Linear Model Theory
Univariate, Multivariate, and Mixed Models
Published on 31. August 2006
480 pages
978-0-470-05213-6 (ISBN)
Description
A precise and accessible presentation of linear model theory,illustrated with data examples
Statisticians often use linear models for data analysis and fordeveloping new statistical methods. Most books on the subject havehistorically discussed univariate, multivariate, and mixed linearmodels separately, whereas Linear Model Theory: Univariate,Multivariate, and Mixed Models presents a unified treatment inorder to make clear the distinctions among the three classes ofmodels.
Linear Model Theory: Univariate, Multivariate, and MixedModels begins with six chapters devoted to providing brief andclear mathematical statements of models, procedures, and notation.Data examples motivate and illustrate the models. Chapters 7-10address distribution theory of multivariate Gaussian variables andquadratic 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 ofexcercises of varying difficulty serve instructors for theirclasses, 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 andapplied univariate linear models from a matrix perspective. Topicscovered include:
* A review of matrix algebra for linear models
* The general linear univariate model
* The general linear multivariate model
* Generalizations of the multivariate linear model
* The linear mixed model
* Multivariate distribution theory
* Estimation in linear models
* Tests in Gaussian linear models
* Choosing a sample size in Gaussian linear models
Filling the need for a text that provides the necessarytheoretical foundations for applying a wide range of methods inreal situations, Linear Model Theory: Univariate, Multivariate,and Mixed Models centers on linear models of interval scaleresponses with finite second moments. Models with complexpredictors, complex responses, or both, motivate thepresentation.
Reviews / Votes
"This text successfully offers a unified context for the theory ofunivariate, multivariate, and mixed modeling settings and may beuseful supplemental text for individuals interested in multivariatemodeling." (Journal of the American Statistician, December2008) "I believe that this text provides an important contribution tothe long-memory time series literature. I feel that itlargely achieves its aims and could be useful for those instructorswishing to teach a semester-long special topics course ... .Istrongly recommend this book to anyone interested in long-memorytime series. Both researchers and beginners alike will findthis text extremely useful." (Journal of the AmericanStatistician, December 2008) "The book will certainly be useful for Ph.D. students andresearchers in biostatistics who want to learn a little bit oftheory of linear models." (Mathematical Reviews, 2007) "...stands out from the others...will certainly have itsenthusiastic supporters." (Biometrics, March 2007) "...an excellent book for graduate students and professionalresearchers." (MAA Reviews, February 2007) "The focus of this book is on linear models with correlatedobservations and Gaussian errors." (Zentralblatt MATH, April2007)More details
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Persons
KEITH E. MULLER, PhD, is Professor and Director of theDivision of Biostatistics in the Department of Epidemiology andHealth Policy Research in the College of Medicine at the Universityof Florida in Gainesville, as well as Professor Emeritus ofBiostatistics at The University of North Carolina at Chapel Hillwhere the book was written.
PAUL W. STEWART, PhD, is Research Associate Professor ofBiostatistics at The University of North Carolina at ChapelHill.
Content
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 ¿.
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 T.
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 ß (Means, Fixed Effects).
5.6 Tests of Covariance Parameters, t (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, vector, 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 ß.
11.7 Unrestricted BLUE Estimation of ß.
11.8 Unrestricted Least Squares Estimation of ß.
11.9 Unrestricted Maximum Likelihood Estimation of ¿.
11.10 Unrestricted BLUE of ¿.
11.11 Related Distributions.
11.12 Formulations of Explicit Restrictions of ß and ¿.
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 Estimability in the Multivariate GLM.
12.3 Unrestricted Likelihood Estimation.
12.4 Estimation of Secondary Parameters.
12.5 Estimation with Multivariate Restrictions.
12.6 Unrestricted Estimation With Compound Symmetry: the "Univariate" Approach to Repeated Measures.
Exercises.
13. Estimation for Generalizations of Multivariate Models.
13.1 Motivation.
13.2 Criteria and Algorithms.
13.3 Weighted Estimation of ß and S.
13.4 Transformations among Growth Curve Designs.
13.5 Within-Individual Design matrices.
13.6 Estimation Methods.
13.7 Relationships to the Univariate and Mixed Models.
Exercises.
14. Estimation for Linear Mixed Models.
14.1 Motivation.
14.2 Statement of the General Linear Mixed Models.
14.3 Estimation and Estimability.
14.4 Some Special Types of Models.
14.5 ML Estimation.
14.6 REML Estimation.
14.7 Small-Sample Properties of Estimators.
14.8 Large-Sample Properties of Variance Estimators.
14.9 Conditional Estimation of di and BLUP Prediction.
Exercises.
PART IV: TESTS IN GAUSSIAN LINEAR MODELS.
15. Tests for Univariate Linear Models.
15.1 Motivation.
15.2 Testability of Univariate Hypotheses.
15.3 Tests of a Priori Hypotheses.
15.4 Related Distributions.
15.5 Transformations and Invariance Properties.
15.6 Confidence Regions for ¿.
Exercises.
16. Tests for Multivariate Linear Models.
16.1 Motivation.
16.2 Testability of Multivariate Hypotheses.
16.3 Tests of a Priori Hypotheses.
16.4 Linear Invariance.
16.5 Four Multivariate Test Statistics.
16.6 Which Multivariate Test Is Best?
16.7 Univariate Approach to Repeated Measures: UNIREP.
16.8 More on Invariance Properties.
16.9 Tests of Hypotheses about S.
16.10 Confidence Regions for T.
Exercises.
17. Tests for Generalizations of Multivariate Linear Models.
17.1 Motivation.
17.2 Doubly Multivariate Models.
17.3 Missing Responses in Multivariate Linear Models.
17.4 Exact and Approximate Weighted Least Squares.
17.5 Seemingly Unrelated Regressions.
17.6 Growth Curve Models (GMANOVA).
17.7 Testing Hypotheses in the GCM.
17.8 Confidence Bands for Growth Curves.
18. Tests for Linear Mixed Models.
18.1 Overview.
18.2 Estimability of ¿ = Cß.
18.3 Likelihood Ratio Tests of Cß.
18.4 Likelihood Ratio Tests Involving t.
18.5 Test Size of Wald-Type Tests of ß Using REML.
18.6 Using Wald-Type Tests of ß with REML.
18.7 Using Wald-Type Tests of {ß,t} with REML.
19. A Review of Multivariate and Univariate Linear Models.
19.1 Matrix Gaussian and Wishart Properties.
19.2 Design Matrix Properties.
19.3 Model Components.
19.4 Primary Parameter and Related Estimators.
19.5 Secondary Parameter Estimators.
19.6 Added-Last and Added-in-Order Tests.
PART V: CHOOSING A SIMPLE SIZE IN GAUSSIAN LINEAR MODELS.
20. Sample Size for Univariate Linear Model.
20.1 Sample Size Consulting: Before You begin.
20.2 The Machinery of a Power Analysis.
20.3 Independent t Example.
20.4 Paired t Example.
20.5 The Impact of s2 or ß in Power analysis.
20.6 Random Predictors.
20.7 Internal Pilot Design.
20.8 Other Criteria for Choosing a Sample Size.
Exercises.
21. Sample Size for Multivariate Linear Model.
21.1 The Machinery of a Power Analysis.
21.2 Paired t Example.
21.3 Time by Treatment Example.
21.4 Comparing between and within Designs.
21.5 Some Invariance Properties.
21.6 Random Predictors.
21.7 Internal Pilot Designs.
Exercises.
22. Sample Size for Generalizations of Multivariate Models.
22.1 Motivation.
22.2 Sample Size Methods for Growth Curve Models.
23. Sample Size for Linear Mixed Models.
23.1 Motivation.
23.2 Methods.
23.3 Internal Pilot Designs.
Appendix: Computing Resources.
References.
Index.
