Brenton R. Clarke
Copyright O 2008 by John Wiley & Sons, Inc. All rights reserved
WILEY SERIES IN PROBABILITY AND STATISTICS

Preface
Acknowledgments
Notation
1 Introduction
1.1 The Linear Model and Examples
1.2 What Are the Objectives?
1.3 Problems
2 Projection Matrices and Vector Space Theory
2.1 Basis of a Vector Space
2.2 Range and Kernel
2.3 Projections
2.3.1 Linear Model Application
2.4 Sums and Differences of Orthogonal Projections
2.5 Problems
3 Least Squares Theory
3.1 The Normal Equations
3.2 The Gauss-Markov Theorem
3.3 The Distribution of So
3.4 Some Simple Significance Tests
3.5 Prediction Intervals
3.6 Problems
4 Distribution Theory
4.1 Motivation
4.2 Noncentral X 2 and F Distributions
4.2.1 Noncentral F-Distribution
4.2.2 Applications to Linear Models
4.2.3 Some Simple Extensions
4.3 Problems
5 Helmert Matrices and Orthogonal Relationships
5.1 Transformations to Independent Normally Distributed Random
Variables
5.2 The Kronecker Product
5.3 Orthogonal Components in Two-Way ANOVA: One Observation
Per Cell
5.4 Orthogonal Components in Two-Way ANOVA with Replications
5.5 The Gauss-Markov Theorem Revisited
5.6 Orthogonal Components for Interaction
5.6.1 Testing for Interaction: One Observation per Cell
5.6.2 Example Calculation of Tukey's One-Degree-of-
Freedom Test Statistic
5.7 Problems
6 Further Discussion of ANOVA
6.1 The Various Representations of Orthogonal Components
6.2 On the Lack of Orthogonality
6.3 Relationship Algebra
6.4 Triple Classification
6.5 Latin Squares
6.6 2" Factorial Design
6.6.1 Yates' Algorithm
6.7 The Function of Randomization
6.8 Brief View of Multiple Comparison Techniques
6.9 Problems
7 Residual Analysis: Diagnostics and Robustness
7.1 Design Diagnostics
7.1.1 Standardized and Studentized Residuals
7.1.2 Combining Design and Residual Effects on Fit: DFITS
7.1.3 Cook's D-Statistic
7.2 Robust Approaches
7.2.1 Adaptive Trimmed Likelihood Algorithm
7.3 Problems
8 Models That Include Variance Components
8.1 The One-Way Random Effects Model
8.2 The Mixed Two-Way Model
8.3 A Split Plot Design
8.3.1 A Traditional Model
8.4 Problems
9 Likelihood Approaches
9.1 Maximum Likelihood Estimation
9.2 REML
9.3 Discussion of Hierarchical Statistical Models
9.3.1 Hierarchy for the Mixed Model (Assuming Normality)
9.4 Problems
10 Uncorrelated Residuals Formed from the Linear Model
10.1 Best Linear Unbiased Error ~stimatest
10.2 The Best Linear Unbiased Scalar Covariance Matrix Approach
10.3 Explicit Solution
10.4 Recursive Residuals
10.4.1 Recursive Residuals and their propertiest+
10.5 Uncorrelated Residuals
10.5.1 The Main Results
10.5.2 Final Remarks
10.6 Problems
11 Further inferential questions relating to ANOVA
References
Index