A Semiparametric Approach for Structural Equation Modeling with Ordinal Data

Figures & data

Table 1. Parameter and corresponding average estimates using AGHQ approximation with 5 quadrature points per latent dimension and 1000 replications. RB, SD, and RMSE stand for relative bias, standard deviation, and root mean squared error, respectively. The sample size is $N=1000$

Parameter	Population value	Average estimate	RB	$10\cdot$SD	$10\cdot$RMSE
${\mathrm{\Psi }}_{11}$	$1.000$	$0.961$	$-3.949$	$1.378$	$1.434$
${\tau }_1^{(2)}$	$-4.000$	$-4.041$	$1.035$	$3.782$	$3.784$
${\mathrm{\Gamma }}_{11}^{(1)}$	$-1.000$	$-0.992$	$-0.761$	$1.630$	$1.632$
${\mathrm{\Gamma }}_{12}^{(1)}$	$-1.000$	$-0.988$	$-1.200$	$1.599$	$1.604$
${\mathrm{\Gamma }}_{11}^{(2)}$	$1.000$	$1.002$	$0.241$	$1.095$	$1.095$
${\mathrm{\Gamma }}_{12}^{(2)}$	$1.000$	$0.997$	$-0.340$	$1.052$	$1.052$
${\mathrm{\Phi }}_{11}^{(1)}$	$0.500$	$0.509$	$1.822$	$0.923$	$0.941$
${\mathrm{\Phi }}_{21}^{(1)}$	$-0.200$	$-0.202$	$1.098$	$0.408$	$0.422$
${\mathrm{\Phi }}_{22}^{(1)}$	$0.500$	$0.513$	$2.529$	$0.907$	$0.942$
${\mathrm{\Phi }}_{11}^{(2)}$	$0.500$	$0.506$	$1.111$	$0.798$	$0.806$
${\mathrm{\Phi }}_{21}^{(2)}$	$-0.200$	$-0.203$	$1.262$	$0.361$	$0.382$
${\mathrm{\Phi }}_{22}^{(2)}$	$0.500$	$0.509$	$1.736$	$0.770$	$0.789$
${\mu }_{11}^{(1)}$	$0.460$	$0.462$	$0.425$	$0.560$	$0.561$
${\mu }_{21}^{(1)}$	$0.460$	$0.463$	$0.642$	$0.582$	$0.585$
${\mu }_{11}^{(2)}$	$2.460$	$2.478$	$0.718$	$1.640$	$1.641$
${\mu }_{21}^{(2)}$	$2.460$	$2.485$	$1.001$	$1.576$	$1.580$
${\pi }_1$	$0.500$	$0.500$	$-0.040$	$0.166$	$0.166$

Table 2. Parameter and corresponding average estimates using AGHQ approximation with 5 quadrature points per latent dimension and 1000 replications. RB, SD, and RMSE stand for relative bias, standard deviation, and root mean squared error, respectively. The sample size is $n=500$

Parameter	Population value	Average estimate	RB	$10\cdot$SD	$10\cdot$RMSE
${\mathrm{\Psi }}_{11}$	$1.000$	$0.963$	$-3.723$	$2.060$	$2.093$
${\tau }_1^{(2)}$	$-4.000$	$-4.101$	$2.532$	$6.539$	$6.544$
${\mathrm{\Gamma }}_{11}^{(1)}$	$-1.000$	$-1.001$	$0.136$	$2.377$	$2.377$
${\mathrm{\Gamma }}_{12}^{(1)}$	$-1.000$	$-1.005$	$0.528$	$2.331$	$2.332$
${\mathrm{\Gamma }}_{11}^{(2)}$	$1.000$	$1.007$	$0.669$	$1.692$	$1.693$
${\mathrm{\Gamma }}_{12}^{(2)}$	$1.000$	$1.007$	$0.662$	$1.614$	$1.615$
${\mathrm{\Phi }}_{11}^{(1)}$	$0.500$	$0.523$	$4.602$	$1.407$	$1.481$
${\mathrm{\Phi }}_{21}^{(1)}$	$-0.200$	$-0.206$	$2.960$	$0.617$	$0.685$
${\mathrm{\Phi }}_{22}^{(1)}$	$0.500$	$0.524$	$4.788$	$1.389$	$1.470$
${\mathrm{\Phi }}_{11}^{(2)}$	$0.500$	$0.511$	$2.132$	$1.190$	$1.208$
${\mathrm{\Phi }}_{21}^{(2)}$	$-0.200$	$-0.203$	$1.490$	$0.524$	$0.545$
${\mathrm{\Phi }}_{22}^{(2)}$	$0.500$	$0.514$	$2.824$	$1.174$	$1.207$
${\mu }_{11}^{(1)}$	$0.460$	$0.466$	$1.219$	$0.819$	$0.828$
${\mu }_{21}^{(1)}$	$0.460$	$0.469$	$4.131$	$0.832$	$0.929$
${\mu }_{11}^{(2)}$	$2.460$	$2.502$	$1.724$	$2.400$	$2.406$
${\mu }_{21}^{(2)}$	$2.460$	$2.501$	$2.089$	$2.481$	$2.490$
${\pi }_1$	$0.500$	$0.501$	$0.103$	$0.233$	$0.233$

Figure 1. The conditional mean of the true data generating process (thick line) and the average estimated conditional mean (thin line) with empirical 95% intervals (filled blue areas) for 2 components (left), 3 components (middle) and when AIC is used for selecting the number of components (right)