This book is designed as a textbook for graduate students and as a resource for researchers seeking a thorough mathematical treatment of its subject. It develops the main results of regression and the analysis of variance, as well as the central results on confounded and fractional factorial experiments. Matrix theory is deemphasized; its role is taken instead by the theory of linear transformations between vector spaces.The text gives a carefully paced and unified presentation of two topics, linear models and experimental design. Students are assumed to have a solid background in linear algebra, basic knowledge of regression and analysis of variance, and some exposure to experimental design, and should be comfortable with reading and constructing mathematical proofs.
The book leads students into the mathematical theory, including many examples both for motivation and for illustration. Over 130 exercises of varying difficulty are included. An extensive mathematical appendix and a detailed index make the text especially accessible.
Linear Models and Design can serve as a textbook for a year-long course in the topics covered, or for a one-semester course in either linear model theory or experimental design. It prepares students for more advanced topics in the field, and assists in developing a thoughtful approach to the existing literature. It includes a guide to terminology as well as discussion of the history and development of ideas, and offers a fresh perspective on the fundamental concepts and results of the subject.
Author(s): Jay H. Beder
Publisher: Springer
Year: 2022
Language: English
Pages: 357
City: Cham
Preface
Contents
List of Figures
List of Tables
1 Linear Models
1.1 Random Vectors
1.2 Some Statistical Concepts
1.2.1 Identifiability
1.2.2 Estimation
1.2.3 Testing: Confidence Sets
1.3 Linear Models
1.4 Regression Models
1.5 Factorial (ANOVA) Models
1.6 Linear Parametric Functions
1.6.1 Identifiability
1.6.2 Contrasts
1.7 Linear Constraints and Hypotheses
1.8 Exercises
2 Effects in a Factorial Experiment
2.1 One-Factor Designs
2.2 Two-Factor Designs
2.2.1 Three Elementary Hypotheses
2.2.2 The Main Effects Hypotheses
2.2.3 Effects in a Two-Factor Design
2.3 Multifactor Designs
2.4 Quantitative Factors
2.5 The Factor-Effects Parametrization
2.5.1 The One-Factor Model
2.5.2 Models with Two Factors
2.5.3 Models with Three or More Factors
2.5.4 The Different Systems of Weights
2.6 The Cell-Means Philosophy
2.7 Random Effects: Components of Variance
2.8 Exercises
3 Estimation
3.1 The Method of Least Squares
3.2 Properties of Least Squares Estimators
3.2.1 Estimability
3.2.2 The Moore-Penrose Inverse of X
3.3 Estimating σ2: Sum of Squares, Mean Square
3.4 t-Tests, t-Intervals
3.5 Exercises
4 Testing
4.1 Testing a Linear Hypothesis
4.1.1 Testing in Constrained Models
4.1.2 Replication. Lack of Fit and Pure Error
4.2 The Wald Statistic
4.3 The Lattice of Hypotheses
4.3.1 The Two-Predictor Regression Model
4.3.2 The Three-Predictor Regression Model
4.3.3 The Two-Factor Design
4.3.4 The Hypothesis of No Model Effect
4.3.5 The Lattice in the Observation Space
4.4 Adjusted SS
4.5 Nested Hypotheses. Sequential SS.
4.5.1 Adjusted or Sequential?
4.5.2 Associated Hypotheses
4.6 Orthogonal Hypotheses
4.6.1 Application: Factorial Designs
X'X Scalar
X' X Diagonal: Proportional Counts
4.6.2 Application: Regression Models
4.7 Affine Hypotheses. Confidence Sets
4.8 Simultaneous Inference
4.8.1 The Bonferroni Method
4.8.2 The Scheffé Method
Affine Hypotheses and Scheffé Confidence Intervals
Application: Confidence Bands
4.9 Inference for Variance Components Models
4.9.1 The One-Factor Design
4.9.2 The Two-Factor Design
4.9.3 Some Challenges
4.10 Exercises
5 Multifactor Designs
5.1 Vectors and Functions: Notation
5.2 The General Theory of Block Effects
5.3 The Multifactor Design: Reprise
5.4 The Kurkjian-Zelen Construction
5.4.1 Multilinear Algebra
5.4.2 Application to Factorial Designs
5.5 Confounding with Blocks
5.5.1 Generalities
5.6 Confounding in Classical Symmetric Factorial Designs
5.6.1 Special Blockings and Block Effects
5.6.2 Components of Interaction
5.6.3 Finding Generalized Interactions
5.6.4 Multiplicative Notation: The Effects Group
5.6.5 Complex Contrasts
5.7 Confounding in Arbitrary Factorial Designs
5.7.1 A Survey of Methods of Confounding
5.7.2 The Problem of Confounding
5.8 Exercises
Section 5.1
Section 5.2
Section 5.3
Section 5.4
Section 5.5
Section 5.6
Section 5.6.5
Section 5.7
6 Fractional Factorial Designs
6.1 Aliasing in Regular Fractions
6.1.1 Multiplicative Notation Again: The Defining Subgroup
6.2 The Resolution of a Regular Fraction
6.3 Construction of Regular Fractions
6.4 An Unexpected Pattern
6.5 Aliasing and Resolution in Arbitrary Fractions
6.5.1 Restriction Maps
6.5.2 Strength and Aliasing
6.5.3 Resolution
6.6 Aliasing and Confounding
6.7 Aliasing, Estimability and Bias
6.8 Relative Aberration
6.8.1 In Nonregular Designs
6.9 Projections
6.9.1 In Non-regular Designs
6.10 Exercises
A Mathematical Background
A.1 Functions
A.2 Relations
A.2.1 Partially Ordered Sets and Lattices
A.2.2 Equivalence Relations and Partitions
A.3 Algebra
A.4 Linear Algebra
A.4.1 Subspaces and Bases
A.4.2 Linear Transformations
A.4.3 Inner Product Spaces
A.4.4 The Transpose. Symmetric Maps
A.4.5 Positive Maps
A.4.6 The Spectral Theorem
A.4.7 Orthogonal Projections
A.4.8 Linear Algebra in Rn. Matrices
Bibliography
Index
