Linear Models

Linear Models

A Mean Model Approach

1st Edition - October 9, 1996

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  • Editors: Zbynek Sidak, David Aldous, Pranab Sen
  • eBook ISBN: 9780080510293

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Description

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

Readership

Graduate students in statistics

Table of Contents

  • Linear Algebra and Related Introductory Topics: Elementary Matrix Concepts. Kronecker Products. Random Vectors. Multivariate Normal Distribution: Multivariate Normal Distribution Function. Conditional Distributionsof Multivariate Normal Vectors. Distributions of Certain Quadratic Forms. Distributions of Quadratic Forms: Quadratic Forms of Normal Random Vectors. Independence. t and F Distributions. Bhats Lemma. Complete, Balanced Factorial Experiments: Models That Admit Restrictions (Finite Models). Models That Do Not Admit Restrictions (Infinite Models). Sum of Squares and Covariance Matrix Algorithms. Expected Mean Squares. Algorithm Applications. Least Squares Regression: Ordinary Least SquaresEstimation. Best Linear Unbiased Estimators. ANOVA Table for the Ordinary Least Squares Regression Function. Weighted Least Squares Regression. Lack of Fit Test. Partitioning the Sum of Squares Regression. The Model Y = X( + E in Complete, BalancedFactorials. Maximum Likelihood Estimation and Related Topics: Maximum Likelihood Estimators (MLEs) of ( and ( + 2. Invariance Property, Sufficiency and Completeness. ANOVA Methods for Finding Maximum Likelihood Estimators. The Likelihood Ratio Test for H( = h. Confidence Bands on Linear Combinations of (. Unbalanced Designs and Missing Data: Replication Matrices. Pattern Matrices and Missing Data. Using Replication and Pattern Matrices Together. Balanced Incomplete Block Designs: General Balanced Incomplete Block Design. Analysis of the General Case. Matrix Derivations of Kempthornes Inter- and Intra-Block Treatment Difference Estimators. Less Than Full Rank Models: Model Assumptions and Examples. The Mean Model Solution. Mean Model Analysis When cov(E) = ( + 2I - n. Estimable Functions. Mean Model Analysis When cov(E) = ( + 2V. The General Mixed Model: The Mixed Model Structure and Assumptions. Random Portion Analysis: Type I Sumof Squares Method. Random Portion Analysis: Restricted Maximum Likelihood Method. Random Portion Analysis: A Numerical Example. Fixed Portion Analysis. Fixed Portion Analysis: A Numerical Example. Appendixes. References. Subject Index.

Product details

  • No. of pages: 228
  • Language: English
  • Copyright: © Academic Press 1996
  • Published: October 9, 1996
  • Imprint: Academic Press
  • eBook ISBN: 9780080510293

About the Series Editors

Zbynek Sidak

Zbynek Šidák was Chairman, Department of Probability and Statistics at the Mathematical Institute, Academy of Sciences, Czech Republic. He is now the principal research worker there. He has worked at various American universities as well. For 30 years, he was Editor of the journal Applications of Mathematics. His interests in statistics were rank tests, multivariate and cluster analysis, ranking and selection procedures, and Markov chains.

Affiliations and Expertise

Mathematical Institute Academy of Sciences, Czech Republic

David Aldous

Pranab Sen

Pranab K. Sen is Cary C. Boshamer Professor of Biostatistics and Statistics at the University of North Carolina, and is a Fellow of the Institute of Mathematical Statistics and of the American Statistical Association. He is also an elected member of the International Statistical Institute.Prenab K. Sen is author or co-author of multiple volumes in Mathematical Statistics, Probability Theory and Biostatistics, and has published extensively in nonparametrics, multivariate and sequential analysis, and reliability and survival analysis.

Affiliations and Expertise

University of North Carolina, Chapel Hill, U.S.A.

About the Author

William Moser

Professor William Moser is a Professor of Chemical Engineering at the Worcester Polytechnic Institute. He is also a member of the Center for Inorganic Membrane Studies at WPI. Professor Moser invented both the Cylindrical Internal Reflectance Reactors (CIR-REACTORS) and Optical Fiber coupled CIR-Reactors (OFCIR-REACTORS), which are now commercial products used for reaction monitoring in homogeneous and heterogeneous catalysis. He was recently issued a patent on remote infrared sensing using optical fiber cables connected to high pressure reactors as well as normal laboratory glassware reactors. Dr. Moser is a Fellow of the New York Academy of Sciences, and is a co-founder of the Organic Reactions Catalysis Society. He is a past Chairman of the ACS Petroleum Division. He is the editor of several books on homogeneous and heterogeneous catalysis, and has a variety of publications and patents in the catalytic and materials science fields.

Affiliations and Expertise

Worcester Polytechnic Institute

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