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SUMMARY Graphs consisting of points, and lines or arrows as connections between selected pairs of points, are used to formulate hypotheses about relations between variables. Points stand for variables, connections represent associations. When a missing connection is interpreted as a conditional independence, the graph characterizes a conditional independence structure as well. Statistical models, called graphical chain models, correspond to special types of graphs which are interpreted in this fashion. Examples are used to illustrate how conditional independences are reflected in summary statistics derived from the models and how the graphs help to identify analogies and equivalences between different models. Graphical chain models are shown to provide a unifying concept for many statistical techniques that in the past have proven to be useful in analyses of data. They also provide tools for new types of analysis.

Statistical aspects of causality are reviewed in simple form and the impact of recent work discussed. Three distinct notions of causality are set out and implications for densities and for linear dependencies explained. The importance of appreciating the possibility of effect modifiers is stressed, be they intermediate variables, background variables or unobserved confounders. In many contexts the issue of unobserved confounders is salient. The difficulties of interpretation when there are joint effects are discussed and possible modifications of analysis explained. The dangers of uncritical conditioning and marginalization over intermediate response variables are set out and some of the problems of generalizing conclusions to populations and individuals explained. In general terms the importance of search for possibly causal variables is stressed but the need for caution is emphasized.

By defining a reducible zero pattern and by using the concept of multiplicative models, we relate linear recursive equations that have been introduced by econometrician Herman Wold (1954) and path analysis as it was proposed by geneticist Sewall Wright (1923) to the statistical theory of covariance selection formulated by Arthur Dempster (1972). We show that a reducible zero pattern is the condition under which parameters as well as least squares estimates in recursive equations are one-to-one transformations of parameters and of maximum likelihood estimates, respectively, in a decomposable covariance selection model. As a consequence, (a) we can give a closed-form expression for the maximum likelihood estimate of a decomposable covariance matrix, (b) we can derive Wright's rule for computing implied correlations in path analysis, and (c) we can describe a search procedure for fitting recursive equations.KEY WORDS : Reducible zero patterns; Multiplicative or decomposable models ; One-to-one transformations of covariance matrices ; Linear recursive equations ; Covariance selection ; Path analysis. path analysis rule (4.1) leads to the MLE's of the equation parameters. We then find a sufficient condition under which both path analysis rules define MLE's. This condition is a reducible zero pattern in the equation parameters or equivalently in the parameters of a covariance selection model.To this end, in Section 2 we derivc linear recursive equations as a triangular reduction of a covariance matrix. We show that a reducible zero pattern for the regression coefficients implies a corresponding covariance selection model in which the inverse cova-' inct matrix has the same reducible zero pattern, anci conversely. I n Section 3 we show that for all multiplicative models the variables can be reordered so that the inverse covariance matrix has a reducible zero pattern. We state closed-form expressions for the MLE of such decomposable covariance matrices. From these we obtain in SW-A renewed interest in studying systems of linear equa-tion 4 Wright's rule for computing implied correlations in tions has been documented in many subject matter areas, path analysis. Finally, we suggest in Section 5 how the in genetics (Li 1975), in sociology and in economics model search among multiplicative models (Wermuth (Blalock 1971; Goldberger and Duncan 1973; Heise 197610) can be modified to become a search for fitting 1975, Duncan 1975), and in psychology (Hodapp 1978). recursive equations. We illustrate this with a set of The purpose of this article is to present a covariance se-sociological data.