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Full Description
This book provides a rigorous development of the foundations of linear models for multiple regression and Analysis of Variance (ANOVA), based on orthogonal projections and relations among linear subspaces. It is appropriate for the linear models course required in most statistics Ph.D. programs.
The presentation is particularly accessible because it is self-contained, general, and taken in logical steps that are linked directly to practicable computations. The broad objective is to provide a path of mastery so that the reader could, if stranded on a desert isle with nothing but pencil, paper, and a computer to perform matrix sums and products, replicate general linear models procedures in extant statistical computing packages.
The primary prerequisite is mathematical maturity, which includes logical thinking and the ability to tell when a proof is a proof. Casual acquaintance with matrices would be helpful but not required. Background in basic statis- tical theory and methods is assumed, mainly for familiarity with terminology and the purposes of statistics in applications.
The material is developed as a series of propositions, each dependent only on those preceding it. The reader is strongly encouraged to prove each one independently. Mastery requires active involvement.
As part of the broad coverage of the mathematics supporting multiple regression and ANOVA, those propositions also establish several new, key results.
There is a unique, best numerator sum of squares for testing an estimable function
The extra residual sum of squares due to imposing a linear hypothesis tests exclusively the estimable part
Models that include exclusively any given set of ANOVA effects can be formulated with contrast coding
Tests of any ANOVA effects in any design and model, including unbalanced and empty cells, can be had with extra residual sum of squares due to deleting predictor variables
Essential properties of Type III methods are identified and proven
Contents
1 Introduction 2 MLR & ANOVA Illustrations I Basics 3 Matrices and Vectors 4 Linear Subspaces 5 Orthogonal Projection 6 The Gram-Schmidt Construction 7 Further Results as Exercises II Inference 8 LMs, LS, and the GM Theorem 9 Estimability 10 Inference on the Mean 11 Restricted Linear Models 12 Special Hypotheses 13 On Methods of Model-Building III ANOVA Models: Linear Models for Effects of Categorical Factors 14 Introduction 15 ANOVA Effects 16 Models with ANOVA Effects 17 Type III 18 ANOVA Exercises and Projects 19 Yates's MWSM 20 Proportional Subclass Numbers 21 ANOVA Comments Appendices Appendix A Proofs and Solutions to Selected Exercises Appendix B Sampling Distributions Bibliography Index
