Preface xiv
1 Introduction 1
1.1 What Is Regression Analysis? 1
1.2 Publicly Available Data Sets 2
1.3 Selected Applications of Regression Analysis 3
1.3.1 Agricultural Sciences 3
1.3.2 Industrial and Labor Relations 4
1.3.3 Government 5
1.3.4 History 5
1.3.5 Environmental Sciences 6
1.3.6 Industrial Production 6
1.3.7 The Space Shuttle Challenger 7
1.3.8 Cost of Health Care 7
1.4 Steps in Regression Analysis 7
1.4.1 Statement of the Problem 9
1.4.2 Selection of Potentially Relevant Variables 9
1.4.3 Data Collection 9
1.4.4 Model Specification 10
1.4.5 Method of Fitting 12
1.4.6 Model Fitting 13
1.4.7 Model Criticism and Selection 14
1.4.8 Objectives of Regression Analysis 15
1.5 Scope and Organization of the Book 16
2 A Brief Introduction to R 19
2.1 What Is R and RStudio? 19
2.2 Installing R and RStudio 20
2.3 Getting Started With R 21
2.3.1 Command Level Prompt 21
2.3.2 Calculations Using R 22
2.3.3 Editing Your R Code 24
2.3.4 Best Practice: Object Names in R 25
2.4 Data Values and Objects in R 25
2.4.1 Types of Data Values in R 25
2.4.2 Types (Structures) of Objects in R 28
2.4.3 Object Attributes 34
2.4.4 Testing (Checking) Object Type 34
2.4.5 Changing Object Type 34
2.5 R Packages (Libraries) 35
2.5.1 Installing R Packages 35
2.5.2 Name Spaces 36
2.5.3 Updating R 37
2.5.4 Datasets in R Packages 37
2.6 Importing (Reading) Data into R Workspace 37
2.6.1 Best Practice: Working Directory 38
2.6.2 Reading ASCII (Text) Files 38
2.6.3 Reading CSV Files 40
2.6.4 Reading Excel Files 40
2.6.5 Reading Files from the Internet 41
2.7 Writing (Exporting) Data to Files 42
2.7.1 Diverting Normal R Output to a File 42
2.7.2 Saving Graphs in Files 42
2.7.3 Exporting Data to Files 43
2.8 Some Arithmetic and Other Operators 43
2.8.1 Vectors 43
2.8.2 Matrix Computations 45
2.9 Programming in R 50
2.9.1 Best Practice: Script Files 50
2.9.2 Some Useful Commands or Functions 50
2.9.3 Conditional Execution 51
2.9.4 Loops 53
2.9.5 Functions and Functionals 54
2.9.6 User Defined Functions 55
2.10 Bibliographic Notes 60
3 Simple Linear Regression 65
3.1 Introduction 65
3.2 Covariance and Correlation Coefficient 65
3.3 Example: Computer Repair Data 69
3.4 The Simple Linear Regression Model 72
3.5 Parameter Estimation 73
3.6 Tests of Hypotheses 77
3.7 Confidence Intervals 82
3.8 Predictions 83
3.9 Measuring the Quality of Fit 84
3.10 Regression Line Through the Origin 88
3.11 Trivial Regression Models 89
3.12 Bibliographic Notes 90
4 Multiple Linear Regression 97
4.1 Introduction 97
4.2 Description of the Data and Model 97
4.3 Example: Supervisor Performance Data 98
4.4 Parameter Estimation 100
4.5 Interpretations of Regression Coefficients 101
4.6 Centering and Scaling 104
4.6.1 Centering and Scaling in Intercept Models 104
4.6.2 Scaling in No-Intercept Models 105
4.7 Properties of the Least Squares Estimators 106
4.8 Multiple Correlation Coefficient 107
4.9 Inference for Individual Regression Coefficients 108
4.10 Tests of Hypotheses in a Linear Model 111
4.10.1 Testing All Regression Coefficients Equal to Zero
4.10.2 Testing a Subset of Regression Coefficients Equal to 113
4.10.3 Testing the Equality of Regression Coefficients
4.10.4 Estimating and Testing of Regression Parameters 118
4.11 Predictions 121
4.12 Summary 122
5 Regression Diagnostics: Detection of Model Violations 131
5.1 Introduction 131
5.2 The Standard Regression Assumptions 132
5.3 Various Types of Residuals 134
5.4 Graphical Methods 136
5.5 Graphs Before Fitting a Model 139
5.5.1 One-Dimensional Graphs 139 5.5.2 Two-Dimensional Graphs 140
5.5.3 Rotating Plots 142
5.5.4 Dynamic Graphs 142
5.6 Graphs After Fitting a Model 143
5.7 Checking Linearity and Normality Assumptions 143
5.8 Leverage, Influence, and Outliers 144
5.8.1 Outliers in the Response Variable 146
5.8.2 Outliers in the Predictors 146
5.8.3 Masking and Swamping Problems 147
5.9 Measures of Influence 148
5.9.1 Cook¢s Distance 150
5.9.2 Welsch and Kuh Measure 151
5.9.3 Hadi¢s Influence Measure 151
5.10 The Potential-Residual Plot 152
5.11 Regression Diagnostics in R 154 5.12 What to Do with the Outliers? 155
5.13 Role of Variables in a Regression Equation 156
5.11.1 Added-Variable Plot 156
5.11.2 Residual Plus Component Plot 157
5.14 Effects of an Additional Predictor 159
5.15 Robust Regression 161
6 Qualitative Variables as Predictors 167
6.1 Introduction 167
6.2 Salary Survey Data 168
6.3 Interaction Variables 171
6.4 Systems of Regression Equations 175
6.4.1 Models with Different Slopes and Different Intercepts 176
6.4.2 Models with Same Slope and Different Intercepts 183
6.4.3 Models with Same Intercept and Different Slopes 184
6.5 Other Applications of Indicator Variables 185
6.6 Seasonality 186
6.7 Stability of Regression Parameters Over Time 187
7 Transformation of Variables 195
7.1 Introduction 195
7.2 Transformations to Achieve Linearity 197
7.3 Bacteria Deaths Due to X-Ray Radiation 199
7.3.1 Inadequacy of a Linear Model 200
7.3.2 Logarithmic Transformation for Achieving Linearity 201
7.4 Transformations to Stabilize Variance 203
7.5 Detection of Heteroscedastic Errors 208
7.6 Removal of Heteroscedasticity 210
7.7 Weighted Least Squares 211
7.8 Logarithmic Transformation of Data 212
7.9 Power Transformation 213
7.10 Summary 216
8 Weighted Least Squares 223
8.1 Introduction 223
8.2 Heteroscedastic Models 224
8.2.1 Supervisors Data 224
8.2.2 College Expense Data 226
8.3 Two-Stage Estimation 227
8.4 Education Expenditure Data 229
8.5 Fitting a Dose-Response Relationship Curve 237
9 The Problem of Correlated Errors 241
9.1 Introduction: Autocorrelation 241
9.2 Consumer Expenditure and Money Stock 242
9.3 Durbin-Watson Statistic 245
9.4 Removal of Autocorrelation by Transformation 246
9.5 Iterative Estimation with Autocorrelated Errors 249
9.6 Autocorrelation and Missing Variables 250
9.7 Analysis of Housing Starts 251
9.8 Limitations of the Durbin-Watson Statistic 253
9.9 Indicator Variables to Remove Seasonality 255
9.10 Regressing Two Time Series 257
10 Analysis of Collinear Data 261
10.1 Introduction 261
10.2 Effects of Collinearity on Inference 262
10.3 Effects of Collinearity on Forecasting 267
CONTENTS
10.4 Detection of Collinearity 271
10.4.1 Simple Signs of Collinearity 271
10.4.2 Variance Inflation Factors 274
10.4.3 The Condition Indices 276
11 Working With Collinear Data 283
11.1 Introduction 283
11.2 Principal Components 283
11.3 Computations Using Principal Components 287
11.4 Imposing Constraints 289
11.5 Searching for Linear Functions of the ²'s 292
11.6 Biased Estimation of Regression Coefficients 295
11.7 Principal Components Regression 296
11.8 Reduction of Collinearity in the Estimation Data 298
11.9 Constraints on the Regression Coefficients 300
11.10 Principal Components Regression: A Caution 301
11.11 Ridge Regression 303
11.12 Estimation by the Ridge Method 305
11.13 Ridge Regression: Some Remarks 308
11.14 Summary 311
11.15 Bibliographic Notes 311
12 Variable Selection Procedures 321
12.1 Introduction 321
12.2 Formulation of the Problem 322
12.3 Consequences of Variables Deletion 322
12.4 Uses of Regression Equations 324
12.4.1 Description and Model Building 324
12.4.2 Estimation and Prediction 324
12.4.3 Control 324
12.5 Criteria for Evaluating Equations 325
12.5.1 Residual Mean Square 325
12.5.2 Mallows Cp 326
12.5.3 Information Criteria 327
12.6 Collinearity and Variable Selection 328
12.7 Evaluating All Possible Equations 328
12.8 Variable Selection Procedures 329
12.8.1 Forward Selection Procedure 329
12.8.2 Backward Elimination Procedure 330
12.8.3 Stepwise Method 330
12.9 General Remarks on Variable Selection Methods 331
12.10 A Study of Supervisor Performance 332
12.11 Variable Selection with Collinear Data 336
12.12 The Homicide Data 336
12.13 Variable Selection Using Ridge Regression 339
12.14 Selection of Variables in an Air Pollution Study 339
12.15 A Possible Strategy for Fitting Regression Models 345
12.16 Bibliographic Notes 347
13 Logistic Regression 353
13.1 Introduction 353
13.2 Modeling Qualitative Data 354
13.3 The Logit Model 354
13.4 Example: Estimating Probability of Bankruptcies 356
13.5 Logistic Regression Diagnostics 358
13.6 Determination of Variables to Retain 359
13.7 Judging the Fit of a Logistic Regression 362
13.8 The Multinomial Logit Model 364
13.8.1 Multinomial Logistic Regression 364
13.8.2 Example: Determining Chemical Diabetes 365
13.8.3 Ordinal Logistic Regression 368
13.8.4 Example: Determining Chemical Diabetes Revisited 368
13.9 Classification Problem: Another Approach 370
14 Further Topics 375
14.1 Introduction 375
14.2 Generalized Linear Model 375
14.3 Poisson Regression Model 376
14.4 Introduction of New Drugs 377
14.5 Robust Regression 378
14.6 Fitting a Quadratic Model 379
14.7 Distribution of PCB in U.S. Bays 381
Exercises 384
References 385
Index
