Plane Answers to Complex Questions: The Theory of Linear Models (Springer Texts in Statistics)

Prefaces
Contents
1 Introduction
	1.1 Random Matrices and Vectors
	1.2 Multivariate Normal Distributions
	1.3 Distributions of Quadratic Forms
		1.3.1 Results for General Covariance Matrices
	1.4 Generalized Linear Models
	1.5 Additional Exercises
	References
2 Estimation
	2.1 Identifiability and Estimability
	2.2 Estimation: Least Squares
	2.3 Estimation: Best Linear Unbiased
	2.4 Estimation: Maximum Likelihood
	2.5 Estimation: Minimum Variance Unbiased
	2.6 Sampling Distributions of Estimates
	2.7 Generalized Least Squares
	2.8 Normal Equations
	2.9 Variance-Bias Tradeoff
		2.9.1 Estimable Functions
	2.10 Bayesian Estimation
		2.10.1 Distribution Theory
	2.11 Additional Exercises
	References
3 Testing
	3.1 More About Models
	3.2 Testing Models
		3.2.1 Small Test Statistics
		3.2.2 A Generalized Test Procedure
	3.3 Testing Linear Parametric Functions
		3.3.1 A Generalized Test Procedure
		3.3.2 Testing an Unusual Class of Hypotheses
	3.4 Discussion
	3.5 Testing Single Degrees of Freedom in a Given Subspace
	3.6 Breaking a Sum of Squares into Independent Components
		3.6.1 General Theory
		3.6.2 Two-Way ANOVA
	3.7 Confidence Regions
	3.8 Tests for Generalized Least Squares Models
		3.8.1 Conditions for Simpler Procedures
	3.9 Additional Exercises
	References
4 One-Way ANOVA
	4.1 Analysis of Variance
	4.2 Estimating and Testing Contrasts
	4.3 Additional Exercises
5 Multiple Comparison Techniques
	5.1 Basic Ideas
	5.2 Scheffé\'s Method
	5.3 Least Significant Difference Method
	5.4 Bonferroni Method
	5.5 Tukey\'s Method
	5.6 Multiple Range Tests: Newman–Keuls and Duncan
	5.7 Summary
		5.7.1 Fisher Versus Neyman–Pearson
	5.8 Additional Exercises
	References
6 Regression Analysis
	6.1 Simple Linear Regression
	6.2 Multiple Regression
		6.2.1 Partitioned Model
		6.2.2 Nonparametric Regression and Generalized Additive Models
	6.3 General Prediction Theory
		6.3.1 Discussion
		6.3.2 General Prediction
		6.3.3 Best Prediction
		6.3.4 Best Linear Prediction
		6.3.5 Inner Products and Orthogonal Projections in General Spaces
	6.4 Multiple Correlation
		6.4.1 Squared Predictive Correlation
	6.5 Partial Correlation Coefficients
	6.6 Best Linear Unbiased Prediction
	6.7 Testing Lack of Fit
		6.7.1 The Traditional Test
		6.7.2 Near Replicate Lack of Fit Tests
		6.7.3 Partitioning Methods
		6.7.4 Nonparametric Methods
	6.8 Polynomial Regression and One-Way ANOVA
	6.9 Additional Exercises
	References
7 Multifactor Analysis of Variance
	7.1 Balanced Two-Way ANOVA Without Interaction
		7.1.1 Contrasts
	7.2 Balanced Two-Way ANOVA with Interaction
		7.2.1 Interaction Contrasts
	7.3 Polynomial Regression and the Balanced  Two-Way ANOVA
	7.4 Two-Way ANOVA with Proportional Numbers
	7.5 Two-Way ANOVA with Unequal Numbers: General Case
		7.5.1 Without Interaction
		7.5.2 Interaction
		7.5.3 Characterizing the Interaction Space
	7.6 Three or More Way Analyses
		7.6.1 Balanced Analyses
		7.6.2 Unbalanced Analyses
	7.7 Additional Exercises
	References
8 Experimental Design Models
	8.1 Completely Randomized Designs
	8.2 Randomized Complete Block Designs: Usual Theory
	8.3 Latin Square Designs
	8.4 Factorial Treatment Structures
	8.5 More on Factorial Treatment Structures
	8.6 Additional Exercises
	References
9 Analysis of Covariance
	9.1 Estimation of Fixed Effects
		9.1.1 Generalized Least Squares
	9.2 Estimation of Error and Tests of Hypotheses
	9.3 Another Adjusted Model and Missing Data
	9.4 Balanced Incomplete Block Designs
	9.5 Testing a Nonlinear Full Model
	9.6 Additional Exercises
	References
10 General Gauss–Markov Models
	10.1 BLUEs with an Arbitrary Covariance Matrix
	10.2 Geometric Aspects of Estimation
	10.3 Hypothesis Testing
	10.4 Least Squares Consistent Estimation
	10.5 Perfect Estimation and More
	References
11 Split Plot Models
	11.1 A Cluster Sampling Model
	11.2 Generalized Split Plot Models
		11.2.1 Estimation and Testing of Estimable Functions
		11.2.2 Testing Models
		11.2.3 Unbalanced Subplots
	11.3 The Split Plot Design
	11.4 Identifying the Appropriate Error
		11.4.1 Subsampling
		11.4.2 Two-Way ANOVA with Interaction
	11.5 Exercise: An Unusual Split Plot Analysis
	References
12 Model Diagnostics
	12.1 Leverage
		12.1.1 Mahalanobis Distances
		12.1.2 Diagonal Elements of the Projection Operator
		12.1.3 Examples
	12.2 Checking Normality
		12.2.1 Other Applications for Normal Plots
	12.3 Checking Independence
		12.3.1 Serial Correlation
	12.4 Heteroscedasticity and Lack of Fit
		12.4.1 Heteroscedasticity
		12.4.2 Huber–White (Robust) Sandwich Estimator
		12.4.3 Lack of Fit
		12.4.4 Residual Plots
	12.5 Updating Formulae and Predicted Residuals
	12.6 Outliers and Influential Observations
	12.7 Transformations
	References
13 Collinearity and Alternative Estimates
	13.1 Defining Collinearity
	13.2 Tolerance and Variance Inflation Factors
	13.3 Regression in Canonical Form and on Principal Components
		13.3.1 Regression in Canonical Form
		13.3.2 Principal Component Regression
		13.3.3 Generalized Inverse Regression
	13.4 Classical Ridge Regression
		13.4.1 Ridge Applied to Principal Components
	13.5 More on Mean Squared Error
	13.6 Robust Estimation and Alternative Distance Measures
	13.7 Orthogonal Regression
	References
14 Variable Selection
	14.1 All Possible Regressions and Best Subset Regression
		14.1.1 R2
		14.1.2 Adjusted R2
		14.1.3 Mallows\'s Cp
		14.1.4 Information Criteria: AIC, BIC
		14.1.5 Cost Complexity Pruning
	14.2 Stepwise Regression
		14.2.1 Traditional Forward Selection
		14.2.2 Backward Elimination
		14.2.3 Other Methods
	14.3 Discussion of Traditional Variable Selection Techniques
		14.3.1 R2
		14.3.2 Influential Observations
		14.3.3 Exploritory Data Analysis
		14.3.4 Multiplicities
		14.3.5 Predictive Models
		14.3.6 Overfitting
	14.4 Modern Forward Selection: Boosting, Bagging,  and Random Forests
		14.4.1 Boosting
		14.4.2 Bagging
		14.4.3 Random Forests
	References
Appendix A Vector Spaces
Appendix B Matrix Results
	B.1  Basic Ideas
	B.2  Eigenvalues and Related Results
	B.3  Projections
	B.4  Miscellaneous Results
	B.5  Properties of Kronecker Products and Vec Operators
	B.6  Tensors
	B.7  Exercises
Appendix C Some Univariate Distributions
Appendix D Multivariate Distributions
	D.1  Identifiability
Appendix E Inference for One Parameter
	E.1  Testing
	E.2  P Values
	E.3  Confidence Intervals
	E.4  Final Comments on Significance Testing
Appendix F Significantly Insignificant Tests
	F.1  Lack of Fit and Small F Statistics
	F.2 The Effect of Correlation and Heteroscedasticity  on F Statistics
Appendix G Randomization Theory Models
	G.1  Simple Random Sampling
	G.2  Completely Randomized Designs
	G.3  Randomized Complete Block Designs
References
Author Index
Subject Index