# Pairwise Likelihood Estimation of the 2 PL Model with Locally Dependent Item Responses
## 3 Simulation Study
### Simulation Study 2: Four Items per Testlet
#### 3.2.2 Results  
Table 2 presents the average absolute bias, average standard deviation, and average coverage rates of item discriminations \(a_{i}\) and item intercepts \(b_{i}\). Again, the condition \(f=0\) corresponds to local independence, while \(f>0\) corresponds to local dependence. All four estimation methods, MML1, PML1, PML2, and PML3, provided approximately unbiased results for item discriminations \(a_{i}\) and item intercepts \(b_{i}\). There was a small bias for a sample size of \(N=500\), but this bias vanished with an increasing sample \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Average Absolute Bias**} & \multicolumn{4}{c}{**Average Standard Deviation**} & \multicolumn{4}{c}{**Average Coverage Rate**} \\ \cline{3-13} \(f\) & \(N\) & **MML1** & **PML1** & **PML2** & **PML3** & **MML1** & **PML1** & **PML2** & **PML3** & **MML1** & **PML1** & **PML2** & **PML3** \\ \hline \multicolumn{13}{c}{_Item discriminations \(a_{i}\)_} \\  & 500 & 0.014 & 0.014 & 0.017 & 0.017 & 0.189 & 0.189 & 0.208 & 0.208 & 95.3 & 95.5 & 95.6 & 95.6 \\
0 & 1000 & 0.008 & 0.008 & 0.010 & 0.010 & 0.131 & 0.132 & 0.144 & 0.144 & 95.2 & 95.2 & 95.4 & 95.4 \\  & 2000 & 0.004 & 0.004 & 0.005 & 0.005 & 0.093 & 0.093 & 0.102 & 0.102 & 94.9 & 95.0 & 94.9 & 94.9 \\  & 500 & **0.220** & **0.200** & 0.025 & 0.025 & 0.207 & 0.204 & 0.230 & 0.230 & **74.3** & **79.5** & 95.9 & 95.9 \\
0.35 & 1000 & **0.216** & **0.196** & 0.010 & 0.010 & 0.144 & 0.142 & 0.159 & 0.159 & **62.6** & **66.4** & 95.3 & 95.3 \\  & 2000 & **0.214** & **0.194** & 0.005 & 0.005 & 0.100 & 0.099 & 0.111 & 0.111 & **57.1** & **58.5** & 95.1 & 95.1 \\  & 500 & **0.827** & **0.710** & 0.028 & 0.028 & 0.296 & 0.290 & 0.260 & 0.260 & **47.1** & **53.2** & 95.5 & 95.5 \\
0.7 & 1000 & **0.818** & **0.703** & 0.012 & 0.012 & 0.195 & 0.187 & 0.173 & 0.173 & **36.1** & **44.7** & 95.5 & 95.5 \\  & 2000 & **0.816** & **0.701** & 0.007 & 0.007 & 0.139 & 0.133 & 0.121 & 0.121 & **25.7** & **34.0** & 95.3 & 95.3 \\  & \multicolumn{13}{c}{_Item intercepts \(b_{i}\)_} \\  & 500 & 0.010 & 0.010 & 0.011 & 0.011 & 0.142 & 0.142 & 0.145 & 0.145 & 95.4 & 95.5 & 95.6 & 95.6 \\
0 & 1000 & 0.005 & 0.005 & 0.005 & 0.005 & 0.100 & 0.100 & 0.102 & 0.102 & 95.3 & 95.3 & 95.4 & 95.4 \\  & 2000 & 0.002 & 0.002 & 0.002 & 0.002 & 0.070 & 0.071 & 0.072 & 0.072 & 95.0 & 95.0 & 95.0 & 95.0 \\  & 500 & **0.035** & **0.032** & 0.012 & 0.012 & 0.148 & 0.148 & 0.152 & 0.152 & 94.1 & 94.4 & 95.4 & 95.4 \\
0.35 & 1000 & **0.037** & **0.033** & 0.006 & 0.006 & 0.103 & 0.103 & 0.105 & 0.105 & 92.7 & 93.2 & 95.4 & 95.4 \\  & 2000 & **0.037** & **0.033** & 0.003 & 0.003 & 0.073 & 0.073 & 0.074 & 0.074 & **90.1** & **91.0** & 95.0 & 95.0 \\  & 500 & **0.132** & **0.114** & 0.016 & 0.016 & 0.169 & 0.166 & 0.158 & 0.159 & **84.8** & **87.5** & 95.6 & 95.6 \\
0.7 & 1000 & **0.135** & **0.116** & 0.007 & 0.007 & 0.119 & 0.116 & 0.109 & 0.109 & **75.7** & **79.8** & 95.2 & 95.2 \\  & 2000 & **0.135** & **0.116** & 0.004 & 0.004 & 0.082 & 0.081 & 0.075 & 0.075 & **66.0** & **71.2** & 95.1 & 95.1 \\ \hline \hline \end{tabular} Note: MML1 = marginal maximum likelihood estimation under local independence; PML1 = pairwise likelihood estimation based on all item pairs under local independence; PML2 = pairwise likelihood estimation under the normal copula model; PML3 = pairwise likelihood estimation based on item pairs with items from different testlets. The residual correlation matrix \(\Sigma^{*}\) as a function of \(f\) is given in (34). Average absolute bias values larger than 0.03 and coverage rates smaller than 92.5 are printed in bold font.  
\end{table} 
Table 2: Simulation Study 2: Average absolute bias and average standard deviations of item parameters as a function of sample size \(N\), and the size of the maximum residual correlation \(f\). size. The average standard deviation of the item parameter estimates decreased with the increasing sample size. In the condition \(f=0\), indicating local independence, the efficiency loss was assessed.