When Trivial Constraints Are Not Trivial: The Choice of Uniqueness Constraints in Confirmatory Factor Analysis

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A Non-arbitrary Method of Identifying and Scaling Latent Variables in SEM and MACS Models (2006, background): “However, Millsap (2001) demonstrated that in certain cases when a model is very unrestricted this may not be the case.”
Measurement Invariance for Latent Constructs in Multiple Populations (2012, methodology): “Although this parameter fixing resolves the identification issue (up to a sign reversal, as treated in this article; e.g., Millsap, 2001) that fixing is not innocuous.”
Multigroup Confirmatory Factor Analysis: Locating the Invariant Referent Sets (2008, methodology): “It should be noted, however, that despite this lack of a systematic selection procedure, different constraint choices can lead to quite different model fit results (Millsap, 2001; Steiger, 2002).”
Assessing Factorial Invariance in Ordered-Categorical Measures (2004, background): “These constraints can take several forms (Bollen & Jöreskog, 1985; Jöreskog, 1979; Millsap, 2001).”