Linear Model Theory: Univariate, Multivariate, and Mixed Models

A precise and accessible presentation of linear model theory, illustrated with data examples
Statisticians often use linear models for data analysis and for developing new statistical methods. Most books on the subject have historically discussed univariate, multivariate, and mixed linear models separately, whereas Linear Model Theory: Univariate, Multivariate, and Mixed Models presents a unified treatment in order to make clear the distinctions among the three classes of models.
Linear Model Theory: Univariate, Multivariate, and Mixed Models begins with six chapters devoted to providing brief and clear mathematical statements of models, procedures, and notation. Data examples motivate and illustrate the models. Chapters 7–10 address distribution theory of multivariate Gaussian variables and quadratic forms. Chapters 11–19 detail methods for estimation, hypothesis testing, and confidence intervals. The final chapters, 20–23, concentrate on choosing a sample size. Substantial sets of excercises of varying difficulty serve instructors for their classes, as well as help students to test their own knowledge.
The reader needs a basic knowledge of statistics, probability, and inference, as well as a solid background in matrix theory and applied univariate linear models from a matrix perspective. Topics covered include:A review of matrix algebra for linear modelsThe general linear univariate modelThe general linear multivariate modelGeneralizations of the multivariate linear modelThe linear mixed modelMultivariate distribution theoryEstimation in linear modelsTests in Gaussian linear modelsChoosing a sample size in Gaussian linear models
Filling the need for a text that provides the necessary theoretical foundations for applying a wide range of methods in real situations, Linear Model Theory: Univariate, Multivariate, and Mixed Models centers on linea r models of interval scale responses with finite second moments. Models with complex predictors, complex responses, or both, motivate the presentation.

Spis treści:
Preface.
PART I: MODELS AND EXAMPLES.
1. Matrix Algebra for Linear Models.
1.1 Notation.
1.2 Some Operators and Special Types of Matrices.
1.3 Five Kinds of Multiplication.
1.4 The Direct Sum.
1.5 Rules of Operations.
1.6 Other Special Types of matrices.
1.7 Quadratic and Bilinear Forms.
1.8 Vector Spaces and Rank.
1.9 Finding Rank.
1.10 Determinants.
1.11 The Inverse and Generalized Inverse.
1.12 Eigenanalysis (Spectral Decomposition).
1.13 Some Factors of Symmetric Matrices.
1.14 Singular Value Decomposition.
1.15 Projections and Other Functions of a Design matrix.
1.16 Special Properties of Patterned Matrices.
1.17 Functional Optimization and Matrix Derivatives.
1.18 Statistical Notation Involving Matrices.
1.19 Statistical Formulas.
1.20 Principal Components.
1.21 Special Covariance Patterns.
2. The General Linear Univariate Model.
2.1 Introduction.
2.2 Model Concepts.
2.3 The General Linear Univariate Linear Model.
2.4 The Univariate General Linear Hypothesis.
2.5 Tests about Variances.
2.6 The Role of the Intercept.
2.7 Population Correlation and Strength of Relationship.
2.8 Statistical Estimates.
2.9 Testing the General Linear Hypothesis.
2.10 Confidence Regions for θ.
2.11 Sufficient Statistics for the Univariate Model.
Exercises.
3. The General Linear Multivariate Model.
3.1 Motivation.
3.2 Definition of the Multivariate Model.
3.3 The Multivariate General Linear Hypothesis.
3.4 Tests About Covariance Matrices.
3.5 Population Correlation.
3.6 Statistical Estimates.
3.7 Overview of Testing Multivariate Hypotheses.
3.8 Computing MULTIREP Tests.
3.9 Computing UNIREP tests.
3.10 Confidence Regions for Θ.3.11 Sufficient Statistics for the Multivariate Model.
3.12 Allowing Missing Data in the Multivariate Model.
Exercises.
4. Generalizations of the Multivariate Linear Model.
4.1 Motivation.
4.2 The Generalized General Linear Univariate Model: Exact and Approximate Weighted Least Squares.
4.3 Doubly Multivariate Models.
4.4 Seemingly Unrelated Regression.
4.5 Growth Curve Models (GMANOVA).
4.6 The Relationship of the GCM to the Multivariate Model.
4.7 Mixed, Hierarchical, and Related Models.
5. The Linear Mixed Model.
5.1 Motivation.
5.2 Definition of the Mixed Model.
5.3 Distribution–Free and Noniterative Estimates.
5.4 Gaussian Likelihood and Iterative Estimates.
5.5 Tests about β (Means, Fixed Effects).
5.6 Tests of Covariance Parameters, τ (random Effects).
Exercises.
6. Choosing the Form of a Linear Model for Analysis.
6.1 The Importance of Understanding Dependence.
6.2 How Many Variables per Independent Sampling Unit?
6.3 What Types of Variables Play a Role?
6.4 What Repeated Sampling Scheme Was Used?
6.5 Analysis Strategies for Multivariate Data.
6.6 cautions and Recommendations.
6.7 Review of Linear Model Notation.
PART II: MULTIVARIATE DISTRIBUTION THEORY.
7. General Theory of Multivariate Distributions.
7.1 Motivation.
7.2 Notation and Concepts.
7.3 Families of Distributions.
7.4 Cumulative Distribution Function.
7.5 Probability Density Function.
7.6 Formulas for Probabilities and Moments.
7.7 Characteristic Function.
7.8 Moment Generating Function.
7.9 Cumulant generating Function.
7.10 Transforming Random Variables.
7.11 Marginal Distributions.
7.12 Independence of Random Vectors.
7.13 Conditional Distributions.
7.14 (Joint) Moments of Multivariate Distributions.
7.15 Conditional Moments of Distributions.
7.16 Special Considerations for Random Matrices.
8. Scalar, vecto r, and Matrix Gaussian Distributions.
8.1 Motivation.
8.2 The Scalar Gaussian Distribution.
8.3 The Vector (“Multivariate”) Gaussian Distribution.
8.4 Marginal Distributions.
8.5 Independence.
8.6 Conditional Distributions.
8.7 Asymptotic Properties.
8.8 The matrix Gaussian Distribution.
8.9 Assessing, Multivariate Gaussian Distribution.
8.10 Tests for Gaussian Distribution.
Exercises.
9. Univariate Quadratic Forms.
9.1 Motivation.
9.2 Chi–Square Distributions.
9.3 General Properties of Quadratic Forms.
9.4 Properties of Quadratic Forms in Gaussian Vectors.
9.5 Independence among Linear and Quadratic Forms.
9.6 The ANOVA Theorem.
9.7 Ratios Involving Quadratic Forms.
Exercises.
10. Multivariate Quadratic Forms.
10.1 The Wishart Distribution.
10.2 The Characteristic Function of the Wishart.
10.3 Properties of the Wishart.
10.4 The Inverse Wishart.
10.5 Related Distributions.
Exercises.
PART III: ESTIMATION IN LINEAR MODELS.
11. Estimation for Univariate and Weighted Linear Models.
11.1 Motivation.
11.2 Statement of the Problem.
11.3 (Unrestricted) Linear Equivalent Linear Models.
11.4 Estimability and Criteria for Checking It.
11.5 Coding Schemes and the Essence matrix.
11.6 Unrestricted Maximum Likelihood Estimation of β.
11.7 Unrestricted BLUE Estimation of β.
11.8 Unrestricted Least Squares Estimation of β.
11.9 Unrestricted Maximum Likelihood Estimation of θ.
11.10 Unrestricted BLUE of θ.
11.11 Related Distributions.
11.12 Formulations of Explicit Restrictions of β and θ.
11.13 Restricted Estimation Via Equivalent Models.
11.14 Fitting Piecewise Polynomial Models Via Splines.
11.15 Estimation for the GGLM: Weighted Least Squares.
Exercises.
12. Estimation for Multivariate Linear Models.
12.1 Alternate Formulations of the Model.
12.2 Esti