# Smooth Information Criterion for Regularized Estimation of Item Response Models
## 5 Simulation Study 3: Mixed Rasch/2 PL Model
### Method  
In this simulation study, we used \(I=20\) items for the DGM of the mixed Rasch/2 PL model. The factor variable \(\theta\) was assumed to be normally distributed with a zero mean and a standard deviation \(\sigma=1.2\). The item difficulties \(b_{i}\) (see the IRF in (20)) of the 20 items were chosen as \(-0.8\), \(0.4\), \(1.2\), \(2.0\), \(-2.0\), \(-0.8\), \(0.4\), \(1.2\), \(2.0\), \(-2.0\), \(-0.8\), \(0.4\), \(1.2\), \(2.0\), \(-2.0\), \(-0.8\), \(0.4\), \(1.2\), \(2.0\), and \(-2.0\). The first 14 items followed the Rasch model (i.e., \(\alpha_{i}=0\) for \(i=1,\ldots,15\)). Items 15 to 20 followed the 2 PL model and had \(\alpha_{i}\) values that equaled \(-\delta\), \(\delta\), \(-\delta\), \(\delta\), and \(-\delta\). The size of \(\delta\) controlled the deviation from the Rasch model. We either chose \(\delta\) as \(\log(1.4)=0.336\) and \(\log(2)=0.693\), indicating small and large deviations from the Rasch model. Moreover, we manipulated the direction of the deviation from the Rasch model. While the previously described conditions had \(\alpha_{i}\) that canceled out on average and resulted in a balanced deviation from the Rasch model (i.e., there was an equal number of items that are smaller and larger than 1, respectively), we also specified an unbalanced deviation from the Rasch model in which Items 15 to 20 all had the value \(\delta\). In this condition, we also studied small (i.e., \(\delta=0.336\)) and large (i.e., \(\delta=0.693\)) deviations from the Rasch model. Hence, in the case of unbalanced deviations from the Rasch model, items had either discriminations of 1 or larger than 1. 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).  
Like in the other two simulation studies, we varied the sample size \(N\) as 500, 1000, and 2000.  
Again, like in Simulation Study 1 and Simulation Study 2, we compared the performance of regularized estimation based on AIC and BIC with the smooth alternatives SAIC and SBIC. A nonequidistant grid of 33 \(\lambda\) values between 0.001 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 \(\sigma\) and \(\alpha_{i}\) (\(i=1,\ldots,I\)), as well as type-I error rates and power rates for logarithms of item discriminations, were assessed.  
Overall, \(R=750\) replications were conducted in each of the 2 (small vs. large deviations) \(\times\) 2 (balanced vs. unbalanced deviations) \(\times\) 3 (sample size) \(=12\) cells of the simulation study. This simulation study was also executed using the statistical software R [44]. Like in the other two simulation studies, the regularized multidimensional logistic IRT model was estimated using the sirt::xxirt() function in the R package sirt [45]. Replication material for this simulation study 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).