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.

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.- 2.2 Multi-Way Tables.- 3 Continuous Models.- 3.1 Graphical Gaussian Models.- 3.2 Regression Models.- 4 Mixed Models.- 4.1 Hierarchical Interaction Models.- 4.2 Breaking Models into Smaller Ones.- 4.3 Mean Linearity.- 4.4 Decomposable Models.- 4.5 CG-Regression Models.- 4.6 Incomplete Data.- 4.7 Discriminant Analysis.- 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.- 6.2 The EH-Procedure.- 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.- 7.2 Chain Graphs.- 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.- 8.3 Pearl's Causal Graphs.- 8.4 Discussion.- 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-HalvingVariant.- D.2 The EM-Algorithm.- D.3 The ME-Algorithm.- References.