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PART ONE SIMPLE LINEAR REGRESSION1

Chapter 1 Linear Regression with One Predictor Variable2

1.1 Relations between Variables2

Functional Relation between Two Variables2

Statistical Relation between Two Variables3

1.2 Regression Models and Their Uses5

Historical Origins5

Basic Concepts5

Construction of Regression Models7

Uses of Regression Analysis8

Regression and Causality8

Use of Computers9

1.3 Simple Linear Regression Model with Distribution of Error Terms Unspecified9

Formal Statement of Model9

Important Features of Model9

Meaning of Regression Parameters11

Alternative Versions of Regression Model12

1.4 Data for Regression Analysis12

Observational Data12

Experimental Data13

Completely Randomized Design13

1.5 Overview of Steps in Regression Analysis13

1.6 Estimation of Regression Function15

Method of Least Squares15

Point Estimation of Mean Response21

Residuals22

Properties of Fitted Regression Line23

1.7 Estimation of Error Terms Variance σ224

Point Estimator of σ224

1.8 Normal Error Regression Model26

Model26

Estimation of Parameters by Method of Maximum Likelihood27

Cited References33

Problems33

Exercises37

Projects38

Chapter 2 Inferences in Regression and Correlation Analysis40

2.1 Inferences Concerning β140

Sampling Distribution of b141

Sampling Distribution of (b1 - β1)/s {b1}44

Confidence Interval for β145

Tests Concerning β147

2.2 Inferences Concerning β048

Sampling Distribution of b048

Sampling Distribution of (bo-βo) /s{bo}49

Confidence Interval for β049

2.3 Some Considerations on Making Inferences Concerning β0 and β150

Effects of Departures from Normality50

Interpretation of Confidence Coefficient and Risks of Errors50

Spacing of the X Levels50

Power of Tests50

2.4 Interval Estimation of E{Yh}52

Sampling Distribution of Yh52

Sampling Distribution of(？h-E{Yh})/s{Yh}54

Confidence Interval for E{Yh}54

2.5 Prediction of New Observation55

Prediction Interval for Yh(new) when Parameters Known56

Prediction Interval for Yh(new) when Parameters Unknown57

Prediction of Mean of m New Observations for Given Xh60

2.6 Confidence Band for Regression Line61

2.7 Analysis of Variance Approach to Regression Analysis63

Partitioning of Total Sum of Squares63

Breakdown of Degrees of Freedom66

Mean Squares66

Analysis of Variance Table67

Expected Mean Squares68

F Test of β1=0 versus β1≠069

2.8 General Linear Test Approach72

Full Model72

Reduced Model72

Test Statistic73

Summary73

2.9 Descriptive Measures of Linear Association between X and Y74

Coefficient of Determination74

Limitations of R275

Coefficient of Correlation76

2.10 Considerations in Applying Regression Analysis77

2.11 Normal Correlation Models78

Distinction between Regression and Correlation Model78

Bivariate Normal Distribution78

Conditional Inferences80

Inferences on Correlation Coefficients83

Spearman Rank Correlation Coefficient87

Cited References89

Problems89

Exercises97

Projects98

Chapter 3 Diagnostics and Remedial Measures100

3.1 Diagnostics for Predictor Variable100

3.2 Residuals102

Properties of Residuals102

Semistudentized Residuals103

Departures from Model to Be Studied by Residuals103

3.3 Diagnostics for Residuals103

Nonlinearity of Regression Function104

Nonconstancy of Error Variance107

Presence of Outliers108

Nonindependence of Error Terms108

Nonnormality of Error Terms110

Omission of Important Predictor Variables112

Some Final Comments114

3.4 Overview of Tests Involving Residuals114

Tests for Randomness114

Tests for Constancy of Variance115

Tests for Outliers115

Tests for Normality115

3.5 Correlation Test for Normality115

3.6 Tests for Constancy of Error Variance116

Brown-Forsythe Test116

Breusch-Pagan Test118

3.7 F Test for Lack of Fit119

Assumptions119

Notation121

Full Model121

Reduced Model123

Test Statistic123

ANOVA Table124

3.8 Overview of Remedial Measures127

Nonlinearity of Regression Function128

Nonconstancy of Error Variance128

Nonindependence of Error Terms128

Nonnormality of Error Terms128

Omission of Important Predictor Variables129

Outlying Observations129

3.9 Transformations129

Transformations for Nonlinear Relation Only129

Transformations for Nonnormality and Unequal Error Variances132

Box-Cox Transformations134

3.10 Exploration of Shape of Regression Function137

Lowess Method138

Use of Smoothed Curves to Confirm Fitted Regression Function139

3.11 Case Example—Plutonium Measurement141

Cited References146

Problems146

Exercises151

Projects152

Case Studies153

Chapter 4 Simultaneous Inferences and Other Topics in Regression Analysis154

4.1 Joint Estimation ofβ0 andβ1154

Need for Joint Estimation154

Bonferroni Joint Confidence Intervals155

4.2 Simultaneous Estimation of Mean Responses157

Working-Hotelling Procedure158

Bonferroni Procedure159

4.3 Simultaneous Prediction Intervals for New Observations160

4.4 Regression through Origin161

Model161

Inferences161

Important Cautions for Using Regression through Origin164

4.5 Effects of Measurement Errors165

Measurement Errors in Y165

Measurement Errors in X165

Berkson Model167

4.6 Inverse Predictions168

4.7 Choice of X Levels170

Cited References172

Problems172

Exercises175

Projects175

Chapter 5 Matrix Approach to Simple Linear Regression Analysis176

5.1 Matrices176

Definition of Matrix176

Square Matrix178

Vector178

Transpose178

Equality of Matrices179

5.2 Matrix Addition and Subtraction180

5.3 Matrix Multiplication182

Multiplication of a Matrix by a Scalar182

Multiplication of a Matrix by a Matrix182

5.4 Special Types of Matrices185

Symmetric Matrix185

Diagonal Matrix185

Vector and Matrix with All Elements Unity187

Zero Vector187

5.5 Linear Dependence and Rank of Matrix188

Linear Dependence188

Rank of Matrix188

5.6 Inverse of a Matrix189

Finding the Inverse190

Uses of Inverse Matrix192

5.7 Some Basic Results for Matrices193

5.8 Random Vectors and Matrices193

Expectation of Random Vector or Matrix193

Variance-Covariance Matrix of Random Vector194

Some Basic Results196

Multivariate Normal Distribution196

5.9 Simple Linear Regression Model in Matrix Terms197

5.10 Least Squares Estimation of Regression Parameters199

Normal Equations199

Estimated Regression Coefficients200

5.11 Fitted Values and Residuals202

Fitted Values202

Residuals203

5.12 Analysis of Variance Results204

Sums of Squares204

Sums of Squares as Quadratic Forms205

5.13 Inferences in Regression Analysis206

Regression Coefficients207

Mean Response208

Prediction of New Observation209

Cited Reference209

Problems209

Exercises212

PART TWO MULTIPLE LINEAR REGRESSION213

Chapter 6 Multiple Regression Ⅰ214

6.1 Multiple Regression Models214

Need for Several Predictor Variables214

First-Order Model with Two Predictor Variables215

First-Order Model with More than Two Predictor Variables217

General Linear Regression Model217

6.2 General Linear Regression Model in Matrix Terms222

6.3 Estimation of Regression Coefficients223

6.4 Fitted Values and Residuals224

6.5 Analysis of Variance Results225

Sums of Squares and Mean Squares225

F Test for Regression Relation226

Coefficient of Multiple Determination226

Coefficient of Multiple Correlation227

6.6 Inferences about Regression Parameters227

Interval Estimation of βk228

Tests for βk228

Joint Inferences228

6.7 Estimation of Mean Response and Prediction of New Observation229

Interval Estimation of E{Yh}229

Confidence Region for Regression Surface229

Simultaneous Confidence Intervals for Several Mean Responses230

Prediction of New Observation Yh(new)230

Prediction of Mean of m New Observations at Xh230

Predictions of g New Observations231

Caution about Hidden Extrapolations231

6.8 Diagnostics and Remedial Measures232

Scatter Plot Matrix232

Three-Dimensional Scatter Plots233

Residual Plots233

Correlation Test for Normality234

Brown-Forsythe Test for Constancy of Error Variance234

Breusch-Pagan Test for Constancy of Error Variance234

F Test for Lack of Fit235

Remedial Measures236

6.9 An Example—Multiple Regression with Two Predictor Variables236

Setting236

Basic Calculations237

Estimated Regression Function240

Fitted Values and Residuals241

Analysis of Appropriateness of Model241

Analysis of Variance243

Estimation of Regression Parameters245

Estimation of Mean Response245

Prediction Limits for New Observations247

Cited Reference248

Problems248

Exercises253

Projects254

Chapter 7 Multiple Regression Ⅱ256

7.1 Extra Sums of Squares256

Basic Ideas256

Definitions259

Decomposition of SSR into Extra Sums of Squares260

ANOVA Table Containing Decomposition of SSR261

7.2 Uses of Extra Sums of Squares in Tests for Regression Coefficients263

Test whether a Single βk=0263

Test whether Several βk=0264

7.3 Summary of Tests Concerning Regression Coefficients266

Test whether All βk=0266

Test whether a Single βk=0267

Test whether Some βk=0267

Other Tests268

7.4 Coefficients of Partial Determination268

Two Predictor Variables269

General Case269

Coefficients of Partial Correlation270

7.5 Standardized Multiple Regression Model271

Roundoff Errors in Normal Equations Calculations271

Lack of Comparability in Regression Coefficients272

Correlation Transformation272

Standardized Regression Model273

X'X Matrix for Transformed Variables274

Estimated Standardized Regression Coefficients275

7.6 Multicollinearity and Its Effects278

Uncorrelated Predictor Variables279

Nature of Problem when Predictor Variables Are Perfectly Correlated281

Effects of Multicollinearity283

Need for More Powerful Diagnostics for Multicollinearity289

Cited Reference289

Problems289

Exercise292

Projects293

Chapter 8 Regression Models for Quantitative and Qualitative Predictors294

8.1 Polynomial Regression Models294

Uses of Polynomial Models294

One Predictor Variable—Second Order295

One Predictor Variable—Third Order296

One Predictor Variable—Higher Orders296

Two Predictor Variables—Second Order297

Three Predictor Variables—Second Order298

Implementation of Polynomial Regression Models298

Case Example300

Some Further Comments on Polynomial Regression305

8.2 Interaction Regression Models306

Interaction Effects306

Interpretation of Interaction Regression Models with Linear Effects306

Interpretation of Interaction Regression Models with Curvilinear Effects309

Implementation of Interaction Regression Models311

8.3 Qualitative Predictors313

Qualitative Predictor with Two Classes314

Interpretation of Regression Coefficients315

Qualitative Predictor with More than Two Classes318

Time Series Applications319

8.4 Some Considerations in Using Indicator Variables321

Indicator Variables versus Allocated Codes321

Indicator Variables versus Quantitative Variables322

Other Codings for Indicator Variables323

8.5 Modeling Interactions between Quantitative and Qualitative Predictors324

Meaning of Regression Coefficients324

8.6 More Complex Models327

More than One Qualitative Predictor Variable328

Qualitative Predictor Variables Only329

8.7 Comparison of Two or More Regression Functions329

Soap Production Lines Example330

Instrument Calibration Study Example334

Cited Reference335

Problems335

Exercises340

Projects341

Case Study342

Chapter 9 Building the Regression Model Ⅰ:Model Selection and Validation343

9.1 Overview of Model-Building Process343

Data Collection343

Data Preparation346

Preliminary Model Investigation346

Reduction of Explanatory Variables347

Model Refinement and Selection349

Model Validation350

9.2 Surgical Unit Example350

9.3 Criteria for Model Selection353

R2 p or SSEp Criterion354

R2 a,p or MSEp Criterion355

Mallows' Cp Criterion357

A1Cp and SBCp Criteria359

PRESSp Criterion360

9.4 Automatic Search Procedures for Model Selection361

"Best" Subsets Algorithm361

Stepwise Regression Methods364

Forward Stepwise Regression364

Other Stepwise Procedures367

9.5 Some Final Comments on Automatic Model Selection Procedures368

9.6 Model Validation369

Collection of New Data to Check Model370

Comparison with Theory, Empirical Evidence,or Simulation Results371

Data Splitting372

Cited References375

Problems376

Exercise380

Projects381

Case Studies382

Chapter 10 Building the Regression Model Ⅱ:Diagnostics384

10.1 Model Adequacy for a Predictor Variable—Added-Variable Plots384

10.2 Identifying Outlying Y Observations—Studentized Deleted Residuals390

Outlying Cases390

Residuals and Semistudentized Residuals392

Hat Matrix392

Studentized Residuals394

Deleted Residuals395

Studentized Deleted Residuals396

10.3 Identifying Outlying X Observations—Hat Matrix Leverage Values398

Use of Hat Matrix for Identifying Outlying X Observations398

Use of Hat Matrix to Identify Hidden Extrapolation400

10.4 Identifying Influential Cases—DFFITS,Cook's Distance,and DFBETAS Measures400

Influence on Single Fitted Value DFFITS401

Influence on All Fitted Values—Cook's Distance402

Influence on the Regression Coefficients DFBETAS404

Influence on Inferences405

Some Final Comments406

10.5 Multicollinearity Diagnostics—Variance Inflation Factor406

Informal Diagnostics407

Variance Inflation Factor408

10.6 Surgical Unit Example Continued410

Cited References414

Problems414

Exercises419

Projects419

Case Studies420

Chapter 11 Building the Regression Model Ⅲ:Remedial Measures421

11.1 Unequal Error Variances Remedial Measures—Weighted Least Squares421

Error Variances Known422

Error Variances Known up to Proportionality Constant424

Error Variances Unknown424

11.2 Multicollinearity Remedial Measures—Ridge Regression431

Some Remedial Measures431

Ridge Regression432

11.3 Remedial Measures for Influential Cases—Robust Regression437

Robust Regression438

IRLS Robust Regression439

11.4 Nonparametric Regression:Lowess Method and Regression Trees449

Lowess Method449

Regression Trees453

11.5 Remedial Measures for Evaluating Precision in Nonstandard Situations—Bootstrapping458

General Procedure459

Bootstrap Sampling459

Bootstrap Confidence Intervals460

11.6 Case Example—MNDOT Traffic Estimation464

The AADT Database464

Model Development465

Weighted Least Squares Estimation468

Cited References471

Problems472

Exercises476

Projects476

Case Studies480

Chapter 12 Autocorrelation in Time Series Data481

12.1 Problems of Autocorrelation481

12.2 First-Order Autoregressive Error Model484

Simple Linear Regression484

Multiple Regression484

Properties of Error Terms485

12.3 Durbin-Watson Test for Autocorrelation487

12.4 Remedial Measures for Autocorrelation490

Addition of Predictor Variables490

Use of Transformed Variables490

Cochrane-Orcutt Procedure492

Hildreth-Lu Procedure495

First Differences Procedure496

Comparison of Three Methods498

12.5 Forecasting with Autocorrelated Error Terms499

Cited References502

Problems502

Exercises507

Projects508

Case Studies508

PART THREE NONLINEAR REGRESSION509

Chapter 13 Introduction to Nonlinear Regression and Neural Networks510

13.1 Linear and Nonlinear Regression Models510

Linear Regression Models510

Nonlinear Regression Models511

Estimation of Regression Parameters514

13.2 Least Squares Estimation in Nonlinear Regression515

Solution of Normal Equations517

Direct Numerical Search—Gauss-Newton Method518

Other Direct Search Procedures525

13.3 Model Building and Diagnostics526

13.4 Inferences about Nonlinear Regression Parameters527

Estimate of Error Term Variance527

Large-Sample Theory528

When Is Large-Sample Theory Applicable?528

Interval Estimation of a Single γk531

Simultaneous Interval Estimation of Several γk532

Test Concerning a Single γk532

Test Concerning Several γk533

13.5 Learning Curve Example533

13.6 Introduction to Neural Network Modeling537

Neural Network Model537

Network Representation540

Neural Network as Generalization of Linear Regression541

Parameter Estimation:Penalized Least Squares542

Example:Ischemic Heart Disease543

Model Interpretation and Prediction546

Some Final Comments on Neural Network Modeling547

Cited References547

Problems548

Exercises552

Projects552

Case Studies554

Chapter 14 Logistic Regression,Poisson Regression,and Generalized Linear Models555

14.1 Regression Models with Binary Response Variable555

Meaning of Response Function when Outcome Variable Is Binary556

Special Problems when Response Variable Is Binary557

14.2 Sigmoidal Response Functions for Binary Responses559

Probit Mean Response Function559

Logistic Mean Response Function560

Complementary Log-Log Response Function562

14.3 Simple Logistic Regression563

Simple Logistic Regression Model563

Likelihood Function564

Maximum Likelihood Estimation564

Interpretation of b1567

Use of Probit and Complementary Log-Log Response Functions568

Repeat Observations—Binomial Outcomes568

14.4 Multiple Logistic Regression570

Multiple Logistic Regression Model570

Fitting of Model571

Polynomial Logistic Regression575

14.5 Inferences about Regression Parameters577

Test Concerning a Single βk:Wald Test578

Interval Estimation of a Single βk579

Test whether Several βk=0:Likelihood Ratio Test580

14.6 Automatic Model Selection Methods582

Model Selection Criteria582

Best Subsets Procedures583

Stepwise Model Selection583

14.7 Tests for Goodness of Fit586

Pearson Chi-Square Goodness of Fit Test586

Deviance Goodness of Fit Test588

Hosmer-Lemeshow Goodness of Fit Test589

14.8 Logistic Regression Diagnostics591

Logistic Regression Residuals591

Diagnostic Residual Plots594

Detection of Influential Observations598

14.9 Inferences about Mean Response602

Point Estimator602

Interval Estimation602

Simultaneous Confidence Intervals for Several Mean Responses603

14.10 Prediction of a New Observation604

Choice of Prediction Rule604

Validation of Prediction Error Rate607

14.11 Polytomous Logistic Regression for Nominal Response608

Pregnancy Duration Data with Polytomous Response609

J—1 Baseline-Category Logitsfor Nominal Response610

Maximum Likelihood Estimation612

14.12 Polytomous Logistic Regression for Ordinal Response614

14.13 Poisson Regression618

Poisson Distribution618

Poisson Regression Model619

Maximum Likelihood Estimation620

Model Development620

Inferences621

14.14 Generalized Linear Models623

Cited References624

Problems625

Exercises634

Projects635

Case Studies640

Appendix A Some Basic Results in Probability and Statistics641

Appendix B Tables659

Appendix C Data Sets677

Appendix D Selected Bibliography687

Index695