# Computational Aspects of \(L_{0}\) Linking in the Rasch Model
## 1 Introduction  
Item response theory (IRT) models [1, 2, 3] are statistical models for multivariate discrete random variables. This article focuses on dichotomous (i.e., binary) random variables and the comparison of two groups through a linking method. Let \(\mathbf{X}=(X_{1},\ldots,X_{l})\) denote a vector of \(I\in\mathbb{N}\) random variables \(X_{i}\in\{0,1\}\), commonly referred to as items or (scored) item responses. A unidimensional IRT model [4] represents the probability distribution \(\mathsf{P}(\mathbf{X}=\mathbf{x})\) for \(\mathbf{x}=(x_{1},\ldots,x_{l})\in\{0,1\}^{l}\),  
\[\mathsf{P}(\mathbf{X}=\mathbf{x};\mathbf{\delta},\mathbf{\gamma})=\int\prod_{i=1}^{l}\Bigl{[}P_ {i}(\theta;\mathbf{\gamma}_{i})^{x_{i}}(1-P_{i}(\theta;\mathbf{\gamma}_{i}))^{1-x_{i}} \Bigr{]}\phi(\theta;\mu,\sigma)\,\mathrm{d}\theta, \tag{1}\]  
where \(\phi\) denotes the density of the normal distribution with mean \(\mu\) and standard deviation (SD) \(\sigma\). The distribution parameters of the latent variable \(\theta\), often labeled as a trait or ability variable, are collected in the vector \(\mathbf{\delta}=(\mu,\sigma)\). The vector \(\mathbf{\gamma}=(\mathbf{\gamma}_{1},\ldots,\mathbf{\gamma}_{l})\) contains the item parameters associated with the item response functions (IRFs) \(P_{i}(\theta;\mathbf{\gamma}_{i})=\mathsf{P}(X_{i}=1|\theta)\) for \(i=1,\ldots,I\). The IRF of the Rasch model [5, 6, 7] is defined as  
\[P_{i}(\theta;\mathbf{\gamma}_{i})=\Psi(\theta-b_{i}), \tag{2}\]  
where \(b_{i}\) denotes the item difficulty \(b_{i}\), and \(\Psi(x)=(1+\exp(-x))^{-1}\) is the logistic distribution function. In this formulation, the item parameter vector is given as \(\mathbf{\gamma}_{i}=(b_{i})\). For a sample of \(N\) individuals with independently and identically distributed observations \(x_{1},\ldots,x_{N}\) from the distribution of the random variable \(\mathbf{X}\), the unknown parameters of the IRT model in (1) can be consistently estimated using maximum likelihood estimation (ML; [8, 9, 10]). IRT models are commonly used to compare the performance of two groups on a test by examining differences in the latent variable \(\theta\) within the framework of the IRT model in (1). This article focuses on a generalization of the mean-mean linking method [11] based on the Rasch model. The purpose of a linking method is to estimate the difference between the distributions of \(\theta\) in the two groups. This difference serves as a summary measure of group performance on the multivariate vector of dichotomous items \(\mathbf{X}\). The linking approach consists of two steps. First, the Rasch model is estimated separately for each group, allowing for potential differential item functioning (DIF), where items may function differently across groups [12; 13; 14]. Second, differences in item parameters are used to estimate group differences in the \(\theta\) variable through a linking method [11; 15; 16]. This article investigates the performance of a generalization of mean-mean (MM; [11]) linking in the presence of fixed uniform DIF [14] in item difficulties. The traditional MM method relies on mean differences in item difficulties, which corresponds to using an \(L_{2}\) loss function. MM linking is a widely used linking or equating method, as discussed in popular textbooks on Rasch modeling [6; 7; 17; 18]. In this study, we investigate a generalization of the MM linking method in the Rasch model using the \(L_{0}\) loss function [19]. Using this robust loss function can essentially remove items with DIF effects from the group comparison directly in the linking method (see also [20; 21; 22; 23; 24; 25; 26; 27]). DIF effects can also be treated through the application of robust procedures in MM linking [19; 28; 29; 30; 31]. The \(L_{0}\) loss function, being nondifferentiable, requires differentiable approximations to ensure feasible and computationally stable estimation.