Description and results

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 factor, 5 variables
1 factor, 8 variables
1 factor, 15 variables
2 factor, 10 variables
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