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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.
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
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.
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.
A precise and accessible presentation of linear model theory, illustrated with data examples Statisticians often use linear models for data analysis and for developing new statistical methods. Most books on the subject have historically discussed univariate, multivariate, and mixed linear models separately, whereas Linear Model Theory: Univariate, Multivariate, and Mixed Models presents a unified treatment in order to make clear the distinctions among the three classes of models. Linear Model Theory: Univariate, Multivariate, and Mixed Models begins with six chapters devoted to providing brief and clear mathematical statements of models, procedures, and notation. Data examples motivate and illustrate the models. Chapters 7-10 address distribution theory of multivariate Gaussian variables and quadratic forms. Chapters 11-19 detail methods for estimation, hypothesis testing, and confidence intervals. The final chapters, 20-23, concentrate on choosing a sample size. Substantial sets of excercises of varying difficulty serve instructors for their classes, as well as help students to test their own knowledge. The reader needs a basic knowledge of statistics, probability, and inference, as well as a solid background in matrix theory and applied univariate linear models from a matrix perspective. Topics covered include: A review of matrix algebra for linear models The general linear univariate model The general linear multivariate model Generalizations of the multivariate linear model The linear mixed model Multivariate distribution theory Estimation in linear models Tests in Gaussian linear models Choosing a sample size in Gaussian linear models Filling the need for a text that provides the necessary theoretical foundations for applying a wide range of methods in real situations, Linear Model Theory: Univariate, Multivariate, and Mixed Models centers on linear models of interval scale responses with finite second moments. Models with complex predictors, complex responses, or both, motivate the presentation.
This innovative, intermediate-level statistics text fills an important gap by presenting the theory of linear statistical models at a level appropriate for senior undergraduate or first-year graduate students. With an innovative approach, the author's introduces students to the mathematical and statistical concepts and tools that form a foundation
Providing a self-contained exposition of the theory of linear models, this treatise strikes a compromise between theory and practice, providing a sound theoretical basis while putting the theory to work in important cases.
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.
A companion volume to Plane answers to complex questions: the theory of linear models (1987), presenting six chapters with shallow treatments of very broad topics showing how the properties of three fundamental ideas from standard linear model theory can be used to examine multivariate, time series,