- ISBN: 9780471754985 | 0471754986
- Cover: Hardcover
- Copyright: 1/2/2008
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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