Estimating, Testing, and Comparing Specific Effects in Structural Equation Models: The Phantom Model Approach Siegfried Macho University of Fribourg Thomas Ledermann University of Connecticut The phantom model approach for estimating, testing, and comparing specific effects within structural equation models (SEMs) is presented. The rationale underlying this novel method consists in representing the specific effect to be assessed as a total effect within a separate latent variable model, the phantom model that is added to the main model. The following favorable features characterize the method: (a) It enables the estimation, testing, and comparison of arbitrary specific effects for recursive and nonrecursive models with latent and manifest variables; (b) it enables the bootstrapping of confidence intervals; and (c) it can be applied with all standard SEM programs permitting latent variables, the specification of equality constraints, and the bootstrapping of total effects. These features along with the fact that no manipulation of matrices and formulas is required make the approach particularly suitable for applied researchers. The method is illustrated by means of 3 examples with real data sets. Keywords: phantom models, specific effects, bootstrapping, SEM In many practical applications, researchers using structural equation modeling (SEM) face the problem of assessing specific effects that cannot be classified as direct, indirect, or total effects, as estimated by most SEM programs. In structural models, the direct effect of variable X on Y is the effect that is not mediated by any other variable in the model. The indirect effect from X to Y is the sum of all mediated effects between the source variable X and the final outcome variable Y. The total effect is the sum of the direct and indirect effect. For example, a scientist might be inter- ested in the specific effect of a source variable X on a final outcome variable Y that is mediated by a subclass of the mediators (inter- vening variables) involved in the effect of X on Y only. The evaluation of such specific effects calls for special techniques that, unfortunately, are not implemented in all SEM programs and require some expertise in matrix algebra to execute. Moreover, not only the estimation and testing of single specific effects but also a comparison of different specific effects (or of direct, indirect, and total effects) might be of substantial interest. For example, a researcher might want to compare the effect of a source on a final outcome variable that is mediated by two groups of intervening variables. Again, most commonly used SEM pro- grams provide no direct solution to this problem. The SEM literature provides two main approaches for address- ing the estimation, testing, and comparison of specific effects. The first one employs matrix methods (Bollen, 1987, 1989). Specific effects are estimated by means of (differences of) matrix products, and the associated standard errors of estimates are computed by means of the delta method in matrix format (Sobel, 1986). The matrix method might not be well suited for all applications and users, for the following reasons: First, the use of complex matrix formulas (involving vector operators and Kronecker products) might have a deterrent effect on users interested primarily in applied research. Second, with complex specific effects, involving many paths, the computation of standard errors of the specific effect becomes quite complex and error prone. Finally, in general the method does not permit the user to bootstrap confidence intervals. At present, the bootstrapping of confidence intervals of arbitrary specific effects and comparison of effects via Bollen’s (1987, 1989) method is only possible with the programs Mplus, Mx, and OpenMx. However, bootstrapped confidence intervals are usually superior to those computed by means of the delta method, especially with small sample sizes (e.g., Bollen & Stine, 1990; MacKinnon, Lockwood, & Williams, 2004). A second approach to the estimation and comparison of specific effects was proposed by Cheung (2007). According to this method, a latent variable, a so-called phantom variable (Rindskopf, 1984), is added to the model as an extra variable with a direct effect from this latent variable to one of the main variables in the model. The variance of this latent variable is fixed to zero, and the structural coefficient assigned to the path from the phantom variable to the main variable is restricted by means of a formula representing the specific effect or a contrast between two or more effects. This strategy forces the program to provide estimates and standard errors for the structural coefficient represented by the formula. The insertion of the additional variable with fixed variance and the direct effect constrained as a function of the other model param- eters does not influence the estimation of parameters for the main This article was published Online First February 7, 2011. Siegfried Macho, Department of Psychology, University of Fribourg, Fribourg, Switzerland; Thomas Ledermann, Department of Psychology, University of Connecticut. We would like to thank Michael Munz for his comments on an earlier draft of the article. Additional material comprising raw data, AMOS graphics files with models, and R source files implementing OpenMx programs of the examples is available online (http://www.unifr.ch/psycho/ site/units/allpsy/team/Macho/research/sem). Correspondence concerning this article should be addressed to Siegfried Macho, Department of Psychology, University of Fribourg, Rue Faucigny 2, CH 1700 Fribourg, Switzerland. E-mail: siegfried.macho@unifr.ch Psychological Methods 2011, Vol. 16, No. 1, 34 – 43 © 2011 American Psychological Association 1082-989X/11/$12.00 DOI: 10.1037/a0021763 34