Touchstones of Equivalence and the Houdini Transformation

Inspired by Peter Molenaar's Houdini transformation, we consider the idea of touchstones between different models. Touchstones represent instances where models that appear different on the surface can have equivalent characteristics. Touchstones can appear as identical tests of model parameters. They can exist in the mean structure, in the covariance structure, or in both. In the latter case, the models will generate identical mean and covariance structures and will fit the data equally well. After showing some examples of touchstones and how they result from constraints on a general model, we show how that idea can suggest Molenaar's Houdini transformation. This transformation allows one to take a latent variable model and derive an equivalent model comprised solely of manifest (observed) variables. As equivalent models, the parameters of one can be transformed into the parameters of the other.