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21
A characterization of Markov equivalence classes for acyclic digraphs
, 1995
"... Undirected graphs and acyclic digraphs (ADGs), as well as their mutual extension to chain graphs, are widely used to describe dependencies among variables in multivariate distributions. In particular, the likelihood functions of ADG models admit convenient recursive factorizations that often allow e ..."
Abstract

Cited by 95 (7 self)
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Undirected graphs and acyclic digraphs (ADGs), as well as their mutual extension to chain graphs, are widely used to describe dependencies among variables in multivariate distributions. In particular, the likelihood functions of ADG models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building Bayesian networks for expert systems. Whereas the undirected graph associated with a dependence model is uniquely determined, there may, however, be many ADGs that determine the same dependence ( = Markov) model. Thus, the family of all ADGs with a given set of vertices is naturally partitioned into Markovequivalence classes, each class being associated with a unique statistical model. Statistical procedures, such as model selection or model averaging, that fail to take into account these equivalence classes, may incur substantial computational or other inefficiencies. Here it is shown that each Markovequivalence class is uniquely determined by a single chain graph, the essential graph, that is itself simultaneously Markov equivalent to all ADGs in the equivalence class. Essential graphs are characterized, a polynomialtime algorithm for their construction is given, and their applications to model selection and other statistical
On the Markov Equivalence of Chain Graphs, Undirected Graphs, and Acyclic Digraphs
 Scandinavian Journal of Statistics
, 1994
"... Graphical Markov models use undirected graphs (UDGs), acyclic directed graphs (ADGs), or (mixed) chain graphs to represent possible dependencies among random variables in a multivariate distribution. Whereas a UDG is uniquely determined by its associated Markov model, this is not true for ADGs or fo ..."
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Cited by 32 (5 self)
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Graphical Markov models use undirected graphs (UDGs), acyclic directed graphs (ADGs), or (mixed) chain graphs to represent possible dependencies among random variables in a multivariate distribution. Whereas a UDG is uniquely determined by its associated Markov model, this is not true for ADGs or for general chain graphs (which include both UDGs and ADGs as special cases). This paper addresses three questions regarding the equivalence of graphical Markov models: when is a given chain graph Markov equivalent (1) to some UDG? (2) to some (at least one) ADG? (3) to some decomposable UDG? The answers are obtained by means of an extension of Frydenberg's (1990) elegant graphtheoretic characterization of the Markov equivalence of chain graphs. 1 Introduction The use of graphs to represent dependence relations among random variables, first introduced by Wright (1921), has generated considerable research activity, especially since the early 1980s. Particular attention has been devoted to gra...
Computing Maximum Likelihood Estimates in loglinear models
, 2006
"... We develop computational strategies for extended maximum likelihood estimation, as defined in Rinaldo (2006), for general classes of loglinear models of widespred use, under Poisson and productmultinomial sampling schemes. We derive numerically efficient procedures for generating and manipulating ..."
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Cited by 11 (3 self)
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We develop computational strategies for extended maximum likelihood estimation, as defined in Rinaldo (2006), for general classes of loglinear models of widespred use, under Poisson and productmultinomial sampling schemes. We derive numerically efficient procedures for generating and manipulating design matrices and we propose various algorithms for computing the extended maximum likelihood estimates of the expectations of the cell counts. These algorithms allow to identify the set of estimable cell means for any given observable table and can be used for modifying traditional goodnessoffit tests to accommodate for a nonexistent MLE. We describe and take advantage of the connections between extended maximum likelihood
Split models for contingency tables
, 2003
"... A framework for loglinear models with context specific independence structures, i.e. conditional independencies holding only for specific values of the conditioning variables is introduced. This framework is constituted by the class of split models. Also a software package named YGGDRASIL which is ..."
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Cited by 8 (1 self)
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A framework for loglinear models with context specific independence structures, i.e. conditional independencies holding only for specific values of the conditioning variables is introduced. This framework is constituted by the class of split models. Also a software package named YGGDRASIL which is designed for statistical inference in split models is presented. Split models are an extension of graphical models for contingency tables. The treatment of split models includes estimation, representation and a Markov property for reading off independencies holding in a specific context. Two examples, including an illustration of the use of YGGDRASIL are
Three Centuries of Categorical Data Analysis: Loglinear Models and Maximum Likelihood Estimation
"... The common view of the history of contingency tables is that it begins in 1900 with the work of Pearson and Yule, but it extends back at least into the 19th century. Moreover it remains an active area of research today. In this paper we give an overview of this history focussing on the development o ..."
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Cited by 6 (3 self)
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The common view of the history of contingency tables is that it begins in 1900 with the work of Pearson and Yule, but it extends back at least into the 19th century. Moreover it remains an active area of research today. In this paper we give an overview of this history focussing on the development of loglinear models and their estimation via the method of maximum likelihood. S. N. Roy played a crucial role in this development with two papers coauthored with his students S. K. Mitra and Marvin Kastenbaum, at roughly the midpoint temporally in this development. Then we describe a problem that eluded Roy and his students, that of the implications of sampling zeros for the existence of maximum likelihood estimates for loglinear models. Understanding the problem of nonexistence is crucial to the analysis of large sparse contingency tables. We introduce some relevant results from the application of algebraic geometry to the study of this statistical problem. 1
Graphical models for inference under outcomedependent sampling
 STAT SCI 2010;25:368–87
, 2010
"... We consider situations where data have been collected such that the sampling depends on the outcome of interest and possibly further covariates, as for instance in casecontrol studies. Graphical models represent assumptions about the conditional independencies among the variables. By including a no ..."
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Cited by 3 (0 self)
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We consider situations where data have been collected such that the sampling depends on the outcome of interest and possibly further covariates, as for instance in casecontrol studies. Graphical models represent assumptions about the conditional independencies among the variables. By including a node for the sampling indicator, assumptions about sampling processes can be made explicit. We demonstrate how to read off such graphs whether consistent estimation of the association between exposure and outcome is possible. Moreover, we give sufficient graphical conditions for testing and estimating the causal effect of exposure on outcome. The practical use is illustrated with a number of examples.
SEQUENTIAL CATEGORY AGGREGATION AND PARTITIONING APPROACHES FOR MULTIWAY CONTINGENCY TABLES BASED ON SURVEY AND CENSUS DATA 1
, 2007
"... Large contingency tables arise in many contexts but especially in the collection of survey and census data by government statistical agencies. Because the vast majority of the variables in this context have a large number of categories, agencies and users need a systematic way of constructing tables ..."
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Cited by 1 (0 self)
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Large contingency tables arise in many contexts but especially in the collection of survey and census data by government statistical agencies. Because the vast majority of the variables in this context have a large number of categories, agencies and users need a systematic way of constructing tables which are summaries of such contingency tables. We propose such an approach in this paper by finding members of a class of restricted loglinear models which maximize the likelihood of the data and use this to find a parsimonious means of representing the table. In contrast with more standard approaches for model search in hierarchical loglinear models (HLLM), our procedure systematically reduces the number of categories of the variables. Through a series of examples, we illustrate the extent to which it can preserve the interaction structure found with HLLMs and be used as a data simplification procedure prior to HLL modeling. A feature of the procedure is that it can easily be applied to many tables with millions of cells, providing a new way of summarizing large data sets in many disciplines. The focus is on information and description rather than statistical testing. The procedure may treat each variable in the table in different ways, preserving full detail, treating it as fully nominal, or preserving ordinality.
QuasiSymmetric Graphical LogLinear Models
"... ABSTRACT. We propose an extension of graphical loglinear models to allow for symmetry constraints on some interaction parameters that represent homologous factors. The conditional independence structure of such quasisymmetric (QS) graphical models is described by an undirected graph with coloured ..."
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Cited by 1 (0 self)
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ABSTRACT. We propose an extension of graphical loglinear models to allow for symmetry constraints on some interaction parameters that represent homologous factors. The conditional independence structure of such quasisymmetric (QS) graphical models is described by an undirected graph with coloured edges, in which a particular colour corresponds to a set of equality constraints on a set of parameters. Unlike standard QS models, the proposed models apply with contingency tables for which only some variables or sets of the variables have the same categories. We study the graphical properties of such models, including conditions for decomposition of model parameters and of maximum likelihood estimates. Key words: conditional independence, decomposition, exchangeability, graphical models, homologous variables
Submitted to the Statistical Science On the Use of Graphical Models for Inference under Outcome Dependent Sampling
"... Abstract. We consider situations where data have been collected such that the sampling depends on the outcome of interest and possibly further covariates, as for instance in case–control studies. Graphical models represent assumptions about the conditional independencies among the variables. By incl ..."
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Abstract. We consider situations where data have been collected such that the sampling depends on the outcome of interest and possibly further covariates, as for instance in case–control studies. Graphical models represent assumptions about the conditional independencies among the variables. By including a node for the sampling indicator, assumptions about sampling processes can be represented. We demonstrate how to read off such graphs whether consistent estimation of the association between exposure and outcome is possible. Moreover, we give sufficient graphical conditions for testing and estimating the causal effect of exposure on outcome. The practical use is illustrated with a number of examples. Key words and phrases: Causal inference; collapsibility; loglinear models; odds ratios; selection bias.. 1.
Dicembre 2005Reference priors for discrete graphical models
"... The combination of graphical models and reference analysis represents a powerful tool for Bayesian inference in highly multivariate settings. It is typically difficult to derive reference priors in complex problems. In this paper we present a suitable mixed parameterisation for a discrete decomposab ..."
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The combination of graphical models and reference analysis represents a powerful tool for Bayesian inference in highly multivariate settings. It is typically difficult to derive reference priors in complex problems. In this paper we present a suitable mixed parameterisation for a discrete decomposable graphical model and derive the corresponding reference prior.