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Structural Equation Modeling: Applications Using Mplus by Xiaoqian Wang, Jichuan Wang

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1.2 Model Identification

A fundamental consideration when specifying a SEM model is model identification. Essentially, model identification concerns whether a unique value for each and every unknown parameter can be estimated from the observed data. For a given free (i.e., unknown) parameter that needs to be model estimated, if it is not possible to express the parameter algebraically as a function of sample variances/covariances, then that parameter is defined to be unidentified. We can get a sense of the problem by considering the example equation Var (y) = Var (η) + Var (ε), where Var (y) is the variance of the observed variable y, Var (η) is the variance of the latent variable η, and Var (ε) is the variance of the measurement error. There are one known [i.e., Var (y)] and two unknowns [i.e., Var (η) and Var (ε)] in the equation; therefore, there is no unique solution for either Var (η) or Var (ε) in this equation. That is, there are an infinite number of combinations of values of Var (η) and Var (ε) that would sum to Var (y), thus rendering this single equation model unidentified. If we wish to solve the problem, we need to impose some constrains in the equation. One such constraint might be to fix the value of Var (ε) to a constant by adding one more equation Var(ε) = C (where C is a constant). Then, Var (η) would be ensured to have a unique estimate, that is, Var (η) = Var (y) − C. In other words, the parameter Var (η) in the equation is identified. The same general principles ...

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