# Pairwise Likelihood Estimation of the 2 PL Model with Locally Dependent Item Responses
## 1 Introduction
### Pairwise Maximum Likelihood (PML) Estimation for the Normal Copula Model for Item Residuals  
The MML estimation of the 2 PL model with a normal copula model for item residuals has the disadvantage that multidimensional normal probabilities must be evaluated. This is particularly computationally demanding for many items within a testlet. Hence, PML can be applied, which only involves bivariate normal probabilities in the estimation of the 2 PL normal copula model.  
The PML estimation described in Section 2.1 can now also involve correlation parameters \(\rho_{ij}\) that capture the dependence of items \(i\) and \(j\). For items \(X_{i}\) and \(X_{j}\) allocated to different testlets, local independence is assumed, and no correlation is estimated.  
For items \(X_{i}\) and \(X_{j}\) in different testlets, bivariate probabilities in PML estimation are evaluated according to (see (5))  
\[L_{2,X_{i}X_{j}}(x,y;\gamma_{i},\gamma_{j})=\int\Psi(2(x-1)(a_{i}\theta-b_{i})) \Psi(2(y-1)(a_{j}\theta-b_{j}))\phi(\theta)\,\mathrm{d}\theta \tag{21}\]  
with \(\gamma_{i}=(a_{i},b_{i})\). For items \(X_{i}\) and \(X_{j}\) in the same testlet, the bivariate probabilities are given as (see (17))  
\[L_{2,X_{i}X_{j}}(x,y;\gamma_{i},\gamma_{j},\rho_{ij})=\int\Phi_{2}\Big{(}\Phi^{ -1}(\Psi((2x-1)(a_{i}\theta-b_{i})))\,,\Phi^{-1}(\Psi((2y-1)(a_{j}\theta-b_{j}) ));\rho_{ij}^{x+y}\Big{)}\phi(\theta)\,\mathrm{d}\theta, \tag{22}\]  
where the term (22) now additionally includes the correlation \(\rho_{ij}\).  
The model parameter \(\gamma\) contains all item parameters \(a_{i}\) and \(b_{i}\), as well as all correlation parameters \(\rho_{ij}\) for the item pairs within a testlet. A parameter estimate is obtained by maximizing the PML optimization function (6) with respect to \(\gamma\). The standard errors of the model parameters can be obtained with Huber-White standard errors by formalizing PML as an M-estimation problem [14; 34]. More formally, PML maximizes  
\[l(\gamma)=\sum_{n=1}^{N}l_{n}(\gamma;\mathbf{x}_{n}), \tag{23}\]  
where \(l_{n}\) are the PML contributions of case \(n\). The parameter estimate \(\widehat{\gamma}\) is obtained by setting the partial derivative of \(l\) with respect to all components \(\gamma_{h}\) of \(\gamma\) equal to 0:  
\[\frac{\partial l}{\partial\gamma_{h}}=\sum_{n=1}^{N}\frac{\partial l_{n}}{ \partial\gamma_{h}}(\gamma;\mathbf{x}_{n})=\mathbf{0}. \tag{24}\]  
The estimated variance matrix \(\widehat{V}\) of \(\widehat{\gamma}\) is obtained with the sandwich formula [34]  
\[\widehat{V}=\widehat{T}^{-1}\widehat{I}\widehat{T}^{-1}\,\ \text{where} \tag{25}\]  
\[\widehat{I}=\left(\sum_{n=1}^{N}\frac{\partial l_{n}}{\partial\gamma_{h}}( \widehat{\gamma};\mathbf{x}_{n})\frac{\partial l_{n}}{\partial\gamma_{k}}( \widehat{\gamma};\mathbf{x}_{n})\right)_{hk}\ \text{and} \tag{26}\]  
\[\widehat{I}=\left(\sum_{n=1}^{N}\frac{\partial^{2}l_{n}}{\partial\gamma_{h} \gamma_{k}}(\widehat{\gamma};\mathbf{x}_{n})\right)_{hk}. \tag{27}\]  
The partial derivatives in (24), (26), and (27) can be obtained by analytical derivations or numerical approximations.  
Like MML, PML estimation relies on the normal distribution assumption for \(\theta\). From the first view, it seems that the normal copula model assumption for item residuals is also crucial. However, note that a single parameter \(\rho_{ij}\) is estimated for all item pairs \((X_{i},X_{j})\) of items within the same testlet.