Linear Models in Statistics

Transcription

1 Linear Models in Statistics ALVIN C. RENCHER Department of Statistics Brigham Young University Provo, Utah A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Weinheim Brisbane Toronto Singapore

2 Contents Preface Acknowledgments xiii xvii 1. Introduction Simple Linear Regression Model, Multiple Linear Regression Model, Analysis of Variance Models, 3 2. Matrix Algebra Matrix and Vector Notation, Matrices, Vectors, and Sealars, Matrix Equality, Transpose, Matrices of Special Form, Operations, Sum of Two Matrices or Two Vectors, Product of Two Matrices or Two Vectors, Partitioned Matrices, Rank, Inverse, Positive Definite Matrices, Systems of Equations, Generalized Inverse, Definition and Properties, Generalized Inverses and Systems of Equations, Determinants, Orthogonal Vectors and Matrices, 37

3 vi CONTENTS 2.11 Trace, Eigenvalues and Eigenvectors, Definition, Functions of a Matrix, Products, Symmetrie Matrices, Positive Definite and Positive Semidefinite Matrices, Idempotent Matrices, Derivatives of Linear Functions and Quadratic Forms, Random Vectors and Matrices Introduction, Means, Variances, Covariances, and Correlations, Mean Vectors and Covariance Matrices for Random Vectors, MeanVector, Covariance Matrix, Generalized Variance, Standardized Distance, Correlation Matrices, Mean Vectors and Covariance Matrices for Partitioned Random Vectors, Linear Functions of Random Vectors, Means, Variances and Covariances, Multivariate Normal Distribution Univariate Normal Density Function, Multivariate Normal Density Function, Moment-Generating Functions, Properties of the Multivariate Normal Distribution, Partial Correlation, Distribution of Quadratic Forms in y 5.1 Sums of Squares, Mean and Variance of Quadratic Forms, Noncentral Chi-Square Distribution, Noncentral F- and f-distributions, Noncentral F-distribution, Noncentral f-distribution, Distribution of Quadratic Forms,

4 CONTENTS vii 5.6 Independence of Linear Forms and Quadratic Forms, 105 Simple Linear Regression The Model, Estimation of ßo, ßi, and CT 2, Hypothesis Test and Confidence Interval for ß\, Coefficient of Determination, 118 Multiple Regression: Estimation Introduction, The Model, Estimation of ß and CT 2, Least Squares Estimator for ß, Properties of the Least Squares Estimator ß, An Estimator for CT 2, Geometry of Least Squares, Variable Space, Sample Space, The Model in Centered Form, Normal Model, Assumptions, Maximum Likelihood Estimators for ß and er 2, Properties ofß and CT 2, R 2 In Fixed-x Regression, Generalized Least Squares: cov(y) = CT 2 V, Estimation of ß andct 2 When cov(y) = CT 2 V, Misspecification of the Error Structure, Model Misspecification, Orthogonalization, 159 Multiple Regression: Tests of Hypotheses and Confidence Intervals Test of Overall Regression, Test onasubset oftheß's, F-Test in Terms of R 2, The General Linear Hypothesis Tests for Ho: Cß = 0 and H 0 :Cß = t, The Test for H 0 :Cß = 0, The Test for H 0 :Cß = t, Tests on ß } ; and a'ß, Testing One ß } ox One a'ß, 191

5 viii CONTENTS Testing Several ß/s or ajß's, Confidence Intervals and Prediction Intervals, Confidence Region for ß, Confidence Interval for ßj, Confidence Interval for a'ß, Confidence Interval for E(y), Prediction Interval for a Future Observation, Confidence Interval for er 2, Simultaneous Intervals, Likelihood Ratio Tests, Multiple Regression: Model Validation and Diagnostics Residuais, The Hat Matrix, Outliers, Influential Observations and Leverage, Multiple Regression: Random x 's Multivariate Normal Regression Model, Estimation in Multivariate Normal Regression, R 2 in Multivariate Normal Regression, Tests and Confidence Intervals, EffectofEachVariableon/? 2, Prediction for Nonnormal Data, Sample Partial Correlations, Analysis of Variance Models Non-Full-Rank Models, One-Way Model, Two-Way Model, Estimation, Estimabilityof/3, Estimable Functions of ß, Estimators, Estimators ofa'ß, Estimatorofff 2, Normal Model, Reparameterization, Side Conditions, Testing Hypotheses, 284

6 CONTENTS ix Testable Hypotheses, Füll and Reduced Model, General Linear Hypothesis, An Illustration of Estimation and Testing, Estimable Functions, Testing a Hypothesis, OrthogonalityofColumnsofX, One-Way Analysis of Variance: Balanced Case The One-Way Model, Estimable Functions, Estimation of Parameters, Solving the Normal Equations, An Estimator for CT 2, Testing the Hypothesis Ho: ßi = A*2 = = M*> Füll and Reduced Model, General Linear Hypothesis, Expected Mean Squares, Füll and Reduced Model, General Linear Hypothesis, Contrasts, Hypothesis Test for a Contrast, Orthogonal Contrasts, Orthogonal Polynomial Contrasts, Two-Way Analysis of Variance: Balanced Case The Two-Way Model, Estimable Functions, Estimatorsof A'ßandCT 2, Solving the Normal Equations and Estimating A'ß, An Estimator for CT 2, Testing Hypotheses, Test for Interaction, Tests for Main Effects, Expected Mean Squares, Sums of Squares Approach, Quadratic Form Approach, Analysis of Variance: Unbalanced Data Introduction, 371

7 X CONTENTS 14.2 One-Way Model, Estimation and Testing, Contrasts, Two-Way Model, Unconstrained Model, Constrained Model, Analysis of Covariance Introduction, Estimation and Testing, The Analysis of Covariance Model, Estimation, Testing Hypotheses, One-Way Model with One Covariate, The Model, Estimation, Testing Hypotheses, Two-Way Model with One Covariate, Tests for Main Effects and Interactions, Test for Slope, Test for Homogeneity of Slopes, One-Way Model with Multiple Covariates, The Model, Estimation, Testing Hypotheses, Analysis of Covariance with Unbalanced Models, Random Effects Models and Mixed Effects Models Introduction, Estimation of K'ß and Prediction of a in y = Xß + Za + e, Best Linear Unbiased Estimatorof A'ß, Best Linear Unbiased Predictor of the Random Vector a, Estimation of Variance Components, Expected Mean Squares, ANOVA Estimators, Hypothesis Tests, Additional Models 17.1 Nonlinear Regression, Logistic Regression,

8 CONTENTS xi 17.3 Loglinear Models, Poisson Regression, Generalized Linear Models, 446 A. Answers and Hints to Selected Problems 449 B. Data Sets and SAS Files 561 Bibliography 563