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
Author(s): Barry Kurt Moser
Edition: 1st
Year: 1996
Language: English
Pages: 228
Front Cover......Page 1
Linear Models: A Mean Model Approach......Page 4
Copyright Page......Page 5
Contents......Page 8
Preface......Page 12
1.1 Elementary Matrix Concepts ......Page 14
1.2 Kronecker Products ......Page 25
1.3 Random Vectors ......Page 29
2.1 Multivariate Normal Distribution Function ......Page 36
2.2 Conditional Distributions of Multivariate Normal Random Vectors ......Page 42
2.3 Distributions of Certain Quadratic Forms ......Page 45
3.1 Quadratic Forms of Normal Random Vectors ......Page 54
3.2 Independence ......Page 58
3.3 The t and F Distributions ......Page 60
3.4 Bhat's Lemma ......Page 62
4.1 Models That Admit Restrictions (Finite Models) ......Page 66
4.2 Models That Do Not Admit Restrictions (Infinite Models) ......Page 69
4.3 Sum of Squares and Covariance Matrix Algorithms ......Page 71
4.4 Expected Mean Squares ......Page 77
4.5 Algorithm Applications ......Page 79
5.1 Ordinary Least-Squares Estimation ......Page 94
5.2 Best Linear Unbiased Estimators ......Page 99
5.3 ANOVA Table for the Ordinary Least-Squares Regression Function ......Page 100
5.4 Weighted Least-Squares Regression ......Page 102
5.5 Lack of Fit Test ......Page 104
5.6 Partitioning the Sum of Squares Regression ......Page 107
5.7 The Model Y = XB + E in Complete, Balanced Factorials......Page 110
6.1 Maximum Likelihood Estimators of B and a2......Page 118
6.2 Invariance Property, Sufficiency, and Completeness......Page 121
6.3 ANOVA Methods for Finding Maximum Likelihood Estimators ......Page 124
6.4 The Likelihood Ratio Test for HB = h ......Page 132
6.5 Confidence Bands on Linear Combinations of B ......Page 139
7.1 Replication Matrices ......Page 144
7.2 Pattern Matrices and Missing Data ......Page 151
7.3 Using Replication and Pattern Matrices Together ......Page 157
8.1 General Balanced Incomplete Block Design ......Page 162
8.2 Analysis of the General Case ......Page 165
8.3 Matrix Derivations of Kempthorne's Interblock and Intrablock Treatment Difference Estimators ......Page 168
9.1 Model Assumptions and Examples ......Page 174
9.2 The Mean Model Solution ......Page 177
9.3 Mean Model Analysis When cov(E) = a2ln ......Page 178
9.4 Estimable Functions ......Page 181
9.5 Mean Model Analysis When cov(E) = a2V ......Page 185
10.1 The Mixed Model Structure and Assumptions ......Page 190
10.2 Random Portion Analysis: Type I Sum of Squares Method ......Page 192
10.3 Random Portion Analysis: Restricted Maximum Likelihood Method ......Page 195
10.4 Random Portion Analysis: A Numerical Example ......Page 196
10.5 Fixed Portion Analysis ......Page 197
10.6 Fixed Portion Analysis: A Numerical Example ......Page 199
Appendix 1 Computer Output for Chapter 5......Page 202
A2.1 Computer Output for Section 7.2 ......Page 206
A2.2 Computer Output for Section 7.3 ......Page 214
Appendix 3 Computer Output for Chapter 8......Page 220
Appendix 4 Computer Output for Chapter 9......Page 222
A5.1 Computer Output for Section 10.2 ......Page 226
A5.2 Computer Output for Section 10.4 ......Page 229
A5.3 Computer Output for Section 10.6 ......Page 231
References and Related Literature ......Page 234
Subject Index ......Page 238
PROBABILITY AND MATHEMATICAL STATISTICS......Page 242
