1 Introduction ................................................................................................ 1
1.1 Building Valid Models ........................................................................ 1
1.2 Motivating Examples .......................................................................... 1
1.2.1 Assessing the Ability of NFL Kickers .................................... 1
1.2.2 Newspaper Circulation ........................................................... 4
1.2.3 Menu Pricing in a New Italian Restaurant
in New York City .................................................................... 5
1.2.4 Effect of Wine Critics’ Ratings on Prices
of Bordeaux Wines .................................................................. 8
1.3 Level of Mathematics ......................................................................... 13
2 Simple Linear Regression .......................................................................... 15
2.1 Introduction and Least Squares Estimates .......................................... 15
2.1.1 Simple Linear Regression Models .......................................... 15
2.2 Inferences About the Slope and the Intercept ..................................... 20
2.2.1 Assumptions Necessary in Order to Make Inferences
About the Regression Model ..................................................... 21
2.2.2 Inferences About the Slope of the Regression Line ............... 21
2.2.3 Inferences About the Intercept of the Regression Line .......... 23
2.3 Confidence Intervals for the Population Regression Line .................. 24
2.4 Prediction Intervals for the Actual Value of Y .................................... 25
2.5 Analysis of Variance ........................................................................... 27
2.6 Dummy Variable Regression .............................................................. 30
2.7 Derivations of Results ......................................................................... 33
2.7.1 Inferences about the Slope of the Regression Line................. 34
2.7.2 Inferences about the Intercept of the Regression Line ........... 35
2.7.3 Confidence Intervals for the Population Regression Line ...... 36
2.7.4 Prediction Intervals for the Actual Value of Y ........................ 37
2.8 Exercises ............................................................................................. 38
xi
xii Contents
3 Diagnostics and Transformations for Simple Linear Regression .......... 45
3.1 Valid and Invalid Regression Models:
Anscombe’s Four Data Sets ............................................................... 45
3.1.1 Residuals ................................................................................. 48
3.1.2 Using Plots of Residuals to Determine Whether
the Proposed Regression Model Is a Valid Model .................. 49
3.1.3 Example of a Quadratic Model ............................................... 50
3.2 Regression Diagnostics: Tools for Checking
the Validity of a Model ....................................................................... 50
3.2.1 Leverage Points ....................................................................... 51
3.2.2 Standardized Residuals ........................................................... 59
3.2.3 Recommendations for Handling Outliers
and Leverage Points ................................................................ 66
3.2.4 Assessing the Influence of Certain Cases ............................... 67
3.2.5 Normality of the Errors ........................................................... 69
3.2.6 Constant Variance ................................................................... 71
3.3 Transformations .................................................................................. 76
3.3.1 Using Transformations to Stabilize Variance ......................... 76
3.3.2 Using Logarithms to Estimate Percentage Effects ................. 79
3.3.3 Using Transformations to Overcome Problems
due to Nonlinearity .................................................................. 83
3.4 Exercises ............................................................................................. 103
4 Weighted Least Squares ............................................................................ 115
4.1 Straight-Line Regression Based on Weighted Least Squares ............. 115
4.1.1 Prediction Intervals for Weighted Least Squares .................... 118
4.1.2 Leverage for Weighted Least Squares .................................... 118
4.1.3 Using Least Squares to Calculate Weighted Least Squares .... 119
4.1.4 Defining Residuals for Weighted Least Squares .................... 121
4.1.5 The Use of Weighted Least Squares ....................................... 121
4.2 Exercises ............................................................................................. 122
5 Multiple Linear Regression ....................................................................... 125
5.1 Polynomial Regression ....................................................................... 125
5.2 Estimation and Inference in Multiple Linear Regression ................... 130
5.3 Analysis of Covariance ....................................................................... 140
5.4 Exercises ............................................................................................. 146
6 Diagnostics and Transformations for Multiple Linear Regression ....... 151
6.1 Regression Diagnostics for Multiple Regression ............................... 151
6.1.1 Leverage Points in Multiple Regression ................................. 152
6.1.2 Properties of Residuals in Multiple Regression ...................... 154
6.1.3 Added Variable Plots .............................................................. 162
Contents xiii
6.2 Transformations .................................................................................. 167
6.2.1 Using Transformations to Overcome Nonlinearity ................. 167
6.2.2 Using Logarithms to Estimate Percentage Effects:
Real Valued Predictor Variables .............................................. 184
6.3 Graphical Assessment of the Mean Function Using
Marginal Model Plots ......................................................................... 189
6.4 Multicollinearity ................................................................................. 195
6.4.1 Multicollinearity and Variance Inflation Factors .................... 203
6.5 Case Study: Effect of Wine Critics’ Ratings on Prices
of Bordeaux Wines ............................................................................. 203
6.6 Pitfalls of Observational Studies Due to Omitted Variables ............... 210
6.6.1 Spurious Correlation Due to Omitted Variables ..................... 210
6.6.2 The Mathematics of Omitted Variables .................................. 213
6.6.3 Omitted Variables in Observational Studies ........................... 214
6.7 Exercises ............................................................................................. 215
7 Variable Selection ....................................................................................... 227
7.1 Evaluating Potential Subsets of Predictor Variables ........................... 228
7.1.1 Criterion 1: R2-Adjusted ......................................................... 228
7.1.2 Criterion 2: AIC, Akaike’s Information Criterion .................. 230
7.1.3 Criterion 3: AICC, Corrected AIC ........................................... 231
7.1.4 Criterion 4: BIC, Bayesian Information Criterion .................. 232
7.1.5 Comparison of AIC, AICC and BIC ........................................ 232
7.2 Deciding on the Collection of Potential Subsets
of Predictor Variables ......................................................................... 233
7.2.1 All Possible Subsets ................................................................ 233
7.2.2 Stepwise Subsets ..................................................................... 236
7.2.3 Inference After Variable Selection .......................................... 238
7.3 Assessing the Predictive Ability of Regression Models ..................... 239
7.3.1 Stage 1: Model Building Using the Training Data Set ........... 239
7.3.2 Stage 2: Model Comparison Using the Test Data Set ............. 247
7.4 Recent Developments in Variable Selection – LASSO ...................... 250
7.5 Exercises ............................................................................................. 252
8 Logistic Regression .................................................................................... 263
8.1 Logistic Regression Based on a Single Predictor ............................... 263
8.1.1 The Logistic Function and Odds ............................................. 265
8.1.2 Likelihood for Logistic Regression with
a Single Predictor .................................................................... 268
8.1.3 Explanation of Deviance ......................................................... 271
8.1.4 Using Differences in Deviance Values
to Compare Models ................................................................. 272
8.1.5 R2 for Logistic Regression ...................................................... 273
8.1.6 Residuals for Logistic Regression .......................................... 274
xiv Contents
8.2 Binary Logistic Regression ............................................................. 277
8.2.1 Deviance for the Case of Binary Data ............................... 280
8.2.2 Residuals for Binary Data ................................................. 281
8.2.3 Transforming Predictors in Logistic Regression
for Binary Data .................................................................. 282
8.2.4 Marginal Model Plots for Binary Data .............................. 286
8.3 Exercises ......................................................................................... 294
9 Serially Correlated Errors ...................................................................... 305
9.1 Autocorrelation ............................................................................... 305
9.2 Using Generalized Least Squares When the Errors Are AR(1) ...... 310
9.2.1 Generalized Least Squares Estimation ............................. 311
9.2.2 Transforming a Model with AR(1) Errors into
a Model with iid Errors ..................................................... 315
9.2.3 A General Approach to Transforming GLS into LS ......... 316
9.3 Case Study ...................................................................................... 319
9.4 Exercises ......................................................................................... 325
10 Mixed Models ........................................................................................... 331
10.1 Random Effects ............................................................................... 331
10.1.1 Maximum Likelihood and Restricted
Maximum Likelihood ........................................................ 334
10.1.2 Residuals in Mixed Models ............................................... 345
10.2 Models with Covariance Structures Which Vary Over Time .......... 353
10.2.1 Modeling the Conditional Mean ....................................... 354
10.3 Exercises ......................................................................................... 368
Appendix: Nonparametric Smoothing ........................................................... 371
References ......................................................................................................... 383
Index .................................................................................................................. 387
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发表于 2013-2-15 14:48:00
