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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 ..."
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Cited by 91 (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
A graphical characterization of lattice conditional independence models
 Ann. Math. and Artificial Intelligence
, 1997
"... Lattice conditional independence (LCI) models for multivariate normal data recently have been introduced for the analysis of nonmonotone missing data patterns and of nonnested dependent linear regression models ( ≡ seemingly unrelated regressions). It is shown here that the class of LCI models coin ..."
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Cited by 9 (2 self)
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Lattice conditional independence (LCI) models for multivariate normal data recently have been introduced for the analysis of nonmonotone missing data patterns and of nonnested dependent linear regression models ( ≡ seemingly unrelated regressions). It is shown here that the class of LCI models coincides with a subclass of the class of graphical Markov models determined by acyclic digraphs (ADGs), namely, the subclass of transitive ADG models. An explicit graphtheoretic characterization of those ADGs that are Markov equivalent to some transitive ADG is obtained. This characterization allows one to determine whether a specific ADG D is Markov equivalent to some transitive ADG, hence to some LCI model, in polynomial time, without an exhaustive search of the (exponentially large) equivalence class [D]. These results do not require the existence or positivity of joint densities. 1. Introduction. The use of directed graphs to represent possible dependencies among statistical variables dates back to Wright (1921) and has generated considerable research activity in the social and natural sciences. Since 1980, particular attention has been directed at
BAYESIAN METHODS TO IMPUTE MISSING COVARIATES FOR CAUSAL INFERENCE AND MODEL SELECTION
, 2008
"... This thesis presents new approaches to deal with missing covariate data in two situations; matching in observational studies and model selection for generalized linear models. In observational studies, inferences about treatment effects are often affected by confounding covariates. Analysts can redu ..."
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This thesis presents new approaches to deal with missing covariate data in two situations; matching in observational studies and model selection for generalized linear models. In observational studies, inferences about treatment effects are often affected by confounding covariates. Analysts can reduce bias due to differences in control and treated units ’ observed covariates using propensity score matching, which results in a matched control group with similar characteristics to the treated group. Propensity scores are typically estimated from the data using a logistic regression. When covariates are partially observed, missing values can be filled in using multiple imputation. Analysts can estimate propensity scores from the imputed data sets to find a matched control set. Typically, in observational studies, covariates are spread thinly over a large space. It is not always clear what an appropriate imputation model for the missing data should be. Implausible imputations can influence the matches selected and hence the estimate of the treatment effect. In propensity score matching, units tend to be selected from among those lying in the treated units ’ covariate space.