Linear Models: A Mean Model Approach
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Table of contents
- Cover
- Contentsvii
- Prefacexi
- Chapter 1. Linear Algebra and Related Introductory Topics1
- 1.1 Elementary Matrix Concepts1
- 1.2 Kronecker Products12
- 1.3 Random Vectors16
- Chapter 2. Multivariate Normal Distribution23
- 2.1 Multivariate Normal Distribution Function23
- 2.2 Conditional Distributions of Multivariate Normal Random Vectors29
- 2.3 Distributions of Certain Quadratic Forms32
- Chapter 3. Distributions of Quadratic Forms41
- 3.1 Quadratic Forms of Normal Random Vectors41
- 3.2 Independence45
- 3.3 The t and F Distributions47
- 3.4 Bhat's Lemma49
- Chapter 4. Complete, Balanced Factorial Experiments53
- 4.1 Models That Admit Restrictions (Finite Models)53
- 4.2 Models That Do Not Admit Restrictions (Infinite Models)56
- 4.3 Sum of Squares and Covariance Matrix Algorithms58
- 4.4 Expected Mean Squares64
- 4.5 Algorithm Applications66
- Chapter 5. Least-Squares Regression81
- 5.1 Ordinary Least-Squares Estimation81
- 5.2 Best Linear Unbiased Estimators86
- 5.3 ANOVA Table for the Ordinary Least-Squares Regression Function87
- 5.4 Weighted Least-Squares Regression89
- 5.5 Lack of Fit Test91
- 5.6 Partitioning the Sum of Squares Regression94
- 5.7 The Model Y = XB + E in Complete, Balanced Factorials97
- Chapter 6. Maximum Likelihood Estimation and Related Topics105
- 6.1 Maximum Likelihood Estimators of B and a2105
- 6.2 Invariance Property, Sufficiency, and Completeness108
- 6.3 ANOVA Methods for Finding Maximum Likelihood Estimators111
- 6.4 The Likelihood Ratio Test for HB = h119
- 6.5 Confidence Bands on Linear Combinations of B126
- Chapter 7. Unbalanced Designs and Missing Data131
- 7.1 Replication Matrices131
- 7.2 Pattern Matrices and Missing Data138
- 7.3 Using Replication and Pattern Matrices Together144
- Chapter 8. Balanced Incomplete Block Designs149
- 8.1 General Balanced Incomplete Block Design149
- 8.2 Analysis of the General Case152
- 8.3 Matrix Derivations of Kempthorne's Interblock and Intrablock Treatment Difference Estimators155
- Chapter 9. Less Than Full Rank Models161
- 9.1 Model Assumptions and Examples161
- 9.2 The Mean Model Solution164
- 9.3 Mean Model Analysis When cov(E) = a2ln165
- 9.4 Estimable Functions168
- 9.5 Mean Model Analysis When cov(E) = a2V172
- Chapter 10. The General Mixed Model177
- 10.1 The Mixed Model Structure and Assumptions177
- 10.2 Random Portion Analysis: Type I Sum of Squares Method179
- 10.3 Random Portion Analysis: Restricted Maximum Likelihood Method182
- 10.4 Random Portion Analysis: A Numerical Example183
- 10.5 Fixed Portion Analysis184
- 10.6 Fixed Portion Analysis: A Numerical Example186
- Appendix 1 Computer Output for Chapter 5189
- Appendix 2 Computer Output for Chapter 7193
- A2.1 Computer Output for Section 7.2193
- A2.2 Computer Output for Section 7.3201
- Appendix 3 Computer Output for Chapter 8207
- Appendix 4 Computer Output for Chapter 9209
- Appendix 5 Computer Output for Chapter 10213
- A5.1 Computer Output for Section 10.2213
- A5.2 Computer Output for Section 10.4216
- A5.3 Computer Output for Section 10.6218
- References and Related Literature221
- Subject Index225
Linear models, normally presented in a highly theoretical and mathematical style, are brought down to earth in this comprehensive textbook. Linear Models examines the subject from a mean model perspective, defining simple and easy-to-learn rules for building mean models, regression models, mean vectors, covariance matrices and sums of squares matrices for balanced and unbalanced data sets. The author includes both applied and theoretical discussions of the multivariate normal distribution, quadratic forms, maximum likelihood estimation, less than full rank models, and general mixed models. The mean model is used to bring all of these topics together in a coherent presentation of linear model theory.
Key Features
* Provides a versatile format for investigating linear model theory, using the mean model
* Uses examples that are familiar to the student:
* design of experiments, analysis of variance, regression, and normal distribution theory
* Includes a review of relevant linear algebra concepts
* Contains fully worked examples which follow the theorem/proof presentation
Key Features
* Provides a versatile format for investigating linear model theory, using the mean model
* Uses examples that are familiar to the student:
* design of experiments, analysis of variance, regression, and normal distribution theory
* Includes a review of relevant linear algebra concepts
* Contains fully worked examples which follow the theorem/proof presentation