USING PATH DIAGRAMS AS A STRUCTURAL EQUATION MODELLING TOOL

P.Spirtes, T.Richardson, C.Meek, R. Scheines, C. Glymour. To appear in Sociological Methods and Research.

ABSTRACT

Linear structural equation models (SEMs) are widely used in sociology, econometrics, biology, and other sciences. A SEM (without free parameters) has two parts: a probability distribution (in the Normal case specified by a set of linear structural equations and a covariance matrix among the error or disturbance terms), and an associated path diagram corresponding to the causal relations among variables specified by the structural equations and the correlations among the error terms. It is often thought that the path diagram is nothing more than a heuristic device for illustrating the assumptions of the model. However, in 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 modelling. These problems include:

• How much do sample data underdetermine the correct model specification? Of course, one must decide how much credence to give alternative explanations that afford different fits to any particular data set. There are a variety of techniques for that purpose, including Bayesian updating, and a variety of fit measures with well understood large sample properties . But what about two or more alternative models that fit a specific data set equally well, or, subject to certain restrictions, fit any data set meeting the restrictions equally well? The number of such equivalents for a given linear structural equation model may be very large. Even if there are sources of knowledge about structure from outside the data set, the number of equivalent models all meeting those knowledge constraints may be considerable, and the structures they postulate may have importantly different implications for policy. Unless we characterize such equivalencies, selection of a particular model can only involve an element of arbitrary choice.
  
  
• Given that there are equivalent models, is it possible to extract the features common to those models? Under some circumstances, every member of a set of equivalent models may share some of the same linear coefficients or correlated errors. If that is the case, then it is possible that even though the data may not help us choose between the different models, the data may provide evidence for features common to all of the best models.
  
  
• When a modeler draws conclusions about coefficients in an unknown underlying structural equation model from a multivariate regression, precisely what assumptions are being made about the structural equation model? For example, when does a non-zero partial regression coefficient correspond to a non-zero coefficient in a structural equation?

These questions have been addressed many times, though usually only for models with special structures, and usually relying on linear algebra, the mathematics that seems most natural for a study of linear models.

The aim of this paper is to 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, issues that have been the focus of intelligent--if, in our judgment, ultimately too sweeping-- criticism of the use of structural equation models. What follows is a report that describes some of what has been learned about these issues by following a different set of mathematical ideas that exploit the graphical structure implicit in structural equation models.

In particular, we will present answers to these questions that depend upon an understanding of the relationship between the path diagram used to represent a structural equation model, and the zero partial correlations entailed by that path diagram (entailed in the sense that every structural equation model that shares the path diagram has a zero partial correlation). We will describe a graphical relation, the Pearl-Geiger-Verma d-separation criterion, among a pair of variables X and Y, and a set of variables Z, that is a necessary and sufficient condition for a structural equation model to entail a zero partial correlation. Such necessary and sufficient conditions have been known for path diagrams without correlated errors, but we will extend the conditions to path diagrams with correlated errors.

In section 2 we will motivate interest in the d-separation relation by describing the problems that it helps to solve in more detail. Then in section 3 we will show how the zero partial correlations entailed by a structural equation model can be read off from its path diagram, and in section 4 use the machinery developed in section 3 to provide some solutions to problems described in section 2. In section 5 we discuss the broader implications of this work for model selection, and illustrate this with two examples in section 6. In section 7 we prove the main theorem, hitherto unpublished, which justifies the use of d-separation in path diagrams representing correlated errors (represented by edges of the form <->, which we call double-headed arrows).

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