Michael E. Sobel

Asymptotic Confidence Intervals for Indirect Effects in Structural Equation Models

For comments on an earlier draft of this chapter and for detailed advice I am indebted to Robert M. Hauser, Halliman H. Winsborough, and Toni Richards, several anonymous reviewers, and the editor of this volume. I also wish to thank John Raisian, Nancy Rytina, and Barbara Mann for their comments and Mark Wilson for able research assistance. The opinions expressed here are the sole responsibility of the author.

Some New Results on Indirect Effects and Their Standard Errors in Covariance Structure Models

I am grateful to Gerhard Arminger, Bruno A. Baldessari, William T. Bielby, George W. Bohrnstedt, Clifford C. Clogg, Hermann Flaschka, Edward H. Freeman, Kenneth J. Singleton, and Nancy B. Tuma for helpful advice. I am also grateful to Clement A. Stone for writing a computer program that provides the standard errors discussed in this chapter. This program can be obtained at nominal cost and is described in Stone (1985).

Handbook of statistical modeling for the social and behavioral sciences

Casual Inference in the Social and Behavioral Sciences. Missing Data. Specification and Estimation of Mean Structures. The Analysis of Contingency Tables. Latent Class Models. Panel Analysis for Metric Data. Panel Analysis for Qualitative Variables. Analysis of Event Histories. Random Coefficient Models. Index.

Direct and Indirect Effects in Linear Structural Equation Models

This article discusses total indirect effects in linear structural equation models. First, I define these effects. Second, I show how the delta method may be used to obtain the standard errors of the sample estimates of these effects and test hypotheses about the magnitudes of the indirect effects. To keep matters simple, I focus throughout on a particularly simple linear structural equation system; for a treatment of the general case, see Sobel (1986). To illustrate the ideas and results, a detailed example is presented.

Effect analysis and causation in linear structural equation models

This paper considers total and direct effects in linear structural equation models. Adopting a causal perspective that is implicit in much of the literature on the subject, the paper concludes that in many instances the effects do not admit the interpretations imparted in the literature. Drawing a distinction between concomitants and factors, the paper concludes that a concomitant has neither total nor direct effects on other variables. When a variable is a factor and one or more intervening variables are concomitants, the notion of a direct effect is not causally meaningful. Even when the notion of a direct effect is meaningful, the usual estimate of this quantity may be inappropriate. The total effect is usually interpreted as an equilibrium multiplier. In the case where there are simultaneity relations among the dependent variables in tghe model, the results in the literature for the total effects of dependent variables on other dependent variables are not equilibrium multipliers, and thus, the usual interpretation is incorrect. To remedy some of these deficiencies, a new effect, the total effect of a factorX on an outcomeY, holding a set of variablesF constant, is defined. When defined, the total and direct effects are a special case of this new effect, and the total effect of a dependent variable on a dependent variable is an equilibrium multiplier.

What Do Randomized Studies of Housing Mobility Demonstrate?: Causal Inference in the Face of Interference

During the past 20 years, social scientists using observational studies have generated a large and inconclusive literature on neighborhood effects. Recent workers have argued that estimates of neighborhood effects based on randomized studies of housing mobility, such as the “Moving to Opportunity” (MTO) demonstration, are more credible. These estimates are based on the implicit assumption of no interference between units; that is, a subjects value on the response depends only on the treatment to which that subject is assigned, not on the treatment assignments of other subjects. For the MTO studies, this assumption is not reasonable. Although little work has been done on the definition and estimation of treatment effects when interference is present, interference is common in studies of neighborhood effects and in many other social settings (e.g., schools and networks), and when data from such studies are analyzed under the “no-interference assumption,” very misleading inferences can result. Furthermore, the consequences of interference (e.g., spillovers) should often be of great substantive interest, even though little attention has been paid to this. Using the MTO demonstration as a concrete context, this article develops a frame-work for causal inference when interference is present and defines a number of causal estimands of interest. The properties of the usual estimators of treatment effects, which are unbiased and/or consistent in randomized studies without interference, are also characterized. When interference is present, the difference between a treatment group mean and a control group mean (unadjusted or adjusted for covariates) estimates not an average treatment effect, but rather the difference between two effects defined on two distinct subpopulations. This result is of great importance, for a researcher who fails to recognize this could easily infer that a treatment is beneficial when in fact it is universally harmful.

Causal Inference in the Social and Behavioral Sciences

The human propensity to think in causal terms is well known (Young 1978), and the manner in which judgments about causation are made in everyday life has been studied extensively by psychologists (Einhorn and Hogarth 1986; White 1990). No doubt this propensity contributes, for better or worse, to the persistence of causal language in scientific discourse, despite some influential attempts (for example, Russell 1913) to banish such talk to the prescientific era.

Identification of Causal Parameters in Randomized Studies With Mediating Variables

Treatments in randomized studies are often targeted to key mediating variables. Researchers want to know if the treatment is effective and how the mediators affect the outcome. The data are often analyzed using structural equation models (SEMs), and model coefficients are interpreted as effects. However, only assignment to treatment groups is randomized, so mediators are self-selected treatments. Thus, the so-called direct effects of mediators on later outcomes do not usually warrant a causal interpretation. Holland (1988) studied the case of a single continuous mediator, criticizing the use of SEMs. He uses treatment assignment as an instrument for the effect of the mediator on the outcome. However, the assumptions he made to justify this approach are overly strong and substantively implausible. This article (a) makes explicit the assumptions needed to justify equating the parameters of SEMs with the effects of mediators, (b) provides weaker and more plausible conditions under which the instrumental variable estimand may be interpreted as an effect, and (c) extends the analysis to include the case of noncompliance.

The robustness of estimates of total indirect effects in covariance structure models estimated by maximum

The large sample distribution of total indirect effects in covariance structure models in well known. Using Monte Carlo methods, this study examines the applicability of the large sample theory to maximum likelihood estimates oftotal indirect effects in sample sizes of 50, 100, 200, 400, and 800. Two models are studied. Model 1 is a recursive model with observable variables and Model 2 is a nonrecursive model with latent variables. For the large sample theory to apply, the results suggest that sample szes of 200 or more and 400 or more are required for models such as Model 1 and Model 2, respectively.

Exchange, Structure, and Symmetry in Occupational Mobility

Previous attempts to related the traditional concepts of exchange and structural mobility to parameters of the log linear model have been flawed. This article reformulated these concepts; introduces a new, more general conceptual distinction between reciprocated and unreciprocated mobility; and matches the concepts of structure and exchange to parameters of the model of quasi symmetry (QS). Specifically, if exchange or reciprocated mobility is defined as that part of the mobility process that results from equal flows between pairs of occupational categories, and if structural mobility is defined as an effect of marginal heterogeneity that operates uniformly on origins, then (if QS or any special case of QS holds) there is a correspondence between the parameter of the model and the concepts of structure and exchange. Furthermore, this correspondence can be used to develop meaningful parametric (as opposed to ad hoc) indexes of structural mobility. However, if QS fails to hold, there is at best a partial correspondence between the concepts of structure and exchange and the parameters of any multiplicative model. Data from Brazil, Great Britain, and the United States are used to illustrate the articles approach.