# Pairwise Likelihood Estimation of the 2 PL Model with Locally Dependent Item Responses
## 1 Introduction  
Item response theory (IRT) models [1; 2] are central to analyzing multivariate dichotomous random variables. IRT models summarize a high-dimensional contingency table by a low-dimensional latent factor variable of interest. In many cases, a reduction to a unidimensional variable is required. The application of IRT models in educational large-scale assessments [3] like the Programme for International Student Assessment (PISA; [4]) and the testing industry is of particular importance. Let \(\mathbf{X}=(X_{1},\ldots,X_{I})\) be the vector of \(I\) dichotomous items \(X_{i}\in\{0,1\}\). A unidimensional IRT model [5; 6] is a statistical model for the probability distribution \(\mathsf{P}(\mathbf{X}=\mathbf{x})\) for \(\mathbf{x}\in\{0,1\}^{I}\), where  
\[\mathsf{P}(\mathbf{X}=\mathbf{x};\mathbf{\gamma})=\int_{-\infty}^{\infty}\prod_{i=1}^{I}P_ {i}(x_{i},\theta;\mathbf{\gamma}_{i})f(\theta)\,\mathrm{d}\theta. \tag{1}\]  
In Equation (1), a latent variable \(\theta\) with density function \(f\) is involved. The variable (also referred to as a trait or ability) can be interpreted as a unidimensional summary of the test items \(\mathbf{X}\). The distribution of \(\theta\) is frequently fixed to a normal distribution (but see [7; 8]). The item response functions (IRF) \(P_{i}(x,\theta;\mathbf{\gamma}_{i})=\mathsf{P}(X_{i}=x|\theta)\) model the relationship of the dichotomous item with \(\theta\). Let \(\mathbf{\gamma}=(\mathbf{\gamma}_{1},\ldots,\mathbf{\gamma}_{I})\). Note that \(P_{i}(0,\theta;\mathbf{\gamma}_{i})+P_{i}(1,\theta;\mathbf{\gamma}_{i})=1\). Importantly, item responses \(X_{i}\) are assumed to be conditionally independent on \(\theta\) in (1). Therefore, pairs of items \(X_{i}\) and \(X_{j}\) are conditionally uncorrelated after controlling the latent ability \(\theta\). More formally, it is assumed that  
\[\mathsf{P}(\mathbf{X}|\theta)=\prod_{i=1}^{I}\mathsf{P}(X_{i}|\theta). \tag{2}\]  
The property (2) is also known as the local independence assumption that can be statistically tested [6; 9]. The vector of item parameters \(\mathbf{\gamma}\) of the IRFs in Equation (1) can be estimated by marginal maximum likelihood (MML) using an expectation-maximization (EM) algorithm [10; 11]. A variety of statistical software packages exist to estimate IRT models [12]. This article focuses on the estimation of IRT models in the violation of the local independence condition in (2). This situation refers to locally dependent item responses. For example, in an educational test, students respond to items referring to several reading texts. However, items that refer to the same reading text are likely more dependent on each other than items from different reading texts. Hence, the local dependence of item responses of items within the same item stimulus (i.e., the reading text) can be expected. In this article, we confine ourselves to estimating the two-parameter logistic (2 PL) IRT model [13]. The IRF for item \(X_{i}\) in the 2 PL model is defined as  
\[P_{i}(x,\theta;a_{i},b_{i})=\Psi((2x-1)(a_{i}\theta-b_{i}))=\frac{\exp(x(a_{i} \theta-b_{i}))}{1+\exp(a_{i}\theta-b_{i})}\text{ for }x=0,1, \tag{3}\]  
where the vector of item parameters \(\mathbf{\gamma}_{i}=(a_{i},b_{i})\) contains the item discrimination \(a_{i}\) and the item intercept \(b_{i}\). The logistic distribution function is denoted by \(\Psi\) and it holds \(1-\Psi(x)=\Psi(-x)\) for real-valued \(x\). This article reviews different estimation techniques for unidimensional IRT models under local dependence. It is demonstrated that MML estimation will lead to biased item parameter estimates if local dependence is ignored in the estimation. A variant of pairwise likelihood estimation is presented that treats the local dependence structure as a nuisance structure in the estimation and effectively removes these pieces of information in model estimation. Hence, our proposed modified estimation approach is unaffected by the presence of local dependence.