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Mathematics | Introduction to Graphical Modelling

Introduction to Graphical Modelling

Edwards, David

2nd ed. 2000, XV, 335 p.

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Graphic modelling is a form of multivariate analysis that uses graphs to represent models. These graphs display the structure of dependencies, both associational and causal, between the variables in the model. This textbook provides an introduction to graphical modelling with emphasis on applications and practicalities rather than on a formal development. It is based on the popular software package for graphical modelling, MIM, a freeware version of which can be downloaded from the Internet. Following an introductory chapter which sets the scene and describes some of the basic ideas of graphical modelling, subsequent chapters describe particular families of models, including log-linear models, Gaussian models, and models for mixed discrete and continuous variables. Further chapters cover hypothesis testing and model selection. Chapters 7 and 8 are new to the second edition. Chapter 7 describes the use of directed graphs, chain graphs, and other graphs. Chapter 8 summarizes some recent work on causal inference, relevant when graphical models are given a causal interpretation. This book will provide a useful introduction to this topic for students and researchers.

Content Level » Research

Keywords » Graph - Graphical Modelling - Graphical Models - Likelihood - Multivariate Analysis - Sim - Variance - algorithms

Related subjects » Mathematics - Statistical Theory and Methods

Table of contents 

1 Preliminaries.- 1.1 Independence and Conditional Independence.- 1.2 Undirected Graphs.- 1.3 Data, Models, and Graphs.- 1.4 Simpson’s Paradox.- 1.5 Overview of the Book.- 2 Discrete Models.- 2.1 Three-Way Tables.- 2.1.1 Example: Lizard Perching Behaviour.- 2.2 Multi-Way Tables.- 2.2.1 Likelihood Equations.- 2.2.2 Deviance.- 2.2.3 Graphs and Formulae.- 2.2.4 Example: Risk Factors for Coronary Heart Disease.- 2.2.5 Example: Chromosome Mapping.- 2.2.6 Example: University Admissions.- 3 Continuous Models.- 3.1 Graphical Gaussian Models.- 3.1.1 Likelihood.- 3.1.2 Maximum Likelihood Estimation.- 3.1.3 Deviance.- 4 3.1.4 Example: Digoxin Clearance.- 3.1.5 Example: Anxiety and Anger.- 3.1.6 Example: Mathematics Marks.- 3.2 Regression Models.- 3.2.1 Example: Determinants of Bone Mineral Content.- 4 Mixed Models.- 4.1 Hierarchical Interaction Models.- 4.1.1 Models with One Discrete and One Continuous Variable.- 4.1.2 A Model with Two Discrete and Two Continuous Variables.- 4.1.3 Model Formulae.- 4.1.4 Formulae and Graphs.- 4.1.5 Maximum Likelihood Estimation.- 4.1.6 Deviance.- 4.1.7 A Simple Example.- 4.1.8 Example: A Drug Trial Using Mice.- 4.1.9 Example: Rats’ Weights.- 4.1.10 Example: Estrogen and Lipid Metabolism.- 4.2 Breaking Models into Smaller Ones.- 4.3 Mean Linearity.- 4.4 Decomposable Models.- 4.5 CG-Regression Models.- 4.5.1 Example: Health Status Indicators.- 4.5.2 Example: Side Effects of an Antiepileptic Drug.- 4.6 Incomplete Data.- 4.6.1 Assumptions for Missing Data.- 4.6.2 Some Latent Variable Models.- 4.6.3 Example: The Components of a Normal Mixture.- 4.6.4 Example: Mathematics Marks, Revisited.- 4.7 Discriminant Analysis.- 4.7.1 Example: Breast Cancer.- 5 Hypothesis Testing.- 5.1 An Overview.- 5.2 X2-Tests.- 5.3 F-Tests.- 5.4 Exact Conditional Tests.- 5.5 Deviance-Based Tests.- 5.6 Permutation F-Test.- 5.7 Pearson x2-Test.- 5.8 Fisher’s Exact Test.- 5.9 Rank Tests.- 5.10 Wilcoxon Test.- 5.11 Kruskal-Wallis Test.- 5.12 Jonckheere-Terpstra Test.- 5.13 Tests for Variance Homogeneity.- 5.14 Tests for Equality of Means Given Homogeneity.- 5.15 Hotelling’s T2.- 6 Model Selection and Criticism.- 6.1 Stepwise Selection.- 6.1.1 Forward Selection.- 6.1.2 Restricting Selection to Decomposable Models.- 6.1.3 Using F-Tests.- 6.1.4 Coherence.- 6.1.5 Other Variants of Stepwise Selection.- 6.2 The EH-Procedure.- 6.2.1 Example: Estrogen and Lipid Metabolism, Continued.- 6.3 Selection Using Information Criteria.- 6.4 Comparison of the Methods.- 6.5 Box-Cox Transformations.- 6.6 Residual Analysis.- 6.7 Dichotomization.- 7 Directed Graphs and Their Models.- 7.1 Directed Acyclic Graphs.- 7.1.1 Markov Properties of DAGs.- 7.1.2 Modelling with DAGs.- 7.1.3 Example: Side Effects of Neuroleptics.- 7.2 Chain Graphs.- 7.2.1 Markov Properties of Chain Graphs.- 7.2.2 Modelling with Chain Graphs.- 7.2.3 Example: Membership of the “Leading Crowd”.- 7.3 Local Independence Graphs.- 7.4 Covariance Graphs.- 7.5 Chain Graphs with Alternative Markov Properties.- 7.6 Reciprocal Graphs.- 8 Causal Inference.- 8.1 Philosophical Aspects.- 8.2 Rubin’s Causal Model.- 8.2.1 Estimating Causal Effects.- 8.2.2 Ignorability.- 8.2.3 Propensity Score.- 8.2.4 Causal Hypothesis Testing.- 8.3 Pearl’s Causal Graphs.- 8.3.1 A Simple Causal model.- 8.3.2 Causal Graphs.- 8.3.3 The Back-Door Criterion.- 8.3.4 The Front-Door Criterion.- 8.4 Discussion.- 8.4.1 Comparison of the Two Approaches.- 8.4.2 Operational Implications.- A The MINI Command Language.- A.1 Introduction.- A.2 Declaring Variables.- A.3 Undirected Models.- A.3.1 Deleting Edges.- A.3.2 Adding Edges.- A.3.3 Other Model-Changing Commands.- A.3.4 Model Properties.- A.4 Block-Recursive Models.- A.4.1 Defining the Block Structure.- A.4.2 Block Mode.- A.4.3 Defining Block-Recursive Models.- A.4.4 Working with Component Models.- A.5 Reading and Manipulating Data.- A.5.1 Reading Casewise Data.- A.5.2 Reading Counts, Means, and Covariances.- A.5.3 Transforming Data.- A.5.4 Restricting Observations.- A.5.5 Generating Raw Data.- A.5.6 Deleting Variables.- A.6 Estimation.- A.6.1 Undirected Models (Complete Data).- A.6.2 Undirected Models (Missing Data).- A.6.3 CG-Regression Models.- A.7 Hypothesis Testing.- A.7.1 x2-Tests.- A.7.2 Test of Homogeneity.- A.7.3 F-Tests.- A.7.4 Edge Deletion Tests.- A.7.5 Edge Deletion F-Tests.- A.7.6 Exact Tests.- A.7.7 Symmetry Tests.- A.7.8 Randomisation Tests.- A.8 Model Selection.- A.8.1 Stepwise Selection.- A.8.2 The EH-Procedure.- A.8.3 Selection Using Information Criteria.- A.9 The Box-Cox Transformation.- A.10 Residuals.- A.11 Discriminant Analysis.- A.12 Utilities.- A.12.1 File Input.- A.12.2 The Workspace.- A.12.3 Printing Information.- A.12.4 Displaying Parameter Estimates.- A.12.5 Displaying Summary Statistics.- A.12.6 Setting the Maximum Model.- A.12.7 Fixing Variables.- A.12.8 Macros.- B Implementation Specifics of MB’!.- B.1 Calling MIM.- B.2 The Main Menu.- B.3 Entering Commands and Navigating the Work Area.- B.4 The Built-In Editor.- B.5 Interactive Data Entry.- B.6 Independence Graphs.- B.7 Simple Data Graphics.- B.7.1 Scatter Plots.- B.7.2 Histograms.- B.7.3 Box Plots.- B.8 Graphics Export Formats.- B.9 Direct Database Access.- B.10 Program Intercommunication.- C On Multivariate Symmetry.- D On the Estimation Algorithms.- D.1 The MIPS Algorithm.- D.1.1 Notation.- D.1.2 The Likelihood Equations.- D.1.3 The General Algorithm.- D.1.4 The A-Collapsible Variant.- D.1.5 The Mean Linear Variant.- D.1.6 The Q-Equivalent Variant.- D.1.7 The Step-Halving Variant.- D.2 The EM-Algorithm.- D.3 The ME-Algorithm.- References.

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