This article describes the use of Bayes factors for comparing Dynamic Causal Models (DCMs). DCMs are used to make inferences about effective connectivity from functional Magnetic Resonance Imaging (fMRI) data. These inferences, however, are contingent upon assumptions about model structure, that is, the connectivity pattern between the regions included in the model. Given the current lack of detailed knowledge on anatomical connectivity in the human brain, there are often considerable degrees of freedom when defining the connectional structure of DCMs. In addition, many plausible scientific hypotheses may exist about which connections are changed by experimental manipulation, and a formal procedure for directly comparing these competing hypotheses is highly desirable. In this article, we show how Bayes factors can be used to guide choices about model structure, both with regard to the intrinsic connectivity pattern and the contextual modulation of individual connections. The combined use of Bayes factors and DCM thus allows one to evaluate competing scientific theories about the architecture of large-scale neural networks and the neuronal interactions that mediate perception and cognition.

Introduction The introduction of Bayesian networks (Pearl 1986b) and associated local computation algorithms (Lauritzen and Spiegelhalter 1988, Shenoy and Shafer 1990, Jensen, Lauritzen and Olesen 1990) has initiated a renewed interest for understanding causal concepts in connection with modelling complex stochastic systems. It has become clear that graphical models, in particular those based upon directed acyclic graphs, have natural causal interpretations and thus form a base for a language in which causal concepts can be discussed and analysed in precise terms. As a consequence there has been an explosion of writings, not primarily within mainstream statistical literature, concerned with the exploitation of this language to clarify and extend causal concepts. Among these we mention in particular books by Spirtes, Glymour and Scheines (1993), Shafer (1996), and Pearl (2000) as well as the collection of papers in Glymour and Cooper (1999). Very briefly, but fundamentally,

The direct effect of one event on another can be defined and measured by holding constant all intermediate variables between the two. Indirect effects present conceptual and practical difficulties (in nonlinear models), because they cannot be isolated by holding certain variables constant. This paper presents a new way of defining the effect transmitted through a restricted set of paths, without controlling variables on the remaining paths. This permits the assessment of a more natural type of direct and indirect effects, one that is applicable in both linear and nonlinear models and that has broader policy-related interpretations. The paper establishes conditions under which such assessments can be estimated consistently from experimental and nonexperimental data, and thus extends path-analytic techniques to nonlinear and nonparametric models.

JEL No. C10 The major contributions of twentieth century econometrics to knowledge were the definition of causal parameters when agents are constrained by resources and markets and causes are interrelated, the analysis of what is required to recover causal parameters from data (the identification problem), and clarification of the role of causal parameters in policy evaluation and in forecasting the effects of policies never previously experienced. This paper summarizes the development of those ideas by the Cowles Commission, the response to their work by structural econometricians and VAR econometricians, and the response to structural and VAR econometrics by calibrators, advocates of natural and social experiments, and by nonparametric econometricians and statisticians.

Chain graphs are a natural generalization of directed acyclic graphs (DAGs) and undirected graphs. However, the apparent simplicity of chain graphs belies the subtlety of the conditional independence hypotheses that they represent. There are a number of simple and apparently plausible, but ultimately fallacious interpretations of chain graphs that are often invoked, implicitly or explicitly. These interpretations also lead to awed 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 as the equilibrium distribution of dynamic models with feedback. These dynamic interpretations lead to a simple theory of intervention, extending the theory developed for DAGs. Finally, we contrast chain graph models under this interpretation with simultaneous equation models which have traditionally been used to model feedback in econometrics. Keywords: Causal model; cha...

this paper, we will show how path diagrams can be used to solve a number of important problems in structural equation modelling. There are a number of problems associated with structural equation modeling. These problems include:

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The brain appears to adhere to two fundamental principles of functional organisation, functional integration and functional specialisation, where the integration within and among specialised areas is mediated by effective connectivity. In this paper we review two different approaches to modelling effective connectivity from fMRI data, Structural Equation Models (SEMs) and Dynamic Causal Models (DCMs). In common to both approaches are model comparison frameworks in which inferences can be made about effective connectivity per se and about how that connectivity can be changed by perceptual or cognitive set. Underlying the two approaches, however, are two very different generative models. In DCM a distinction is made between the ‘neuronal level ’ and the ‘hemodynamic level’. Experimental inputs cause changes in effective connectivity expressed at the level of neurodynamics which in turn cause changes in the observed hemodynamics. In SEM changes in effective connectivity lead directly to changes in the covariance structure of the observed hemodynamics. Because changes in effective connectivity in the brain occur at a neuronal level DCM is the preferred model for fMRI data. This review focuses on the underlying assumptions and limitations of each model and demonstrates their application to data from a study of attention to visual motion.

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This article examines the theoretical status of latent variables as used in modern test theory models. First, it is argued that a consistent interpretation of such models requires a realist ontology for latent variables. Second, the relation between latent variables and their indicators is discussed. It is maintained that this relation can be interpreted as a causal one but that in measurement models for interindividual differences the relation does not apply to the level of the individual person. To substantiate intraindividual causal conclusions, one must explicitly represent individual level processes in the measurement model. Several research strategies that may be useful in this respect are discussed, and a typology of constructs is proposed on the basis of this analysis. The need to link individual processes to latent variable models for interindividual differences is emphasized. Consider the following sentence: “Einstein would not have been able to come up with his e � mc 2 had he not possessed such an extraordinary intelligence. ” What does this sentence express? It relates observable behavior (Einstein’s writing e � mc 2)toan unobservable attribute (his extraordinary intelligence), and it does so by assigning to the unobservable attribute a causal role in