The Theory of Linear Models and Multivariate Analysis

The Theory of Linear Models and Multivariate Analysis
Author: Steven F. Arnold
Publisher: John Wiley & Sons
Total Pages: 502
Release: 1981
Genre: Mathematics
ISBN:

Basic statistical definitions and theorems. Subspaces and projections. Properties of the multivariate and spherical normal distributions. Introduction to linear models. A sufficient statistic. Estimation. Tests about the mean. Simultaneous confidence intervals - scheffe type. Tests about the variance. Asymptotic validity of procedures under nonnormal distributions. James-Stein and Ridge estimators. Inference based on the studentized range distribution and bonferroni's inequality. The generalized linear model. The repeated measures model. Random effects and mixed models. The correlation model. The distribution theory for multivariate analysis. The multivariate one-and two-sample models - inference about the mean vector. The multivariate linear model. Discriminant analysis. Testing hypotheses about the covariance matrix. Simplifying the structure of the covariance matrix.

Linear Models in Statistics

Linear Models in Statistics
Author: Alvin C. Rencher
Publisher: John Wiley & Sons
Total Pages: 690
Release: 2008-01-07
Genre: Mathematics
ISBN: 0470192607

The essential introduction to the theory and application of linear models—now in a valuable new edition Since most advanced statistical tools are generalizations of the linear model, it is neces-sary to first master the linear model in order to move forward to more advanced concepts. The linear model remains the main tool of the applied statistician and is central to the training of any statistician regardless of whether the focus is applied or theoretical. This completely revised and updated new edition successfully develops the basic theory of linear models for regression, analysis of variance, analysis of covariance, and linear mixed models. Recent advances in the methodology related to linear mixed models, generalized linear models, and the Bayesian linear model are also addressed. Linear Models in Statistics, Second Edition includes full coverage of advanced topics, such as mixed and generalized linear models, Bayesian linear models, two-way models with empty cells, geometry of least squares, vector-matrix calculus, simultaneous inference, and logistic and nonlinear regression. Algebraic, geometrical, frequentist, and Bayesian approaches to both the inference of linear models and the analysis of variance are also illustrated. Through the expansion of relevant material and the inclusion of the latest technological developments in the field, this book provides readers with the theoretical foundation to correctly interpret computer software output as well as effectively use, customize, and understand linear models. This modern Second Edition features: New chapters on Bayesian linear models as well as random and mixed linear models Expanded discussion of two-way models with empty cells Additional sections on the geometry of least squares Updated coverage of simultaneous inference The book is complemented with easy-to-read proofs, real data sets, and an extensive bibliography. A thorough review of the requisite matrix algebra has been addedfor transitional purposes, and numerous theoretical and applied problems have been incorporated with selected answers provided at the end of the book. A related Web site includes additional data sets and SAS® code for all numerical examples. Linear Model in Statistics, Second Edition is a must-have book for courses in statistics, biostatistics, and mathematics at the upper-undergraduate and graduate levels. It is also an invaluable reference for researchers who need to gain a better understanding of regression and analysis of variance.

Regression Analysis and Linear Models

Regression Analysis and Linear Models
Author: Richard B. Darlington
Publisher: Guilford Publications
Total Pages: 689
Release: 2016-08-22
Genre: Social Science
ISBN: 1462527981

Emphasizing conceptual understanding over mathematics, this user-friendly text introduces linear regression analysis to students and researchers across the social, behavioral, consumer, and health sciences. Coverage includes model construction and estimation, quantification and measurement of multivariate and partial associations, statistical control, group comparisons, moderation analysis, mediation and path analysis, and regression diagnostics, among other important topics. Engaging worked-through examples demonstrate each technique, accompanied by helpful advice and cautions. The use of SPSS, SAS, and STATA is emphasized, with an appendix on regression analysis using R. The companion website (www.afhayes.com) provides datasets for the book's examples as well as the RLM macro for SPSS and SAS. Pedagogical Features: *Chapters include SPSS, SAS, or STATA code pertinent to the analyses described, with each distinctively formatted for easy identification. *An appendix documents the RLM macro, which facilitates computations for estimating and probing interactions, dominance analysis, heteroscedasticity-consistent standard errors, and linear spline regression, among other analyses. *Students are guided to practice what they learn in each chapter using datasets provided online. *Addresses topics not usually covered, such as ways to measure a variable’s importance, coding systems for representing categorical variables, causation, and myths about testing interaction.

Linear Models

Linear Models
Author: Brenton R. Clarke
Publisher: John Wiley & Sons
Total Pages: 288
Release: 2008-09-19
Genre: Mathematics
ISBN: 9780470377970

An insightful approach to the analysis of variance in the study of linear models Linear Models explores the theory of linear models and the dynamic relationships that these models have with Analysis of Variance (ANOVA), experimental design, and random and mixed-model effects. This one-of-a-kind book emphasizes an approach that clearly explains the distribution theory of linear models and experimental design starting from basic mathematical concepts in linear algebra. The author begins with a presentation of the classic fixed-effects linear model and goes on to illustrate eight common linear models, along with the value of their use in statistics. From this foundation, subsequent chapters introduce concepts pertaining to the linear model, starting with vector space theory and the theory of least-squares estimation. An outline of the Helmert matrix is also presented, along with a thorough explanation of how the ANOVA is created in both typical two-way and higher layout designs, ultimately revealing the distribution theory. Other important topics covered include: Vector space theory The theory of least squares estimation Gauss-Markov theorem Kronecker products Diagnostic and robust methods for linear models Likelihood approaches to estimation A discussion of Bayesian theory is also included for purposes of comparison and contrast, and numerous illustrative exercises assist the reader with uncovering the nature of the models, using both classic and new data sets. Requiring only a working knowledge of basic probability and statistical inference, Linear Models is a valuable book for courses on linear models at the upper-undergraduate and graduate levels. It is also an excellent reference for practitioners who use linear models to conduct research in the fields of econometrics, psychology, sociology, biology, and agriculture.

Multivariate, Multilinear and Mixed Linear Models

Multivariate, Multilinear and Mixed Linear Models
Author: Katarzyna Filipiak
Publisher: Springer Nature
Total Pages: 357
Release: 2021-10-01
Genre: Mathematics
ISBN: 3030754944

This book presents the latest findings on statistical inference in multivariate, multilinear and mixed linear models, providing a holistic presentation of the subject. It contains pioneering and carefully selected review contributions by experts in the field and guides the reader through topics related to estimation and testing of multivariate and mixed linear model parameters. Starting with the theory of multivariate distributions, covering identification and testing of covariance structures and means under various multivariate models, it goes on to discuss estimation in mixed linear models and their transformations. The results presented originate from the work of the research group Multivariate and Mixed Linear Models and their meetings held at the Mathematical Research and Conference Center in Będlewo, Poland, over the last 10 years. Featuring an extensive bibliography of related publications, the book is intended for PhD students and researchers in modern statistical science who are interested in multivariate and mixed linear models.

Linear Models And Regression With R: An Integrated Approach

Linear Models And Regression With R: An Integrated Approach
Author: Debasis Sengupta
Publisher: World Scientific
Total Pages: 773
Release: 2019-07-30
Genre: Mathematics
ISBN: 9811200424

Starting with the basic linear model where the design and covariance matrices are of full rank, this book demonstrates how the same statistical ideas can be used to explore the more general linear model with rank-deficient design and/or covariance matrices. The unified treatment presented here provides a clearer understanding of the general linear model from a statistical perspective, thus avoiding the complex matrix-algebraic arguments that are often used in the rank-deficient case. Elegant geometric arguments are used as needed.The book has a very broad coverage, from illustrative practical examples in Regression and Analysis of Variance alongside their implementation using R, to providing comprehensive theory of the general linear model with 181 worked-out examples, 227 exercises with solutions, 152 exercises without solutions (so that they may be used as assignments in a course), and 320 up-to-date references.This completely updated and new edition of Linear Models: An Integrated Approach includes the following features:

Beyond Multiple Linear Regression

Beyond Multiple Linear Regression
Author: Paul Roback
Publisher: CRC Press
Total Pages: 436
Release: 2021-01-14
Genre: Mathematics
ISBN: 1439885400

Beyond Multiple Linear Regression: Applied Generalized Linear Models and Multilevel Models in R is designed for undergraduate students who have successfully completed a multiple linear regression course, helping them develop an expanded modeling toolkit that includes non-normal responses and correlated structure. Even though there is no mathematical prerequisite, the authors still introduce fairly sophisticated topics such as likelihood theory, zero-inflated Poisson, and parametric bootstrapping in an intuitive and applied manner. The case studies and exercises feature real data and real research questions; thus, most of the data in the textbook comes from collaborative research conducted by the authors and their students, or from student projects. Every chapter features a variety of conceptual exercises, guided exercises, and open-ended exercises using real data. After working through this material, students will develop an expanded toolkit and a greater appreciation for the wider world of data and statistical modeling. A solutions manual for all exercises is available to qualified instructors at the book’s website at www.routledge.com, and data sets and Rmd files for all case studies and exercises are available at the authors’ GitHub repo (https://github.com/proback/BeyondMLR)

Univariate and Multivariate General Linear Models

Univariate and Multivariate General Linear Models
Author: Kevin Kim
Publisher: CRC Press
Total Pages: 576
Release: 2006-10-11
Genre: Mathematics
ISBN: 9781584886341

Reviewing the theory of the general linear model (GLM) using a general framework, Univariate and Multivariate General Linear Models: Theory and Applications with SAS, Second Edition presents analyses of simple and complex models, both univariate and multivariate, that employ data sets from a variety of disciplines, such as the social and behavioral sciences. With revised examples that include options available using SAS 9.0, this expanded edition divides theory from applications within each chapter. Following an overview of the GLM, the book introduces unrestricted GLMs to analyze multiple regression and ANOVA designs as well as restricted GLMs to study ANCOVA designs and repeated measurement designs. Extensions of these concepts include GLMs with heteroscedastic errors that encompass weighted least squares regression and categorical data analysis, and multivariate GLMs that cover multivariate regression analysis, MANOVA, MANCOVA, and repeated measurement data analyses. The book also analyzes double multivariate linear, growth curve, seeming unrelated regression (SUR), restricted GMANOVA, and hierarchical linear models. New to the Second Edition Two chapters on finite intersection tests and power analysis that illustrates the experimental GLMPOWER procedure Expanded theory of unrestricted general linear, multivariate general linear, SUR, and restricted GMANOVA models to comprise recent developments Expanded material on missing data to include multiple imputation and the EM algorithm Applications of MI, MIANALYZE, TRANSREG, and CALIS procedures A practical introduction to GLMs, Univariate and Multivariate General Linear Models demonstrates how to fully grasp the generality of GLMs by discussing them within a general framework.

The Linear Model and Hypothesis

The Linear Model and Hypothesis
Author: George Seber
Publisher: Springer
Total Pages: 208
Release: 2015-10-08
Genre: Mathematics
ISBN: 3319219308

This book provides a concise and integrated overview of hypothesis testing in four important subject areas, namely linear and nonlinear models, multivariate analysis, and large sample theory. The approach used is a geometrical one based on the concept of projections and their associated idempotent matrices, thus largely avoiding the need to involvematrix ranks. It is shown that all the hypotheses encountered are either linear or asymptotically linear, and that all the underlying models used are either exactly or asymptotically linear normal models. This equivalence can be used, for example, to extend the concept of orthogonality to other models in the analysis of variance, and to show that the asymptotic equivalence of the likelihood ratio, Wald, and Score (Lagrange Multiplier) hypothesis tests generally applies.