# Pairwise Likelihood Estimation of the 2 PL Model with Locally Dependent Item Responses
## 3 Simulation Study
### Simulation Study 2: Four Items per Testlet
#### 3.2.1 Method
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c} \hline  & & & & & & & & & & & & & & & & & & & & & & & & \\ \cline{3-16} \(f\) & \(\mathbf{N}\) & **MML1** & **PML1** & **MML2** & **PML2** & **PML3** & **MML1** & **PML1** & **MML2** & **PML2** & **PML3** & **MML1** & **PML1** & **MML2** & **PML2** & **PML3** \\ \hline \multicolumn{16}{c}{_Item discriminations \(a_{i}\)_} \\  & & 500 & 0.016 & 0.016 & 0.016 & 0.016 & 0.016 & 0.189 & 0.190 & 0.194 & 0.194 & 0.194 & 0.194 & 95.4 & 95.6 & 95.4 & 95.5 & 95.5 \\
0 & 1000 & 0.008 & 0.008 & 0.008 & 0.008 & 0.132 & 0.132 & 0.135 & 0.135 & 0.135 & 95.0 & 95.1 & 95.1 & 95.1 & 95.1 \\  & 2000 & 0.005 & 0.005 & 0.005 & 0.005 & 0.005 & 0.093 & 0.093 & 0.095 & 0.095 & 95.1 & 95.1 & 95.0 & 95.0 & 95.0 \\  & 500 & **0.075** & **0.076** & 0.020 & 0.021 & 0.021 & 0.195 & 0.195 & 0.200 & 0.201 & 0.201 & 94.2 & 95.1 & 95.2 & 95.5 & 95.5 \\
0.35 & 1000 & **0.063** & **0.064** & 0.007 & 0.008 & 0.008 & 0.136 & 0.136 & 0.139 & 0.140 & 0.140 & **92.2** & 93.2 & 95.2 & 95.3 & 95.3 \\  & 2000 & **0.060** & **0.061** & 0.004 & 0.004 & 0.004 & 0.095 & 0.095 & 0.097 & 0.098 & 0.098 & **87.8** & **89.5** & 95.1 & 95.2 & 95.2 \\  & 500 & **0.163** & **0.148** & 0.021 & 0.021 & 0.021 & 0.234 & 0.217 & 0.209 & 0.212 & 0.212 & **85.0** & **90.9** & 95.1 & 95.4 & 95.4 \\
0.7 & 1000 & **0.154** & **0.139** & 0.008 & 0.008 & 0.008 & 0.160 & 0.149 & 0.143 & 0.145 & 0.145 & **76.4** & **82.5** & 95.1 & 95.2 & 95.2 \\  & 2000 & **0.152** & **0.138** & 0.005 & 0.005 & 0.005 & 0.111 & 0.104 & 0.099 & 0.101 & 0.101 & **66.1** & **69.4** & 95.1 & 95.2 & 95.2 \\  & \multicolumn{16}{c}{_Item intercepts \(b_{i}\)_} \\  & & 500 & 0.009 & 0.009 & 0.009 & 0.009 & 0.009 & 0.009 & 0.143 & 0.144 & 0.144 & 0.144 & 0.144 & 95.3 & 95.3 & 95.3 & 95.3 \\
0 & 1000 & 0.005 & 0.005 & 0.005 & 0.005 & 0.005 & 0.100 & 0.101 & 0.101 & 0.101 & 0.101 & 95.1 & 95.2 & 95.1 & 95.2 \\  & 2000 & 0.003 & 0.003 & 0.003 & 0.003 & 0.003 & 0.071 & 0.071 & 0.071 & 0.071 & 94.9 & 95.0 & 94.9 & 94.9 & 94.9 \\  & 500 & 0.020 & 0.022 & 0.012 & 0.012 & 0.012 & 0.145 & 0.146 & 0.146 & 0.146 & 0.146 & 95.2 & 95.3 & 95.2 & 95.3 & 95.3 \\
0.35 & 1000 & 0.013 & 0.015 & 0.005 & 0.005 & 0.005 & 0.101 & 0.102 & 0.101 & 0.102 & 0.102 & 95.0 & 95.0 & 95.1 & 95.1 & 95.1 \\  & 2000 & 0.012 & 0.013 & 0.003 & 0.003 & 0.003 & 0.071 & 0.072 & 0.071 & 0.071 & 94.7 & 94.6 & 95.0 & 95.0 & 95.0 \\  & 500 & **0.031** & 0.029 & 0.013 & 0.013 & 0.013 & 0.150 & 0.150 & 0.149 & 0.150 & 0.150 & 94.3 & 94.7 & 95.1 & 95.2 & 95.2 \\
0.7 & 1000 & 0.028 & 0.026 & 0.005 & 0.005 & 0.005 & 0.103 & 0.103 & 0.102 & 0.102 & 0.102 & 93.9 & 94.4 & 95.2 & 95.2 & 95.2 \\  & 2000 & 0.028 & 0.025 & 0.003 & 0.003 & 0.003 & 0.073 & 0.073 & 0.071 & 0.072 & **92.2** & 92.9 & 95.1 & 95.0 & 95.0 \\ \hline \end{tabular} Note.: MML1 = marginal maximum likelihood estimation under local independence; PML1 = pairwise likelihood estimation based on all item pairs under local independence; MML2 = marginal maximum likelihood estimation under the normal copula model; PML3 = pairwise likelihood estimation based on item pairs with items from different testlets. The residual correlation matrix \(\mathbf{\Sigma}^{*}\) as a function of \(f\) is given in (29). Average absolute bias values larger than 0.03 and coverage rates smaller than 92.5 are printed in bold font.  
\end{table} 
Table 1: Simulation Study 1: 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 \(f\). This specification induced a dependency structure in the residuals for items within testlets. Specifically, the first four items were allocated to the first testlet, and there was a constant residual correlation \(f\). Items 5 to 8 referred to the second testlet, and the residual correlations varied depending on the item pair. The third testlet was composed of Items 9 to 12. Note that Item 12 was locally independent of Items 9, 10, and 11. The maximum residual correlation \(f\) was varied as 0, 0.35, and 0.7.