Linear Models and Design

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