## TABLE OF CONTENTS

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 S_{Ω}.

3.4 Some Simple Significance Tests.

3.5 Prediction Intervals.

3.6 Problems.

**4. Distribution Theory.**

4.1 Motivation.

4.2 Non-Central X^{2} and F Distributions.

4.2.1 Non-Central 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’s Degree of Freedom Statistic.

5.7 Problems.

**6. Further Discussion of ANOVA.**

6.1 The Different Representations of Orthogonal Components.

6.2 On the Lack of Orthogonality.

6.3 The Relationship Algebra.

6.4 The Triple Classification.

6.5 Latin Squares.

6.6 2^{k} Factorial Designs.

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 The Cook-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 Estimates.

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 Properties.

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.