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A simulation study was carried out to investigate the performance of the proposed limited information goodness-of-fit tests. The data was generated using an underlying variable framework. The true parameter values were

  • Loadings: \(\boldsymbol\lambda = (0.8, 0.7, 0.47, 0.38, 0.34, \dots)\)
  • Factor correlations: \(\phi = 0.3\) (1 factor) or \({\boldsymbol\phi}= (0.2, 0.3, 0.4)\) (2 factors)
  • Thresholds: \(\boldsymbol\tau = (-1.43, -0.55, -0.13, -0.82, -1.13,\dots)\)

Five scenarios were investigated:

  1. 1 factor, 5 variables
  2. 1 factor, 8 variables
  3. 1 factor, 15 variables
  4. 2 factor, 10 variables
  5. 3 factor, 15 variables

For each scenario, \(B=1000\) data were generated either according to simple random sample or a complex sampling procedure using true parameter values, and the rejection rate (Type I error) were calculated.

To conduct a power analysis, an extra latent variable \(x \sim \mathop{\mathrm{N}}(0,1)\) independent to the latent factor \(\eta\) was added to the \(y^*\) variables. The loadings of \(x\) are similar to the true values except that some noise (\(\mathop{\mathrm{N}}(0,0.1^2)\)) was added. This means that the fitted model is misspecified because a missing factor was not accounted for.

The simulations took roughly 20 hours to complete. No convergence issue reported when the sample size is in the range \(500 \leq n \leq 3000\) (previous versions of this simulation study saw lavaan having difficulty converging when \(n=100\) or \(n=250\)–these simulation results are not trustworthy so have been omitted).

Path diagrams

SRS type I errors

SRS power plots

Complex sampling type I errors

Complex sampling power plots

Distribution of test statistics