In this note we describe the DIFFTEST command implemented in Mplus for testing nested models with a mean and variance adjusted chi-square statistics. The DIFFTEST command is available for both the MLMV and the WLSMV estimators. In this note we will discuss the implementation only for the WLSMV estimator, but the MLMV implementation is similar. Suppose that there are two nested SEM models H0 and H1 where the parameters in each of the models are θ0 and θ1. Let di be the number of parameters in model Hi. Let’s assume that H0 is nested in H1. We want to test the hypothesis that θ1 = f(θ0). The WLSMV estimates are obtained by minimizing the fit functions T0 = (σ(θ0) − s) ′ W −1 (σ(θ0) − s) (1) T1 = (σ(θ1) − s) ′ W −1 (σ(θ1) − s), (2) where s represents all sample statistics in the unrestricted model and σ(θi) are the Hi model estimated sample statistics, see Muthen (1998-2004). Let Γ be an estimate of the variance covariance matrix of the sample statistics s. Define also the following matrices ∂σ(θi) ∂θi = ∆i (3
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