The Houdini Transformation: True, but Illusory

Molenaar (2003, 2011) showed that a common factor model could be transformed into an equivalent model without factors, involving only observed variables and residual errors. He called this invertible transformation the Houdini transformation. His derivation involved concepts from time series and state space theory. This article verifies the Houdini transformation on a general latent variable model using algebraic methods. The results show that the Houdini transformation is illusory in the sense that the Houdini-transformed model remains a latent variable model. Contrary to common knowledge, a model that is a path model with only observed variables and residual errors may, in fact, be a latent variable model.