# Smooth Information Criterion for Regularized Estimation of Item Response Models
## 5 Simulation Study 3: Mixed Rasch/2 PL Model
### Results  
Table 5 contains the (average) absolute bias and (average) RMSE for the estimated model parameters. Notably, there was a different pattern of findings in the conditions of balanced and unbalanced deviations from the Rasch model. In general, SAIC performed well for the estimation of \(\sigma\), except for small balanced deviations from the Rasch model with a sample size of \(N=500\). In most of the conditions, estimation based on SBIC performed similarly, if not better, than BIC for the estimation of \(\sigma\) in terms of RMSE.  
Table 6 displays type-I error rates and power rates for estimated logarithms of item discriminations. In contrast to the estimation based on the AIC, SAIC had acceptable type-I error rates. Moreover, power rates for detecting deviations from the Rasch model were much higher for SAIC then BIC or SBIC.  
\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** & **Dev** & \(N\) & **AIC** & **SAIC** & **BIC** & **SBIC** & **AIC** & **SAIC** & **BIC** & **SBIC** \\ \hline  & & & \multicolumn{4}{c}{_Balanced deviations from the Rasch model_} \\  & & & 500 & **0.053** & **0.038** & 0.004 & 0.010 & 0.099 & 0.082 & 0.070 & 0.072 \\  & small & 1000 & **0.062** & 0.005 & 0.012 & 0.004 & 0.107 & 0.054 & 0.051 & 0.056 \\ \(\sigma\) & & 2000 & **0.109** & 0.001 & 0.025 & 0.009 & 0.167 & 0.032 & 0.044 & 0.037 \\  & & 500 & 0.018 & 0.014 & 0.004 & 0.006 & 0.069 & 0.065 & 0.064 & 0.064 \\  & large & 1000 & 0.016 & 0.002 & 0.002 & 0.002 & 0.050 & 0.044 & 0.041 & 0.041 \\  & & 2000 & 0.007 & 0.000 & 0.000 & 0.000 & 0.035 & 0.033 & 0.030 & 0.030 \\  & & 500 & **0.035** & 0.017 & 0.001 & 0.002 & 0.124 & 0.091 & 0.048 & 0.047 \\  & small & 1000 & **0.048** & 0.003 & 0.002 & 0.001 & 0.119 & 0.060 & 0.027 & 0.020 \\ \(\alpha_{i}=0\) & & 2000 & **0.096** & 0.001 & 0.003 & 0.000 & 0.166 & 0.042 & 0.023 & 0.003 \\  & & 500 & 0.018 & 0.012 & 0.002 & 0.002 & 0.101 & 0.082 & 0.046 & 0.044 \\  & large & 1000 & 0.017 & 0.003 & 0.001 & 0.000 & 0.066 & 0.058 & 0.023 & 0.014 \\  & & 2000 & 0.007 & 0.001 & 0.000 & 0.000 & 0.033 & 0.040 & 0.010 & 0.003 \\  & & 500 & **0.071** & **0.070** & **0.127** & **0.137** & 0.224 & 0.239 & 0.283 & 0.289 \\  & small & 1000 & **0.081** & 0.024 & **0.075** & **0.101** & 0.184 & 0.154 & 0.212 & 0.237 \\ \(\alpha_{i}\
eq 0\) & & 2000 & **0.120** & 0.003 & **0.072** & **0.045** & 0.199 & 0.085 & 0.159 & 0.156 \\  & & 500 & 0.014 & 0.017 & 0.015 & 0.018 & 0.207 & 0.215 & 0.230 & 0.231 \\  & large & 1000 & 0.019 & 0.006 & 0.006 & 0.008 & 0.139 & 0.136 & 0.138 & 0.143 \\  & & 2000 & 0.004 & 0.004 & 0.003 & 0.004 & 0.095 & 0.095 & 0.093 & 0.094 \\ \hline  & & & \multicolumn{4}{c}{_Unbalanced deviations from the Rasch model_} \\  & & & 500 & 0.008 & **0.032** & **0.084** & **0.083** & 0.076 & 0.081 & 0.110 & 0.109 \\  & small & 1000 & 0.015 & 0.008 & 0.028 & **0.055** & 0.051 & 0.050 & 0.061 & 0.078 \\ \(\sigma\) & & 2000 & 0.011 & 0.003 & 0.002 & 0.015 & 0.036 & 0.032 & 0.031 & 0.040 \\  & & 500 & 0.001 & 0.001 & 0.001 & 0.001 & 0.069 & 0.062 & 0.061 & 0.060 \\  & large & 1000 & 0.005 & 0.001 & 0.000 & 0.001 & 0.046 & 0.043 & 0.040 & 0.040