# Smooth Information Criterion for Regularized Estimation of Item Response Models
## 3 Simulation Study 1: Differential Item Functioning
### Method  
In this simulation study, we use a data-generating model (DGM) similar to the one used in the simulation study in [38]. The factor variable \(\theta\) was assumed to be univariatenormally distributed. We fixed the mean \(\mu_{1}\) and the standard deviation \(\sigma_{1}\) of the factor variable \(\theta\) in the first group to 0 and 1, respectively. The factor variable \(\theta\) had a mean \(\mu_{2}\) of 0.5 and a standard deviation \(\sigma_{2}\) of 0.8 in the second group. In total, \(I=25\) items were used in this simulation study. We now describe the item parameters used for the IRF defined in (14). The common item discriminations \(a_{i}\) of the 25 items were chosen as 1.3, 1.4, 1.5, 1.7, 1.6, 1.3, 1.4, 1.5, 1.7, 1.6, 1.3, 1.4, 1.5, 1.7, 1.6, 1.3, 1.4, 1.5, 1.7, 1.6, 1.3, 1.4, 1.5, 1.7, and 1.6. The item difficulties \(b_{i}\) were chosen as \(-0.8\), 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, \(-2.0\), \(-0.8\), 0.4, 1.2, 2.0, \(-0.8\), 0.4, 1.2, 2.0, and \(-2.0\). The DIF effects \(\delta_{i}\) were zero for the first 15 items. Items 16 to 25 had non-zero DIF effects. In the condition of small DIF effects (see [38]), we chose \(\delta_{i}\) values of \(-0.60\), 0.60, \(-0.65\), 0.70, 0.65, \(-0.70\), 0.60, \(-0.65\), 0.70, and \(-0.65\) for Items 16 to 25. In the condition of large DIF effects, we multiplied these effects by 2. These two conditions are referred to as balanced DIF conditions because the DIF effects \(\delta_{i}\) average to zero. In line with other studies, we also considered unbalanced DIF [43], in which we took absolute DIF effects in the small DIF and large DIF conditions. In the unbalanced DIF conditions, all DIF effects \(\delta_{i}\) were assumed positive and did not average to zero. 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). Moreover, we varied the sample size \(N\) in this simulation study by 500, 1000, and 2000. There were \(N/2\) subjects in each of the two groups. The regularized 2 PL model with DIF was estimated with the regularized likelihood function using the SCAD penalty on a nonequidistant grid of 37 \(\lambda\) values between 0.0001 and 1 (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). We approximated the nondifferentiable SCAD penalty function by its differentiable approximating function using the tuning \(\varepsilon=0.001\). We saved parameter estimates that minimized AIC and BIC. Item parameters that did not exceed the threshold \(\tau=0.02\) in its absolute value were regularized. In the direct minimization of SAIC and SBIC, we tried the values 0.01, 0.001, and 0.0001 of the tuning parameters \(\varepsilon\). It was found that \(\varepsilon=0.001\) performed best, which is the reason why we only reported this solution. As the outcome of the simulation study, we studied (average) absolute bias and (average) root mean square error (RMSE) of model parameter estimates as well as type-I error rates and power rates. Absolute bias and RMSE were computed for estimates of distribution parameters \(\mu_{2}\) and \(\sigma_{2}\). Moreover, absolute bias and RMSE were computed for all estimates of DIF effects \(\delta_{i}\). Formally, let \(\gamma_{h}\) be the \(h\)th parameter (\(h=1,\ldots,H\)) in the model parameter vector \(\gamma\). Let \(\widehat{\gamma}_{hr}\) be the parameter estimate of \(\gamma_{h}\) in replication \(r\) (\(r=1,\ldots,R\)).