# Smooth Information Criterion for Regularized Estimation of Item Response Models
## 4 Simulation Study 2: Multidimensional Logistic Item Response Model
### Method  
In this simulation study, we used a DGM with \(I=20\) items and \(D=2\) factor variables \(\theta_{1}\) and \(\theta_{2}\). The first 10 items loaded on the first dimension, while Items 11 to 20 loaded on the second dimension. The factor variable \((\theta_{1},\theta_{2})\) was bivariate normally distributed with standardized normally distributed components and a fixed correlation \(\rho\) of 0.5. Moreover, we specified five cross-loadings. Items 2 and 6 had a cross-loading of size \(\delta\) on the second dimension, while Items 13, 16, and 19 had a cross-loading of size \(\delta\) on the first dimension. The DGM is visualized in Figure 1. In more detail, the loading matrix \(A\) that contains the item discriminations \(a_{id}\) (see (18)) is given by \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & \multicolumn{6}{c}{**Type-I Error Rate**} & \multicolumn{6}{c}{**Power Rate**} \\ \cline{3-10}
**DIF** & \(N\) & **AIC** & **SAIC** & **BIC** & **SBIC** & **AIC** & **SAIC** & **BIC** & **SBIC** \\ \hline \multirow{3}{*}{large} & 500 & **15.3** & **14.2** & 1.2 & 5.8 & 99.7 & 99.8 & 97.5 & 99.4 \\  & 1000 & **15.0** & 9.0 & 0.8 & 2.0 & 100 & 100 & 99.9 & 100 \\  & 2000 & **14.5** & 6.2 & 0.6 & 0.6 & 100 & 100 & 100 & 100 \\ \hline \multirow{3}{*}{small} & \multicolumn{6}{c}{_Unbalanced DIF_} \\  & 500 & **25.8** & **23.8** & 4.8 & **11.3** & **68.9** & **65.6** & **30.6** & **54.0** \\  & 1000 & **21.0** & **15.8** & 5.1 & 3.7 & 89.4 & 85.4 & **71.3** & **79.9** \\  & 2000 & **13.6** & 9.1 & 2.7 & 0.4 & 97.7 & 99.6 & 95.9 & 98.1 \\  & 500 & **14.5** & **28.9** & 3.4 & 6.3 & 98.0 & 99.0 & 96.7 & 98.9 \\  & 1000 & 9.6 & **22.4** & 0.6 & 2.0 & 99.8 & 100 & 99.7 & 100 \\  & 2000 & 3.4 & **21.5** & 0.3 & 0.6 & 100 & 100 & 100 & 100 \\ \hline \hline \end{tabular} Note. Type-I error rates larger than 10.0 and power rates smaller than 80.0 are printed in bold font.  
\end{table} 
Table 2: _Cont._  
Figure 1: Simulation Study 2: Data-generating model with \(I=20\) items \(X_{i}\) (\(i=1,\ldots,20\)) and two factor variables \(\theta_{1}\) and \(\theta_{2}\). Cross-loadings are depicted by red dashed lines. \[A=\left( \begin{array}{ccc}0.6&0\\ 0.8&\delta\\ 1.0&0\\ 1.4&0\\ 1.2&0\\ 0.6&\delta\\ 0.8&0\\ 1.0&0\\ 1.4&0\\ 1.2&0\\ 0&0.6\\ 0&0.6\\ 0&0.8\\ 0&0.8\\ 0&0.8\\ 0&0.8\\ 0&1.0\\ \delta&1.4\\ 0&1.2\end{array} \right). \tag{19}\]  
The size of the cross-loading \(\delta\) was chosen as 0.3, indicating a small cross-loading), or 0.5, indicating a large cross-loading. The item difficulties \(b_{i}\) (see (18)) of the 20 items were \(-0.8\), 0.4, 1.2, 2.0, \(-2.0\), \(-0.8\), 0.4, 1.2, 2.0, \(-2.0\), \(-0.8\), 0.4, 1.2, 2.0, \(-2.0\), \(-0.8\), 0.4, 1.2, 2.0, and \(-2.0\). The item parameters can also be found at [[https://osf.io/ykew6](https://osf.io/ykew6)]([https://osf.io/ykew6](https://osf.io/ykew6)) (accessed on 2 April 2024). We varied the sample size \(N\) as 500, 1000, and 2000, which may be interpreted as a small, moderate, and large sample size. Like in Simulation Study 1, we compared the performance of regularized estimation based on AIC and BIC with the smooth alternatives SAIC and SBIC. A nonequidistant grid of 37 \(\lambda\) values between 0.0001 and 1 was chosen (see the R simulation code at [[https://osf.io/ykew6](https://osf.io/ykew6)]([https://osf.io/ykew6](https://osf.io/ykew6)); accessed on 2 April 2024). The optimization functions were specified with the same tuning parameters for differentiable approximations as in Simulation Study 1 (see Section 3.1). (Average) absolute bias and (average) RMSE of model parameters, as well as type-I error rates and power rates for cross-loadings, were assessed for the four estimation methods. In total, \(R=750\) replications were conducted in each of the 2 (small vs. large cross-loadings) \(\times\) 3 (sample size) \(=6\) cells of the simulation study. The whole simulation study was conducted using the statistical software R [44].