Introduction to Linear Models and Statistical Inference

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Frederick C. Tinsley
Colorado College
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

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关键词：introduction Statistical statistica troduction statistic Statistical models introduction Inference Linear

INTRODUCTION: STATISTICAL QUESTIONS 1
1. DATA: PLOTS AND LOCATION 13
1.1 Plot the Data 13
1.2 Measures of Location: Single Observations 16
1.3 Measures of Location: Paired Observations 21
1.4 Robust Measures of Location: Paired Observations 31
1.5 Linear Algebra for Least Squares (Optional) 34
Exercises 37
2. DATA: DISPERSION AND CORRELATION 43
2.1 Measures of Dispersion: Single Observations 43
2.2 Measures of Dispersion: Paired Observations 47
2.3 Robust Measures of Dispersion: Paired Observations 51
2.4 Analysis of Variance 54
2.5 Measures of Linear Relationship 58
2.6 Analysis of Variance using Linear Algebra (Optional) 65
Exercises 74
3. RANDOM VARIABLES: PROBABILITY AND DENSITY 79
3.1 Random Variables 79
3.2 Probability 81
3.3 Finding Probabilities 84
3.4 Densities: Discrete Random Variables 90
3.5 Densities: Continuous Random Variables 92
3.6 Binomial Random Variables 97
3.7 Normal Random Variables 102
Exercises 108
4. RANDOM VARIABLES: EXPECTATION AND VARIANCE 113
4.1 Expectation of a Random Variable 113
4.2 Properties of Expectation 118
4.3 Independent Random Variables 123
4.4 Variance of a Random Variable 128
4.5 Correlation Coefficient 137
4.6 Properties of Normal Random Variables 143
VI
CONTENTS
4.7 Linear Algebra for Random Vectors (Optional) 145
Exercises 151
5. STATISTICAL INFERENCE 155
5.1 Populations and Samples 156
5.2 Unbiases Estimators 161
5.3 Distribution of X 165
5.4 Confidence Intervals 174
5.5 Hypothesis Testing 180
5.6 General Inference Problem 185
5.7 The Runs Test for Randomness 191
5.8 Testing for Normality 194
5.9 Linear Algebra for Inference (Optional) 204
Exercises 211
6. SIMPLE LINEAR MODELS 217
6.1 Basics of the Simple Linear Model 217
6.2 Estimators for the Simple Linear Model 219
6.3 Inference for the Slope 221
6.4 Testing the Hypothesis 6=0 224
6.5 Coefficient of Determination 230
6.6 Inference for the Intercept 232
6.7 Inference for the Variance 235
6.8 Prediction Intervals 236
6.9 Regression Through the Origin 241
6.10 Earthquake Example 244
6.11 Linear Algebra: The Simple Linear Model (Optional) 246
Exercises 252
7. LINEAR MODEL DIAGNOSTICS 259
7.1 Residual Plots 261
7.2 Standardized Residuals 264
7.3 Testing Assumption 1: Is X a Valid Predictor? 268
7.4 Testing Assumption 2: Does Efa) = 0 for all i? 269
7.5 Testing Assumption 2: Does Varfa) = o2 for all i? 284
7.6 Testing Assumption 3: Are the Errors Independent? 290
7.7 Testing Assumption 4: Are the Errors Normal? 299
7.8 Distribution of the Residuals 303
7.9 Linear Algebra for Residuals (Optional) 307
Exercises 319
8. LINEAR MODELS: TWO INDEPENDENT VARIABLES 331
8.1 Calculating Parameters 333
8.2 Analysis of Variance 337
CONTENTS
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8.3 The Effects of Independent Variables 341
8.4 Inference for the Bivariate Linear Model 345
8.5 Diagnostics for the Bivariate Linear Model 351
8.6 Linear Algebra: Bivariate Linear Model (Optional) 362
Exercises 365
9. LINEAR MODELS: SEVERAL INDEPENDENT VARIABLES 369
9.1 A Multivariate Example 370
9.2 Analysis of Variance 372
9.3 Inference for the Multivariate Linear Model 374
9.4 Selecting Predictors 377
9.5 Diagnostics for the Multivariate Model 388
9.6 A Larger Example 390
9.7 A Curious Example 396
9.8 Linear Algebra: Multivariate Linear Model (Optional) 399
Exercises 407
10. MODEL BUILDING 411
10.1 Transformations 413
10.2 Indicator Variables 429
10.3 Using R? Carefully 434
10.4 Selection of Predictors 437
10.5 Outliers and Influence 442
10.6 Comprehensive Example: College Presidents 447
10.7 Linear Algebra for Model Building (Optional) 457
Exercises 465
11. EXTENDED LINEAR MODELS 469
11.1 Analysis of Variance Models 469
11.2 Analysis of Covariance Models 480
11.3 Diagnostics for ANOVA and ANCOVA 485
11.4 Binary Logistic Regression Models 495
11.5 Robust Regression Methods 504
11.6 Total Least Squares 508
11.7 Linear Algebra for ANOVA (Optional) 517
Exercises 522
APPENDIX A: Data References 531
APPENDIX B: MINITAB Reference 541
APPENDIX C: Introduction to Linear Algebra 551
APPENDIX D: Statistical Tables 567
REFERENCES 575
INDEX 581