Multivariate Statistical Inference and Applications

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ISBN: 9780471571513 | 0471571512
Cover: Hardcover
Copyright: 12/29/1997

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The most accessible introduction to the theory and practice of multivariate analysis Multivariate Statistical Inference and Applications is a user-friendly introduction to basic multivariate analysis theory and practice for statistics majors as well as nonmajors with little or no background in theoretical statistics. Among the many special features of this extremely accessible first text on multivariate analysis are: * Clear, step-by-step explanations of all key concepts and procedures along with original, easy-to-follow proofs * Numerous problems, examples, and tables of distributions * Many real-world data sets drawn from a wide range of disciplines * Reviews of univariate procedures that give rise to multivariate techniques * An extensive survey of the world literature on multivariate analysis * An in-depth review of matrix theory * A disk including all the data sets and SAS command files for all examples and numerical problems found in the book These same features also make Multivariate Statistical Inference and Applications an excellent professional resource for scientists and clinicians who need to acquaint themselves with multivariate techniques. It can be used as a stand-alone introduction or in concert with its more methods-oriented sibling volume, the critically acclaimed Methods of Multivariate Analysis.

1. Some Properties of Random Vectors and Matrices

(36)

1.1 Introduction

(1)

1.2 Univariate and Bivariate Random Variables

(4)

1.2.1 Univariate Random Variables

(2)

1.2.2 Bivariate Random Variables

(2)

1.3 Mean Vectors and Covariance Matrices for Random Vectors

(5)

1.4 Correlation Matrices

(1)

1.5 Partitioned Mean Vectors and Covariance Matrices

(2)

1.6 Linear Functions of Random Variables

(6)

1.6.1 Sample Means, Variances, and Covariances

(5)

1.6.2 Population Means, Variances, and Covariances

(1)

1.7 Measuring Intercorrelation

(2)

1.8 Mahalanobis Distance

(1)

1.9 Missing Data

(4)

1.10 Robust Estimators of XXX and XXX

(10)

2. The Multivariate Normal Distribution

(23)

2.1 Univariate and Multivariate Normal Density Functions

(6)

2.1.1 Univariate Normal

(1)

2.1.2 Multivariate Normal

(2)

2.1.3 Constant Density Ellipsoids

(2)

2.1.4 Generating Multivariate Normal Data

(1)

2.1.5 Moments

(1)

2.2 Properties of Multivariate Normal Random Vectors

(6)

2.3 Estimation of Parameters in the Multivariate Normal Distribution

(7)

2.3.1 Maximum Likelihood Method

(3)

2.3.2 Properties of y and S

(1)

2.3.3 Wishart Distribution

(3)

2.4 Additional Topics

(4)

3. Hotelling's T(2)-Tests

(61)

3.1 Introduction

(1)

3.2 Test for H(0): XXX = XXX(0) with XXX Known

(1)

3.3 Hotelling's T(2)-test for H(0): XXX = XXX(0) with XXX Unknown

(13)

3.3.1 Univariate t-Test for H(0): XXX = XXX(0) with XXX(2) Unknown

(1)

3.3.2 Likelihood Ratio Method of Test Construction

(3)

3.3.3 One-Sample T(2)-Test

(1)

3.3.4 Formal Definition of T(2) and Relationship to F

(1)

3.3.5 Effect on T(2) of Adding a Variable

(3)

3.3.6 Properties of the T(2)-Test

(1)

3.3.7 Likelihood Ratio Test

(1)

3.3.8 Union-Intersection Test

(2)

3.4 Confidence Intervals and Tests for Linear Functions of XXX

(11)

3.4.1 Confidence Region for XXX

(1)

3.4.2 Confidence Interval for a Single Linear Combination a'XXX

(1)

3.4.3 Simultaneous Confidence Intervals for XXX(j) and a'XXX

(3)

3.4.4 Bonferroni Confidence Intervals for XXX(j) and a'XXX

(2)

3.4.5 Tests for H(0): a'XXX = a'XXX(0) and H(0): XXX(j) = XXX(0j)

(4)

3.4.6 Tests for H(0): CXXX = 0

(2)

3.5 Tests of H(0): XXX(1) = XXX(2) Assuming XXX(1) = XXX(2)

(7)

3.5.1. Review of Univariate Likelihood Ratio Test for H(0): XXX(1) = XXX(2) When XXX(2)(1) = XXX(2)(2)

(2)

3.5.2. Test for H(0): XXX(1) = XXX(2) When XXX(1) = XXX(2)

(1)

3.5.3. Effect on T(2) of Adding a Variable

(4)

3.5.4. Properties of the Two-Sample T(2)-Statistic

(1)

3.5.5. Likelihood Ratio and Union-Intersection Tests

(1)

3.6. Confidence Intervals and Tests for Linear Functions of Two Mean Vectors

(4)

3.6.1. Confidence Region for XXX(1) - XXX(2)

(1)

3.6.2. Simultaneous Confidence Intervals for a'(XXX(1) - XXX(2)) and XXX(1j) - XXX(2j)

(1)

3.6.3. Bonferroni Confidence Intervals for a'(XXX(1) - XXX(2)) and XXX(1j) - XXX(2j)

(1)

3.6.4. Tests for H(0): a'(XXX(1) - XXX(2)) = a'XXX(0) and H(0j): XXX(1j) - XXX(2j) = 0

(1)

3.6.5. Test for H(0): C(XXX(1) - XXX(2)) = 0

(1)

3.7. Robustness of the T(2) -test

(1)

3.7.1. Robustness to XXX(1) is not equal to XXX(2)

(1)

3.7.2. Robustness to Nonnormality

(1)

3.8. Paired Observation Test

(2)

3.9. Testing H(0): XXX(1) = XXX(2) When XXX(1) is not equal to XXX(2)

(5)

3.9.1. Univariate Case

(1)

3.9.2. Multivariate Case

100

(4)

3.10. Power and Sample Size

104

(4)

3.11. Tests on a Subvector

108

(4)

3.11.1. Two-Sample Case

108

(2)

3.11.2. Step-Down Test

110

(1)

3.11.3. Selection of Variables

111

(1)

3.11.4. One-Sample Case

112

(1)

3.12. Nonnormal Approaches to Hypothesis Testing

112

(2)

3.12.1. Elliptically Contoured Distributions

112

(1)

3.12.2. Nonparametric Tests

113

(1)

3.12.3. Robust Versions of T(2)

114

(1)

3.13. Application of T(2) In Multivariate Quality Control

114

(7)

4. Multivariate Analysis of Variance

121

(80)

4.1. One-Way Classification

121

(14)

4.1.1. Model for One-Way Multivariate Analysis of Variance

121

(1)

4.1.2. Wilks' Likelihood Ratio Test

122

(5)

4.1.3. Roy's Union-Intersection Test

127

(3)

4.1.4. The Pillai and Lawley-Hotelling Test Statistics

130

(1)

4.1.5. Summary of the Four Test Statistics

131

(1)

4.1.6. Effect of an Additional Variable on Wilks' XXX

132

(2)

4.1.7. Tests on Individual Variables

134

(1)

4.2. Power and Robustness Comparisons for the Four MANOVA Test Statistics

135

(3)

4.3. Tests for Equality of Covariance Matrices

138

(2)

4.4. Power and Sample Size for the Four MANOVA Tests

140

(2)

4.5. Contrasts Among Mean Vectors

142

(6)

4.5.1. Univariate Contrasts

142

(3)

4.5.2. Multivariate Contrasts

145

(3)

4.6. Two-Way Multivariate Analysis of Variance

148

(3)

4.7. Higher Order Models

151

(1)

4.8. Unbalanced Data

152

(22)

4.8.1. Introduction

152

(1)

4.8.2. Univariate One-Way Model

153

(2)

4.8.3. Multivariate One-Way Model

155

(5)

4.8.4. Univariate Two-Way Model

160

(8)

4.8.5. Multivariate Two-Way Model

168

(6)

4.9. Tests on a Subvector

174

(4)

4.9.1. Testing a Single Subvector

174

(3)

4.9.2. Step-Down Test

177

(1)

4.9.3. Stepwise Selection of Variables

177

(1)

4.10. Multivariate Analysis of Covariance

178

(16)

4.10.1. Introduction

178

(1)

4.10.2. Univariate Analysis of Covariance: One-Way Model with One Covariate

179

(4)

4.10.3. Univariate Anlaysis of Covariance: Two-Way Model with One Covariate

183

(3)

4.10.4. Additional Topics in Univariate Analysis of Covariance

186

(1)

4.10.5. Multivariate Analysis of Covariance

187

(7)

4.11. Alternative Approaches to Testing H(0): XXX(1) = XXX(2) = ... = XXX(k)

194

(7)

5. Discriminant Functions for Descriptive Group Separation

201

(29)

5.1. Introduction

201

(1)

5.2. Two Groups

201

(1)

5.3. Several Groups

202

(4)

5.3.1. Discriminant Functions

202

(3)

5.3.2. Assumptions

205

(1)

5.4. Standardized Coefficients

206

(1)

5.5. Tests of Hypotheses

207

(3)

5.5.1. Two Groups

207

(2)

5.5.2. Several Groups

209

(1)

5.6. Discriminant Analysis for Higher Order Designs

210

(1)

5.7. Interpretation of Discriminant Functions

210

(6)

5.7.1. Standardized Coefficients and Partial F-Values

211

(1)

5.7.2. Correlations between Variables and Discriminant Functions

211

(4)

5.7.3. Other Approaches

215

(1)

5.8. Confidence Intervals

216

(1)

5.9. Subset Selection

216

(3)

5.9.1. Discriminant Function Approach to Selection

217

(1)

5.9.2. Stepwise Selection

217

(1)

5.9.3. All Possible Subsets

218

(1)

5.9.4. Selection in Higher Order Designs

218

(1)

5.10. Bias in Subset Selection

219

(2)

5.11. Other Estimators of Discriminant Functions

221

(9)

5.11.1. Ridge Discriminant Analysis and Related Techniques

222

(1)

5.11.2. Robust Discriminant Analysis

223

(7)

6. Classification of Observations into Groups

230

(36)

6.1. Introduction

230

(1)

6.2. Two Groups

230

(6)

6.2.1. Equal Population Covariance Matrices

230

(2)

6.2.2. Unequal Population Covariance Matrices

232

(1)

6.2.3. Unequal Costs of Misclassification

233

(1)

6.2.4. Posterior Probability Approach

234

(1)

6.2.5. Robustness to Departures from the Assumptions

234

(1)

6.2.6. Robust Procedures

235

(1)

6.3. Several Groups

236

(4)

6.3.1. Equal Population Covariance Matrices

236

(1)

6.3.2. Unequal Population Covariance Matrices

237

(2)

6.3.3. Use of Linear Discriminant Functions for Classification

239

(1)

6.4. Estimation of Error Rates

240

(4)

6.5. Correcting for Bias in the Apparent Error Rate

244

(3)

6.5.1. Partitioning the Sample

244

(1)

6.5.2. Holdout Method

244

(1)

6.5.3. Bootstrap Estimator

245

(1)

6.5.4. Comparison of Error Estimators

245

(2)

6.6. Subset Selection

247

(4)

6.6.1. Selection Based on Separation of Groups

247

(2)

6.6.2. Selection Based on Allocation

249

(1)

6.6.3. Selection in the Heteroscedastic Case

250

(1)

6.7. Bias in Stepwise Classification Analysis

251

(3)

6.8. Logistic and Probit Classification

254

(7)

6.8.1. The Logistic Model for Two Groups with XXX(1) = XXX(2)

254

(2)

6.8.2. Comparison of Logistic Classification with Linear Classification Functions

256

(2)

6.8.3. Quadratic Logistic Functions When XXX(1) is not equal to XXX(2)

258

(1)

6.8.4. Logistic Classification for Several Groups

258

(1)

6.8.5. Additional Topics in Logistic Classification

259

(1)

6.8.6. Probit Classification

259

(2)

6.9. Additional Topics in Classification

261

(5)

7. Multivariate Regression

266

(46)

7.1. Introduction

266

(1)

7.2. Multiple Regression: Fixed x's

267

(9)

7.2.1. Least Squares Estimators and Properties

267

(1)

7.2.2. An Estimator for XXX(2)

268

(1)

7.2.3. The Model in Centered Form

269

(2)

7.2.4. Hypothesis Tests and Confidence Intervals

271

(4)

7.2.5. R(2) in Fixed-x Regression

275

(1)

7.2.6. Model Validation

275

(1)

7.3. Multiple Regression: Random x's

276

(3)

7.3.1. Model for Random x's

276

(1)

7.3.2. Estimation of XXX(0), XXX(1) and XXX(2)

277

(1)

7.3.3. R(2) in Random-x Regression

278

(1)

7.3.4. Tests and Confidence Intervals

279

(1)

7.4. Estimation in the Multivariate Multiple Regression Model: Fixed x's

279

(10)

7.4.1. The Multivariate Model

279

(2)

7.4.2. Least Squares Estimator for B

281

(2)

7.4.3. Properties of B

283

(2)

7.4.4. An Estimator for XXX

285

(1)

7.4.5. Normal Model for the Y(i)'s

286

(2)

7.4.6. The Multivariate Model in Centered Form

288

(1)

7.4.7. Measures of Multivariate Association

289

(1)

7.5. Hypothesis Tests in the Multivariate Multiple Regression Model: Fixed x's

289

(11)

7.5.1. Test for Significance of Regression

289

(4)

7.5.2. Test on a Subset of the Rows of B

293

(2)

7.5.3. General Linear Hypotheses CB = O and CBM = O

295

(2)

7.5.4. Tests and Confidence Intervals for a Single XXX(jk) and a Bilinear Function a'Bb

297

(1)

7.5.5. Simultaneous Tests and Confidence Intervals for the XXX(jk)'s and Bilinear Functions a'Bb

298

(1)

7.5.6. Tests in the Presence of Missing Data

299

(1)

7.6. Multivariate Model Validation: Fixed x's

300

(3)

7.6.1. Lack-of-Fit Tests

300

(1)

7.6.2. Residuals

300

(1)

7.6.3. Influence and Outliers

301

(2)

7.6.4. Measurement Errors

303

(1)

7.7. Multivariate Regression: Random x's

303

(2)

7.7.1. Multivariate Normal Model for Random x's

303

(1)

7.7.2. Estimation of XXX(0), B(1), and XXX

304

(1)

7.7.3. Tests and Confidence Intervals in the Multivariate Random-x Case

305

(1)

7.8. Additional Topics

305

(7)

7.8.1. Correlated Response Methods

305

(1)

7.8.2. Categorical Data

306

(1)

7.8.3. Subset Selection

307

(1)

7.8.4. Other Topics

307

(5)

8. Canonical Correlation

312

(25)

8.1. Introduction

312

(1)

8.2. Canonical Correlations and Canonical Variates

312

(6)

8.3. Properties of Canonical Correlations and Variates

318

(2)

8.3.1. Properties of Canonical Correlations

318

(2)

8.3.2. Properties of Canonical Variates

320

(1)

8.4. Tests of Significance for Canonical Correlations

320

(6)

8.4.1. Tests of Independence of y and x

320

(4)

8.4.2. Test of Dimension of Relationship between the y's and the x's

324

(2)

8.5. Validation

326

(2)

8.6. Interpretation of Canonical Variates

328

(3)

8.6.1. Standardized Coefficients

328

(1)

8.6.2. Rotation of Canonical Variate Coefficients

328

(1)

8.6.3. Correlations between Variables and Canonical Variates

329

(2)

8.7. Redundancy Analysis

331

(2)

8.8. Additional Topics

333

(4)

9. Principal Component Analysis

337

(40)

9.1. Introduction

337

(1)

9.2. Definition and Properties of Principal Components

338

(5)

9.2.1. Maximum Variance Property

338

(2)

9.2.2. Principal Components as Projections

340

(1)

9.2.3. Properties of Principal Components

341

(2)

9.3. Principal Components as a Rotation of Axes

343

(1)

9.4. Principal Components from the Correlation Matrix

344

(3)

9.5. Methods for Discarding Components

347

(5)

9.5.1. Percent of Variance

347

(1)

9.5.2. Average Eigenvalue

348

(1)

9.5.3. Scree Graph

348

(1)

9.5.4. Significance Tests

349

(3)

9.5.5. Other Methods

352

(1)

9.6. Information in the Last Few Principal Components

352

(1)

9.7. Interpretation of Principal Components

353

(10)

9.7.1. Special Patterns in S or R

353

(4)

9.7.2. Testing H(0): XXX = XXX(2) [(1 - p)I + pJ] and P(p) = (1 - p)I + pJ

357

(2)

9.7.3. Additional Rotation

359

(2)

9.7.4. Correlations between Variables and Principal Components

361

(2)

9.8. Relationship Between Principal Components and Regression

363

(8)

9.8.1. Principal Component Regression

364

(6)

9.8.2. Latent Root Regression

370

(1)

9.9. Principal Component Analysis with Grouped Data

371

(1)

9.10. Additional Topics

372

(5)

10. Factor Analysis

377

(22)

10.1. Introduction

377

(1)

10.2. Basic Factor Model

377

(4)

10.2.1. Model and Assumptions

377

(2)

10.2.2. Scale Invariance of the Model

379

(1)

10.2.3. Rotation of Factor Loadings in the Model

380

(1)

10.3. Estimation of Loadings and Communalities

381

(5)

10.3.1. Principal Component Method

381

(2)

10.3.2. Principal Factor Method

383

(1)

10.3.3. Iterated Principal Factor Method

384

(1)

10.3.4. Maximum Likelihood Method

384

(1)

10.3.5. Other Methods

385

(1)

10.3.6. Comparison of Methods

385

(1)

10.4. Determining the Number of Factors, m

386

(1)

10.5. Rotation of Factor Loadings

387

(4)

10.5.1. Introduction

387

(1)

10.5.2. Orthogonal Rotation

388

(1)

10.5.3. Oblique Rotations

389

(1)

10.5.4. Interpretation of the Factors

390

(1)

10.6. Factor Scores

391

(1)

10.7. Applicability of the Factor Analysis Model

392

(1)

10.8. Factor Analysis and Grouped Data

393

(1)

10.9. Additional Topics

394

(5)

Appendix A. Review of Matrix Algebra

399

(18)

A.1. Introduction

399

(2)

A.1.1. Basic Definitions

399

(1)

A.1.2. Matrices with Special Patterns

400

(1)

A.2. Properties of Matrix Addition and Multiplication

401

(3)

A.3. Partitioned Matrices

404

(2)

A.4. Rank of Matrices

406

(1)

A.5. Inverse Matrices

407

(1)

A.6. Positive Definite and Positive Semidefinite Matrices

408

(1)

A.7. Determinants

409

(1)

A.8. Trace of a Matrix

410

(1)

A.9. Orthogonal Vectors and Matrices

410

(1)

A.10. Eigenvalues and Eigenvectors

411

(1)

A.11. Eigenstructure of Symmetric and Positive Definite Matrices

412

(2)

A.12. Idempotent Matrices

414

(1)

A.13. Differentiation

414

(3)

Appendix B. Tables

417

(32)

Appendix C. Answers and Hints to Selected Problems

449

(56)

Appendix D. About the Diskette

505

(2)

Bibliography

507

(42)

Index

549

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Multivariate Statistical Inference and Applications

Summary

Author Biography

Table of Contents

Supplemental Materials