.5 Method OP

6.14.6 Method PPCA

6.14.7 R Functions outpca, robpca, robpcaS, SPCA, Ppca, Ppca.summary

6.14.8 Comments on Choosing the Number of Components

6.15 Cluster Analysis

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

6.16 Multivariate Discriminate Analysis

6.16.1 R Function KNNdist

6.17 Exercises

7 One-Way and Higher Designs for Independent Groups

7.1 Trimmed Means and a One-Way Design

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

A Robust, Heteroscedastic Measure of Effect Size

7.1.2 R Functions t1way, t1wayv2, esmcp, fac2list, t1wayF

Data Management

7.1.3 A Generalization of Box’s Method

7.1.4 R Function box1way

7.1.5 Comparing Medians and Other Quantiles

7.1.6 R Functions med1way and Qanova

7.1.7 A Bootstrap-t Method

7.1.8 R Functions t1waybt and btrim

7.2 Two-Way Designs and Trimmed Means

7.2.1 R Function t2way

7.2.2 Comparing Medians

7.2.3 R Functions med2way and Q2anova

7.3 Three-Way Designs and Trimmed Means Including Medians

7.3.1 R Functions t3way, fac2list and Q3anova

7.4 Multiple Comparisons Based on Medians and Other Trimmed Means

7.4.1 Basic Methods Based on Trimmed Means

A Step-Down Multiple Comparison Procedure

7.4.2 R Functions lincon, conCON and stepmcp

7.4.3 Multiple Comparisons for Two-Way and Three-Way Designs

7.4.4 R Functions bbmcp, mcp2med, bbbmcp, mcp3med, con2way and con3way

7.4.5 A Bootstrap-t Procedure

7.4.6 R Functions linconb, bbtrim and bbbtrim

7.4.7 Controlling the Familywise Error Rate: Improvements on the Bonferroni Method

Rom’s Method

Hochberg’s Method

Hommel’s Method

Benjamini-Hochberg Method

The k-FWER Procedures

7.4.8 R Functions p.adjust and mcpKadjp

7.4.9 Percentile Bootstrap Methods for Comparing Medians, Other Trimmed Means and Quantiles

7.4.10 R Functions linconpb, bbmcppb, bbbmcppb, medpb, Qmcp, med2mcp, med3mcp and q2by2

7.4.11 Judging Sample Sizes

7.4.12 R Function hochberg

7.4.13 Explanatory Measure of Effect Size

7.4.14 R Functions ESmainMCP and esImcp

7.4.15 Comparing Curves (Functional Data)

7.4.16 R Functions funyuenpb and Flplot2g

7.5 A Random Effects Model for Trimmed Means

7.5.1 A Winsorized Intraclass Correlation

7.5.2 R Function rananova

7.6 Global Tests Based on M-Measures of Location

Method SHB

Method LSB

7.6.1 R Functions b1way and pbadepth

7.6.2 M-Estimators and Multiple Comparisons

Variation of a Bootstrap-t Method

A Percentile Bootstrap Method: Method SR

7.6.3 R Functions linconm and pbmcp

7.6.4 M-Estimators and the Random Effects Model

7.6.5 Other Methods for One-Way Designs

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

7.7.1 R Functions pbad2way and mcp2a

7.8 Ranked-Based Methods for a One-Way Design

7.8.1 The Rust-Fligner Method

7.8.2 R Function rfanova

7.8.3 A Heteroscedastic Rank-Based Method That Allows Tied Values

7.8.4 R Function bdm

7.8.5 Inferences About a Probabilistic Measure of Effect Size

Method CHMCP

Method WMWAOV

Method DBH

7.8.6 R Functions cidmulv2, wmwaov and cidM

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

7.9.1 R Function bdm2way

7.9.2 The Patel-Hoel Approach to Interactions

7.9.3 R Function rimul

7.10 MANOVA Based on Trimmed Means

7.10.1 R Functions MULtr.anova, MULAOVp, bw2list and YYmanova

7.10.2 Linear Contrasts

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

Data Management

7.11 Nested Designs

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

7.12 Exercises

8 Comparing Multiple Dependent Groups

8.1 Comparing Trimmed Means

8.1.1 Omnibus Test Based on the Trimmed Means of the Marginal Distributions

8.1.2 R Function rmanova

8.1.3 Pairwise Comparisons and Linear Contrasts Based on Trimmed Means

8.1.4 Linear Contrasts Based on the Marginal Random Variables

8.1.5 R Functions rmmcp, rmmismcp and trimcimul

8.1.6 Judging the Sample Size

8.1.7 R Functions stein1.tr and stein2.tr

8.2 Bootstrap Methods Based on Marginal Distributions

8.2.1 Comparing Trimmed Means

8.2.2 R Function rmanovab

8.2.3 Multiple Comparisons Based on Trimmed Means

8.2.4 R Functions pairdepb and bptd

8.2.5 Percentile Bootstrap Methods

Method RMPB3

Method RMPB4

Missing Values

8.2.6 R Functions bd1way, ddep and ddepGMC_C

8.2.7 Multiple Comparisons Using M-Estimators or Skipped Estimators

8.2.8 R Functions lindm and mcpOV

8.3 Bootstrap Methods Based on Difference Scores

8.3.1 R Function rmdzero

8.3.2 Multiple Comparisons

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

8.4 Comments on Which Method to Use

8.5 Some Rank-Based Methods

Method AP

Method BPRM

Decision Rule

8.5.1 R Functions apanova and bprm

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

8.6.1 Analyzing a Between-by-Within Design Based on Trimmed Means

8.6.2 R Functions bwtrim and tsplit

8.6.3 Data Management: R Function bw2list

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

8.6.5 R Functions bwtrimbt and tsplitbt

8.6.6 Percentile Bootstrap Methods for a Between-by-Within Design

8.6.7 R Functions sppba, sppbb and sppbi

8.6.8 Multiple Comparisons

Method BWMCP

Method BWAMCP: Comparing Levels of Factor A for Each Level of Factor B

Method BWBMCP: Dealing with Factor B

Method BWIMCP: Interactions

Methods SPMCPA, SPMCPB and SPMCPI

8.6.9 R Functions bwmcp, bwamcp, bwbmcp, bwimcp, bwimcpES, spmcpa, spmcpb and spmcpi

8.6.10 Within-by-Within Designs

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

8.6.12 A Rank-Based Approach

8.6.13 R Function bwrank

8.6.14 Rank-Based Multiple Comparisons

8.6.15 R Function bwrmcp

8.6.16 Multiple Comparisons when Using a Patel-Hoel Approach to Interactions

8.6.17 R Function sisplit

8.7 Some Rank-Based Multivariate Methods

8.7.1 The Munzel-Brunner Method

8.7.2 R Function mulrank

8.7.3 The Choi-Marden Multivariate Rank Test

8.7.4 R Function cmanova

8.8 Three-Way Designs

8.8.1 Global Tests Based on Trimmed Means

8.8.2 R Functions bbwtrim, bwwtrim, wwwtrim, bbwtrimbt, bwwtrimbt and wwwtrimbt

8.8.3 Data Management: R Functions bw2list and bbw2list

8.8.4 Multiple Comparisons

8.8.5 R Function wwwmcp

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

Bootstrap-t Methods

Percentile Bootstrap Methods

8.9 Exercises

9 Correlation and Tests of Independence

9.1 Problems with Pearson’s Correlation

9.1.1 Features of Data That Affect r and T

9.1.2 Heteroscedasticity and the Classic Test that rho=0

9.2 Two Types of Robust Correlations

9.3 Some Type M Measures of Correlation

9.3.1 The Percentage Bend Correlation

9.3.2 A Test of Independence Based on rhopb

9.3.3 R Function pbcor

9.3.4 A Test of Zero Correlation Among p Random Variables

9.3.5 R Function pball

9.3.6 The Winsorized Correlation

9.3.7 R Functions wincor and winall

9.3.8 The Biweight Midcovariance and Correlation

9.3.9 R Functions bicov and bicovm

9.3.10 Kendall’s tau

9.3.11 Spearman’s rho

9.3.12 R Functions tau, spear, cor and taureg

9.3.13 Heteroscedastic Tests of Zero Correlation

9.3.14 R Functions corb, pcorb and pcorhc4

9.4 Some Type O Correlations

9.4.1 MVE and MCD Correlations

9.4.2 Skipped Measures of Correlation

9.4.3 The OP Correlation

9.4.4 Inferences Based on Multiple Skipped Correlations

9.4.5 R Functions scor, mscor and scorci

9.5 A Test of Independence Sensitive to Curvature

Method INDT

Method MEDIND

9.5.1 R Functions indt, indtall and medind

9.6 Comparing Correlations: Independent Case

9.6.1 Comparing Pearson Correlations

9.6.2 Comparing Robust Correlations

9.6.3 R Functions twopcor, twohc4cor and twocor

9.7 Exercises

10 Robust Regression

10.1 Problems with Ordinary Least Squares

10.1.1 Computing Conﬁdence Intervals Under Heteroscedasticity

Method HC4WB-D

Method HC4WB-C

10.1.2 An Omnibus Test

10.1.3 R Functions lsﬁtci, olshc4, hc4test and hc4wtest

10.1.4 Comments on Comparing Means via Dummy Coding

10.1.5 Salvaging the Homoscedasticity Assumption

10.2 Theil-Sen Estimator

10.2.1 R Functions tsreg, tshdreg, correg, regplot and regp2plot

10.3 Least Median of Squares

10.3.1 R Function lmsreg

10.4 Least Trimmed Squares Estimator

10.4.1 R Functions ltsreg and ltsgreg

10.5 Least Trimmed Absolute Value Estimator

10.5.1 R Function ltareg

10.6 M-Estimators

10.7 The Hat Matrix

10.8 Generalized M-Estimators

10.8.1 R Function bmreg

10.9 The Coakley-Hettmansperger and Yohai Estimators

10.9.1 MM-Estimator

10.9.2 R Functions chreg and MMreg

10.10 Skipped Estimators

10.10.1 R Functions mgvreg and opreg

10.11 Deepest Regression Line

10.11.1 R Functions rdepth and mdepreg

10.12 A Criticism of Methods with a High Breakdown Point

10.13 Some Additional Estimators

10.13.1 S-Estimators and tau-Estimators

10.13.2 R Functions snmreg and stsreg

10.13.3 E-Type Skipped Estimators

10.13.4 R Functions mbmreg, tstsreg, tssnmreg and gyreg

10.13.5 Methods Based on Robust Covariances

10.13.6 R Functions bireg, winreg and COVreg

10.13.7 L-Estimators

10.13.8 L1 and Quantile Regression

10.13.9 R Functions qreg, rqﬁt, qplotreg

10.13.10 Methods Based on Estimates of the Optimal Weights

10.13.11 Projection Estimators

10.13.12 Methods Based on Ranks

10.13.13 R Functions Rﬁt and Rﬁt.est

10.13.14 Empirical Likelihood Type and Distance-Constrained Maximum Likelihood Estimators

10.14 Comments About Various Estimators

10.14.1 Contamination Bias

10.15 Outlier Detection Based on a Robust Fit

10.15.1 Detecting Regression Outliers

10.15.2 R Functions reglev and rmblo

10.16 Logistic Regression and the General Linear Model

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

10.16.2 The General Linear Model

10.16.3 R Function glmrob

10.17 Multivariate Regression

10.17.1 The RADA Estimator

10.17.2 The Least Distance Estimator

10.17.3 R Functions MULMreg, mlrreg and Mreglde

10.17.4 Multivariate Least Trimmed Squares Estimator

10.17.5 R Function MULtsreg

10.17.6 Other Robust Estimators

10.18 Exercises

11 More Regression Methods

11.1 Inferences About Robust Regression Parameters

11.1.1 Omnibus Tests for Regression Parameters

11.1.2 R Function regtest

11.1.3 Inferences About Individual Parameters

11.1.4 R Functions regci, regciMC and wlogregci

11.1.5 Methods Based on the Quantile Regression Estimator

11.1.6 R Functions rqtest, qregci and qrchk

11.1.7 Inferences Based on the OP Estimator

11.1.8 R Functions opregpb and opregpbMC

11.1.9 Hypothesis Testing when Using a Multivariate Regression Estimator RADA

11.1.10 R Function mlrGtest

11.1.11 Robust ANOVA via Dummy Coding

11.1.12 Conﬁdence Bands for the Typical Value of y Given x

11.1.13 R Functions regYhat, regYci, and regYband

11.1.14 R Function regse

11.2 Comparing the Regression Parameters of J >=2 Groups

11.2.1 Methods for Comparing Independent Groups

Methods Based on the Least Squares Regression Estimator

Multiple Comparisons

Methods Based on Robust Estimators

11.2.2 R Functions reg2ci, reg1way, reg1wayISO, ancGpar, ols1way, ols1wayISO, olsJmcp, olsJ2, reg1mcp and olsWmcp

11.2.3 Methods for Comparing Two Dependent Groups

Methods Based on a Robust Estimator

Methods Based on the Least Squares Estimator

11.2.4 R Functions DregG, difreg, DregGOLS

11.3 Detecting Heteroscedasticity

11.3.1 A Quantile Regression Approach

11.3.2 Koenker-Bassett Method

11.3.3 R Functions qhomt and khomreg

11.4 Curvature and Half-Slope Ratios

11.4.1 R Function hratio

11.5 Curvature and Nonparametric Regression

11.5.1 Smoothers

11.5.2 Kernel Estimators and Cleveland’s LOWESS

Kernel Smoothing

Cleveland’s LOWESS

11.5.3 R Functions lplot, lplot.pred and kerreg

11.5.4 The Running-Interval Smoother

11.5.5 R Functions rplot and runYhat

11.5.6 Smoothers for Estimating Quantiles

11.5.7 R Function qhdsm

11.5.8 Special Methods for Binary Outcomes

11.5.9 R Functions logSM, logSMpred, bkreg and rplot.bin

11.5.10 Smoothing with More than One Predictor

11.5.11 R Functions rplot, runYhat, rplotsm and runpd

11.5.12 LOESS

11.5.13 Other Approaches

11.5.14 R Functions adrun, adrunl, gamplot, gamplotINT

11.6 Checking the Speciﬁcation of a Regression Model

11.6.1 Testing the Hypothesis of a Linear Association

11.6.2 R Function lintest

11.6.3 Testing the Hypothesis of a Generalized Additive Model

11.6.4 R Function adtest

11.6.5 Inferences About the Components of a Generalized Additive Model

11.6.6 R Function adcom

11.6.7 Detecting Heteroscedasticity Based on Residuals

11.6.8 R Function rhom

11.7 Regression Interactions and Moderator Analysis

11.7.1 R Functions kercon, riplot, runsm2g, ols.plot.inter, olshc4.inter, reg.plot.inter and regci.inter

11.7.2 Mediation Analysis

11.7.3 R Functions ZYmediate, regmed2 and regmediate

11.8 Comparing Parametric, Additive and Nonparametric Fits

11.8.1 R Functions adpchk and pmodchk

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

11.9.1 R Functions RobRsq, qcorp1 and qcor

11.9.2 Comparing Two Independent Groups via the LOWESS Version of Explanatory Power

11.9.3 R Functions smcorcom and smstrcom

11.10 Comparing Predictors

11.10.1 Comparing Correlations

11.10.2 R Functions TWOpov, TWOpNOV, corCOMmcp, twoDcorR, and twoDNOV

11.10.3 Methods Based on Prediction Error

The 0.632 Estimator

The Leave-One-Out Cross-Validation Method

11.10.4 R Functions regpre and regpreCV

11.10.5 R Function larsR

11.10.6 Inferences About Which Predictors Are Best

Method IBS

Method BTS

Method SM

11.10.7 R Functions regIVcom, ts2str and sm2strv7

11.11 Marginal Longitudinal Data Analysis: Comments on Comparing Groups

11.11.1 R Functions long2g, longreg, longreg.plot and xyplot

11.12 Exercises

12 ANCOVA

12.1 Methods Based on Speciﬁc Design Points and a Linear Model

12.1.1 Method S1

12.1.2 Method S2

12.1.3 Dealing with Two Covariates

12.1.4 R Functions ancJN, ancJNmp, ancJNmpcp, anclin, reg2plot and reg2g.p2plot

12.2 Methods when There Is Curvature and a Single Covariate

12.2.1 Method Y

12.2.2 Method BB: Bootstrap Bagging

12.2.3 Method UB

12.2.4 Method TAP

12.2.5 Method G

12.2.6 R Functions ancova, ancovaWMW, ancpb, rplot2g, runmean2g, lplot2g, ancdifplot, ancboot, ancbbpb, qhdsm2g, ancovaUB, ancovaUB.pv, ancdet, ancmg1 and ancGLOB

12.3 Dealing with Two Covariates when There Is Curvature

12.3.1 Method MC1

12.3.2 Method MC2

12.3.3 Method MC3

12.3.4 R Functions ancovamp, ancovampG, ancmppb, ancmg, ancov2COV, ancdes and ancdet2C

12.4 Some Global Tests

12.4.1 Method TG

12.4.2 R Functions ancsm and Qancsm

12.5 Methods for Dependent Groups

12.5.1 Methods Based on a Linear Model

12.5.2 R Functions Dancts and Dancols

12.5.3 Dealing with Curvature: Methods DY, DUB and DTAP

12.5.4 R Functions Dancova, Dancovapb, DancovaUB and Dancdet

12.6 Exercises

Index

References