1 Introduction

Path analysis was invented by the geneticist Sewall Wright (1921a, 1934, 1960, 1968), and has been widely applied to problems in genetics and the behavioral sciences. It is a technique which allows us to represent, in diagrammatic form, linear `structural' models and hence derive predictions for the variances and covariances (the covariance structure) of our variables under that model. The books by Kenny (1979), Li (1975), or Wright (1968) supply good introductory treatments of path analysis, and general descriptions of structural equation modeling can be found in Bollen (1989) and Loehlin (1987). In this chapter we provide only the basic background necessary to understand models used in the genetic analyses presented in this text. A path diagram is a useful heuristic tool to graphically display causal and correlational relations or the paths between variables. Used correctly, it is one of several mathematically complete descriptions of a linear model, which include less visually immediate forms such as (i) structural equations and (ii) expected covariances derived in terms of the parameters of the model. Since all three forms are mathematically complete, it is possible to translate from one to another for such purposes as applying it to data, increasing understanding of the model, verifying its identification, or presenting results. The advantage of the path method is that it goes beyond measuring the degree of association by the correlation coefficient or determining the best prediction by the regression coefficient. Instead, the user makes explicit hypotheses about relationships between the variables which are quantified by path coefficients. Better still, the model's predictions may be statistically compared with the observed data, as we shall go on to discuss in Chapters  and . Path models are in fact extremely general, subsuming a large number of multivariate methods, including (but not limited to) multiple regression, principle component or factor analysis, canonical correlation, discriminant analysis and multivariate analysis of variance and covariance. Therefore those that take exception to `path analysis' in its broadest sense, should be aware that they dismiss a vast array of multivariate statistical methods. We begin by considering the conventions used to draw and read a path diagram, and explain the difference between correlational paths and causal paths (Section 5.2). In Sections 5.3 and 5.4 we briefly describe assumptions of the method and tracing rules for path diagrams. Then, to illustrate their use, we present simple linear regression models familiar to most readers (Section 5.5). We define these both as path diagrams and as structural equations -- some individuals handle path diagrams more easily, others respond better to equations! We also apply the method to two basic representations of a simple genetic model for covariation in twins (Section 5.6), with special reference to the identity between the matrix specification of a model and its graphical representation. Finally we discuss identification of models and parameters in Section 5.7.