1st Edition

A Primer on Linear Models

By John F. Monahan Copyright 2008
    304 Pages 6 B/W Illustrations
    by Chapman & Hall

    304 Pages
    by Chapman & Hall

    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.

    Examples of the General Linear Model
    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
    The Linear Least Squares Problem
    The Normal Equations
    The Geometry of Least Squares
    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
    Multivariate Normal Distribution
    Chi-Square and Related Distributions
    Distribution of Quadratic Forms
    Cochran’s Theorem
    Regression Models with Joint Normality
    Statistical Inference
    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
    Further Topics in Testing
    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
    Variance Components: One Way
    Variance Components: Two-Way Mixed ANOVA
    Variance Components: General Case
    The Split Plot
    Predictions and BLUPs
    The Multivariate Linear Model
    The Multivariate Gauss–Markov Model
    Inference under Normality Assumptions
    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
    A Summary, Notes, and Exercises appear at the end of most chapters.


    John F. Monahan

    "… I found the book very helpful. … the result is very nice, very readable, and in particular I like the idea of avoiding leaps in the development and proofs, or referring to other sources for the details of the proofs. This is a useful well-written instructive book."
    International Statistical Review

    "This work provides a brief, and also complete, foundation for the theory of basic linear models . . . can be used for graduate courses on linear models."
    – Nicoleta Breaz, Zentralblatt Math

    ". . . well written . . . would serve well as the textbook for an introductory course in linear models, or as references for researchers who would like to review the theory of linear models."
    Justine Shults, Department of Biostatistics, University of Pennsylvania School of Medicine, Journal of Biopharmaceutical Statistics