Introduction to Modern Nonparametric Statistics

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ISBN: 9780534387754 | 0534387756
Cover: Hardcover
Copyright: 5/5/2003

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Guided by problems that frequently arise in actual practice, James Higgins? book presents a wide array of nonparametric methods of data analysis that researchers will find useful. It discusses a variety of nonparametric methods and, wherever possible, stresses the connection between methods. For instance, rank tests are introduced as special cases of permutation tests applied to ranks. The author provides coverage of topics not often found in nonparametric textbooks, including procedures for multivariate data, multiple regression, multi-factor analysis of variance, survival data, and curve smoothing. This truly modern approach teaches non-majors how to analyze and interpret data with nonparametric procedures using today?s computing technology.

Preliminaries

(10)

Cumulative Distributions and Probability Density Functions

(1)

Common Continuous Probability Distributions

(3)

The Binomial Distribution

(1)

Confidence Intervals and Tests of Hypotheses

(2)

Parametric versus Nonparametric Methods

(1)

Classes of Nonparametric Methods

(4)

One-Sample Methods

(12)

A Nonparametric Test of Hypothesis and Confidence Interval for the Median

(3)

Binomial Test

(2)

Confidence Interval

(1)

Computer Analysis

(1)

Estimating the Population cdf and Percentiles

(3)

Confidence Interval for the Population cdf

(2)

Inferences for Percentiles

(1)

Computer Analysis

(1)

A Comparison of Statistical Tests

(6)

Type I Errors

(1)

Power

(2)

Derivations

(1)

Exercises

(2)

Two-Sample Methods

(56)

A Two-Sample Permutation Test

(7)

The Permutation Test

(3)

Summary of Steps Used in a Two-Sample Permutation Test

(1)

Hypotheses for the Two-Sample Permutation Test

(1)

Computer Analysis

(1)

Permutation Tests Based on the Median and Trimmed Means

(1)

A Permutation Test Based on Medians

(1)

Trimmed Means

(1)

Random Sampling the Permutations

(4)

An Approximate p-Value Based on Random Sampling the Permutations

(1)

Computer Analysis Using Resampling Stats

(2)

Computer Analysis Using StatXact

(1)

Wilcoxon Rank-Sum Test

(6)

Steps in Conducting the Wilcoxon Rank-Sum Test

(1)

Comments on the Use of the Wilcoxon Rank-Sum Test

(1)

A Statistical Table for the Wilcoxon Rank-Sum Test

(1)

Computer Analysis

(2)

Wilcoxon Rank-Sum Test Adjusted for Ties

(2)

Steps in Conducting the Wilcoxon Rank-Sum Test Adjusted for Ties

(1)

Computer Analysis

(1)

Mann--Whitney Test and a Confidence Interval

(6)

The Mann--Whitney Statistic

(1)

Equivalence of Mann--Whitney and Wilcoxon Rank-Sum Statistics

(1)

A Confidence Interval for a Shift Parameter and the Hodges--Lehmann Estimate

(3)

Computer Analysis

(1)

Scoring Systems

(2)

Three Common Scoring Systems

(2)

Computer Analysis

(1)

Tests for Equality of Scale Parameters and an Omnibus Test

(7)

Siegel--Tukey and Ansari--Bradley Test

(1)

Tests on Deviances

(4)

Kolmogorov--Smirnov Test

(1)

Computer Analysis

(1)

Selecting Among Two-Sample Tests

(7)

The t-Test

(1)

The Wilcoxon Rank-Sum Test versus the t-Test

(1)

Relative Efficiency

(1)

Power of Permutation Tests

(2)

Large-Sample Approximations

(14)

Sampling Formulas

(1)

Application to the Wilcoxon Rank-Sum Test

(2)

Wilcoxon Rank-Sum Test with Ties

(2)

Explicit Formulas for E(Wties) and var(Wties)

(1)

Large-Sample Confidence Interval Based on Mann--Whitney Test

(1)

A Normal Approximation for the Permutation Distribution

(1)

Exercises

(6)

K-Sample Methods

(30)

K-Sample Permutation Tests

(7)

The F Statistic

(1)

Steps in Carrying Out the Permutation F-Test

(2)

Alternative Forms of the Permutation F Statistic

(1)

Computer Analysis

(2)

The Kruskal--Wallis Test

(6)

The Kruskal--Wallis Statistic

(2)

Adjustment for Ties

(2)

An Intuitive Derivation of the Chi-Square Approximation for KW

(1)

Tests on General Scores

(1)

Computer Analysis

(1)

Multiple Comparisons

(7)

Three Rank-Based Procedures for Controlling Experiment-Wise Error Rate Assuming No Ties in the Data

(4)

Multiple Comparisons for General Scores (Including Ties)

(1)

Multiple Comparison Permutation Tests

(2)

Variance of a Difference of Means When Sampling from a Finite Population

(1)

Computer Analysis

(1)

Ordered Alternatives

(10)

Jonckheere--Terpstra Test

101

(1)

Large-Sample Approximation

101

(2)

Computer Analysis

103

(2)

Exercises

105

(4)

Paired Comparisons and Blocked Designs

109

(36)

Paired-Comparison Permutation Test

109

(8)

Steps for a Paired-Comparison Permutation Test

110

(3)

Randomly Selected Permutations

113

(1)

Large-Sample Approximations

113

(1)

A Test for the Median of a Symmetric Population

114

(1)

Computer Analysis

114

(3)

Signed-Rank Test

117

(8)

The Wilcoxon Signed-Rank Test without Ties in the Data

118

(1)

Large-Sample Approximation

118

(2)

Adjustment for Ties

120

(4)

Computer Analysis

124

(1)

Other Paired-Comparison Tests

125

(2)

Sign Test

125

(1)

General Scoring Systems

126

(1)

Selecting Among Paired-Comparison Tests

127

(1)

A Permutation Test for a Randomized Complete Block Design

127

(5)

F Statistic for Randomized Complete Block Designs

128

(1)

Permutation F-Test for Randomized Complete Block Designs

129

(1)

Multiple Comparisons

130

(1)

Computer Analysis

131

(1)

Friedman's Test for a Randomized Complete Block Design

132

(6)

Friedman's Test without Ties

133

(1)

Adjustment for Ties

134

(2)

Cochran's Q and Kendall's W

136

(1)

Chi-Square Approximation for FM or FMties

136

(1)

Computer Analysis

137

(1)

Ordered Alternatives for a Randomized Complete Block Design

138

(7)

Page's Test

139

(1)

Computer Analysis

140

(1)

Exercises

141

(4)

Tests for Trends and Association

145

(50)

A Permutation Test for Correlation and Slope

145

(8)

The Correlation Coefficient

145

(1)

Slope of Least Squares Line

146

(1)

The Permutation Test

147

(4)

Large-Sample Approximation for the Permutation Distribution of r

151

(1)

Computer Analysis

152

(1)

Spearman Rank Correlation

153

(5)

Statistical Test for Spearman Rank Correlation

154

(1)

Large-Sample Approximation

154

(4)

Computer Analysis

158

(1)

Kendall's Tau

158

(5)

Estimating Kendall's Tau with No Ties in the Data

159

(1)

Large-Sample Approximation

160

(1)

Adjustment for Ties in the Data

161

(2)

Computer Analysis

163

(1)

Permutation Tests for Contingency Tables

163

(9)

Hypotheses to Be Tested and the Chi-Square Statistic

163

(2)

Permutation Chi-Square Test

165

(2)

Multiple Comparisons in Contingency Tables

167

(2)

Computer Analysis

169

(3)

Fisher's Exact Test for a 2 x 2 Contingency Table

172

(4)

Probability Distribution Under the Null Hypothesis

173

(2)

Computer Analysis

175

(1)

Contingency Tables with Ordered Categories

176

(3)

Singly Ordered Tables

176

(2)

Doubly Ordered Tables

178

(1)

Computer Analysis

179

(1)

Mantel--Haenszel Test

179

(7)

Hypotheses to Be Tested

179

(2)

Testing Hypotheses for Stratified Contingency Tables

181

(2)

Estimation of the Odds Ratio

183

(1)

Computer Analysis

184

(2)

McNemar's Test

186

(9)

Notation and Hypotheses

187

(1)

Test Statistic and Permutation Distribution

187

(1)

Computer Analysis

188

(1)

Exercises

189

(6)

Multivariate Tests

195

(30)

Two-Sample Multivariate Permutation Tests

195

(7)

Notation and Assumptions

196

(1)

The Permutation Version of Hotelling's T2

197

(1)

Other Multivariate Statistics

198

(2)

Computer Analysis

200

(2)

Two-Sample Multivariate Rank Tests

202

(4)

A Sum of Wilcoxon Statistics

203

(2)

Computer Analysis

205

(1)

Multivariate Paired Comparisons

206

(4)

A Permutation Test Based on T2

207

(1)

Random Sampling the Permutations

207

(2)

Other Multivariate Statistics

209

(1)

Computer Analysis

210

(1)

Multivariate Rank Tests for Paired Comparisons

210

(6)

Testing Medians of a Symmetric Distribution

212

(2)

A Sum of Signed-Rank Statistics

214

(2)

Computer Analysis

216

(1)

Multiresponse Categorical Data

216

(9)

Hypotheses for Multiresponse Data

217

(1)

A Statistical Test

218

(2)

Computer Analysis

220

(2)

Exercises

222

(3)

Analysis of Censored Data

225

(24)

Estimating the Survival Function

225

(4)

Censored Data

225

(1)

Kaplan-Meier Estimate

226

(2)

Computer Analysis

228

(1)

Permutation Tests for Two-Sample Censored Data

229

(3)

Censoring Mechanism

229

(2)

Examples of Tests Based on Medians and Ranks

231

(1)

Computer Analysis

232

(1)

Gehan's Generalization of the Mann--Whitney Test

232

(4)

An Alternative Form of Uc Based on Gehan Scores

233

(2)

Large-Sample Approximations

235

(1)

Computer Analysis

236

(1)

Scoring Systems for Censored Data

236

(6)

Prentice--Wilcoxon Scores

236

(2)

Log-Rank Scores

238

(1)

The Likelihood Approach

239

(2)

Computer Analysis

241

(1)

Tests Using Scoring Systems for Censored Data

242

(7)

Two-Sample and K-Sample Tests

242

(2)

Paired-Comparison and Blocked Designs

244

(1)

Computer Analysis

245

(1)

Exercises

245

(4)

Nonparametric Bootstrap Methods

249

(52)

The Basic Bootstrap Method

249

(5)

Mean Squared Error and Margin of Error

250

(1)

The Bootstrap Estimate of MSE

250

(3)

Computer Analysis

253

(1)

Bootstrap Intervals for Location-Scale Models

254

(7)

Interval Estimate of the Mean

255

(2)

Confidence Intervals for the Variance and Standard Deviation

257

(1)

Coverage Percentages

258

(1)

Derivation of Pivotal Quantities

259

(1)

Computer Analysis

260

(1)

BCA and Other Bootstrap Intervals

261

(5)

Percentile and Residual Methods

261

(1)

BCA Method

262

(3)

Computer Analysis

265

(1)

Correlation and Regression

266

(10)

Bivariate Bootstrap Sampling

266

(2)

Fixed-X Bootstrap Sampling

268

(2)

Bootstrap Inferences for the Slope of a Regression Line

270

(4)

Computer Analysis

274

(2)

Two-Sample Inference

276

(7)

t-Pivot Method Assuming Equality of Error Distributions

276

(2)

Unequal Distributions of the Errors

278

(3)

Computer Analysis

281

(2)

Bootstrap Sampling from Several Populations

283

(5)

Equal Error Distributions

283

(2)

Unequal Error Distributions

285

(1)

Regression with Unequal Error Variances

285

(3)

Computer Analysis

288

(1)

Bootstrap Sampling for Multiple Regression

288

(13)

The Bootstrap Procedure for Testing H0(M) and H0(j)

289

(1)

A Confidence Interval for βj

290

(2)

Theoretical Development

292

(3)

Other Methods for Regression Analysis

295

(1)

Computer Analysis

296

(2)

Exercises

298

(3)

Multifactor Experiments

301

(22)

Analysis of Variance Models

301

(8)

The Bootstrap Approach to the Analysis of the Fixed-Effects Model

302

(3)

A Permutation Test for the Overall Effect of a Factor

305

(2)

Computer Analysis

307

(2)

Aligned-Rank Transform

309

(5)

Aligning the Data in a Completely Random Design

310

(1)

Aligning the Data in a Split-Plot Design

311

(3)

Properties of the Aligned-Rank Procedure

314

(1)

Computer Analysis

314

(1)

Testing for Lattice-Ordered Alternatives

314

(9)

A Test Statistic for Lattice-Ordered Alternatives

315

(1)

Lattice Correlation

316

(2)

References for Lattice-Ordered Alternatives

318

(1)

Computer Analysis

319

(1)

Exercises

319

(4)

Smoothing Methods and Robust Model Fitting

323

(14)

Estimating the Probability Density Function

323

(3)

Kernel Method

324

(2)

Computer Analysis

326

(1)

Nonparametric Curve Smoothing

326

(3)

Loess Method

326

(1)

Kernel Method

327

(2)

Computer Analysis

329

(1)

Robust and Rank-Based Regression

329

(8)

M-Estimation

329

(2)

Rank-Based Regression

331

(3)

Computer Analysis

334

(1)

Exercises

334

(3)

Appendix

337

(14)

References

351

(5)

Index

356

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Introduction to Modern Nonparametric Statistics

Summary

Table of Contents

Supplemental Materials