Given the importance of linear models in statistical theory and experimental research, a good understanding of their fundamental principles and theory is essential. Supported by a large number of examples, Linear Model Methodology provides a strong foundation in the theory of linear models and explores the latest developments in data analysis.
After presenting the historical evolution of certain methods and techniques used in linear models, the book reviews vector spaces and linear transformations and discusses the basic concepts and results of matrix algebra that are relevant to the study of linear models. Although mainly focused on classical linear models, the next several chapters also explore recent techniques for solving well-known problems that pertain to the distribution and independence of quadratic forms, the analysis of estimable linear functions and contrasts, and the general treatment of balanced random and mixed-effects models. The author then covers more contemporary topics in linear models, including the adequacy of Satterthwaite's approximation, unbalanced fixed- and mixed-effects models, heteroscedastic linear models, response surface models with random effects, and linear multiresponse models. The final chapter introduces generalized linear models, which represent an extension of classical linear models.
Linear models provide the groundwork for analysis of variance, regression analysis, response surface methodology, variance components analysis, and more, making it necessary to understand the theory behind linear modeling. Reflecting advances made in the last thirty years, this book offers a rigorous development of the theory underlying linear models.
Is sum, this is a carefully written and reliable book that reflects the experience of the author in teaching graduate level courses on linear models. I will certainly add it to the list of reference textbooks for the graduate one-quarter course on linear model theory taught at UC Santa Cruz. --Raquel Prado, University of California-Santa Cruz The material is well chosen and well organized, and includes many results that are not found in other textbooks. ! Throughout the book, the presentation is very clear and well organized, with a focus on mathematical developments. Most results are stated with proofs, some material is based on the author's own contributions to the field. Generally, many important special cases are treated in detail, which will make the book also highly useful as a reference. There are also many worked-out examples from different subject areas to illustrate the methods. Later chapters also include some instructions on how to use the methods in SAS. Furthermore, there are lots of exercises at the end of each chapter. ! The book is very accessible and encompassing ! the book will be an excellent choice both as a text and as a reference book. --T. Mildenberger, Statistical Papers, April 2011 The material on which this book is based has been taught in a couple of courses at the University of Florida for about 20 years and the author's skills and experience in doing this are superbly represented in this fine text. ! there are numerous exercises that reinforce both the theoretical and the practical aspects of regression! This is an excellent, reliable, and comprehensive text. --International Statistical Review (2010), 78 This book provides a thorough overview which is similar to other available texts but in a very different way. The choice of topics covered, their organization and presentation are the unique features that distinguish this book. ! This book is well structured as a textbook as well as a reference with every chapter explaining the definitions, principles and methods of the subject matter illustrated by data-based examples with the details on use of SAS software, wherever possible. ! the topics that are covered in Chapters 7--12 are not generally found in a single book. ! The book would make an excellent textbook for a course on linear models at masters and graduate levels. Moreover, some parts of the book can also be a part of a course on analysis of variance. Overall, the book is a valuable reference for those involved in research and teaching in this area. --Journal of the Royal Statistical Society, Series A, 2010
Preface | p. xv |
Author | p. xix |
Linear Models: Some Historical Perspectives | p. 1 |
The Invention of Least Squares | p. 3 |
The Gauss-Markov Theorem | p. 4 |
Estimability | p. 4 |
Maximum Likelihood Estimation | p. 5 |
Analysis of Variance | p. 6 |
Balanced and Unbalanced Data | p. 7 |
Quadratic Forms and Craig's Theorem | p. 8 |
The Role of Matrix Algebra | p. 9 |
The Geometric Approach | p. 10 |
Basic Elements of Linear Algebra | p. 13 |
Introduction | p. 13 |
Vector Spaces | p. 13 |
Vector Subspaces | p. 14 |
Bases and Dimensions of Vector Spaces | p. 16 |
Linear Transformations | p. 17 |
Exercises | p. 20 |
Basic Concepts in Matrix Algebra | p. 23 |
Introduction and Notation | p. 23 |
Notation | p. 24 |
Some Particular Types of Matrices | p. 24 |
Basic Matrix Operations | p. 25 |
Partitioned Matrices | p. 27 |
Determinants | p. 28 |
The Rank of a Matrix | p. 31 |
The Inverse of a Matrix | p. 33 |
Generalized Inverse of a Matrix | p. 34 |
Eigenvalues and Eigenvectors | p. 34 |
Idempotent and Orthogonal Matrices | p. 36 |
Parameterization of Orthogonal Matrices | p. 36 |
Quadratic Forms | p. 39 |
Decomposition Theorems | p. 40 |
Some Matrix Inequalities | p. 43 |
Function of Matrices | p. 46 |
Matrix Differentiation | p. 48 |
Exercises | p. 52 |
The Multivariate Normal Distribution | p. 59 |
History of the Normal Distribution | p. 59 |
The Univariate Normal Distribution | p. 60 |
The Multivariate Normal Distribution | p. 61 |
The Moment Generating Function | p. 63 |
The General Case | p. 63 |
The Case of the Multivariate Normal | p. 65 |
Conditional Distribution | p. 67 |
The Singular Multivariate Normal Distribution | p. 69 |
Related Distributions | p. 69 |
The Central Chi-Squared Distribution | p. 70 |
The Noncentral Chi-Squared Distribution | p. 70 |
The t-Distribution | p. 73 |
The F-Distribution | p. 74 |
The Wishart Distribution | p. 75 |
Examples and Additional Results | p. 75 |
Some Misconceptions about the Normal Distribution | p. 77 |
Characterization Results | p. 78 |
Exercises | p. 80 |
Quadratic Forms in Normal Variables | p. 89 |
The Moment Generating Function | p. 89 |
Distribution of Quadratic Forms | p. 94 |
Independence of Quadratic Forms | p. 103 |
Independence of Linear and Quadratic Forms | p. 108 |
Independence and Chi-Squaredness of Several Quadratic Forms | p. 111 |
Computing the Distribution of Quadratic Forms | p. 118 |
Distribution of a Ratio of Quadratic Forms | p. 119 |
Positive Definiteness of the Matrix W_{t}_{-1} in (5.2) | p. 120 |
A is Idempotent if and Only If ^{1/2} A^{1/2}is Idempotent | p. 121 |
Exercises | p. 121 |
Full-Rank Linear Models | p. 127 |
Least-Squares Estimation | p. 128 |
Estimation of the Mean Response | p. 130 |
Properties of Ordinary Least-Squares Estimation | p. 132 |
Distributional Properties | p. 132 |
Properties under the Normality Assumption | p. 133 |
The Gauss-Markov Theorem | p. 134 |
Generalized Least-Squares Estimation | p. 137 |
Least-Squares Estimation under Linear Restrictions on ÃŸ | p. 137 |
Maximum Likelihood Estimation | p. 140 |
Properties of Maximum Likelihood Estimators | p. 141 |
Inference Concerning ÃŸ | p. 146 |
Confidence Regions and "Confidence Intervals | p. 148 |
Simultaneous Confidence Intervals | p. 148 |
The Likelihood Ratio Approach to Hypothesis Testing | p. 149 |
Examples and Applications | p. 151 |
Confidence Region for the Location of the Optimum | p. 151 |
Confidence Interval on the True Optimum | p. 154 |
Confidence Interval for a Ratio | p. 157 |
Demonstrating the Gauss-Markov Theorem | p. 159 |
Comparison of Two Linear Models | p. 162 |
Exercises | p. 169 |
Less-Than-Full-Rank Linear Models | p. 179 |
Parameter Estimation | p. 179 |
Some Distributional Properties | p. 180 |
Reparameterized Model | p. 181 |
Estimable Linear Functions | p. 184 |
Properties of Estimable Functions | p. 185 |
Testable Hypotheses | p. 187 |
Simultaneous Confidence Intervals on Estimable Linear Functions | p. 192 |
The Relationship between ScheffÃ©'s Simultaneous Confidence Intervals and the F-Test Concerning H_{0} : AÃŸ = 0 | p. 194 |
Determination of an Influential Set of Estimable Linear Functions | p. 196 |
Bonferroni's Intervals | p. 199 |
idÃ¡k's Intervals | p. 200 |
Simultaneous Confidence Intervals on All Contrasts among the Means with Heterogeneous Group Variances | p. 202 |
The Brown-Forsythe Intervals | p. 202 |
SpjÃ¸tvoll's Intervals | p. 203 |
The Special Case of Contrasts | p. 205 |
Exact Conservative Intervals | p. 206 |
Further Results Concerning Contrasts and Estimable Linear Functions | p. 209 |
A Geometrical Representation of Contrasts | p. 209 |
Simultaneous Confidence Intervals for Two Estimable Linear Functions and their Ratio | p. 213 |
Simultaneous Confidence Intervals Based on ScheffÃ©'s Method | p. 213 |
Simultaneous Confidence Intervals Based on the Bonferroni Inequality | p. 214 |
Conservative Simultaneous Confidence Intervals | p. 214 |
Exercises | p. 216 |
Balanced Linear Models | p. 225 |
Notation and Definitions | p. 225 |
The General Balanced Linear Model | p. 229 |
Properties of Balanced Models | p. 232 |
Balanced Mixed Models | p. 237 |
Distribution of Sums of Squares | p. 238 |
Estimation of Fixed Effects | p. 240 |
Complete and Sufficient Statistics | p. 249 |
ANOVA Estimation of Variance Components | p. 254 |
The Probability of a Negative ANOVA Estimator | p. 254 |
Confidence Intervals on Continuous Functions of the Variance Components | p. 257 |
Confidence Intervals on Linear Functions of the Variance Components | p. 259 |
Confidence Intervals on Ratios of Variance Components | p. 263 |
Exercises | p. 266 |
The Adequacy of Satterthwaite's Approximation | p. 271 |
Satterthwaite's Approximation | p. 271 |
A Special Case: The Behrens-Fisher Problem | p. 274 |
Adequacy of Satterthwaite's Approximation | p. 278 |
Testing Departure from Condition (9.35) | p. 282 |
Measuring the Closeness of Satterthwaite's Approximation | p. 287 |
Determination of _{sup} | p. 290 |
Examples | p. 290 |
The Behrens-Fisher Problem | p. 291 |
A Confidence Interval on the Total Variation | p. 293 |
A Linear Combination of Mean Squares | p. 296 |
Determination of the Matrix G in Section 9.2.1 | p. 297 |
Exercises | p. 297 |
Unbalanced Fixed-Effects Models | p. 301 |
The R-Notation | p. 301 |
Two-Way Models without Interaction | p. 304 |
Estimable Linear Functions for Model (10.10) | p. 305 |
Testable Hypotheses for Model (10.10) | p. 306 |
Type I Testable Hypotheses | p. 309 |
Type II Testable Hypotheses | p. 310 |
Two-Way Models with Interaction | p. 314 |
Tests of Hypotheses | p. 315 |
Testing the Interaction Effect | p. 318 |
Type III Analysis in SAS | p. 322 |
Other Testable Hypotheses | p. 324 |
Higher-Order Models | p. 327 |
A Numerical Example | p. 331 |
The Method of Unweighted Means | p. 336 |
Distributions of SS_{Au}, SS_{Bu}, and SS_{ABu} | p. 338 |
Approximate Distributions of SS_{Au}, SS_{Bu} and SS_{ABu} | p. 340 |
Exercises | p. 342 |
Unbalanced Random and Mixed Models | p. 349 |
Estimation of Variance Components | p. 350 |
ANOVA Estimation-Henderson's Methods | p. 350 |
Henderson's Method III | p. 351 |
Maximum Likelihood Estimation | p. 357 |
Restricted Maximum Likelihood Estimation | p. 362 |
Properties of REML Estimators | p. 366 |
Estimation of Estimable Linear Functions | p. 369 |
Inference Concerning the Random One-Way Model | p. 373 |
Adequacy of the Approximation | p. 376 |
Confidence Intervals on _{}^{2} and _{}^{2}/_{ϵ}^{2} | p. 379 |
Inference Concerning the Random Two-Way Model | p. 380 |
Approximate Tests Based on the Method of Unweighted Means | p. 380 |
Adequacy of the Approximation | p. 384 |
Exact Tests | p. 385 |
Exact Test Concerning H_{0} : ÃŸ = 0 | p. 386 |
Exact Tests Concerning _{}^{2} and _{ÃŸ}^{2} | p. 388 |
Exact Tests for Random Higher-Order Models | p. 397 |
Inference Concerning the Mixed Two-Way Model | p. 398 |
Exact Tests Concerning _{ÃŸ}^{2} and _{}_{ÃŸ}^{2} | p. 398 |
An Exact Test for the Fixed Effects | p. 401 |
Inference Concerning the Random Two-Fold Nested Model | p. 406 |
An Exact Test Concerning _{ÃŸ}^{2} | p. 407 |
Inference Concerning the Mixed Two-Fold Nested Model | p. 411 |
An Exact Test Concerning _{ÃŸ}(_{})^{2} | p. 411 |
An Exact Test for the Fixed Effects | p. 412 |
Inference Concerning the General Mixed Linear Model | p. 415 |
Estimation and Testing of Fixed Effects | p. 416 |
Tests Concerning the Random Effects | p. 417 |
Appendix 11.A | p. 421 |
Appendix 11.B | p. 422 |
Exercises | p. 422 |
Additional Topics in Linear Models | p. 427 |
Heteroscedastic Linear Models | p. 427 |
The Random One-Way Model with Heterogeneous Error Variances | p. 428 |
An Approximate Test Concerning H_{0} : &sigma_{}^{2} = 0 | p. 430 |
Point and Interval Estimation of &sigma_{}^{2} | p. 433 |
Detecting Heterogeneity in Error Variances | p. 435 |
A Mixed Two-Fold Nested Model with Heteroscedastic Random Effects | p. 437 |
Tests Concerning the Fixed Effects | p. 438 |
Tests Concerning the Random Effects | p. 441 |
Response Surface Models | p. 443 |
Response Surface Models with Random Block Effects | p. 446 |
Analysis Concerning the Fixed Effects | p. 448 |
Analysis Concerning the Random Effects | p. 449 |
Linear Multiresponse Models | p. 453 |
Parameter Estimation | p. 454 |
Hypothesis Testing | p. 456 |
Hypothesis of Concurrence | p. 457 |
Hypothesis of Parallelism | p. 458 |
Testing for Lack of Fit | p. 459 |
Responses Contributing to LOF | p. 462 |
Exercises | p. 467 |
Generalized Linear Models | p. 473 |
Introduction | p. 473 |
The Exponential Family | p. 474 |
Likelihood Function | p. 478 |
Estimation of Parameters | p. 479 |
Estimation of the Mean Response | p. 483 |
Asymptotic Distribution of ÃŸ | p. 484 |
Computation of ÃŸ in SAS | p. 485 |
Goodness of Fit | p. 487 |
The Deviance | p. 487 |
Pearson's Chi-Square Statistic | p. 490 |
Residuals | p. 491 |
Hypothesis Testing | p. 497 |
Wald Inference | p. 497 |
Likelihood Ratio Inference | p. 498 |
Confidence Intervals | p. 499 |
Wald's Confidence Intervals | p. 499 |
Likelihood Ratio-Based Confidence Intervals | p. 500 |
Gamma-Distributed Response | p. 504 |
Deviance for the Gamma Distribution | p. 506 |
Variance-Covariance Matrix of ÃŸ | p. 506 |
Exercises | p. 509 |
Bibliography | p. 515 |
Index | p. 535 |
Table of Contents provided by Ingram. All Rights Reserved. |
ISBN: 9781584884811
ISBN-10: 1584884819
Audience:
Professional
Format:
Hardcover
Language:
English
Number Of Pages: 562
Published: 27th October 2009
Publisher: Taylor & Francis Inc
Country of Publication: US
Dimensions (cm): 23.5 x 15.6
x 3.18
Weight (kg): 0.94
Edition Number: 1