Univariate and Multivariate General Linear Models

Theory and Applications with SAS, Second Edition

Kevin Kim, Neil Timm

• Publisher: Chapman and Hall/CRC
• Year: 2006
• Print ISBN: 9780367453442
• eBook ISBN: 9781000738254
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

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

Chapter 1

Overview of General Linear Model

1.1 Introduction

In this chapter, we introduce the structure of the general linear model (GLM) and use the structure to classify the linear models discussed in this book. The multivariate normal distribution which forms the basis for most of the hypothesis testing theory of the linear model is reviewed, along with a general approach to hypothesis testing. Graphical methods and tests for assessing univariate and multivariate normality are also reviewed. The generation of multivariate normal data, the construction of Quantile-Quantile (Q-Q) plots, chi-square plots, scatter plots, and data transformation procedures are reviewed and illustrated to evaluate normality.

1.2 General Linear Model

Data analysis in the social and behavioral sciences and numerous other disciplines is associated with a model known as the GLM. Employing matrix notation, univariate and multivariate linear models may be represented using the general form

${\Omega }_0:y=X\beta +e(1.1)$

where y_{n ×1} is a vector of n observations, X_n×k is a known design matrix of full column rank k, β_{k ×1} is a vector of k fixed parameters, e_{n ×1} is a random vector of errors with mean zero, E(e) = 0, and covariance matrix Ω = cov(e). If the design matrix is not of full rank, one may reparameterize the model to create an equivalent model of full rank. In this book, we systematically discuss the GLM specified by (1.1) with various structures for X and Ω.

Depending on the structure of X and Ω, the model in (1.1) has many names in the literature. To illustrate, if Ω = σ²I_n in (1.1), the model is called the classical linear regression model or the standard linear regression model. If we partition X to have the form X = (X₁, X₂) where X₁ is associated with fixed effects and X₂ is associated with random effects, and if covariance matrix Ω has the form

$\Omega =X_{2}V{X}_2\prime +\Psi (1.2)$

where V and Ψ are covariance matrices, then (1.1) becomes the general linear mixed model (GLMM). If we let X and Ω take the general form

$X=\left(\begin{array}{cccc}\hfill X_1& \hfill 0& \hfill ...& \hfill 1\\ \hfill 0& \hfill X_2& \hfill ...& \hfill 0\\ \hfill ⋮& \hfill ⋮& \hfill \ddots & \hfill ⋮\\ \hfill 0& \hfill 0& \hfill ...& \hfill X_p\end{array}\right)={\oplus }_{i=1}^p{X}_i(1.3)$

$\Omega =\Sigma \otimes I_n(1.4)$

where Σ_p×p is a covariance matrix, A ⊗ B denotes the Kronecker product of two matrices A and B (A⊗ = a_ijB), and ${\oplus }_{i=1}^p$ represents the direct sum of the matrices X_i, then (1.1) is Zellner’s seemingly unrelated regression (SUR) model or a multiple design multivariate (MDM) model. The SUR model may also be formulated as p separate linear regression models that are not independent

$y_i=X_i{\beta }_{ii}+e_i(1...$

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Table of Contents
• List of Tables
• Preface
• 1 Overview of General Linear Model
• 2 Unrestricted General Linear Models
• 3 Restricted General Linear Models
• 4 Weighted General Linear Models
• 5 Multivariate General Linear Models
• 6 Doubly Multivariate Linear Model
• 7 Restricted MGLM and Growth Curve Model
• 8 SUR Model and Restricted GMANOVA Model
• 9 Simultaneous Inference Using Finite Intersection Tests
• 10 Computing Power for Univariate and Multivariate GLM
• 11 Two-Level Hierarchical Linear Models
• 12 Incomplete Repeated Measurement Data
• 13 Structural Equation Modeling
• References
• Author Index
• Subject Index