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Linear Model Methodology - Andre I. Khuri

Linear Model Methodology


Published: 27th October 2009
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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

Prefacep. xv
Authorp. xix
Linear Models: Some Historical Perspectivesp. 1
The Invention of Least Squaresp. 3
The Gauss-Markov Theoremp. 4
Estimabilityp. 4
Maximum Likelihood Estimationp. 5
Analysis of Variancep. 6
Balanced and Unbalanced Datap. 7
Quadratic Forms and Craig's Theoremp. 8
The Role of Matrix Algebrap. 9
The Geometric Approachp. 10
Basic Elements of Linear Algebrap. 13
Introductionp. 13
Vector Spacesp. 13
Vector Subspacesp. 14
Bases and Dimensions of Vector Spacesp. 16
Linear Transformationsp. 17
Exercisesp. 20
Basic Concepts in Matrix Algebrap. 23
Introduction and Notationp. 23
Notationp. 24
Some Particular Types of Matricesp. 24
Basic Matrix Operationsp. 25
Partitioned Matricesp. 27
Determinantsp. 28
The Rank of a Matrixp. 31
The Inverse of a Matrixp. 33
Generalized Inverse of a Matrixp. 34
Eigenvalues and Eigenvectorsp. 34
Idempotent and Orthogonal Matricesp. 36
Parameterization of Orthogonal Matricesp. 36
Quadratic Formsp. 39
Decomposition Theoremsp. 40
Some Matrix Inequalitiesp. 43
Function of Matricesp. 46
Matrix Differentiationp. 48
Exercisesp. 52
The Multivariate Normal Distributionp. 59
History of the Normal Distributionp. 59
The Univariate Normal Distributionp. 60
The Multivariate Normal Distributionp. 61
The Moment Generating Functionp. 63
The General Casep. 63
The Case of the Multivariate Normalp. 65
Conditional Distributionp. 67
The Singular Multivariate Normal Distributionp. 69
Related Distributionsp. 69
The Central Chi-Squared Distributionp. 70
The Noncentral Chi-Squared Distributionp. 70
The t-Distributionp. 73
The F-Distributionp. 74
The Wishart Distributionp. 75
Examples and Additional Resultsp. 75
Some Misconceptions about the Normal Distributionp. 77
Characterization Resultsp. 78
Exercisesp. 80
Quadratic Forms in Normal Variablesp. 89
The Moment Generating Functionp. 89
Distribution of Quadratic Formsp. 94
Independence of Quadratic Formsp. 103
Independence of Linear and Quadratic Formsp. 108
Independence and Chi-Squaredness of Several Quadratic Formsp. 111
Computing the Distribution of Quadratic Formsp. 118
Distribution of a Ratio of Quadratic Formsp. 119
Positive Definiteness of the Matrix Wt-1 in (5.2)p. 120
A is Idempotent if and Only If 1/2 A1/2is Idempotentp. 121
Exercisesp. 121
Full-Rank Linear Modelsp. 127
Least-Squares Estimationp. 128
Estimation of the Mean Responsep. 130
Properties of Ordinary Least-Squares Estimationp. 132
Distributional Propertiesp. 132
Properties under the Normality Assumptionp. 133
The Gauss-Markov Theoremp. 134
Generalized Least-Squares Estimationp. 137
Least-Squares Estimation under Linear Restrictions on ßp. 137
Maximum Likelihood Estimationp. 140
Properties of Maximum Likelihood Estimatorsp. 141
Inference Concerning ßp. 146
Confidence Regions and "Confidence Intervalsp. 148
Simultaneous Confidence Intervalsp. 148
The Likelihood Ratio Approach to Hypothesis Testingp. 149
Examples and Applicationsp. 151
Confidence Region for the Location of the Optimump. 151
Confidence Interval on the True Optimump. 154
Confidence Interval for a Ratiop. 157
Demonstrating the Gauss-Markov Theoremp. 159
Comparison of Two Linear Modelsp. 162
Exercisesp. 169
Less-Than-Full-Rank Linear Modelsp. 179
Parameter Estimationp. 179
Some Distributional Propertiesp. 180
Reparameterized Modelp. 181
Estimable Linear Functionsp. 184
Properties of Estimable Functionsp. 185
Testable Hypothesesp. 187
Simultaneous Confidence Intervals on Estimable Linear Functionsp. 192
The Relationship between Scheffé's Simultaneous Confidence Intervals and the F-Test Concerning H0 : Aß = 0p. 194
Determination of an Influential Set of Estimable Linear Functionsp. 196
Bonferroni's Intervalsp. 199
idák's Intervalsp. 200
Simultaneous Confidence Intervals on All Contrasts among the Means with Heterogeneous Group Variancesp. 202
The Brown-Forsythe Intervalsp. 202
Spjøtvoll's Intervalsp. 203
The Special Case of Contrastsp. 205
Exact Conservative Intervalsp. 206
Further Results Concerning Contrasts and Estimable Linear Functionsp. 209
A Geometrical Representation of Contrastsp. 209
Simultaneous Confidence Intervals for Two Estimable Linear Functions and their Ratiop. 213
Simultaneous Confidence Intervals Based on Scheffé's Methodp. 213
Simultaneous Confidence Intervals Based on the Bonferroni Inequalityp. 214
Conservative Simultaneous Confidence Intervalsp. 214
Exercisesp. 216
Balanced Linear Modelsp. 225
Notation and Definitionsp. 225
The General Balanced Linear Modelp. 229
Properties of Balanced Modelsp. 232
Balanced Mixed Modelsp. 237
Distribution of Sums of Squaresp. 238
Estimation of Fixed Effectsp. 240
Complete and Sufficient Statisticsp. 249
ANOVA Estimation of Variance Componentsp. 254
The Probability of a Negative ANOVA Estimatorp. 254
Confidence Intervals on Continuous Functions of the Variance Componentsp. 257
Confidence Intervals on Linear Functions of the Variance Componentsp. 259
Confidence Intervals on Ratios of Variance Componentsp. 263
Exercisesp. 266
The Adequacy of Satterthwaite's Approximationp. 271
Satterthwaite's Approximationp. 271
A Special Case: The Behrens-Fisher Problemp. 274
Adequacy of Satterthwaite's Approximationp. 278
Testing Departure from Condition (9.35)p. 282
Measuring the Closeness of Satterthwaite's Approximationp. 287
Determination of supp. 290
Examplesp. 290
The Behrens-Fisher Problemp. 291
A Confidence Interval on the Total Variationp. 293
A Linear Combination of Mean Squaresp. 296
Determination of the Matrix G in Section 9.2.1p. 297
Exercisesp. 297
Unbalanced Fixed-Effects Modelsp. 301
The R-Notationp. 301
Two-Way Models without Interactionp. 304
Estimable Linear Functions for Model (10.10)p. 305
Testable Hypotheses for Model (10.10)p. 306
Type I Testable Hypothesesp. 309
Type II Testable Hypothesesp. 310
Two-Way Models with Interactionp. 314
Tests of Hypothesesp. 315
Testing the Interaction Effectp. 318
Type III Analysis in SASp. 322
Other Testable Hypothesesp. 324
Higher-Order Modelsp. 327
A Numerical Examplep. 331
The Method of Unweighted Meansp. 336
Distributions of SSAu, SSBu, and SSABup. 338
Approximate Distributions of SSAu, SSBu and SSABup. 340
Exercisesp. 342
Unbalanced Random and Mixed Modelsp. 349
Estimation of Variance Componentsp. 350
ANOVA Estimation-Henderson's Methodsp. 350
Henderson's Method IIIp. 351
Maximum Likelihood Estimationp. 357
Restricted Maximum Likelihood Estimationp. 362
Properties of REML Estimatorsp. 366
Estimation of Estimable Linear Functionsp. 369
Inference Concerning the Random One-Way Modelp. 373
Adequacy of the Approximationp. 376
Confidence Intervals on 2 and 2/ϵ2p. 379
Inference Concerning the Random Two-Way Modelp. 380
Approximate Tests Based on the Method of Unweighted Meansp. 380
Adequacy of the Approximationp. 384
Exact Testsp. 385
Exact Test Concerning H0 : ß = 0p. 386
Exact Tests Concerning 2 and ß2p. 388
Exact Tests for Random Higher-Order Modelsp. 397
Inference Concerning the Mixed Two-Way Modelp. 398
Exact Tests Concerning ß2 and ß2p. 398
An Exact Test for the Fixed Effectsp. 401
Inference Concerning the Random Two-Fold Nested Modelp. 406
An Exact Test Concerning ß2p. 407
Inference Concerning the Mixed Two-Fold Nested Modelp. 411
An Exact Test Concerning ß()2p. 411
An Exact Test for the Fixed Effectsp. 412
Inference Concerning the General Mixed Linear Modelp. 415
Estimation and Testing of Fixed Effectsp. 416
Tests Concerning the Random Effectsp. 417
Appendix 11.Ap. 421
Appendix 11.Bp. 422
Exercisesp. 422
Additional Topics in Linear Modelsp. 427
Heteroscedastic Linear Modelsp. 427
The Random One-Way Model with Heterogeneous Error Variancesp. 428
An Approximate Test Concerning H0 : &sigma2 = 0p. 430
Point and Interval Estimation of &sigma2p. 433
Detecting Heterogeneity in Error Variancesp. 435
A Mixed Two-Fold Nested Model with Heteroscedastic Random Effectsp. 437
Tests Concerning the Fixed Effectsp. 438
Tests Concerning the Random Effectsp. 441
Response Surface Modelsp. 443
Response Surface Models with Random Block Effectsp. 446
Analysis Concerning the Fixed Effectsp. 448
Analysis Concerning the Random Effectsp. 449
Linear Multiresponse Modelsp. 453
Parameter Estimationp. 454
Hypothesis Testingp. 456
Hypothesis of Concurrencep. 457
Hypothesis of Parallelismp. 458
Testing for Lack of Fitp. 459
Responses Contributing to LOFp. 462
Exercisesp. 467
Generalized Linear Modelsp. 473
Introductionp. 473
The Exponential Familyp. 474
Likelihood Functionp. 478
Estimation of Parametersp. 479
Estimation of the Mean Responsep. 483
Asymptotic Distribution of ßp. 484
Computation of ß in SASp. 485
Goodness of Fitp. 487
The Deviancep. 487
Pearson's Chi-Square Statisticp. 490
Residualsp. 491
Hypothesis Testingp. 497
Wald Inferencep. 497
Likelihood Ratio Inferencep. 498
Confidence Intervalsp. 499
Wald's Confidence Intervalsp. 499
Likelihood Ratio-Based Confidence Intervalsp. 500
Gamma-Distributed Responsep. 504
Deviance for the Gamma Distributionp. 506
Variance-Covariance Matrix of ßp. 506
Exercisesp. 509
Bibliographyp. 515
Indexp. 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