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Prefacep. xiii
Introductionp. 1
Simple Linear Regression Modelp. 1
Multiple Linear Regression Modelp. 2
Analysis-of-Variance Modelsp. 3
Matrix Algebrap. 5
Matrix and Vector Notationp. 5
Matrices, Vectors, and Scalarsp. 5
Matrix Equalityp. 6
Transposep. 7
Matrices of Special Formp. 7
Operationsp. 9
Sum of Two Matrices or Two Vectorsp. 9
Product of a Scalar and a Matrixp. 10
Product of Two Matrices or Two Vectorsp. 10
Hadamard Product of Two Matrices or Two Vectorsp. 16
Partitioned Matricesp. 16
Rankp. 19
Inversep. 21
Positive Definite Matricesp. 24
Systems of Equationsp. 28
Generalized Inversep. 32
Definition and Propertiesp. 33
Generalized Inverses and Systems of Equationsp. 36
Determinantsp. 37
Orthogonal Vectors and Matricesp. 41
Tracep. 44
Eigenvalues and Eigenvectorsp. 46
Definitionp. 46
Functions of a Matrixp. 49
Productsp. 50
Symmetric Matricesp. 51
Positive Definite and Semidefinite Matricesp. 53
Idempotent Matricesp. 54
Vector and Matrix Calculusp. 56
Derivatives of Functions of Vectors and Matricesp. 56
Derivatives Involving Inverse Matrices and Determinantsp. 58
Maximization or Minimization of a Function of a Vectorp. 60
Random Vectors and Matricesp. 69
Introductionp. 69
Means, Variances, Covariances, and Correlationsp. 70
Mean Vectors and Covariance Matrices for Random Vectorsp. 75
Mean Vectorsp. 75
Covariance Matrixp. 75
Generalized Variancep. 77
Standardized Distancep. 77
Correlation Matricesp. 77
Mean Vectors and Covariance Matrices for Partitioned Random Vectorsp. 78
Linear Functions of Random Vectorsp. 79
Meansp. 80
Variances and Covariancesp. 81
Multivariate Normal Distributionp. 87
Univariate Normal Density Functionp. 87
Multivariate Normal Density Functionp. 88
Moment Generating Functionsp. 90
Properties of the Multivariate Normal Distributionp. 92
Partial Correlationp. 100
Distribution of Quadratic Forms in yp. 105
Sums of Squaresp. 105
Mean and Variance of Quadratic Formsp. 107
Noncentral Chi-Square Distributionp. 112
Noncentral F and t Distributionsp. 114
Noncentral F Distributionp. 114
Noncentral t Distributionp. 116
Distribution of Quadratic Formsp. 117
Independence of Linear Forms and Quadratic Formsp. 119
Simple Linear Regressionp. 127
The Modelp. 127
Estimation of [beta subscript 0], [beta subscript 1], and [sigma superscript 2]p. 128
Hypothesis Test and Confidence Interval for [beta subscript 1]p. 132
Coefficient of Determinationp. 133
Multiple Regression: Estimationp. 137
Introductionp. 137
The Modelp. 137
Estimation of [beta] and [sigma superscript 2]p. 141
Least-Squares Estimator for [beta]p. 145
Properties of the Least-Squares Estimator [beta]p. 141
An Estimator for [sigma superscript 2]p. 149
Geometry of Least-Squaresp. 151
Parameter Space, Data Space, and Prediction Spacep. 152
Geometric Interpretation of the Multiple Linear Regression Modelp. 153
The Model in Centered Formp. 154
Normal Modelp. 157
Assumptionsp. 157
Maximum Likelihood Estimators for [beta] and [sigma superscript 2]p. 158
Properties of [beta] and [sigma superscript 2]p. 159
R[superscript 2] in Fixed-x Regressionp. 161
Generalized Least-Squares: cov(y) = [sigma superscript 2]Vp. 164
Estimation of [beta] and [sigma superscript 2] when cov(y) = [sigma superscript 2]Vp. 164
Misspecification of the Error Structurep. 167
Model Misspecificationp. 169
Orthogonalizationp. 174
Multiple Regression: Tests of Hypotheses and Confidence Intervalsp. 185
Test of Overall Regressionp. 185
Test on a Subset of the [beta] Valuesp. 189
F Test in Terms of R[superscript 2]p. 196
The General Linear Hypothesis Tests for H[subscript 0]: C[beta] = 0 and H[subscript 0]: C[beta] = tp. 198
The Test for H[subscript 0]: C[beta] = 0p. 198
The Test for H[subscript 0]: C[beta] = tp. 203
Tests on [beta subscript j] and a' [beta]p. 204
Testing One [beta subscript j] or One a' [beta]p. 204
Testing Several [beta subscript j] or a'[subscript i beta] Valuesp. 205
Confidence Intervals and Prediction Intervalsp. 209
Confidence Region for [beta]p. 209
Confidence Interval for [beta subscript j]p. 210
Confidence Interval for a'[beta]p. 211
Confidence Interval for E(y)p. 211
Prediction Interval for a Future Observationp. 213
Confidence Interval for [sigma superscript 2]p. 215
Simultaneous Intervalsp. 215
Likelihood Ratio Testsp. 217
Multiple Regression: Model Validation and Diagnosticsp. 227
Residualsp. 227
The Hat Matrixp. 230
Outliersp. 232
Influential Observations and Leveragep. 235
Multiple Regression: Random x'sp. 243
Multivariate Normal Regression Modelp. 244
Estimation and Testing in Multivariate Normal Regressionp. 245
Standardized Regression Coefficientsp. 249
R[superscript 2] in Multivariate Normal Regressionp. 254
Tests and Confidence Intervals for R[superscript 2]p. 258
Effect of Each Variable on R[superscript 2]p. 262
Prediction for Multivariate Normal or Nonnormal Datap. 265
Sample Partial Correlationsp. 266
Multiple Regression: Bayesian Inferencep. 277
Elements of Bayesian Statistical Inferencep. 277
A Bayesian Multiple Linear Regression Modelp. 279
A Bayesian Multiple Regression Model with a Conjugate Priorp. 280
Marginal Posterior Density of [beta]p. 282
Marginal Posterior Densities of [tau] and [sigma superscript 2]p. 284
Inference in Bayesian Multiple Linear Regressionp. 285
Bayesian Point and Interval Estimates of Regression Coefficientsp. 285
Hypothesis Tests for Regression Coefficients in Bayesian Inferencep. 286
Special Cases of Inference in Bayesian Multiple Regression Modelsp. 286
Bayesian Point and Interval Estimation of [sigma superscript 2]p. 287
Bayesian Inference through Markov Chain Monte Carlo Simulationp. 288
Posterior Predictive Inferencep. 290
Analysis-of-Variance Modelsp. 295
Non-Full-Rank Modelsp. 295
One-Way Modelp. 295
Two-Way Modelp. 299
Estimationp. 301
Estimation of [beta]p. 302
Estimable Functions of [beta]p. 305
Estimatorsp. 309
Estimators of [lambda]'[beta]p. 309
Estimation of [sigma superscript 2]p. 313
Normal Modelp. 314
Geometry of Least-Squares in the Overparameterized Modelp. 316
Reparameterizationp. 318
Side Conditionsp. 320
Testing Hypothesesp. 323
Testable Hypothesesp. 323
Full-Reduced-Model Approachp. 324
General Linear Hypothesisp. 326
An Illustration of Estimation and Testingp. 329
Estimable Functionsp. 330
Testing a Hypothesisp. 331
Orthogonality of Columns of Xp. 333
One-Way Analysis-of-Variance: Balanced Casep. 339
The One-Way Modelp. 339
Estimable Functionsp. 340
Estimation of Parametersp. 341
Solving the Normal Equationsp. 341
An Estimator for [sigma superscript 2]p. 343
Testing the Hypothesis H[subscript 0]: [mu subscript 1] = [mu subscript 2] = ... = [mu subscript k]p. 344
Full-Reduced-Model Approachp. 344
General Linear Hypothesisp. 348
Expected Mean Squaresp. 351
Full-Reduced-Model Approachp. 352
General Linear Hypothesisp. 354
Contrastsp. 357
Hypothesis Test for a Contrastp. 357
Orthogonal Contrastsp. 358
Orthogonal Polynomial Contrastsp. 363
Two-Way Analysis-of-Variance: Balanced Casep. 377
The Two-Way Modelp. 377
Estimable Functionsp. 378
Estimators of [lambda]'[beta] and [sigma superscript 2]p. 382
Solving the Normal Equations and Estimating [lambda]'[beta]p. 382
An Estimator for [sigma superscript 2]p. 384
Testing Hypothesesp. 385
Test for Interactionp. 385
Tests for Main Effectsp. 395
Expected Mean Squaresp. 403
Sums-of-Squares Approachp. 403
Quadratic Form Approachp. 405
Analysis-of-Variance: The Cell Means Model for Unbalanced Datap. 413
Introductionp. 413
One-Way Modelp. 415
Estimation and Testingp. 415
Contrastsp. 417
Two-Way Modelp. 421
Unconstrained Modelp. 421
Constrained Modelp. 428
Two-Way Model with Empty Cellsp. 432
Analysis-of-Covariancep. 443
Introductionp. 443
Estimation and Testingp. 444
The Analysis-of-Covariance Modelp. 444
Estimationp. 446
Testing Hypothesesp. 448
One-Way Model with One Covariatep. 449
The Modelp. 449
Estimationp. 449
Testing Hypothesesp. 450
Two-Way Model with One Covariatep. 457
Tests for Main Effects and Interactionsp. 458
Test for Slopep. 462
Test for Homogeneity of Slopesp. 463
One-Way Model with Multiple Covariatesp. 464
The Modelp. 464
Estimationp. 465
Testing Hypothesesp. 468
Analysis-of-Covariance with Unbalanced Modelsp. 473
Linear Mixed Modelsp. 479
Introductionp. 479
The Linear Mixed Modelp. 479
Examplesp. 481
Estimation of Variance Componentsp. 486
Inference for [beta]p. 490
An Estimator for [beta]p. 490
Large-Sample Inference for Estimable Functions of [beta]p. 491
Small-Sample Inference for Estimable Functions of [beta]p. 491
Inference for the a[subscript i] Termsp. 497
Residual Diagnosticsp. 501
Additional Modelsp. 507
Nonlinear Regressionp. 507
Logistic Regressionp. 508
Loglinear Modelsp. 511
Poisson Regressionp. 512
Generalized Linear Modelsp. 513
Answers and Hints to the Problemsp. 517
Referencesp. 653
Indexp. 663
Table of Contents provided by Ingram. All Rights Reserved.p. 313Normal Modelp. 314Geometry of Least-Squares in the Overparameterized Modelp. 316Reparameterizationp. 318Side Conditionsp. 320Testing Hypothesesp. 323Testable Hypothesesp. 323Full-Reduced-Model Approachp. 324General Linear Hypothesisp. 326An Illustration of Estimation and Testingp. 329Estimable Functionsp. 330Testing a Hypothesisp. 331Orthogonality of Columns of Xp. 333One-Way Analysis-of-Variance: Balanced Casep. 339The One-Way Modelp. 339Estimable Functionsp. 340Estimation of Parametersp. 341Solving the Normal Equationsp. 341An Estimator for [sigma superscript 2]p. 343Testing the Hypothesis H[subscript 0]: [mu subscript 1] = [mu subscript 2] = ... = [mu subscript k]p. 344Full-Reduced-Model Approachp. 344General Linear Hypothesisp. 348Expected Mean Squaresp. 351Full-Reduced-Model Approachp. 352General Linear Hypothesisp. 354Contrastsp. 357Hypothesis Test for a Contrastp. 357Orthogonal Contrastsp. 358Orthogonal Polynomial Contrastsp. 363Two-Way Analysis-of-Variance: Balanced Casep. 377The Two-Way Modelp. 377Estimable Functionsp. 378Estimators of [lambda]'[beta] and [sigma superscript 2]p. 382Solving the Normal Equations and Estimating [lambda]'[beta]p. 382An Estimator for [sigma superscript 2]p. 384Testing Hypothesesp. 385Test for Interactionp. 385Tests for Main Effectsp. 395Expected Mean Squaresp. 403Sums-of-Squares Approachp. 403Quadratic Form Approachp. 405Analysis-of-Variance: The Cell Means Model for Unbalanced Datap. 413Introductionp. 413One-Way Modelp. 415Estimation and Testingp. 415Contrastsp. 417Two-Way Modelp. 421Unconstrained Modelp. 421Constrained Modelp. 428Two-Way Model with Empty Cellsp. 432Analysis-of-Covariancep. 443Introductionp. 443Estimation and Testingp. 444The Analysis-of-Covariance Modelp. 444Estimationp. 446Testing Hypothesesp. 448One-Way Model with One Covariatep. 449The Modelp. 449Estimationp. 449Testing Hypothesesp. 450Two-Way Model with One Covariatep. 457Tests for Main Effects and Interactionsp. 458Test for Slopep. 462Test for Homogeneity of Slopesp. 463One-Way Model with Multiple Covariatesp. 464The Modelp. 464Estimationp. 465Testing Hypothesesp. 468Analysis-of-Covariance with Unbalanced Modelsp. 473Linear Mixed Modelsp. 479Introductionp. 479The Linear Mixed Modelp. 479Examplesp. 481Estimation of Variance Componentsp. 486Inference for [beta]p. 490An Estimator for [beta]p. 490Large-Sample Inference for Estimable Functions of [beta]p. 491Small-Sample Inference for Estimable Functions of [beta]p. 491Inference for the a[subscript i] Termsp. 497Residual Diagnosticsp. 501Additional Modelsp. 507Nonlinear Regressionp. 507Logistic Regressionp. 508Loglinear Modelsp. 511Poisson Regressionp. 512Generalized Linear Modelsp. 513Answers and Hints to the Problemsp. 517Referencesp. 653Indexp. 663Table of Contents provided by Ingram. All Rights Reserved.

This completely revised and updated edition develops the basic theory of linear models for regression, analysis of variance, analysis of covariance, and linear mixed models. Through the expansion of relevant material and the inclusion of the latest technological developments in the field, this book provides readers with the theoretical foundation to correctly interpret computer software output as well as effectively use, customize, and understand linear models.