Linear Models in Statistics

by Rencher, Alvin C.; Schaalje, G. Bruce

Edition: 2nd

ISBN13: 9780471754985

ISBN10: 0471754986

Format: Hardcover

Pub. Date: 2008-01-02

Publisher(s): Wiley-Interscience

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Summary

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.

Author Biography

Alvin C. Rencher, PhD, is Professor of Statistics at Brigham Young University. Dr. Rencher is a Fellow of the American Statistical Association and the author of Methods of Multivariate Analysis and Multivariate Statistical Inference and Applications, both published by Wiley.

G. Bruce Schaalje, PhD, is Professor of Statistics at Brigham Young University. He has authored over 120 journal articles in his areas of research interest, which include mixed linear models, small sample inference, and design of experiments.

Preface	p. xiii
Introduction	p. 1
Simple Linear Regression Model	p. 1
Multiple Linear Regression Model	p. 2
Analysis-of-Variance Models	p. 3
Matrix Algebra	p. 5
Matrix and Vector Notation	p. 5
Matrices, Vectors, and Scalars	p. 5
Matrix Equality	p. 6
Transpose	p. 7
Matrices of Special Form	p. 7
Operations	p. 9
Sum of Two Matrices or Two Vectors	p. 9
Product of a Scalar and a Matrix	p. 10
Product of Two Matrices or Two Vectors	p. 10
Hadamard Product of Two Matrices or Two Vectors	p. 16
Partitioned Matrices	p. 16
Rank	p. 19
Inverse	p. 21
Positive Definite Matrices	p. 24
Systems of Equations	p. 28
Generalized Inverse	p. 32
Definition and Properties	p. 33
Generalized Inverses and Systems of Equations	p. 36
Determinants	p. 37
Orthogonal Vectors and Matrices	p. 41
Trace	p. 44
Eigenvalues and Eigenvectors	p. 46
Definition	p. 46
Functions of a Matrix	p. 49
Products	p. 50
Symmetric Matrices	p. 51
Positive Definite and Semidefinite Matrices	p. 53
Idempotent Matrices	p. 54
Vector and Matrix Calculus	p. 56
Derivatives of Functions of Vectors and Matrices	p. 56
Derivatives Involving Inverse Matrices and Determinants	p. 58
Maximization or Minimization of a Function of a Vector	p. 60
Random Vectors and Matrices	p. 69
Introduction	p. 69
Means, Variances, Covariances, and Correlations	p. 70
Mean Vectors and Covariance Matrices for Random Vectors	p. 75
Mean Vectors	p. 75
Covariance Matrix	p. 75
Generalized Variance	p. 77
Standardized Distance	p. 77
Correlation Matrices	p. 77
Mean Vectors and Covariance Matrices for Partitioned Random Vectors	p. 78
Linear Functions of Random Vectors	p. 79
Means	p. 80
Variances and Covariances	p. 81
Multivariate Normal Distribution	p. 87
Univariate Normal Density Function	p. 87
Multivariate Normal Density Function	p. 88
Moment Generating Functions	p. 90
Properties of the Multivariate Normal Distribution	p. 92
Partial Correlation	p. 100
Distribution of Quadratic Forms in y	p. 105
Sums of Squares	p. 105
Mean and Variance of Quadratic Forms	p. 107
Noncentral Chi-Square Distribution	p. 112
Noncentral F and t Distributions	p. 114
Noncentral F Distribution	p. 114
Noncentral t Distribution	p. 116
Distribution of Quadratic Forms	p. 117
Independence of Linear Forms and Quadratic Forms	p. 119
Simple Linear Regression	p. 127
The Model	p. 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 Determination	p. 133
Multiple Regression: Estimation	p. 137
Introduction	p. 137
The Model	p. 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-Squares	p. 151
Parameter Space, Data Space, and Prediction Space	p. 152
Geometric Interpretation of the Multiple Linear Regression Model	p. 153
The Model in Centered Form	p. 154
Normal Model	p. 157
Assumptions	p. 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 Regression	p. 161
Generalized Least-Squares: cov(y) = [sigma superscript 2]V	p. 164
Estimation of [beta] and [sigma superscript 2] when cov(y) = [sigma superscript 2]V	p. 164
Misspecification of the Error Structure	p. 167
Model Misspecification	p. 169
Orthogonalization	p. 174
Multiple Regression: Tests of Hypotheses and Confidence Intervals	p. 185
Test of Overall Regression	p. 185
Test on a Subset of the [beta] Values	p. 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] = t	p. 198
The Test for H[subscript 0]: C[beta] = 0	p. 198
The Test for H[subscript 0]: C[beta] = t	p. 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] Values	p. 205
Confidence Intervals and Prediction Intervals	p. 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 Observation	p. 213
Confidence Interval for [sigma superscript 2]	p. 215
Simultaneous Intervals	p. 215
Likelihood Ratio Tests	p. 217
Multiple Regression: Model Validation and Diagnostics	p. 227
Residuals	p. 227
The Hat Matrix	p. 230
Outliers	p. 232
Influential Observations and Leverage	p. 235
Multiple Regression: Random x's	p. 243
Multivariate Normal Regression Model	p. 244
Estimation and Testing in Multivariate Normal Regression	p. 245
Standardized Regression Coefficients	p. 249
R[superscript 2] in Multivariate Normal Regression	p. 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 Data	p. 265
Sample Partial Correlations	p. 266
Multiple Regression: Bayesian Inference	p. 277
Elements of Bayesian Statistical Inference	p. 277
A Bayesian Multiple Linear Regression Model	p. 279
A Bayesian Multiple Regression Model with a Conjugate Prior	p. 280
Marginal Posterior Density of [beta]	p. 282
Marginal Posterior Densities of [tau] and [sigma superscript 2]	p. 284
Inference in Bayesian Multiple Linear Regression	p. 285
Bayesian Point and Interval Estimates of Regression Coefficients	p. 285
Hypothesis Tests for Regression Coefficients in Bayesian Inference	p. 286
Special Cases of Inference in Bayesian Multiple Regression Models	p. 286
Bayesian Point and Interval Estimation of [sigma superscript 2]	p. 287
Bayesian Inference through Markov Chain Monte Carlo Simulation	p. 288
Posterior Predictive Inference	p. 290
Analysis-of-Variance Models	p. 295
Non-Full-Rank Models	p. 295
One-Way Model	p. 295
Two-Way Model	p. 299
Estimation	p. 301
Estimation of [beta]	p. 302
Estimable Functions of [beta]	p. 305
Estimators	p. 309
Estimators of [lambda]'[beta]	p. 309
Estimation of [sigma superscript 2]	p. 313
Normal Model	p. 314
Geometry of Least-Squares in the Overparameterized Model	p. 316
Reparameterization	p. 318
Side Conditions	p. 320
Testing Hypotheses	p. 323
Testable Hypotheses	p. 323
Full-Reduced-Model Approach	p. 324
General Linear Hypothesis	p. 326
An Illustration of Estimation and Testing	p. 329
Estimable Functions	p. 330
Testing a Hypothesis	p. 331
Orthogonality of Columns of X	p. 333
One-Way Analysis-of-Variance: Balanced Case	p. 339
The One-Way Model	p. 339
Estimable Functions	p. 340
Estimation of Parameters	p. 341
Solving the Normal Equations	p. 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 Approach	p. 344
General Linear Hypothesis	p. 348
Expected Mean Squares	p. 351
Full-Reduced-Model Approach	p. 352
General Linear Hypothesis	p. 354
Contrasts	p. 357
Hypothesis Test for a Contrast	p. 357
Orthogonal Contrasts	p. 358
Orthogonal Polynomial Contrasts	p. 363
Two-Way Analysis-of-Variance: Balanced Case	p. 377
The Two-Way Model	p. 377
Estimable Functions	p. 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 Hypotheses	p. 385
Test for Interaction	p. 385
Tests for Main Effects	p. 395
Expected Mean Squares	p. 403
Sums-of-Squares Approach	p. 403
Quadratic Form Approach	p. 405
Analysis-of-Variance: The Cell Means Model for Unbalanced Data	p. 413
Introduction	p. 413
One-Way Model	p. 415
Estimation and Testing	p. 415
Contrasts	p. 417
Two-Way Model	p. 421
Unconstrained Model	p. 421
Constrained Model	p. 428
Two-Way Model with Empty Cells	p. 432
Analysis-of-Covariance	p. 443
Introduction	p. 443
Estimation and Testing	p. 444
The Analysis-of-Covariance Model	p. 444
Estimation	p. 446
Testing Hypotheses	p. 448
One-Way Model with One Covariate	p. 449
The Model	p. 449
Estimation	p. 449
Testing Hypotheses	p. 450
Two-Way Model with One Covariate	p. 457
Tests for Main Effects and Interactions	p. 458
Test for Slope	p. 462
Test for Homogeneity of Slopes	p. 463
One-Way Model with Multiple Covariates	p. 464
The Model	p. 464
Estimation	p. 465
Testing Hypotheses	p. 468
Analysis-of-Covariance with Unbalanced Models	p. 473
Linear Mixed Models	p. 479
Introduction	p. 479
The Linear Mixed Model	p. 479
Examples	p. 481
Estimation of Variance Components	p. 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] Terms	p. 497
Residual Diagnostics	p. 501
Additional Models	p. 507
Nonlinear Regression	p. 507
Logistic Regression	p. 508
Loglinear Models	p. 511
Poisson Regression	p. 512
Generalized Linear Models	p. 513
Answers and Hints to the Problems	p. 517
References	p. 653
Index	p. 663
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Linear Models in Statistics

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