On the Estimation of Path Coefficients for Unmeasured Variables from Correlations Among Observed Variables

This paper discusses the use of unmeasured variables in path models. Problems of estimation of the path coefficients of a path model are explored when unmeasured variables are utilized as both causes and effects (intervening variables). The paper concludes with a discussion of conditions for the identification of a path model containing unmeasured variables and some remarks on the substantive interpretation of unmeasured variables. B eginning with H. M. Blalock's (1960; 1961a; 1961b) extension of Herbert Simon's (1957) seminal essay on spurious correlation, sociologists have become increasingly aware of the utility of linear causal models as a procedure for bridging the gap between sociological theory on the one hand and the results of classical statistical analyses on the other. That is, the practices of: (1) writing a causal model as a set of simultaneous equations which are assumed to represent the structural relations among a given set of variables, and (2) engaging in statistical analysis to estimate the values of the parameters of the model, are rapidly becoming standard aspects of sociological research methodology. Finally, the conditions under which the researcher can infer that the model adequately reflects the empirical phenomena under investigation (make causal inferences) have been explicated and synthesized. There have been a number of contributions * This is a revision of a paper presented at the annual meeting of the Southern Sociological Society, 1969. It is based on Kenneth C. Land, "Explorations in Mathematical Sociology," unpublished Ph.D. dissertation, University of Texas at Austin, 1969. The author is grateful to the anonymous reviewers for helpful comments on an earlier draft of the paper. This content downloaded from 157.55.39.162 on Thu, 11 Aug 2016 05:45:38 UTC All use subject to http://about.jstor.org/terms PATH COEFFICIENTS FOR UNMEASURED VARIABLES 507 by sociologists to the development of causal modeling methodology. Blalock, for example, has been most active in the formalization of criteria for making causal inferences (e.g., Blalock, 1961c; 1961d; 1962a; 1962b; 1963a; 1965; 1967a; 1968a; 1968b). Recently, he (Blalock, 1966; 1967b; 1967c) has also studied some particular cases of the identification problem in sociology. Duncan (1966) introduced sociologists to path analysis, a technique developed by the geneticist Sewall Wright (1960). Finally, Boudon (1968) extended path analysis to the cases of dichotomous variables and nonadditive relationships. The technique of path analysis seems particularly useful to sociologists as a linear modeling device. The primary advantage of path analysis accrues from the fact that it deals with standardized structural coefficients (cf. Boudon, 1968:202-208). This property yields a theorem which facilitates the decomposition of the zero-order correlation coefficient for two variables into parameter estimates of the direct and indirect effects of one variable on the other through the postulated causal structure (cf. Duncan, 1966:5-6). The technique has been well-developed for the case of one-way, i.e., recursive, causal systems (e.g., Heise, 1969a; Land, 1969). Boudon (1968:227-233) has also shown how the method can be applied to a sociological example of lagged reciprocal interaction. A second advantage of path analysis lies in its capacity to treat hypothetical (unmeasured) variables in a postulated causal structure (cf. Land, 1969). There are two general cases for which sociologists will be interested in utilizing unmeasured variables in a causal model. The first of these arises in the study of measurement error. This application of path analysis has been explored by Siegel and Hodge (1968), Heise (1969b), and Costner (1969), and the reader is referred to those sources for a treatment of the problem. Briefly, Siegel and Hodge utilize path analysis to study discrepancies between census and sample survey measures of the same variables. On the other hand, Heise develops formulas for estimating the reliability and stability of a measure from data collected at three or more points in time. Finally, Costner derives conditions under which the parameters of a path model can be estimated if one has multiple indicators of each variable. The common characteristic of all of these applications of path analysis is that the hypothetical (unmeasured) variables enter the path models only as causes of the observed variables. A second case in which sociologists will utilize unmeasured variables is as variables which intervene between measured variables in a causal model. The methodology for path analysis in this case has not as yet been developed for sociologists, and it is that goal to which this paper is de-