# Pairwise Likelihood Estimation of the 2 PL Model with Locally Dependent Item Responses
## 5 Discussion
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{\(a_{i}\)} & \multicolumn{4}{c}{\(b_{i}\)} \\ \cline{3-10}
**Item** & **Testlet** & **MML1** & **PML1** & **PML2** & **PML3** & **MML1** & **PML1** & **PML2** & **PML3** \\ \hline M192Q01 & 1 & 1.26 (0.16) & 1.27 (0.17) & 1.32 (0.18) & 1.32 (0.18) & 0.23 (0.11) & 0.23 (0.11) & 0.23 (0.11) & 0.23 (0.11) \\ M406Q01 & 2 & 1.77 (0.22) & 1.74 (0.23) & 1.45 (0.20) & 1.45 (0.20) & 0.37 (0.13) & 0.36 (0.13) & 0.33 (0.12) & 0.33 (0.12) \\ M406Q02 & 2 & 2.27 (0.32) & 2.22 (0.33) & 1.83 (0.26) & 1.83 (0.26) & 1.66 (0.22) & 1.63 (0.21) & 1.45 (0.18) & 1.45 (0.18) \\ M423Q01 & 3 & 0.53 (0.12) & 0.52 (0.13) & 0.53 (0.13) & 0.53 (0.13) & \(-1.11\) (0.10) & \(-1.11\) (0.10) & \(-1.11\) (0.11) & \(-1.11\) (0.11) \\ M496Q01 & 4 & 1.38 (0.18) & 1.39 (0.18) & 1.24 (0.17) & 1.24 (0.17) & \(-0.29\) (0.12) & \(-0.30\) (0.12) & \(-0.28\) (0.11) & \(-0.28\) (0.11) \\ M496Q02 & 4 & 1.13 (0.16) & 1.12 (0.17) & 0.94 (0.15) & 0.94 (0.15) & \(-1.15\) (0.13) & \(-1.15\) (0.13) & \(-1.09\) (0.12) & \(-1.09\) (0.12) \\ M564Q01 & 5 & 0.82 (0.13) & 0.82 (0.13) & 0.85 (0.13) & \(-0.07\) (0.10) & \(-0.07\) (0.10) & \(-0.07\) (0.10) & \(-0.07\) (0.10) \\ M564Q02 & 5 & 0.79 (0.12) & 0.79 (0.12) & 0.82 (0.13) & 0.82 (0.13) & \(-0.12\) (0.10) & \(-0.12\) (0.10) & \(-0.12\) (0.10) & \(-0.12\) (0.10) \\ M571Q01 & 6 & 1.42 (0.18) & 1.43 (0.19) & 1.53 (0.21) & 1.53 (0.21) & \(-0.26\) (0.12) & \(-0.26\) (0.12) & \(-0.27\) (0.12) & \(-0.27\) (0.12) \\ M603Q01 & 7 & 1.17 (0.15) & 1.18 (0.16) & 1.23 (0.17) & 1.23 (0.17) & \(-0.28\) (0.11) & \(-0.29\) (0.11) & \(-0.29\) (0.11) & \(-0.29\) (0.11) \\ M603Q02 & 7 & 1.42 (0.18) & 1.42 (0.18) & 1.51 (0.21) & 1.51 (0.21) & 0.14 (0.12) & 0.14 (0.12) & 0.15 (0.12) & 0.15 (0.12) \\ \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.  
\end{table} 
Table 5: Dataset data.pisaMath: Estimated item discriminations \(a_{i}\) and item intercepts \(b_{i}\) (standard errors in parentheses) for different estimation methods. this case, the local dependence vanishes for infinitely long tests [52]. It is likely that the negative impacts of local dependence in reporting from tests should not be exaggerated unless confronted with very short tests or an excessive dependence structure. This research received no external funding. Not applicable. The datasets used in this article are contained in the R package sirt [36]. Available online: [[https://CRAN.R-project.org/package=sirt](https://CRAN.R-project.org/package=sirt)]([https://CRAN.R-project.org/package=sirt](https://CRAN.R-project.org/package=sirt)) (accessed on 6 February 2024). The authors declare no conflicts of interest. The following abbreviations are used in this manuscript:  
Two-parameter logistic  
expectation-maximization  
item response theory  
marginal maximum likelihood  
Programme for International Student Assessment  
pairwise maximum likelihood