this paper are based on random samples. In practice, however, the tests will also be applied to regression residuals and residuals from time series models.

This paper provides conditions and procedures for deciding if patterns of independencies found in covariance and concentration matrices can be generated by a stepwise recursive process represented by some directed acyclic graph. If such an agreement is found, we know that one or several causal processes could be responsible for the observed independencies, and our procedures could then be used to elucidate the graphical structure common to these processes, so as to evaluate their compatibility against substantive knowledge of the domain. If we find that the observed pattern of independencies does not agree with any stepwise recursive process, then there are a number of different possibilities. For instance, -- some weak dependencies could have been mistaken for independencies and led to the wrong omission of edges from the covariance or concentration graphs. -- some of the observed linear dependencies reflect accidental cancellations or hide actual nonlinear relations, or -- the process responsible for the data is non-recursive, involving aggregated variables, simultenous reciprocal interactions, or mixtures of several causal processes. In order to recognize accidental independencies it would be helpful to conduct several longitudinal studies under slightly varying conditions. In such studies the covariances for the same set of variables is estimated under different conditions and the variations in the conditions would typically affect the numerical values of the parameters. But, if the data were generated by a causal process represented by some directed acyclic graph, then the basic structural properties reflected in the missing edges of that graph should remain unchanged. Under such assumptions, the pattern of independencies that is "implied" by the dag (see Definitio...

This paper objects to the arising problems due to fitting graphical chain models to multidimensional data sets. This multivariate statistical tool is used to cope with complex research questions concerning not only direct, but also indirect associations between the variables of interest. Due to this high complexity sensible strategies for fitting such models are required. Here, a data--driven selection strategy is discussed. Its application is illustrated for an empirical data example in detail. 1 Introduction In empirical studies, typically a large number of different variables is collected for each individual. Often the researcher has a certain idea concerning the underlying association structure which implies direct as well as indirect influences on the main responses. Such influences cannot be captured by conventional regression models. Here, graphical chain models are the adequate tool since they allow for intermediate variables additional to the pure explanatories and pure respo...

Several different types of statistical interaction are defined and distinguished, primarily on the basis of the nature of the factors defining the interaction. Illustrative examples, mostly epidemiological, are given. The emphasis is primarily on interpretation rather than on methods for detecting interactions. 1. Introduction. Interaction

Fitting a graphical chain model to a multivariate data set consists of di erent steps some of which being rather tedious. The paper outlines the basic features and overall architecture of the computer program GraphFitI which provides the application of a selection strategy for tting graphical chain models and for visualising the resulting models as a graph. It additionally supports the user at the di erent steps of the analysis by an integrated help system.

this paper are based on random samples. In practice, however, the tests will also be applied to regression residuals and residuals from time series models

A crucial task in modern genetic medicine is the understanding of complex genetic diseases. The main complicating features are that a combination of genetic and environmental risk factors is involved, and the phenotype of interest may be complex. Traditional statistical techniques based on lod-scores fail when the disease is no longer monogenic and the underlying disease transmission model is not defined. Different kinds of association tests have been proved to be an appropriate and powerful statistical tool to detect a “candidate gene ” for a complex disorder. However, statistical techniques able to investigate direct and indirect influences among phenotypes, genotypes and environmental risk factors, are required to analyse the association structure of complex diseases. In this paper we propose graphical models as a natural tool to analyse the multifactorial structure of complex genetic diseases. An application of this model to primary hypertension data set is illustrated.

ABSTRACT: In this paper, we study sequences of regressions in joint or single responses given a set of context variables, where a dependence structure of interest is captured by a regression graph. These graphs have nodes representing random variables and three types of edge. Their set of missing edges defines the independence structure of the graph provided two properties are used that are not common to all probability distributions, named the intersection and the composition property. We derive the additionally needed properties for tracing the effects of single active paths and for excluding any canceling of effects due to several paths connecting the same pair of nodes. For this, we use the notion of a generating process for the joint distribution and derive new properties of an edge matrix calculus for transforming graphs. One key is the M-matrix property of each regularized square edge matrix, others are the proposed notions of traceable regressions and of singleton transitivity.