# Pairwise Likelihood Estimation of the 2 PL Model with Locally Dependent Item Responses
## 4 Empirical Examples
### Dataset data.read  
The first dataset, data.read, contains the item responses of 328 subjects on 12 items that stem from a reading comprehension test of Austrian students. The 12 items are arranged into three testlets, each testlet containing of four items. Table 3 contains the estimated item discriminations \(a_{i}\) and estimated item intercepts \(b_{i}\) and their standard error estimates for the four different estimation methods MML1, PML1, PML2, and PML3. The average item discrimination was higher for the methods that ignored the testlet structure (MML: 1.241, PML1: 1.184) than PML2 or PML3 with 0.983, which took the testlet structure into account in the estimation.  
The average absolute difference in the estimated item discriminations was 0.454 for PML3 and MML, and it was 0.345 for PML3 and PML1. The average absolute difference between PML1 and MML was substantially smaller at 0.130. Note that PML2 and PML3 resulted in practically identical item parameter estimates.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline  & & & \multicolumn{4}{c}{\(a_{i}\)} & \multicolumn{4}{c}{\(b_{i}\)} \\ \cline{3-11}
**Item** & **Testlet** & **MML1** & **PML1** & **PML2** & **PML3** & **MML1** & **PML1** & **PML2** & **PML3** \\ \hline A1 & 1 & 0.96 (0.25) & 0.97 (0.26) & 0.85 (0.30) & 0.85 (0.30) & \(-\)2.09 (0.23) & \(-\)2.04 (0.21) & \(-\)1.97 (0.22) & \(-\)1.97 (0.22) \\ A2 & 1 & 1.44 (0.28) & 1.36 (0.31) & 1.54 (0.51) & 1.54 (0.51) & \(-\)1.45 (0.21) & \(-\)1.38 (0.20) & \(-\)1.46 (0.28) & \(-\)1.46 (0.28) \\ A3 & 1 & 1.14 (0.21) & 1.08 (0.27) & 1.07 (0.33) & 1.07 (0.33) & \(-\)0.28 (0.14) & \(-\)0.33 (0.14) & \(-\)0.33 (0.14) & \(-\)0.33 (0.14) \\ A4 & 1 & 0.83 (0.18) & 0.85 (0.21) & 0.88 (0.28) & 0.88 (0.28) & 0.19 (0.13) & 0.18 (0.13) & 0.19 (0.13) & 0.19 (0.13) \\ B1 & 2 & 0.58 (0.17) & 0.60 (0.17) & 0.67 (0.23) & 0.67 (0.23) & \(-\)1.04 (0.14) & \(-\)0.98 (0.14) & \(-\)1.00 (0.14) & \(-\)1.00 (0.14) \\ B2 & 2 & 0.67 (0.17) & 0.67 (0.16) & 0.80 (0.24) & 0.80 (0.24) & \(-\)0.09 (0.12) & \(-\)0.03 (0.12) & \(-\)0.03 (0.13) & \(-\)0.03 (0.13) \\ B3 & 2 & 1.22 (0.32) & 1.11 (0.29) & 1.18 (0.42) & 1.17 (0.42) & \(-\)2.85 (0.33) & \(-\)2.76 (0.30) & \(-\)2.80 (0.37) & \(-\)2.80 (0.37) \\ B4 & 2 & 1.11 (0.22) & 1.11 (0.23) & 1.51 (0.45) & 1.51 (0.45) & \(-\)0.93 (0.16) & \(-\)0.95 (0.16) & \(-\)1.08 (0.21) & \(-\)1.08 (0.21) \\ C1 & 3 & 2.40 (0.63) & 2.31 (0.90) & 0.91 (0.36) & 0.91 (0.36) & \(-\)4.30 (0.75) & \(-\)4.37 (1.00) & \(-\)2.97 (0.33) & \(-\)2.97 (0.33) \\ C2 & 3 & 1.53 (0.29) & 1.48 (0.31) & 1.05 (0.25) & 1.05 (0.25) & \(-\)1.32 (0.21) & \(-\)1.27 (0.20) & \(-\)1.11 (0.16) & \(-\)1.11 (0.16) \\ C3 & 3 & 1.95 (0.49) & 1.56 (0.48) & 0.72 (0.26) & 0.72 (0.26) & \(-\)3.07 (0.49) & \(-\)2.65 (0.40) & \(-\)2.10 (0.21) & \(-\)2.10 (0.21) \\ C4 & 3 & 1.08 (0.23) & 1.11 (0.28) & 0.63 (0.20) & 0.63 (0.20) & \(-\)1.36 (0.18) & \(-\)1.26 (0.18) & \(-\)1.11 (0.14) & \(-\)1.11 (0.14) \\ \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 3: Dataset data.read: Estimated item discriminations \(a_{i}\) and item intercepts \(b_{i}\) (standard errors in parentheses) for different estimation methods.