# Pairwise Likelihood Estimation of the 2 PL Model with Locally Dependent Item Responses
## 4 Empirical Examples
### Dataset data.pisaMath  
The third dataset, data.pisaMath, contains the item responses of 565 Austrian students on 11 mathematics items to an item cluster in a PISA study [40]. The 11 items are arranged into four testlets with two items each, and there are three single items without a testlet structure included in the dataset.  
Table 5 contains the estimated item parameters for data.pisaMath. The average item discrimination was slightly larger for MML1 and PML1, with 1.269 and 1.263, compared to the PML3 estimation method, with a value of 1.205. Interestingly, the average absolute difference in the item discriminations between PML1 and MML1 was very small at 0.012, but they were substantially larger between PML3 and MML1, with 0.125, and between PML3 and PML1, with 0.132. Again, PML2 and PML3 provided practically identical item parameter estimates.  
\begin{table}
\begin{tabular}{l c c c c c c c c c} \hline  & & & \multicolumn{3}{c}{\(a_{i}\)} & \multicolumn{3}{c}{\(b_{i}\)} \\ \cline{3-10}
**Item** & **Testlet** & **MML1** & **PML1** & **PML2** & **PML3** & **MML1** & **PML1** & **PML2** & **PML3** \\ \hline R432Q01 & 1 & 2.33 (0.34) & 2.64 (0.43) & 2.83 (0.54) & 2.83 (0.54) & \(-\)3.39 (0.36) & \(-\)3.67 (0.46) & \(-\)3.86 (0.56) & \(-\)3.86 (0.56) \\ R432Q05 & 1 & 2.04 (0.26) & 2.02 (0.26) & 2.13 (0.31) & 2.13 (0.31) & \(-\)1.70 (0.19) & \(-\)1.72 (0.20) & \(-\)1.78 (0.22) & \(-\)1.78 (0.22) \\ R432Q06 & 1 & 1.22 (0.25) & 1.09 (0.27) & 1.07 (0.28) & 1.07 (0.28) & 2.99 (0.26) & 2.90 (0.26) & 2.88 (0.26) & 2.88 (0.26) \\ R456Q01 & 2 & 1.19 (0.30) & 1.59 (0.49) & 1.38 (0.43) & 1.38 (0.43) & \(-\)4.30 (0.41) & \(-\)4.81 (0.68) & \(-\)4.56 (0.57) & \(-\)4.56 (0.57) \\ R456Q02 & 2 & 0.88 (0.15) & 0.94 (0.17) & 0.83 (0.16) & 0.83 (0.16) & \(-\)1.89 (0.14) & \(-\)1.94 (0.15) & \(-\)1.88 (0.14) & \(-\)1.88 (0.14) \\ R456Q06 & 2 & 1.90 (0.26) & 2.00 (0.30) & 1.87 (0.27) & 1.87 (0.27) & \(-\)2.82 (0.26) & \(-\)2.90 (0.30) & \(-\)2.78 (0.27) & \(-\)2.78 (0.27) \\ R460Q01 & 3 & 1.53 (0.19) & 1.51 (0.19) & 1.48 (0.20) & 1.48 (0.20) & \(-\)0.71 (0.12) & \(-\)0.76 (0.13) & \(-\)0.75 (0.12) & \(-\)0.75 (0.12) \\ R460Q05 & 3 & 1.71 (0.24) & 1.79 (0.27) & 1.67 (0.27) & 1.67 (0.27) & \(-\)2.68 (0.24) & \(-\)2.74 (0.27) & \(-\)2.64 (0.26) & \(-\)2.64 (0.26) \\ R460Q06 & 3 & 1.24 (0.16) & 1.24 (0.16) & 1.16 (0.16) & 1.16 (0.16) & \(-\)0.50 (0.11) & \(-\)0.49 (0.11) & \(-\)0.47 (0.11) & \(-\)0.47 (0.11) \\ R466Q02 & 4 & 1.32 (0.17) & 1.29 (0.15) & 1.29 (0.16) & 1.29 (0.16) & 0.09 (0.11) & 0.11 (0.11) & 0.11 (0.11) & 0.11 (0.11) \\ R466Q03 & 4 & 0.97 (0.17) & 1.00 (0.16) & 0.97 (0.17) & 0.97 (0.17) & 1.66 (0.14) & 1.66 (0.14) & 1.64 (0.14) & 1.64 (0.14) \\ R466Q06 & 4 & 1.52 (0.20) & 1.53 (0.21) & 1.55 (0.22) & 1.55 (0.22) & \(-\)1.76 (0.16) & \(-\)1.79 (0.17) & \(-\)1.80 (0.18) & \(-\)1.80 (0.18) \\ \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.  
\end{table}
Table 4: Dataset data.pisaRead: Estimated item discriminations \(a_{i}\) and item intercepts \(b_{i}\) (standard errors in parentheses) for different estimation methods.