×

zbMATH — the first resource for mathematics

Introduction to graphical modelling. Incl. 1 disk. (English) Zbl 0856.62004
Springer Texts in Statistics. New York, NY: Springer-Verlag. xii, 274 p. (1995).
The book begins with definitions of conditional independence and independence graph of a model. The mood of the book is then set by two examples of Simpson’s Paradox to illustrate the inadequacy of studying marginal associations between variables when the real interest is in a direct effect (conditional association) of one variable on another. The emphasis in this book is on model structure, more precisely, conditional independence (CI) structure. It examines various families of models that can be specified in terms of pairwise CI and describe the models through independence graphs.
Chapter 2 describes one such family – the log-linear models for discrete data. Here, pairwise CI is equivalent to zero two-factor interactions. The focus is on aspects relevant to graphical modeling. Four examples – lizard perching behavior, risk factors for coronary heart disease, chromosome mapping, and university admissions – are worked out in detail. Chapter 3 is a continuous variable version of the previous chapter – with examples. Also considered are models in the multivariate regression framework with an example from data arising in a clinical study of the effects of estrogen therapy on bone mineral content in 150 post-menopausal women.
Chapter 4 discusses the construction of hierarchical interaction models by combining log-linear models for discrete variables with graphical Gaussian models for continuous variables. Beginning with a conditional Gaussian distribution and models with one discrete and one continuous variable, the chapter proceeds to construct models for two discrete and two continuous variables and then to the MANOVA framework. Examples are worked out on the way. The chapter ends with a discussion of breaking models into smaller models.
Chapter 5 deals with significance tests in the framework of hierarchical interaction models. First the large sample chi-squared test and small sample \(F\)-test are described. Then a variety of exact conditional tests are described. These “permutation” tests are nonparametric and have the advantage of being able to compute the exact conditional distribution of any test statistic. A disadvantage, however, is that they are computation-intensive. The chapter closes with some standard multivariate tests which can be formulated for these models.
Chapter 6 introduces some techniques for model selection and criticism and a final chapter treats miscellaneous topics such as: missing data, discrimination, models for directed graphs, and statistical notions of causality.
The large number of interesting examples that appear throughout the book are analyzed using MIM, a command driven PC-program designed for graphical modeling. A student version of MIM is included with the book with a reference guide provided in an appendix.

MSC:
62-09 Graphical methods in statistics (MSC2010)
62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
62J99 Linear inference, regression
62G10 Nonparametric hypothesis testing
Software:
MIM
PDF BibTeX XML Cite