Thomas S. Richardson

Chain graph models and their causal interpretations

Chain graphs are a natural generalization of directed acyclic graphs and undirected graphs. However, the apparent simplicity of chain graphs belies the subtlety of the conditional independence hypotheses that they represent. There are many simple and apparently plausible, but ultimately fallacious, interpretations of chain graphs that are often invoked, implicitly or explicitly. These interpretations also lead to flawed methods for applying background knowledge to model selection. We present a valid interpretation by showing how the distribution corresponding to a chain graph may be generated from the equilibrium distributions of dynamic models with feed-back. These dynamic interpretations lead to a simple theory of intervention, extending the theory developed for directed acyclic graphs. Finally, we contrast chain graph models under this interpretation with simultaneous equation models which have traditionally been used to model feed-back in econometrics.

Markov Properties for Acyclic Directed Mixed Graphs

We consider acycfic directed mixed graphs, in which directed edges (x->y) and bi-directed edges (x 4-+ y) may occur. A simple extension of Pearls d-separation criterion, called m-separation, is applied to these graphs. We introduce a local Markov property which is equivalent to the global property resulting from the m-separation criterion for arbitrary distributions.

The TETRAD Project: Constraint Based Aids to Causal Model Specification

The statistical community has brought logical rigor and mathematical precision to the problem of using data to make inferences about a models parameter values. The TETRAD project, and related work in computer science and statistics, aims to apply those standards to the problem of using data and background knowledge to make inferences about a models specification. We begin by drawing the analogy between parameter estimation and model specification search. We then describe how the specification of a structural equation model entails familiar constraints on the covariance matrix for all admissible values of its parameters; we survey results on the equivalence of structural equation models, and we discuss search strategies for model specification. We end by presenting several algorithms that are implemented in the TETRAD I1 program.

An Evaluation of Machine-Learning Methods for Predicting Pneumonia Mortality

This paper describes the application of eight statistical and machine-learning methods to derive computer models for predicting mortality of hospital patients with pneumonia from their findings at initial presentation. The eight models were each constructed based on 9847 patient cases and they were each evaluated on 4352 additional cases. The primary evaluation metric was the error in predicted survival as a function of the fraction of patients predicted to survive. This metric is useful in assessing a models potential to assist a clinician in deciding whether to treat a given patient in the hospital or at home. We examined the error rates of the models when predicting that a given fraction of patients will survive. We examined survival fractions between 0.1 and 0.6. Over this range, each models predictive error rate was within 1% of the error rate of every other model. When predicting that approximately 30% of the patients will survive, all the models have an error rate of less than 1.5%. The models are distinguished more by the number of variables and parameters that they contain than by their error rates; these differences suggest which models may be the most amenable to future implementation as paper-based guidelines.

Learning high-dimensional directed acyclic graphs with latent and selection variables

We consider the problem of learning causal information between random variables in directed acyclic graphs (DAGs) when allowing arbitrarily many latent and selection variables. The FCI (Fast Causal Inference) algorithm has been explicitly designed to infer conditional independence and causal information in such settings. However, FCI is computationally infeasible for large graphs. We therefore propose the new RFCI algorithm, which is much faster than FCI. In some situations the output of RFCI is slightly less informative, in particular with respect to conditional independence information. However, we prove that any causal information in the output of RFCI is correct in the asymptotic limit. We also define a class of graphs on which the outputs of FCI and RFCI are identical. We prove consistency of FCI and RFCI in sparse high-dimensional settings, and demonstrate in simulations that the estimation performances of the algorithms are very similar. All software is implemented in the R-package pcalg.

Using Path Diagrams as a Structural Equation Modeling Tool

A linear structural equation model (SEM) without free parameters has two parts: a probability distribution and an associated path diagram corresponding to the causal relations among variables specified by the structural equations and the correlations among the error terms. This article shows how path diagrams can be used to solve a number of important problems in structural equation modeling; for example, How much do sample data underdetermine the correct model specification? Given that there are equivalent models, is it possible to extract the features common to those models? When a modeler draws conclusions about coefficients in an unknown underlying SEM from a multivariate regression, precisely what assumptions are being made about the SEM? The authors explain how the path diagram provides much more than heuristics for special cases; the theory of path diagrams helps to clarify several of the issues just noted.

Estimation of a covariance matrix with zeros

We consider estimation of the covariance matrix of a multivariate random vector under the constraint that certain covariances are zero. We first present an algorithm, which we call iterative conditional fitting, for computing the maximum likelihood estimate of the constrained covariance matrix, under the assumption of multivariate normality. In contrast to previous approaches, this algorithm has guaranteed convergence properties. Dropping the assumption of multivariate normality, we show how to estimate the covariance matrix in an empirical likelihood approach. These approaches are then compared via simulation and on an example of gene expression. Copyright 2007, Oxford University Press.

MARKOV EQUIVALENCE FOR ANCESTRAL GRAPHS

Ancestral graphs can encode conditional independence relations that arise in directed acyclic graph (DAG) models with latent and selection variables. However, for any ancestral graph, there may be several other graphs to which it is Markov equivalent. We state and prove conditions under which two maximal ancestral graphs are Markov equivalent to each other, thereby extending analogous results for DAGs given by other authors. These conditions lead to an algorithm for determining Markov equivalence that runs in time that is polynomial in the number of vertices in the graph.

Binary models for marginal independence

Log-linear models are a classical tool for the analysis of contingency tables. In particular, the subclass of graphical log-linear models provides a general framework for modelling conditional independences. However, with the exception of special structures, marginal independence hypotheses cannot be accommodated by these traditional models. Focusing on binary variables, we present a model class that provides a framework for modelling marginal independences in contingency tables. The approach that is taken is graphical and draws on analogies with multivariate Gaussian models for marginal independence. For the graphical model representation we use bidirected graphs, which are in the tradition of path diagrams. We show how the models can be parameterized in a simple fashion, and how maximum likelihood estimation can be performed by using a version of the iterated conditional fitting algorithm. Finally we consider combining these models with symmetry restrictions. Copyright (c) 2008 The Authors.

Marginal log‐linear parameters for graphical Markov models

Marginal log-linear (MLL) models provide a flexible approach to multivariate discrete data. MLL parametrizations under linear constraints induce a wide variety of models, including models defined by conditional independences. We introduce a subclass of MLL models which correspond to Acyclic Directed Mixed Graphs (ADMGs) under the usual global Markov property. We characterize for precisely which graphs the resulting parametrization is variation independent. The MLL approach provides the first description of ADMG models in terms of a minimal list of constraints. The parametrization is also easily adapted to sparse modelling techniques, which we illustrate using several examples of real data.