Linear Models

A Mean Model Approach

William R. Moser, Zbynek Sidak, David Aldous, Pranab K. Sen

• Publisher: Academic Press
• Year: 1996
• Print ISBN: 9780125084659
• eBook ISBN: 9780080510293
• Topic: Matemáticas
• Subtopic: Matemáticas discretas

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. - 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

1

Linear Algebra and Related Introductory Topics

A summary of relevant linear algebra concepts is presented in this chapter. Throughout the text boldfaced letters such as A, U, T, X, Y, t, g, u are used to represent matrices and vectors, italicized capital letters such as Y, U, T, E, F are used to represent random variables, and lowercase italicized letters such as r, s, t, n, c are used as constants.

1.1 ELEMENTARY MATRIX CONCEPTS

The following list of definitions provides a brief summary of some useful matrix operations.

Definition 1.1.1

Matrix: An r × s matrix A is a rectangular array of elements with r rows and s columns. An r × 1 vector Y is a matrix with r rows and 1 column. Matrix elements are restricted to real numbers throughout the text.

Definition 1.1.2

Transpose: If A is an n × s matrix, then the transpose of A, denoted by A′, is an s × n matrix formed by interchanging the rows and columns of A.

Definition 1.1.3

Identity Matrix, Matrix of Ones and Zeros: I_n represents an n × n identity matrix, J_n is an n × n matrix of ones, 1_n is an n × 1 vector of ones, and 0_m×n is an m × n matrix of zeros.

Definition 1.1.4

Multiplication of Matrices: Let a_ij represent the ij^th element of an r × s matrix A with i = 1,…, r rows and j = 1,…, s columns. Likewise, let b_jk represent the jk^th element of an s × t matrix B with j = 1,…, s rows and k = 1,…, t columns. The matrix multiplication of A and B is represented by AB = C where C is an r × t matrix whose ik^th e...

Table of contents

• Cover image
• Title page
• Table of Contents
• Inside Front Cover
• Copyright
• Dedication
• Preface
• Chapter 1: Linear Algebra and Related Introductory Topics
• Chapter 2: Multivariate Normal Distribution
• Chapter 3: Distributions of Quadratic Forms
• Chapter 4: Complete, Balanced Factorial Experiments
• Chapter 5: Least-Squares Regression
• Chapter 6: Maximum Likelihood Estimation and Related Topics
• Chapter 7: Unbalanced Designs and Missing Data
• Chapter 8: Balanced Incomplete Block Designs
• Chapter 9: Less Than Full Rank Models
• Chapter 10: The General Mixed Model
• APPENDIX 1: COMPUTER OUTPUT FOR CHAPTER 5
• APPENDIX 2: COMPUTER OUTPUT FOR CHAPTER 7
• APPENDIX 3: COMPUTER OUTPUT FOR CHAPTER 8
• APPENDIX 4: COMPUTER OUTPUT FOR CHAPTER 9
• APPENDIX 5: COMPUTER OUTPUT FOR CHAPTER 10
• References and Related Literature
• Subject Index
• PROBABILITY AND MATHEMATICAL STATISTICS