A Primer on Linear Models (Chapman & Hall/CRC Texts in Statistical Science)
By: John F. Monahan (author)Paperback
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A Primer on Linear Models presents a unified, thorough, and rigorous development of the theory behind the statistical methodology of regression and analysis of variance (ANOVA). It seamlessly incorporates these concepts using non-full-rank design matrices and emphasizes the exact, finite sample theory supporting common statistical methods. With coverage steadily progressing in complexity, the text first provides examples of the general linear model, including multiple regression models, one-way ANOVA, mixed-effects models, and time series models. It then introduces the basic algebra and geometry of the linear least squares problem, before delving into estimability and the Gauss-Markov model. After presenting the statistical tools of hypothesis tests and confidence intervals, the author analyzes mixed models, such as two-way mixed ANOVA, and the multivariate linear model. The appendices review linear algebra fundamentals and results as well as Lagrange multipliers. This book enables complete comprehension of the material by taking a general, unifying approach to the theory, fundamentals, and exact results of linear models.
Preface Examples of the General Linear Model Introduction One-Sample Problem Simple Linear Regression Multiple Regression One-Way ANOVA First Discussion The Two-Way Nested Model Two-Way Crossed Model Analysis of Covariance Autoregression Discussion The Linear Least Squares Problem The Normal Equations The Geometry of Least Squares Reparameterization Gram-Schmidt Orthonormalization Estimability and Least Squares Estimators Assumptions for the Linear Mean Model Confounding, Identifiability, and Estimability Estimability and Least Squares Estimators First Example: One-Way ANOVA Second Example: Two-Way Crossed without Interaction Two-Way Crossed with Interaction Reparameterization Revisited Imposing Conditions for a Unique Solution to the Normal Equations Constrained Parameter Space Gauss-Markov Model Model Assumptions The Gauss-Markov Theorem Variance Estimation Implications of Model Selection The Aitken Model and Generalized Least Squares Application: Aggregation Bias Best Estimation in a Constrained Parameter Space Addendum: Variance of Variance Estimator Distributional Theory Introduction Multivariate Normal Distribution Chi-Square and Related Distributions Distribution of Quadratic Forms Cochran's Theorem Regression Models with Joint Normality Statistical Inference Introduction Results from Statistical Theory Testing the General Linear Hypothesis The Likelihood Ratio Test and Change in SSE First Principles Test and LRT Confidence Intervals and Multiple Comparisons Identifiability Further Topics in Testing Introduction Reparameterization Applying Cochran's Theorem for Sequential SS Orthogonal Polynomials and Contrasts Pure Error and the Lack-of-Fit Test Heresy: Testing Nontestable Hypotheses Variance Components and Mixed Models Introduction Variance Components: One Way Variance Components: Two-Way Mixed ANOVA Variance Components: General Case The Split Plot Predictions and BLUPs The Multivariate Linear Model Introduction The Multivariate Gauss-Markov Model Inference under Normality Assumptions Testing Repeated Measures Confidence Intervals Appendix A: Review of Linear Algebra Notation and Fundamentals Rank, Column Space, and Nullspace Some Useful Results Solving Equations and Generalized Inverses Projections and Idempotent Matrices Trace, Determinants, and Eigenproblems Definiteness and Factorizations Appendix B: Lagrange Multipliers Main Results Bibliography A Summary, Notes, and Exercises appear at the end of most chapters.
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- ID: 9781420062014
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