Linear models: the theory and application of analysis of variance

271 Pages·2008·8.26 MB·English

by  Brenton R. Clarke

LINEAR MODELS
The Theory and Application of
Analysis of Variance
Brenton R. Clarke
Wl LEY
A JOHN WlLEY & SONS, INC., PUBLICATION
This Page Intentionally Left Blank
LINEAR MODELS
WILEY SERIES IN PROBABILITY AND STATISTICS
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A complete list of the titles in this series appears at the end of this volume.
LINEAR MODELS
The Theory and Application of
Analysis of Variance
Brenton R. Clarke
Wl LEY
A JOHN WlLEY & SONS, INC., PUBLICATION
Copyright O 2008 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data:
Clarke, Brenton R.
Linear models : the theory and application of analysis of variance I Brenton
R. Clarke.
p. cm. - (Wiley series in probability and statistics)
Includes bibliographical references and index.
ISBN 978-0-470-02566-6 (cloth)
1. Linear models (Statistics) 2. Analysis of variance. I. Title.
QA279.C55 2008
5 19.5'384~22
Printed in the United States of America.
To my w@, Erica
and
my sons, Andrew and Stephen
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CONTENTS
Preface
Acknowledgments
Notation xvii
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
viii CONTENTS
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 X2 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