Linear Model Methodology

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ISBN: 9781584884811 | 1584884819
Cover: Hardcover
Copyright: 10/21/2009

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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.

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¿^1/2is 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 ß	p. 137
Maximum Likelihood Estimation	p. 140
Properties of Maximum Likelihood Estimators	p. 141
Inference Concerning ß	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 Scheffé's Simultaneous Confidence Intervals and the F-Test Concerning H₀ : Aß = 0	p. 194
Determination of an Influential Set of Estimable Linear Functions	p. 196
Bonferroni's Intervals	p. 199
¿idák'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
Spjøtvoll'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 Scheffé'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/¿_ϵ²	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 H₀ : ¿¿ß = 0	p. 386
Exact Tests Concerning ¿_¿² and ¿_ß²	p. 388
Exact Tests for Random Higher-Order Models	p. 397
Inference Concerning the Mixed Two-Way Model	p. 398
Exact Tests Concerning ¿_ß² and ¿_¿_ß²	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 ¿_ß²	p. 407
Inference Concerning the Mixed Two-Fold Nested Model	p. 411
An Exact Test Concerning ¿_ß(_¿)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 H₀ : &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 ß	p. 484
Computation of ß 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 ß	p. 506
Exercises	p. 509
Bibliography	p. 515
Index	p. 535
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