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Linear Regression Analysis
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285
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G. A. Seber, A. J. Lee
On this page
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1 · Vectors of Random Variables
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2 · Multivariate Normal Distribution
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3 · Linear Regression: Estimation and Distribution Theory
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4 · Hypothesis Testing
5 · Confidence Intervals and Regions
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6 · Straight-Line Regression
7 · Polynomial Regression
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8 · Analysis of Variance
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9 · Departures from Underlying Assumptions
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10 · Departures from Assumptions: Diagnosis and Remedies
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11 · Computational Algorithms for Fitting a Regression
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12 · Prediction and Model Selection
Chapter 1
Vectors of Random Variables
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Section 1a
1.1 Notation, 1.2 Statistical Models, 1.3 Linear Regression Models, 1.4 Expectation and Covariance Operators
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Section 1b
1.5 Mean and Variance of Quadratic Forms
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Section 1c
1.6 Moment Generating Functions and Independence
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Chapter 2
Multivariate Normal Distribution
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Section 2a
2.1 Density Function
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Section 2b
2.2 Moment Generating Functions
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Section 2c
2.3 Statistical Independence
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Section 2d
2.4 Distribution of Quadratic Forms
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Chapter 3
Linear Regression: Estimation and Distribution Theory
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Section 3a
3.1 Least Squares Estimation
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Section 3b
3.2 Properties of Least Squares Estimates
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Section 3c
3.3 Unbiased Estimation of sigma^2
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Section 3d
3.4 Distribution Theory
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Section 3e
3.5 Maximum Likelihood Estimation, 3.6 Orthogonal Columns in the Regression Matrix
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Section 3f
3.7 Introducing Further Explanatory Variables: 3.7.1 General Theory, 3.7.2 One Extra Variable
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Section 3g
3.8 Estimation with Linear Restrictions: 3.8.1 Method of Lagrange Multipliers, 3.8.2 Method of Orthogonal Projections
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Section 3h
3.9 Design Matrix of Less Than Full Rank: 3.9.1 Least Squares Estimation
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Section 3i
3.9.2 Estimable Functions
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Section 3j
3.9.3 Introducing Further Explanatory Variables, 3.9.4 Introducing Linear Restrictions
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Section 3k
3.10 Generalized Least Squares
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Section 3l
3.11 Centering and Scaling the Explanatory Variables: 3.11.1 Centering, 3.11.2 Scaling
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Section 3m
3.12 Bayesian Estimation
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Section 3n
3.13 Robust Regression: 3.13.1 M-Estimates, 3.13.2 Estimates Based on Robust Location and Scale Measures, 3.13.3 Measuring Robustness, 3.13.4 Other Robust Estimates
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Chapter 4
Hypothesis Testing
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Section 4a
4.1 Introduction, 4.2 Likelihood Ratio Test, 4.3 F-Test: 4.3.1 Motivation, 4.3.2 Derivation
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Section 4b
4.3.3 Some Examples, 4.3.4 The Straight Line
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Section 4c
4.4 Multiple Correlation Coefficient
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Section 4d
4.5 Canonical Form for H
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Chapter 5
Confidence Intervals and Regions
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Chapter 6
Straight-Line Regression
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Section 6a
6.1 The Straight Line: 6.1.1 Confidence Intervals for the Slope and Intercept, 6.1.2 Confidence Interval for the X-Intercept, 6.1.3 Prediction Intervals and Bands, 6.1.4 Prediction Intervals for the Response, 6.1.5 Inverse Prediction (Calibration)
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Section 6b
6.2 Straight Line through the Origin, 6.3 Weighted Least Squares for the Straight Line: 6.3.1 Known Weights, 6.3.2 Unknown Weights
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Section 6c
6.4 Comparing Straight Lines: 6.4.1 General Model, 6.4.2 Use of Dummy Explanatory Variables
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Chapter 7
Polynomial Regression
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Chapter 8
Analysis of Variance
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Section 8a
8.1 Introduction, 8.2 One-Way Classification: 8.2.1 General Theory, 8.2.2 Confidence Intervals, 8.2.3 Underlying Assumptions
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Section 8b
8.3 Two-Way Classification (Unbalanced): 8.3.1 Representation as a Regression Model, 8.3.2 Hypothesis Testing, 8.3.3 Procedures for Testing the Hypotheses, 8.3.4 Confidence Intervals
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Section 8c
8.4 Two-Way Classification (Balanced)
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Section 8d
8.5 Two-Way Classification (One Observation Per Mean): 8.5.1 Underlying Assumptions, 8.6 Higher-Way Classifications with Equal Numbers Per Mean: 8.6.1 Definition of Interactions, 8.6.2 Hypothesis Testing, 8.6.3 Missing Observations
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Section 8e
8.7 Designs with Simple Block Structure, 8.8 Analysis of Covariance
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Chapter 9
Departures from Underlying Assumptions
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Section 9a
9.1 Introduction, 9.2 Bias: 9.2.1 Bias Due to Underfitting, 9.2.2 Bias Due to Overfitting
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Section 9b
9.3 Incorrect Variance Matrix
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Section 9c
9.4 Effect of Outliers, 9.5 Robustness of the F-Test to Nonnormality: 9.5.1 Effect of the Regressor Variables, 9.5.2 Quadratically Balanced F-Tests
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Section 9d
9.6 Effect of Random Explanatory Variables (9.6.1-9.6.6), 9.7 Collinearity: 9.7.1 Effect on the Variances of the Estimated Coefficients, 9.7.2 Variance Inflation Factors, 9.7.3 Variances and Eigenvalues, 9.7.4 Perturbation Theory, 9.7.5 Collinearity and Prediction
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Chapter 10
Departures from Assumptions: Diagnosis and Remedies
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Section 10a
10.1 Introduction, 10.2 Residuals and Hat Matrix Diagonals
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Section 10b
10.3 Dealing with Curvature: 10.3.1 Visualizing Regression Surfaces, 10.3.2 Transforming to Remove Curvature, 10.3.3 Adding and Deleting Variables
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Section 10c
10.4 Nonconstant Variance and Serial Correlation: 10.4.1 Detecting Nonconstant Variance, 10.4.2 Estimating Variance Functions, 10.4.3 Transforming to Equalize Variances, 10.4.4 Serial Correlation and the Durbin-Watson Test
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Section 10d
10.5 Departures from Normality: 10.5.1 Normal Plotting, 10.5.2 Transforming the Response, 10.5.3 Transforming Both Sides
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Section 10e
10.6 Detecting and Dealing with Outliers: 10.6.1 Types of Outliers, 10.6.2 Identifying High-Leverage Points, 10.6.3 Leave-One-Out Case Diagnostics, 10.6.4 Test for Outliers, 10.6.5 Other Methods
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Section 10f
10.7 Diagnosing Collinearity: 10.7.1 Drawbacks of Centering, 10.7.2 Detection of Points Influencing Collinearity, 10.7.3 Remedies for Collinearity
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Chapter 11
Computational Algorithms for Fitting a Regression
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Section 11a
11.1 Introduction: 11.1.1 Basic Methods, 11.2 Direct Solution of the Normal Equations: 11.2.1 Calculation of the Matrix X^T X, 11.2.2 Solving the Normal Equations
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Section 11b
11.3 QR Decomposition: 11.3.1 Calculation of Regression Quantities, 11.3.2 Algorithms for the QR and WU Decompositions
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Section 11c
11.4 Singular Value Decomposition: 11.4.1 Regression Calculations Using the SVD, 11.4.2 Computing the SVD, 11.5 Weighted Least Squares, 11.6 Adding and Deleting Cases and Variables: 11.6.1 Updating Formulas, 11.6.2 Connection with the Sweep Operator, 11.6.3 Adding and Deleting Cases and Variables Using QR, 11.7 Centering the Data, 11.8 Comparing Methods: 11.8.1 Resources, 11.8.2 Efficiency, 11.8.3 Accuracy, 11.8.4 Two Examples, 11.8.5 Summary
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Section 11d
11.9 Rank-Deficient Case (11.9.1-11.9.4), 11.10 Computing the Hat Matrix Diagonals: 11.10.1 Using the Cholesky Factorization, 11.10.2 Using the Thin QR Decomposition, 11.11 Calculating Test Statistics, 11.12 Robust Regression Calculations: 11.12.1 Algorithms for L1 Regression, 11.12.2 Algorithms for M- and GM-Estimation, 11.12.3 Elemental Regressions, 11.12.4 Algorithms for High-Breakdown Methods
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Chapter 12
Prediction and Model Selection
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Section 12a
12.1 Introduction, 12.2 Why Select?
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Section 12b
12.3 Choosing the Best Subset: 12.3.1 Goodness-Of-Fit Criteria, 12.3.2 Criteria Based on Prediction Error, 12.3.3 Estimating Distributional Discrepancies, 12.3.4 Approximating Posterior Probabilities
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Section 12c
12.4 Stepwise Methods: 12.4.1 Forward Selection, 12.4.2 Backward Elimination, 12.4.3 Stepwise Regression
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Section 12d
12.5 Shrinkage Methods: 12.5.1 Stein Shrinkage, 12.5.2 Ridge Regression, 12.5.3 Garrote and Lasso Estimates
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Section 12e
12.6 Bayesian Methods: 12.6.1 Predictive Densities, 12.6.2 Bayesian Prediction, 12.6.3 Bayesian Model Averaging
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Section 12f
12.7 Effect of Model Selection on Inference: 12.7.1 Conditional and Unconditional Distributions, 12.7.2 Bias, 12.7.3 Conditional Means and Variances, 12.7.4 Estimating Coefficients Using Conditional Likelihood, 12.7.5 Other Effects of Model Selection
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Section 12g
12.8 Computational Considerations: 12.8.1 Methods for All Possible Subsets, 12.8.2 Generating the Best Regressions, 12.8.3 All Possible Regressions Using QR Decompositions
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Section 12h
12.9 Comparison of Methods: 12.9.1 Identifying the Correct Subset, 12.9.2 Using Prediction Error as a Criterion
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