Using Phantom Variables in Structural Equation Modeling to Assess Model Sensitivity to External Misspecification

External misspecification, the omission of key variables from a structural model, can fundamentally alter the inferences one makes without such variables present. This article presents 2 strategies for dealing with omitted variables, the first a fixed parameter approach incorporating the omitted variable into the model as a phantom variable where all associated parameter values are fixed, and the other a random parameter approach specifying prior distributions for all of the phantom variable's associated parameter values under a Bayesian framework. The logic and implementation of these methods are discussed and demonstrated on an applied example from the educational psychology literature. The argument is made that such external misspecification sensitivity analyses should become a routine part of measured and latent variable modeling where the inclusion of all salient variables might be in question. Translational Abstract A model can be thought of as a comprehensive mechanism for understanding the behavior of data in a population. From this perspective, any extent to which that mechanism is incorrect constitutes a misspecification. This article focuses on external misspecifications-the omission of key variables from a structural model (e.g., covariate, mediator). A variable may be absent because its importance was not brought to light until after a study had been conducted (e.g., as suggested by a reviewer) or it simply might be unavailable within an existing dataset being used in a secondary data analysis. Left unchecked, however, external misspecification can fundamentally alter the inferences one might make without such variables present. Two strategies are presented for investigating sensitivity to omitted variables. The first is a fixed parameter approach which integrates an omitted variable into the model as a phantom variable where all associated parameter values are fixed. The second strategy is a random parameter approach within a Bayesian framework in which prior distributions are specified for all of the phantom variables associated parameter values. Both methodologies allow researchers to bring expert knowledge of the substantive domain to bear on the analytic model through specifying likely candidates for the parameter values linked to the missing variables. The reasoning behind each method is discussed more generally before giving way to their implementation on an applied example from the educational psychology literature. Our conclusion is that such an external misspecification sensitivity analysis ought to become a systematic part of any modeling endeavor where the inclusion of all salient variables might be in question.