# Smooth Information Criterion for Regularized Estimation of Item Response Models
## 3 Simulation Study 1: Differential Item Functioning
### Results
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & & \multicolumn{4}{c}{**(Average) Absolute Bias**} & \multicolumn{4}{c}{**(Average) RMSE**} \\ \cline{3-10}
**Par** & **DIF** & \(N\) & **AIC** & **SAIC** & **BIC** & **SBIC** & **AIC** & **SAIC** & **BIC** & **SBIC** \\ \hline \multirow{8}{*}{\(\mu_{2}\)} & \multirow{8}{*}{small} & \multicolumn{4}{c}{_Unbalanced DIF_} & & & & & & & \\  & & 500 & **0.093** & **0.099** & **0.115** & **0.091** & 0.157 & 0.139 & 0.163 & 0.144 \\  & & 1000 & **0.050** & **0.047** & **0.049** & **0.033** & 0.111 & 0.087 & 0.110 & 0.085 \\  & & 2000 & 0.025 & 0.008 & 0.013 & 0.004 & 0.069 & 0.048 & 0.072 & 0.048 \\  & & 500 & **0.053** & **0.072** & 0.024 & 0.020 & 0.145 & 0.122 & 0.143 & 0.100 \\  & & 1000 & 0.024 & 0.030 & 0.003 & 0.002 & 0.084 & 0.076 & 0.077 & 0.068 \\  & & 2000 & 0.004 & 0.012 & 0.001 & 0.000 & 0.053 & 0.059 & 0.048 & 0.048 \\  & & 500 & 0.004 & 0.004 & 0.001 & 0.003 & 0.067 & 0.067 & 0.067 & 0.067 \\  & & small & 1000 & 0.000 & 0.000 & 0.001 & 0.000 & 0.048 & 0.048 & 0.048 & 0.048 \\  & & 2000 & 0.000 & 0.000 & 0.001 & 0.000 & 0.032 & 0.031 & 0.031 & 0.031 \\ \(\sigma_{2}\) & & 500 & 0.000 & 0.001 & 0.000 & 0.001 & 0.065 & 0.065 & 0.064 & 0.065 \\  & & large & 1000 & 0.000 & 0.001 & 0.001 & 0.000 & 0.046 & 0.046 & 0.045 & 0.045 \\  & & 2000 & 0.001 & 0.002 & 0.001 & 0.001 & 0.031 & 0.032 & 0.031 & 0.032 \\  & & 500 & **0.114** & **0.108** & **0.035** & **0.062** & 0.296 & 0.251 & 0.170 & 0.207 \\  & & small & 1000 & **0.072** & **0.055** & 0.028 & 0.016 & 0.207 & 0.150 & 0.143 & 0.095 \\ \(\delta_{i}\) & & 2000 & **0.037** & 0.013 & 0.015 & 0.001 & 0.122 & 0.079 & 0.098 & 0.023 \\ (no DIF) & & 500 & **0.079** & **0.114** & **0.032** & 0.030 & 0.255 & 0.236 & 0.206 & 0.144 \\  & & large & 1000 & **0.038** & **0.051** & 0.005 & 0.003 & 0.133 & 0.145 & 0.072 & 0.065 \\  & & 2000 & 0.010 & 0.025 & 0.001 & 0.001 & 0.055 & 0.107 & 0.023 & 0.030 \\  & & 500 & **0.185** & **0.221** & **0.399** & **0.262** & 0.405 & 0.409 & 0.553 & 0.462 \\  & & small & 1000 & **0.089** & **0.111** & **0.158** & **0.117** & 0.270 & 0.285 & 0.368 & 0.319 \\ \(\delta_{i}\) & & 2000 & **0.036** & 0.010 & 0.020 & 0.010 & 0.167 & 0.135 & 0.179 & 0.151 \\ (DIF) & & 500 & **0.075** & **0.112** & **0.037** & 0.029 & 0.339 & 0.312 & 0.354 & 0.303 \\  & & large & 1000 & **0.036** & **0.050** & 0.005 & 0.004 & 0.212 & 0.199 & 0.207 & 0.196 \\  & & 2000 & 0.011 & 0.020 & 0.004 & 0.004 & 0.138 & 0.144 & 0.137 & 0.138 \\ \hline \hline \end{tabular} Note. Par = parameter; \(\mu_{2}\) = mean of \(\theta\) in second group; \(\sigma_{2}\) = standard deviation of \(\theta\) in second group; \(\delta_{i}\) (no DIF) = DIF parameters with zero population values; \(\delta_{i}\) (DIF) = DIF parameters with non-zero population values; Absolute bias values larger than 0.03 are printed in bold font.  
\end{table} 
Table 2: Simulation Study 1: Type-I error rate and power rate for DIF effects \(\delta_{i}\) as a function of the extent of differential item functioning (DIF) and sample size \(N\) for balanced and unbalanced DIF.