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Contributions: AbstractIdentification issues in latent trait models
Some interesting issues in identification are evident even in a simple two orthogonal factor model in correlation metric. Fixing one loading to zero identifies the model up to reversals of the factors. Jöreskog (1969) originally thought that fixing one loading to a nonzero constant would also identify the model. Jennrich (1978) showed that alternate modes still exist when loadings are fixed to nonzero constants, and the equivalent modes are less "transparent". Nevertheless, both concluded that nonzero constraints "will probably obtain solutions that are at least locally unique". Jennrich did add, however, that "more work needs to be done to clarify these issues and resolve identification problems" (p.426). In a simple twofactor model, fixing loadings to certain nonzero values produces overlapping modes in the likelihood that can affect inference. Furthermore, for certain specific choices of nonzero constraints, common software packages crash and the standard errors are inestimable. With simple rotations of the factors, maintaining the constraints, equivalent modes in the likelihood can be graphically demonstrated. Implications exist for other more complex models. For example, the rotation technique shows the existence of minor modes in the latent growth curve model with estimated slope coefficients, and thus correctly predicts choices of start values that will fail to converge. Furthermore, in Bayesian inference for factor models, the MCMC algorithm may oscillate between two adjacent modes, distorting posterior summaries. The problem of visualizing high dimensional models complicates the identification issues discussed by Jöreskog and Jennrich. Models with fixed nonzero loadings can indeed be very problematic, but simple graphing techniques can bring greater "transparency" to the topic. References
