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

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).