2.1 Measurement Model

The two-parameter logistic (2PL) model (Birnbaum, Reference Birnbaum, Lord and Novick1968) is widely used to model binary item responses (e.g., wrong/right or absent/present). In the absence of DIF, the 2PL model assumes a logistic relationship between $Y_{\mathrm{ij}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{ij}$$\end{document} and ${\theta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _i$$\end{document} , which is independent of the value of $x_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_i$$\end{document} . That is,

(1) $\begin{array}{c}P(Y_{\mathrm{ij}}=1|{\theta }_i=\theta )=\frac{exp(a_j\theta +d_j)}{1+exp(a_j\theta +d_j)},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P(Y_{ij} = 1\vert \theta _i = \theta ) = \frac{\exp (a_j\theta + d_j)}{1+\exp (a_j\theta + d_j)}, \end{aligned}$$\end{document}

where the slope parameter $a_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_j$$\end{document} and intercept parameter $d_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_j$$\end{document} are typically known as the discrimination and easiness parameters, respectively. The right-hand side of (1) as a function of $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} is known as the 2PL item response function. When the items potentially suffer from DIF, the item response functions may depend on the group membership $x_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_i$$\end{document} . In that case, the item response function can be modelled as:

(2) $\begin{array}{c}P(Y_{\mathrm{ij}}=1|{\theta }_i=\theta ,x_i)=\frac{exp(a_j\theta +d_j+{\gamma }_j{x}_i)}{1+exp(a_j\theta +d_j+{\gamma }_j{x}_i)},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P(Y_{ij} = 1\vert \theta _i = \theta , x_i) = \frac{\exp (a_j\theta + d_j + \gamma _j x_i)}{1+\exp (a_j\theta + d_j + \gamma _j x_i)}, \end{aligned}$$\end{document}

where ${\gamma }_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j$$\end{document} is an item-specific parameter that characterizes its DIF effect. More precisely,

$\begin{array}{c}\frac{P(Y_{\mathrm{ij}}=1|{\theta }_i=\theta ,x_i=1)/(1-P(Y_{\mathrm{ij}}=1|{\theta }_i=\theta ,x_i=1))}{P(Y_{\mathrm{ij}}=1|{\theta }_i=\theta ,x_i=0)/(1-P(Y_{\mathrm{ij}}=1|{\theta }_i=\theta ,x_i=0))}=exp\left({\gamma }_j\right).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{P(Y_{ij} = 1\vert \theta _i = \theta , x_i = 1)/(1- P(Y_{ij} = 1\vert \theta _i = \theta , x_i = 1))}{P(Y_{ij} = 1\vert \theta _i = \theta , x_i=0)/(1-P(Y_{ij} = 1\vert \theta _i = \theta , x_i=0))} = \exp (\gamma _j). \end{aligned}$$\end{document}

That is, $exp\left({\gamma }_j\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exp (\gamma _j)$$\end{document} is the odds ratio for comparing two individuals from two groups who have the same latent trait level. Item j is DIF-free under this model when ${\gamma }_j=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j = 0$$\end{document} . We further make the local independence assumption as in most IRT models. That is, $Y_{i1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{i1}$$\end{document} , ..., $Y_{\mathrm{iJ}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{iJ}$$\end{document} are assumed to be conditionally independent, given ${\theta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _i$$\end{document} and $x_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_i$$\end{document} .

2.2 Structural Model

A structural model specifies the distribution of ${\theta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _i$$\end{document} , which may depend on the group membership. Specifically, we assume the conditional distribution of ${\theta }_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _i$$\end{document} given $x_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_i$$\end{document} to follow a normal distribution,

$\begin{array}{c}{\theta }_i|x_i\sim N(\beta x_i,1_{\{x_i=0\}}+{\sigma }^2{1}_{\{x_i=1\}}).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \theta _i \vert x_i \sim N(\beta x_i, 1_{\{x_i = 0\}} + \sigma ^2 1_{\{x_i = 1\}}). \end{aligned}$$\end{document}

Note that the latent trait distribution for the reference group is set to a standard normal distribution to identify the location and scale of the latent trait. A similar assumption is typically adopted in IRT models for a single group of individuals.

The MIMIC model for DIF combines the above measurement and structural models, for which a path diagram is given in Fig. 1. The marginal likelihood function for this MIMIC model takes the form

(3) $\begin{array}{c}L\left(\Xi \right)=\prod _{i=1}^N\int \left(\prod _{j=1}^J,\frac{exp\left(y_{\mathrm{ij}}(a_j\theta +d_j+{\gamma }_j{x}_i)\right)}{1+exp(a_j\theta +d_j+{\gamma }_j{x}_i)}\right)\frac{1}{\sqrt{2\pi }}exp\left(\frac{-{(\theta -\beta x_i)}^2}{2(1_{\{x_i=0\}}+{\sigma }^2{1}_{\{x_i=1\}})}\right)d\theta ,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} L(\Xi ) = \prod _{i=1}^N \int \left( \prod _{j=1}^J \frac{\exp (y_{ij}(a_j\theta + d_j + \gamma _j x_i))}{1+\exp (a_j\theta + d_j + \gamma _j x_i)}\right) \frac{1}{\sqrt{2\pi }} \exp \left( \frac{-(\theta - \beta x_i)^2}{2(1_{\{x_i = 0\}} + \sigma ^2 1_{\{x_i = 1\}})}\right) d\theta ,\qquad \end{aligned}$$\end{document}

where $\Xi =\{\beta ,{\sigma }^2,a_j,d_j,{\gamma }_j,j=1,\dots ,J\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi = \{\beta , \sigma ^2, a_j, d_j, \gamma _j, j = 1, \ldots , J\}$$\end{document} denotes all the fixed model parameters.

The goal of DIF analysis is to detect the DIF items, i.e., the items for which ${\gamma }_j\ne 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j \ne 0$$\end{document} . Unfortunately, without further assumptions, this problem is ill-posed due to the non-identifiability of the model. We discuss this identifiability issue below.

Figure 1. The path diagram of a MIMIC model for DIF analysis. The subscript i is omitted for simplicity. The dashed lines from x to $Y_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_j$$\end{document} indicate the DIF effects.

2.3 Model Identifiability

Without further assumptions, the above MIMIC model is not identifiable. That is, for any constant c, the model remains equivalent, if we simultaneously replace $\beta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} and ${\gamma }_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j$$\end{document} by $\beta +c$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta +c$$\end{document} and ${\gamma }_j-a_{j}c$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j -a_jc$$\end{document} , respectively, and keep $a_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_j$$\end{document} unchanged. This identifiability issue is due to that all the items are allowed to suffer from DIF, resulting in an unidentified latent trait. In other words, without further assumptions, it is impossible to disentangle the DIF effects and the difference between the latent trait distributions of the two groups.

According to Theorem 8.3 of San Martín (Reference San Martín and van der Linden2016), the location shift described above is the only source of indeterminacy for this MIMIC model when $J\ge 3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J\ge 3$$\end{document} and the sizes of both groups go to infinity. Let ${\Xi }^{\ast }=\{{\beta }^{\ast },{\left({\sigma }^{\ast }\right)}^2,a_j^{\ast },d_j^{\ast },{\gamma }_j^{\ast },j=1,\dots ,J\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi ^* = \{\beta ^*, (\sigma ^*)^2, a_j^*, d_j^*, \gamma _j^*, j = 1, \ldots , J\}$$\end{document} be a set of parameters for the true model. Then for any constant c, the set of parameters ${\Xi }^{\ast }\left(c\right)=\{{\beta }^{\ast }+c,{\left({\sigma }^{\ast }\right)}^2,a_j^{\ast },d_j^{\ast },{\gamma }_j^{\ast }-a_j^{\ast }c,j=1,\dots ,J\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi ^*(c) = \{\beta ^*+c, (\sigma ^*)^2, a_j^*, d_j^*, \gamma _j^* - a_j^*c, j = 1, \ldots , J\}$$\end{document} gives the same data distribution. Moreover, if a set of parameters implies the same data distribution, then it has to take the form of ${\Xi }^{\ast }\left(c\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi ^*(c)$$\end{document} for some constant c. Knowing one or more anchor items means that the corresponding ${\gamma }_j^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j^*$$\end{document} s are known to be zero, which fixes the location indeterminacy. However, if no anchor item is known, we need to answer the question: which member of this equivalent class should be used to define DIF effects? We address it in Sect. 3.

3.1 Model Identifiability, Sparsity, and Minimal $L_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1$$\end{document} Condition

We now address the model identifiability problem. The most natural idea is to choose ${\Xi }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi ^*$$\end{document} as the true parameter vector when the corresponding ${\gamma }^{\ast }={({\gamma }_1^{\ast },\dots ,{\gamma }_J^{\ast })}^{\top }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\gamma }^* = (\gamma _1^*, \ldots , \gamma _J^*)^\top $$\end{document} is the sparsest in the equivalent class $\{{\Xi }^{\ast }\left(c\right):c\in R\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\Xi ^*(c): c \in {\mathbb {R}}\}$$\end{document} . In other words, we say ${\Xi }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi ^*$$\end{document} is the true model parameter when

(4) $\begin{array}{c}\Vert {\gamma }^{\ast }{\Vert }_0<{\Vert {\gamma }^{\ast }\left(c\right)\Vert }_0\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \varvec{\gamma }^*\Vert _0 < \Vert \varvec{\gamma }^*(c)\Vert _0 \end{aligned}$$\end{document}

for any $c\ne 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\ne 0$$\end{document} , where ${\gamma }^{\ast }\left(c\right)={({\gamma }_1^{\ast }-a_1^{\ast }c,\dots ,{\gamma }_J^{\ast }-a_J^{\ast }c)}^{\top }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\gamma }^*(c) = (\gamma _1^* - a_1^*c, \ldots , \gamma _J^* - a_J^*c)^\top $$\end{document} and ${\Vert ·\Vert }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \cdot \Vert _0$$\end{document} denotes the $L_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_0$$\end{document} norm, i.e., the number of nonzero entries in a vector. This definition requires the ${\gamma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\gamma }^*$$\end{document} to be unique, which further implies that ${\Xi }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi ^*$$\end{document} is unique. We note that this sparsity assumption is essential if one wants to formulate the DIF detection problem as a model selection problem, i.e., using statistical criteria to decide which DIF effect parameters are zero. It is explicitly or implicitly made by most DIF detection methods that do not require anchor items, including item purification and regularised estimation methods. Note that our notion of true model parameter requires the true DIF parameter vector ${\gamma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\gamma }^*$$\end{document} to have at least two zero elements, i.e., $\Vert {\gamma }^{\ast }{\Vert }_0\le J-2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \varvec{\gamma }^*\Vert _0 \le J-2$$\end{document} . If $\Vert {\gamma }^{\ast }{\Vert }_0\ge J-1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \varvec{\gamma }^*\Vert _0 \ge J-1$$\end{document} , then one can easily find a value of $c\ne 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\ne 0$$\end{document} such that $\Vert {\gamma }^{\ast }{\left(c\right)\Vert }_0\le {\Vert {\gamma }^{\ast }\Vert }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \varvec{\gamma }^*(c)\Vert _0 \le \Vert \varvec{\gamma }^*\Vert _0$$\end{document} by setting $c={\gamma }_j^{\ast }/a_j^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c = \gamma _j^*/a_j^*$$\end{document} for any j satisfying ${\gamma }_j^{\ast }\ne 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j^* \ne 0$$\end{document} . In that case, the definition of ${\Xi }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi ^*$$\end{document} is violated.

We also notice that the uniqueness of ${\gamma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\gamma }^*$$\end{document} is guaranteed when it is sufficiently sparse. In particular, when $\Vert {\gamma }^{\ast }{\Vert }_0<J/2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \varvec{\gamma }^*\Vert _0 < J/2$$\end{document} , then $\Vert {\gamma }^{\ast }{\left(c\right)\Vert }_0\ge J/2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \varvec{\gamma }^*(c)\Vert _0 \ge J/2$$\end{document} for any $c\ne 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\ne 0$$\end{document} (assuming that $a_j^{\ast }\ne 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_j^* \ne 0$$\end{document} for all j), and thus, the uniqueness of ${\gamma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\gamma }^*$$\end{document} is guaranteed. In the rest, we consider settings when ${\gamma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\gamma }^*$$\end{document} is sufficiently sparse. Further discussions will be provided in the sequel regarding the sparsity level. We note that this “sufficiently sparse" assumption aligns well with the practical utility of DIF analysis in educational testing (e.g., Holland & Wainer, Reference Holland and Wainer1993) as well as certain settings of psychological measurement (e.g., Chapter 1, Millsap, Reference Millsap2012) and health-related measurement (e.g., Scott et al., Reference Scott, Fayers, Aaronson, Bottomley, de Graeff, Groenvold, Gundy, Koller, Petersen and Sprangers2010). For example, in educational testing, DIF analysis is conducted to ensure the fairness of a test form. In this application, the test operator aims to identify a small number of DIF items that cause a bias in the test result. The identified items will be reviewed by domain experts and then revised or removed from the item pool. For this process to be operationally feasible, one typically needs to assume that most items are DIF-free, i.e., ${\gamma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\gamma }^*$$\end{document} is sufficiently sparse.

The $L_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_0$$\end{document} norm is not easy to work with from a statistical perspective. Due to the randomness in the data, likelihood-based estimation methods almost never give us a truly sparse solution. Consequently, one essentially needs to search over $O\left(2^J\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(2^J)$$\end{document} all possible models to find the sparest model (e.g., using a suitable information criterion). Item purification and regularized estimation methods narrow the search by stepwise procedures and regularized estimation procedures, respectively. Even with these methods, the computation can still be intensive, and consistent selection of the true model is not always guaranteed.

To develop our method, we consider a surrogate to (4). Specifically, we require the following minimal $L_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1$$\end{document} (ML1) condition to hold

(5) $\begin{array}{c}\sum _{j=1}^J|{\gamma }_j^{\ast }|<\sum _{j=1}^J|{\gamma }_j^{\ast }-a_j^{\ast }c|,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{j=1}^J \vert \gamma _j^* \vert < \sum _{j=1}^J \vert \gamma _j^* -a_j^*c\vert , \end{aligned}$$\end{document}

or equivalently, $\Vert {\gamma }^{\ast }{\Vert }_1<{\Vert {\gamma }^{\ast }\left(c\right)\Vert }_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \varvec{\gamma }^*\Vert _1 < \Vert \varvec{\gamma }^*(c)\Vert _1$$\end{document} for all $c\ne 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c \ne 0$$\end{document} . This assumption implies that, among all models that are equivalent to the true model, the true parameter vector ${\gamma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\gamma }^*$$\end{document} has the smallest $L_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1$$\end{document} norm. Equivalently, we can rewrite (5) as

(6) $\begin{array}{c}arg\underset{c}{min}h\left(c\right)=0,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \arg \min _c h(c) = 0, \end{aligned}$$\end{document}

where $h\left(c\right)={\sum }_{j=1}^J|{\gamma }_j^{\ast }-a_j^{\ast }c|$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(c) = \sum _{j=1}^J \vert \gamma _j^* -a_j^*c\vert $$\end{document} . We give an example of h(c) in Fig. 2, where h(c) is constructed with a sparse ${\gamma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\gamma }^*$$\end{document} . More specifically, we construct h(c) with $J=10$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J = 10$$\end{document} , $a_j^{\ast }=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_j^* = 1$$\end{document} for all j, ${\gamma }_j^{\ast }=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j^* = 0$$\end{document} and 1 when $j=1,\dots ,8$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j = 1, \ldots , 8$$\end{document} and $j=9,10$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j = 9, 10$$\end{document} , respectively. In this example, we note that h(c) has a unique minumum at $c=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=0$$\end{document} , i.e., (5), or equivalently, (6) holds.

Although (4) and (5) are different in general, they coincide with each other when ${\gamma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\gamma }^*$$\end{document} is sufficiently sparse. The following proposition provides a sufficient and necessary condition for the ML1 condition (5) (or equivalently (6)) to hold. The proof is given in the Supplementary Materials.

Figure 2. Function $h\left(c\right)={\sum }_{j=1}^J|{\gamma }_j^{\ast }-a_j^{\ast }c|$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(c) = \sum _{j=1}^J \vert \gamma _j^* -a_j^*c\vert $$\end{document} , where $J=10$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J = 10$$\end{document} , $a_j^{\ast }=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_j^* = 1$$\end{document} for all j, ${\gamma }_j^{\ast }=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j^* = 0$$\end{document} and 1 for $j=1,\dots ,8$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j = 1, \ldots , 8$$\end{document} and $j=9,10$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j = 9, 10$$\end{document} , respectively. The minimal value of h(c) is achieved when $c=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c = 0$$\end{document} .

Proposition 1

Assume that $a_j^{\ast }\ne 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_j^* \ne 0$$\end{document} for all j. Condition (5) holds if and only if

(7) $\begin{array}{c}\sum _{j=1}^J\left|a_j^{\ast }\right|\left(-,I,(,\frac{{\gamma }_j^{\ast }}{a_j^{\ast }},\ge ,0,),+,I,(,\frac{{\gamma }_j^{\ast }}{a_j^{\ast }},<,0,)\right)<0\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{j=1}^J |a_j^*| \left( - I\bigg (\frac{\gamma _j^*}{a_j^*} \ge 0\bigg ) + I\bigg (\frac{\gamma _j^*}{a_j^*}< 0\bigg )\right) < 0 \end{aligned}$$\end{document}

and

(8) $\begin{array}{c}\sum _{j=1}^J\left|a_j^{\ast }\right|\left(-,I,(,\frac{{\gamma }_j^{\ast }}{a_j^{\ast }},>,0,),+,I,(,\frac{{\gamma }_j^{\ast }}{a_j^{\ast }},\le ,0,)\right)>0,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{j=1}^J |a_j^*| \left( - I\bigg (\frac{\gamma _j^*}{a_j^*}> 0\bigg ) + I\bigg (\frac{\gamma _j^*}{a_j^*} \le 0\bigg )\right) > 0, \end{aligned}$$\end{document}

where $I(·)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(\cdot )$$\end{document} is the indicator function.

We note that inequalities (7) and (8) hold for the example in Fig. 2, where ${\sum }_{j=1}^{J}I({\gamma }_j^{\ast }\ge 0)=10$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{j=1}^J I(\gamma _j^* \ge 0) = 10$$\end{document} , ${\sum }_{j=1}^{J}I({\gamma }_j^{\ast }<0)=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{j=1}^J I(\gamma _j^* < 0) =0$$\end{document} , ${\sum }_{j=1}^{J}I({\gamma }_j^{\ast }\le 0)=8$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{j=1}^J I(\gamma _j^* \le 0) = 8$$\end{document} and ${\sum }_{j=1}^{J}I({\gamma }_j^{\ast }>0)=2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{j=1}^J I(\gamma _j^* > 0) = 2$$\end{document} . To elaborate on the results of Proposition 1, we first consider a special case when $a_j^{\ast }=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_j^* = 1$$\end{document} for all j, i.e., the measurement model is a one-parameter logistic model when there is no DIF. Then according to Proposition 1, the ML1 condition holds if and only if ${\sum }_{j=1}^{J}I({\gamma }_j^{\ast }\ge 0)>{\sum }_{j=1}^{J}I({\gamma }_j^{\ast }<0)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{j=1}^J I(\gamma _j^* \ge 0) > \sum _{j=1}^J I(\gamma _j^* < 0) $$\end{document} and ${\sum }_{j=1}^{J}I({\gamma }_j^{\ast }\le 0)>{\sum }_{j=1}^{J}I({\gamma }_j^{\ast }>0).$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{j=1}^J I(\gamma _j^* \le 0)> \sum _{j=1}^J I(\gamma _j^* > 0).$$\end{document} Suppose that more than half of the items are DIF-free, i.e., ${\sum }_{j=1}^{J}I({\gamma }_j^{\ast }=0)>J/2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{j=1}^J I(\gamma _j^* = 0) > J/2$$\end{document} . Then, the ML1 condition holds, because ${\sum }_{j=1}^{J}I({\gamma }_j^{\ast }\ge 0)>J/2>{\sum }_{j=1}^{J}I({\gamma }_j^{\ast }<0)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{j=1}^J I(\gamma _j^* \ge 0)> J/2 > \sum _{j=1}^J I(\gamma _j^* < 0) $$\end{document} and ${\sum }_{j=1}^{J}I({\gamma }_j^{\ast }\le 0)>J/2>{\sum }_{j=1}^{J}I({\gamma }_j^{\ast }>0).$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{j=1}^J I(\gamma _j^* \le 0)> J/2> \sum _{j=1}^J I(\gamma _j^* > 0).$$\end{document} In this case, as discussed previously, (4) also holds. More generally, let ${\gamma }_{\left(1\right)}^{\ast }\le {\gamma }_{\left(2\right)}^{\ast }\le \cdots \le {\gamma }_{\left(J\right)}^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{(1)}^* \le \gamma _{(2)}^* \le \cdots \le \gamma _{(J)}^*$$\end{document} be the order statistics of ${\gamma }_1^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _1^*$$\end{document} , ..., ${\gamma }_J^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _J^*$$\end{document} . The ML1 condition holds when ${\gamma }_{\left(\right(J+1)/2)}^{\ast }=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{((J+1)/2)}^* = 0$$\end{document} if J is an odd number, and when ${\gamma }_{(J/2)}^{\ast }={\gamma }_{\left(\right(J/2)+1)}^{\ast }=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{(J/2)}^* = \gamma _{((J/2)+1)}^* = 0$$\end{document} if J is an even number. That is, the ML1 condition holds when we have similar numbers of positive and negative DIF items and a few non-DIF items, in which case the ML1 condition can hold even if ${\sum }_{j=1}^{J}I({\gamma }_j^{\ast }=0)\le J/2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{j=1}^J I(\gamma _j^* = 0) \le J/2$$\end{document} . However, if all the DIF items are of the same direction (all positive or all negative), then it is easy to show that the ML1 condition does not hold if ${\sum }_{j=1}^{J}I({\gamma }_j^{\ast }=0)\le J/2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{j=1}^J I(\gamma _j^* = 0) \le J/2$$\end{document} .

We then extend the above discussion to the general setting where the discrimination parameters vary across items. Based on Proposition 1, we provide a sufficient condition for the ML1 condition, which suggests that the ML1 condition holds when ${\gamma }_j^{\ast }=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j^* = 0$$\end{document} for a sufficient number of items.

Corollary 1

Assume that $a_j^{\ast }\ne 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_j^* \ne 0$$\end{document} for all j. Let ${\rho }^{\ast }={max}_j\left\{\right|a_j^{\ast }\left|\right\}/{min}_j\left\{\right|a_j^{\ast }\left|\right\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho ^* = \max _j\{\vert a_j^*\vert \}/\min _j\{\vert a_j^*\vert \}$$\end{document} . Then Condition (5) holds if

(9) $\begin{array}{c}\sum _{j=1}^{J}I({\gamma }_j^{\ast }/a_j^{\ast }\le 0)>{\rho }^{\ast }\sum _{j=1}^{J}I({\gamma }_j^{\ast }/a_j^{\ast }>0)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{j=1}^J I(\gamma _j^*/a_j^* \le 0)> \rho ^* \sum _{j=1}^J I(\gamma _j^*/a_j^* > 0) \end{aligned}$$\end{document}

and

(10) $\begin{array}{c}\sum _{j=1}^{J}I({\gamma }_j^{\ast }/a_j^{\ast }<0)>{\rho }^{\ast }\sum _{j=1}^{J}I({\gamma }_j^{\ast }/a_j^{\ast }\ge 0).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{j=1}^J I(\gamma _j^*/a_j^* < 0) > \rho ^* \sum _{j=1}^J I(\gamma _j^*/a_j^* \ge 0). \end{aligned}$$\end{document}

We note (9) and (10) are not a necessary condition, meaning that the ML1 condition can still hold even if (9) and (10) are not satisfied. Here, ${\rho }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho ^*$$\end{document} quantifies the variation of the absolute discrimination parameters, where a larger value of ${\rho }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho ^*$$\end{document} indicates a higher variation. Corollary 1 suggests that ML1 condition holds if ${\sum }_{j=1}^{J}I({\gamma }_j^{\ast }=0)>({\rho }^{\ast }/(1+{\rho }^{\ast }))J$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{j=1}^J I(\gamma _j^* = 0) > (\rho ^*/(1+\rho ^*))J$$\end{document} , in which case (4) also holds. For instance, when ${\rho }^{\ast }=2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho ^* = 2$$\end{document} , then the ML1 condition is guaranteed if ${\sum }_{j=1}^{J}I({\gamma }_j^{\ast }=0)>(2/3)J$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{j=1}^J I(\gamma _j^* = 0) > (2/3)J$$\end{document} , i.e., at least two-thirds of the items are DIF-free. This sparsity requirement can be relaxed if the sizes of items with ${\gamma }_j^{\ast }/a_j^{\ast }>0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j^*/a_j^* > 0$$\end{document} and those with ${\gamma }_j^{\ast }/a_j^{\ast }<0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j^*/a_j^* < 0$$\end{document} are balanced.

3.2 Parameter Estimation

Suppose that the true model parameters satisfy the ML1 condition. Then, these parameters can be estimated by finding the ML1 estimate $\hat{\Xi }=\{\hat{\beta },{\hat{\sigma }}^2,{\hat{\gamma }}_j,{\hat{d}}_j,{\hat{a}}_j,j=1,\dots ,J\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\Xi }} = \{{\hat{\beta }}, {\hat{\sigma }}^2, {\hat{\gamma }}_j, {\hat{d}}_j, {\hat{a}}_j, j=1, \ldots ,J\} $$\end{document} satisfying

(11) $\begin{array}{c}logL\left(\hat{\Xi }\right)=\underset{\Xi }{max}logL\left(\Xi \right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \log L({\hat{\Xi }}) = \max _{\Xi } \log L(\Xi ) \end{aligned}$$\end{document}

and for any $\tilde{\Xi }=\{\tilde{\beta },{\tilde{\sigma }}^2,{\tilde{\gamma }}_j,{\tilde{d}}_j,{\tilde{a}}_j,j=1,\dots ,J\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Xi }= \{\tilde{\beta }, \tilde{\sigma }^2, \tilde{\gamma }_j, {\tilde{d}}_j, {\tilde{a}}_j, j=1, \ldots ,J\} $$\end{document} satisfying $logL\left(\tilde{\Xi }\right)={max}_{\Xi }logL\left(\Xi \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log L(\tilde{\Xi }) = \max _{\Xi } \log L(\Xi )$$\end{document} ,

(12) $\begin{array}{c}\sum _{j=1}^J|{\hat{\gamma }}_j|\le \sum _{j=1}^J\left|{\tilde{\gamma }}_j\right|.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{j=1}^J \vert {\hat{\gamma }}_j \vert \le \sum _{j=1}^J \vert \tilde{\gamma }_j\vert . \end{aligned}$$\end{document}

That is, $\hat{\Xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\Xi }}$$\end{document} is a maximum likelihood estimate whose DIF parameter vector has the smallest $L_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1$$\end{document} norm. We adopt a two-stage estimator to find $\hat{\Xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\Xi }}$$\end{document} . First, we find an estimator $\tilde{\Xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Xi }$$\end{document} that maximizes $logL\left(\Xi \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log L(\Xi )$$\end{document} , but the corresponding $\tilde{\gamma }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\varvec{\gamma }}$$\end{document} not necessarily has the minimum $L_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1$$\end{document} norm in its equivalent class defined by location shift. Second, we find $\hat{\Xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\Xi }}$$\end{document} within the equivalent class of $\tilde{\Xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Xi }$$\end{document} such that the corresponding $\hat{\gamma }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\varvec{\gamma }}$$\end{document} has the minimum $L_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1$$\end{document} norm.

In principle, $\tilde{\Xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Xi }$$\end{document} in the first stage can be obtained by maximizing $logL\left(\Xi \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log L(\Xi )$$\end{document} without imposing constraints on model parameters. However, due to the location indeterminacy, the Hessian matrix of $logL\left(\Xi \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log L(\Xi )$$\end{document} degenerates, and thus, the optimization often suffers from slow convergence. To avoid this issue, we fix the location indeterminacy issue by constraining ${\tilde{\gamma }}_1=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\gamma }_1 = 0$$\end{document} . Due to the location indeterminacy of the model, one can always make this constraint without sacrificing the likelihood function value, even if item 1 is not DIF-free. We also remark that the constraint ${\tilde{\gamma }}_1=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\gamma }_1 = 0$$\end{document} can be replaced by any equivalent constraint, for example, ${\tilde{\gamma }}_2=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\gamma }_2 = 0$$\end{document} , while not affecting the final estimation result. The two-stage estimator is summarized in Algorithm 1.

Algorithm 1

We provide some remarks about the optimisation in Step 2. This step finds the transformation that leads to the ML1 solution among all the models equivalent to the estimated model from Step 1. The optimization problem (14) is convex that takes the same form as the Least Absolute Deviations (LAD) objective function in median regression (Koenker, Reference Koenker2005). Specifically, the LAD function is a statistical optimization function measuring the sum of absolute residuals. Given a set of data $(x_i,y_i)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_i, y_i)$$\end{document} for $i=1,\cdots ,n$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i = 1,\dots , n$$\end{document} , the LAD function is defined as $S\left(f\right)={\sum }_{i=1}^n|y_i-f\left(x_i\right)|$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(f) = \sum _{i=1}^{n} \vert y_i - f(x_i) \vert $$\end{document} , and we seek to find f that minimizes LAD function S. Our problem (14) is convex since we are minimizing a convex LAD function over a set of real numbers, which gives us a unique global optimum. Consequently, it can be solved using standard statistical packages/software for quantile regression. The R package “quantreg” (Koenker, Reference Koenker2022) is used in our simulation study and real data analysis.

The ML1 condition (5), together with some additional regularity conditions, guarantees the consistency of the above ML1 estimator. That is, $\hat{\Xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\Xi }}$$\end{document} will converge to ${\Xi }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi ^*$$\end{document} as the sample size N grows to infinity. This result is formalized in Theorem 1, with its proof given in the Supplementary Materials.

Theorem 1

Let ${\Xi }^{\ast }=\{{\beta }^{\ast },{\left({\sigma }^{\ast }\right)}^2,{\gamma }_j^{\ast },d_j^{\ast },a_j^{\ast },j=1,\dots ,J\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi ^*=\{\beta ^*, (\sigma ^*)^2, \gamma _j^*, d_j^*, a_j^*, j=1, \ldots , J\}$$\end{document} be the true model parameters, and ${\Xi }^{†}=\{{\beta }^{†},{\left({\sigma }^2\right)}^{†},{\gamma }_j^{†},d_j^{†},a_j^{†},j=1,\dots ,J\}={\Xi }^{\ast }({\gamma }_1^{\ast }/a_1^{\ast })$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi ^\dagger =\{\beta ^\dagger , (\sigma ^2)^\dagger , \gamma _j^\dagger , d_j^\dagger , a_j^\dagger , j=1, \ldots ,J\} = \Xi ^*(\gamma _1^*/a_1^*)$$\end{document} be the true parameter values of the equivalent MIMIC model with constraint ${\gamma }_1^{†}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _1^\dagger =0$$\end{document} . Assume this equivalent model satisfies the standard regularity conditions in Theorem 5.14 of van der Vaart (Reference van der Vaart2000) that concerns the consistency of maximum likelihood estimation. Further, assume that the ML1 condition (5) holds. Then $|\hat{\beta }-{\beta }^{\ast }|=o_P\left(1\right),|{\hat{\sigma }}^2-{\left({\sigma }^2\right)}^{\ast }|=o_P\left(1\right),|{\hat{\gamma }}_j-{\gamma }_j^{\ast }|=o_P\left(1\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert {\hat{\beta }} - \beta ^*\vert =o_P(1), {\vert \hat{\sigma }^2 -(\sigma ^2)^*\vert =o_P(1)}, \vert {\hat{\gamma }}_j - \gamma _j^*\vert =o_P(1)$$\end{document} , $|{\hat{a}}_j-a_j^{\ast }|=o_P\left(1\right),\text{and}|{\hat{d}}_j-d_j^{\ast }|=o_P\left(1\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert {\hat{a}}_j - a_j^*\vert =o_P(1), \text{ and } \vert {\hat{d}}_j - d_j^*\vert =o_P(1)$$\end{document} as $N\to \infty$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\rightarrow \infty $$\end{document} .

With a consistent point estimator, one can consistently select the true model, i.e., identifying the zeros and nonzeros in ${\gamma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\gamma }^*$$\end{document} , using a hard-thresholding procedure (see, e.g. Meinshausen & Yu, Reference Meinshausen and Yu2009). As our focus is on the statistical inference of DIF parameters, we skip the details of the hard-thresholding procedure here. Once the final model is selected, it may be possible to verify the ML1 condition by checking whether the sufficient conditions in Corollary 1 hold for the selected model.

3.3 Statistical Inference

The statistical inference of individual ${\gamma }_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j$$\end{document} parameters is of particular interest in the DIF analysis. With the proposed estimator, we can draw valid statistical inference on the DIF parameters ${\gamma }_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j$$\end{document} .

Note that the uncertainty of ${\hat{\gamma }}_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\gamma }}_j$$\end{document} is inherited from $\tilde{\Xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Xi }$$\end{document} , where $\sqrt{N}(\tilde{\Xi }-{\Xi }^{†})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{N}(\tilde{\Xi }-\Xi ^\dagger )$$\end{document} asymptotically follows a mean-zero multivariate normal distributionFootnote 1 by the large-sample theory for maximum likelihood estimation; see Supplementary Materials for more details. We denote this multivariate normal distribution by $N(0,{\Sigma }^{\ast })$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N({\textbf{0}}, \Sigma ^*)$$\end{document} , where a consistent estimator of ${\Sigma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma ^*$$\end{document} , denoted by ${\hat{\Sigma }}_N$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\Sigma }}_N$$\end{document} , can be obtained based on the marginal likelihood. We define a function

$\begin{array}{c}{G}_j\left(\Xi \right)={\gamma }_j-a_j\times arg\underset{c}{min}\sum _{l=1}^J|{\gamma }_l-a_{l}c|,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} G_j(\Xi ) = \gamma _j - a_j \times \arg \min _{c} \sum _{l=1}^J \vert \gamma _l - a_lc\vert , \end{aligned}$$\end{document}

where $\Xi =\{\beta ,{\sigma }^2,a_l,d_l,{\gamma }_l,l=1,\dots ,J\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi = \{\beta , \sigma ^2, a_l, d_l, \gamma _l, l = 1, \ldots , J\}$$\end{document} . Note that the function $G_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_j$$\end{document} maps an arbitrary parameter vector of the MIMC model to the ${\gamma }_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j$$\end{document} parameter of the equivalent ML1 parameter vector. To draw statistical inference, we need the distribution of

$\begin{array}{c}{\hat{\gamma }}_j-{\gamma }_j^{\ast }=G_j\left(\tilde{\Xi }\right)-G_j\left({\Xi }^{†}\right).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\hat{\gamma }}_j - \gamma _j^* = G_j(\tilde{\Xi }) - G_j(\Xi ^\dagger ). \end{aligned}$$\end{document}

By the asymptotic distribution of $\sqrt{N}(\tilde{\Xi }-{\Xi }^{†})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{N}(\tilde{\Xi }-\Xi ^\dagger )$$\end{document} , we know that the distribution of $G_j\left(\tilde{\Xi }\right)-G_j\left({\Xi }^{†}\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_j(\tilde{\Xi }) - G_j(\Xi ^\dagger )$$\end{document} can be approximated by that of $G_j({\Xi }^{†}+Z/\sqrt{N})-G_j\left({\Xi }^{†}\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_j(\Xi ^\dagger + {\textbf{Z}}/\sqrt{N}) - G_j(\Xi ^\dagger )$$\end{document} , and the latter can be further approximated by $G_j(\tilde{\Xi }+Z/\sqrt{N})-G_j\left(\tilde{\Xi }\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_j(\tilde{\Xi }+ \textbf{Z}/\sqrt{N}) - G_j(\tilde{\Xi })$$\end{document} , where $Z$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{Z}}$$\end{document} follows a normal distribution $N(0,{\hat{\Sigma }}_N)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N({\textbf{0}}, {\hat{\Sigma }}_N)$$\end{document} . Therefore, although function $G_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_j$$\end{document} does not have an analytic form, we can approximate the distribution of ${\hat{\gamma }}_j-{\gamma }_j^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\gamma }}_j - \gamma _j^*$$\end{document} by Monte Carlo simulation. We summarize this procedure in Algorithm 2. It can be viewed as a parametric bootstrap procedure (Davison and Hinkley, Reference Davison and Hinkley1997; Zhang, Reference Zhang2018).

Algorithm 2

Algorithm 2 only involves sampling from a multivariate normal distribution and solving a convex optimization problem based on the LAD objective function, both of which are computationally efficient. The value of M is set to 10,000 in our simulation study and 50,000 in the real data example below.

The P-values can be used to control the type-I error rate, i.e., the probability of mistakenly detecting a non-DIF item as a DIF one. To control the item-specific type-I errors to be below a pre-specified threshold $\alpha$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} (e.g., $\alpha =0.05$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = 0.05$$\end{document} ), we detect the items for which the corresponding P-values are below $\alpha$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} . Besides the type-I error, we may also consider the false discovery rate (FDR) for DIF detection (Bauer et al., Reference Bauer, Belzak and Cole2020) to account for multiple comparisons, where the FDR is defined as the expected ratio of the number of non-DIF items to the total number of detections. To control the FDR, the Benjamini–Hochberg (B-H) procedure (Benjamini and Hochberg, Reference Benjamini and Hochberg1995) can be employed to the P-values. Other compound risks may also be considered, such as the familywise error rate.

4.1 Related Works

Many of the IRT-based DIF analyses (Thissen et al., Reference Thissen, Steinberg and Gerrard1986; Thissen, Reference Thissen, Wainer and Braun1988; Thissen et al., Reference Thissen, Steinberg, Wainer, Holland and Wainer1993) require prior knowledge about a subset of DIF-free items, which are known as the anchor items. More precisely, we denote this known subset by A. Under the MIMIC model described above, it implies that the constraints ${\gamma }_j=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j = 0$$\end{document} are imposed for all $j\in A$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in A$$\end{document} in the estimation. With these zero constraints, the ${\gamma }_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j$$\end{document} parameters cannot be freely transformed, and thus, the above MIMIC model becomes identifiable. Therefore, for each non-anchor item $j\notin A$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \notin A$$\end{document} , the hypothesis of ${\gamma }_j=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j = 0$$\end{document} can be tested, for example, by a likelihood ratio test. The DIF items can then be detected based on the statistical inference of these hypothesis tests.

The validity of the anchor-item-based analyses relies on the assumption that the anchor items are truly DIF-free. If the anchor set includes one or more DIF items, then the results can be misleading (Kopf et al., Reference Kopf, Zeileis and Strobl2015b). To address the issue brought by the mis-specification of the anchor set, item purification methods (Candell and Drasgow, Reference Candell and Drasgow1988; Clauser et al., Reference Clauser, Mazor and Hambleton1993; Fidalgo et al., Reference Fidalgo, Mellenbergh and Muñiz2000; Wang and Yeh, Reference Wang and Yeh2003; Wang and Su, Reference Wang and Su2004; Wang et al., Reference Wang, Shih and Yang2009; Kopf et al., Reference Kopf, Zeileis and Strobl2015b, Reference Kopf, Zeileis and Strobla) have been proposed that iteratively purify the anchor set. These methods conduct model selection using a stepwise procedure to select the anchor set, implicitly assuming that there exists a reasonably large set of DIF-free items. Then, DIF is assessed by hypothesis testing given the selected anchor set. This type of method also has several limitations. First, the model selection results may be sensitive to the choice of the initial set of anchor items and the specific stepwise procedure (e.g., forward or backward selection), which is a common issue with stepwise model selection procedures (e.g., stepwise variable selection for linear regression). Second, the model selection step has uncertainty. As a result, there is no guarantee that the selected anchor set is completely DIF-free, and furthermore, the post-selection statistical inference of items may not be valid in the sense that the type-I error may not be controlled at the targeted significance level.

Bechger and Maris (Reference Bechger and Maris2015) and Yuan et al. (Reference Yuan, Liu and Han2021) proposed DIF detection methods based on the idea of differential item pair functioning. They considered a one-parameter logistic model setting, which corresponds to the case when $a_1=\cdots =a_J$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_1 = \cdots = a_J$$\end{document} in the current MIMIC model. Their idea is that the difference ${\gamma }_j-{\gamma }_{j^{\prime }}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j - \gamma _{j'}$$\end{document} is identifiable for any $j\ne j^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\ne j'$$\end{document} , though each individual ${\gamma }_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j$$\end{document} is not identifiable due to location indeterminacy. Bechger and Maris (Reference Bechger and Maris2015) focused on testing ${\gamma }_j-{\gamma }_{j^{\prime }}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j - \gamma _{j'} = 0$$\end{document} for all item pairs, and Yuan et al. (Reference Yuan, Liu and Han2021) proposed data visualization methods and a Monte Carlo test to identify individual DIF items. However, they did not provide statistical inferences for the DIF effects of individual items. In particular, Yuan et al. (Reference Yuan, Liu and Han2021) constructed item-specific confidence intervals for the DIF effect parameters. However, their confidence intervals are constructed for an order statistic considering information from all the items and, thus, can only test the DIF effect of the item ranked in the jth place by their procedure. Moreover, their construction of confidence intervals requires a strong assumption that all the items are DIF-free, which does not hold in DIF detection problems. Our procedure does not require such an assumption.

Regularized estimation methods (Magis et al., Reference Magis, Tuerlinckx and De Boeck2015; Tutz and Schauberger, Reference Tutz and Schauberger2015; Huang, Reference Huang2018; Belzak and Bauer, Reference Belzak and Bauer2020; Bauer et al., Reference Bauer, Belzak and Cole2020; Schauberger and Mair, Reference Schauberger and Mair2020) have also been proposed for identifying the anchor items, which also implicitly assumes that many items are DIF-free. These methods do not require prior knowledge about anchor items and simultaneously select the DIF-free items and estimate the model parameters using a LASSO-type penalty (Tibshirani, Reference Tibshirani1996). Under the above MIMIC model, a regularized estimation procedure solves the following optimization problem,

(15) $\begin{array}{c}{\hat{\Xi }}^{\lambda }=arg\underset{\Xi }{min}-logL\left(\Xi \right)+\lambda \sum _{j=1}^J\left|{\gamma }_j\right|,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\hat{\Xi }}^\lambda = \arg \min _{\Xi } - \log L(\Xi ) + \lambda \sum _{j=1}^J \vert \gamma _j\vert , \end{aligned}$$\end{document}

where $\lambda >0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda > 0$$\end{document} is a tuning parameter that determines the sparsity level of the estimated ${\gamma }_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j$$\end{document} parameters. Generally speaking, a larger value of $\lambda$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} leads to a more sparse vector ${\hat{\gamma }}^{\lambda }=({\hat{\gamma }}_1^{\lambda },\dots ,{\hat{\gamma }}_J^{\lambda }).$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\varvec{\gamma }}^\lambda = ({{\hat{\gamma }}}_1^\lambda , \ldots , {\hat{\gamma }}_J^\lambda ).$$\end{document} A regularization method (e.g. Belzak & Bauer, Reference Belzak and Bauer2020) solves the optimization problem (15) for a sequence of $\lambda$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} values and then selects the tuning parameter $\lambda$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} based on the Bayesian information criterion (BIC; Schwarz, Reference Schwarz1978). Let $\hat{\lambda }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\lambda }}$$\end{document} be the selected tuning parameter. Items for which ${\hat{\gamma }}_j^{\hat{\lambda }}\ne 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\hat{\gamma }}}_j^{{\hat{\lambda }}} \ne 0$$\end{document} are classified as DIF items and the rest are classified as DIF-free items. While the regularization methods are computationally more stable than stepwise model selection in the item purification methods, they also suffer from some limitations. First, they involve solving non-smooth optimization problems like (15) for different tuning parameter values, which is not only computationally intensive but also requires tailored computation code that is not available in most statistical packages/software for DIF analysis. Second, these methods may be sensitive to the choice of the tuning parameter. Although methods and theories have been developed in the statistics literature to guide the selection of the tuning parameter, there is no consensus on how the tuning parameter should be chosen, leaving ambiguity in the application of these methods. Third, from the theoretical perspective, it is not clear whether these methods can guarantee model selection consistency. In particular, the model selection consistency of the LASSO procedure almost always requires a strong assumption called the irrepresentable condition (Zhao and Yu, Reference Zhao and Yu2006; van de Geer and Bühlmann, Reference van de Geer and Bühlmann2009). It is not clear when this assumption holds for the current problem. On the other hand, the proposed ML1 condition is much easier to understand and check, as discussed in Sect. 3.1. Finally, as a common issue of regularized estimation methods, obtaining valid statistical inference from these methods is not straightforward. That is, regularized estimation like (15) does not provide a valid p-value for testing the null hypothesis ${\gamma }_j=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _j = 0$$\end{document} . In fact, post-selection inference after regularized estimation was conducted in Bauer et al. (Reference Bauer, Belzak and Cole2020), where the type I error cannot be controlled at the targeted level under some simulation scenarios.

We notice that there is a connection between the proposed estimator and the regularized estimator (15). Note that $\hat{\Xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\Xi }}$$\end{document} is the one with the smallest ${\sum }_{j=1}^J\left|{\gamma }_j\right|$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{j=1}^J \vert \gamma _j\vert $$\end{document} among all equivalent estimators that maximize the likelihood function (3). When the solution path of (15) is smooth and the solution to the ML1 problem (14) is unique, it is easy to see that $\hat{\Xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\Xi }}$$\end{document} is the limit of ${\hat{\Xi }}^{\lambda }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\Xi }}^{\lambda }$$\end{document} when the positive tuning parameter $\lambda$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} converges to zero. In other words, the proposed estimator can be viewed as a limiting version of the LASSO estimator (15). According to Theorem 1, this limiting version of the LASSO estimator is statistically consistent under the ML1 condition and some reasonable regularity conditions.

We clarify that the proposed method may not always outperform other methods in terms of accuracy in classifying items, such as the LASSO procedure. From the simulation results in Sect. 5, we see that the proposed method and the LASSO procedure have similar accuracy in item classification when the DIF parameters are large. The key advantage of the proposed method is that the proposed one provides valid statistical inference (e.g., P-values) when anchor items are not available. The inference results allow us to tackle the uncertainty in the decisions of DIF detection, which can be useful in many applications of DIF analysis where high-stake decisions need to be made.

4.2 Extensions

While we focus on the two-group setting and uniform-DIF (i.e., only the intercepts depend on the groups) in the previous discussion, the proposed framework is very general that can be easily generalised to other settings. In what follows, we discuss the ML1 condition under different settings. The proposed methods for point estimation and statistical inference can be extended accordingly.

Non-uniform DIF. Under the 2PL measurement model, non-uniform DIF happens when the discrimination parameter also differs across groups. To model non-uniform DIF, we extend the current measurement model (2) to

(16) $\begin{array}{c}P(Y_{\mathrm{ij}}=1|{\theta }_i=\theta ,x_i)=\frac{exp(a_{j}exp\left({\zeta }_j{x}_i\right)\theta +d_j+{\gamma }_j{x}_i)}{1+exp(a_{j}exp\left({\zeta }_j{x}_i\right)\theta +d_j+{\gamma }_j{x}_i)},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P(Y_{ij} = 1\vert \theta _i = \theta , x_i) = \frac{\exp (a_j\exp (\zeta _j x_i)\theta + d_j + \gamma _j x_i)}{1+\exp (a_j\exp (\zeta _j x_i)\theta + d_j + \gamma _j x_i)}, \end{aligned}$$\end{document}

while keeping the structural model the same as in Sect. 2.2. This extended model has both location and scale indeterminacies. Let ${\Xi }^{\ast }=\{{\beta }^{\ast },{\left({\sigma }^{\ast }\right)}^2,a_j^{\ast },d_j^{\ast },{\zeta }_j^{\ast },{\gamma }_j^{\ast },j=1,\dots ,J\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi ^* = \{\beta ^*, (\sigma ^*)^2, a_j^*, d_j^*, \zeta _j^*, \gamma _j^*, j = 1, \ldots , J\}$$\end{document} be a set of parameters for the true model. Then, a set of parameters yields the same data distribution as the true model if there exist constants m and c such that ${\Xi }^{\ast }(m,c)=\{({\beta }^{\ast }-c)\times exp(-m),exp(-2m)\times {\left({\sigma }^{\ast }\right)}^2,a_j^{\ast },d_j^{\ast },{\zeta }_j^{\ast }+m,{\gamma }_j^{\ast }-c{a}_j^{\ast }exp\left({\zeta }_j^{\ast }\right),j=1,\dots ,J\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi ^*(m,c) = \{(\beta ^*-c)\times \exp (-m), \exp (-2\,m)\times (\sigma ^*)^2, a_j^*, d_j^*, \zeta _j^* + m, \gamma _j^* - ca_j^*\exp (\zeta _j^*), j = 1, \ldots , J\}$$\end{document} . Note that an item j is DIF-free if ${\zeta }_j={\gamma }_j=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta _j = \gamma _j =0$$\end{document} . Under the same spirit as the ML1 condition (5), we may assume the true model parameters ${\Xi }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi ^*$$\end{document} to satisfy

$\begin{array}{c}\sum _{j=1}^J|{\zeta }_j^{\ast }|<\sum _{j=1}^J|{\zeta }_j^{\ast }+m|,\text{and}\sum _{j=1}^J|{\gamma }_j^{\ast }|<\sum _{j=1}^J|{\gamma }_j^{\ast }+c{a}_j^{\ast }exp\left({\zeta }_j^{\ast }\right)|\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\sum _{j=1}^J \vert \zeta _j^* \vert< \sum _{j=1}^J \vert \zeta _j^* +m \vert }, \text{ and } \sum _{j=1}^J\vert \gamma _j^*\vert < \sum _{j=1}^J\vert \gamma _j^* + ca_j^*\exp (\zeta _j^*)\vert \end{aligned}$$\end{document}

when $m\ne 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ne 0$$\end{document} and $c\ne 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\ne 0$$\end{document} . These conditions tend to be satisfied when the proportion of DIF-free items is sufficiently large.

Multi-group setting. There may be more than two groups in some DIF applications. Suppose that there are $K+1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K+1$$\end{document} groups—one reference group and K focal groups. Let $x_i\in \{0,\dots ,K\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_i \in \{0, \ldots , K\}$$\end{document} indicate the group membership.

For simplicity, we focus on the uniform DIF setting. Then, the measurement model becomes:

(17) $\begin{array}{c}P(Y_{\mathrm{ij}}=1|{\theta }_i=\theta ,x_i=k)=\frac{exp(a_j\theta +d_j+{\gamma }_{\mathrm{jk}})}{1+exp(a_j\theta +d_j+{\gamma }_{\mathrm{jk}})},k=1,\dots ,K,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P(Y_{ij} = 1\vert \theta _i = \theta , x_i =k) = \frac{\exp (a_j \theta + d_j + \gamma _{jk} )}{1+\exp (a_j \theta + d_j + \gamma _{jk})}, k = 1, \ldots , K, \end{aligned}$$\end{document}

and

(18) $\begin{array}{c}P(Y_{\mathrm{ij}}=1|{\theta }_i=\theta ,x_i=0)=\frac{exp(a_j\theta +d_j)}{1+exp(a_j\theta +d_j)}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P(Y_{ij} = 1\vert \theta _i = \theta , x_i =0) = \frac{\exp (a_j \theta + d_j )}{1+\exp (a_j \theta + d_j ) }. \end{aligned}$$\end{document}

The structural model becomes ${\theta }_i|x_i=k\sim N({\beta }_k,{\sigma }_k^2),k=1,\dots ,K,$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _i \vert x_i=k \sim N(\beta _k, \sigma _k^2), k = 1, \ldots , K,$$\end{document} and ${\theta }_i|x_i=0\sim N(0,1)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _i \vert x_i=0 \sim N(0, 1)$$\end{document} . Under this model, an item j is DIF-free if ${\gamma }_{\mathrm{jk}}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{jk} = 0$$\end{document} for all k. The location indeterminacy under this model leads to the following ML1 condition for identifying the true model parameters ${\Xi }^{\ast }=\{{\beta }_k^{\ast },{\left({\sigma }_k^{\ast }\right)}^2,a_j^{\ast },d_j^{\ast },{\gamma }_{\mathrm{jk}}^{\ast },k=1,\dots ,K,j=1,\dots ,J,\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi ^* = \{\beta ^*_k, (\sigma _k^*)^2, a_j^*, d_j^*, \gamma _{jk}^*, k = 1, \ldots , K, j = 1, \ldots , J,\}$$\end{document} :

$\begin{array}{c}\sum _{j=1}^J|{\gamma }_{\mathrm{jk}}^{\ast }|<\sum _{j=1}^J|{\gamma }_{\mathrm{jk}}^{\ast }-a_j^{\ast }{c}_k|,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{j=1}^{J} \vert \gamma _{jk}^* \vert < \sum _{j=1}^{J} \vert \gamma _{jk}^* -a_j^* c_k\vert , \end{aligned}$$\end{document}

for $c_k\ne 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_k\ne 0$$\end{document} , $k=1,\dots ,K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k = 1, \ldots , K$$\end{document} .

We note that this ML1 condition for the multi-group setting allows the majority of the items to be DIF items as long as the vector ${({\gamma }_{1k}^{\ast },\dots ,{\gamma }_{\mathrm{Jk}}^{\ast })}^{\top }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\gamma _{1k}^*, \ldots , \gamma _{Jk}^*)^\top $$\end{document} is sufficiently sparse for each focal group. Similar to the discussion in Sect. 3.1, in the special case of the one-parameter logistic model, the ML1 condition is guaranteed to hold if ${\sum }_{j=1}^{J}I({\gamma }_{\mathrm{jk}}^{\ast }=0)>J/2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{j=1}^J I(\gamma _{jk}^* = 0) > J/2$$\end{document} , for all k. Note that the set of items satisfying ${\gamma }_{\mathrm{jk}}^{\ast }=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{jk}^* = 0$$\end{document} can vary across focal groups.

Continuous covariates. In some applications, DIF might be caused by continuous covariates, such as age. Suppose that we have K continuous covariates $x_i={(x_{i1},\dots ,x_{\mathrm{iK}})}^{\top }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{x}}_i = (x_{i1}, \ldots , x_{iK})^\top $$\end{document} , rather than discrete groups. Then, we may consider the following measurement model:

(19) $\begin{array}{c}P(Y_{\mathrm{ij}}=1|{\theta }_i=\theta ,x_i)=\frac{exp(a_j\theta +d_j+{\gamma }_j^{\top }{x}_i)}{1+exp(a_j\theta +d_j+{\gamma }_j^{\top }{x}_i)},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P(Y_{ij} = 1\vert \theta _i = \theta , {\textbf{x}}_i) = \frac{\exp (a_j \theta + d_j +\varvec{\gamma }_j^\top {\textbf{x}}_{i} )}{1+\exp (a_j \theta + d_j +\varvec{\gamma }_j^\top {\textbf{x}}_{i})}, \end{aligned}$$\end{document}

where ${\gamma }_j={({\gamma }_{j1},\dots ,{\gamma }_{\mathrm{jK}})}^{\top }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\gamma }_j = (\gamma _{j1}, \ldots , \gamma _{jK})^\top $$\end{document} be the corresponding DIF parameters. We may assume the structural model takes a homoscedastic latent regression form $\theta |x_i\sim N(\beta x_i,1)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \vert {\textbf{x}}_i \sim N(\varvec{\beta }{\textbf{x}}_i,1)$$\end{document} , where the variance is fixed to 1 to avoid scale indeterminacy.Footnote 2 Under this MIMIC model, an item j is DIF-free if ${\gamma }_{\mathrm{jk}}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{jk} = 0$$\end{document} for all k. The location indeterminacy under this model leads to the following ML1 condition for identifying the true model parameters ${\Xi }^{\ast }=\{{\beta }_k^{\ast },a_j^{\ast },d_j^{\ast },{\gamma }_{\mathrm{jk}}^{\ast },k=1,\dots ,K,j=1,\dots ,J,\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi ^* = \{\beta ^*_k, a_j^*, d_j^*, \gamma _{jk}^*, k = 1, \ldots , K, j = 1, \ldots , J,\}$$\end{document} :

$\begin{array}{c}\sum _{j=1}^J|{\gamma }_{\mathrm{jk}}^{\ast }|<\sum _{j=1}^J|{\gamma }_{\mathrm{jk}}^{\ast }-a_j^{\ast }{c}_k|,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{j=1}^{J} \vert \gamma _{jk}^* \vert < \sum _{j=1}^{J} \vert \gamma _{jk}^* -a_j^* c_k\vert , \end{aligned}$$\end{document}

for $c_k\ne 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_k\ne 0$$\end{document} , $k=1,\dots ,K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k = 1, \ldots , K$$\end{document} .

We note that this ML1 condition is similar to that under the multi-group setting. This is because the multi-group setting can be written in a very similar form as the current MIMIC model (by representing the groups using a covariate vector with dummy variables), except that the structural model under the multi-group setting allows heteroscedasticity. We also note that the current model assumes that a DIF effect is a linear combination of the covariates, which may seem inflexible, especially when comparing with the tree-based methods (Strobl et al., Reference Strobl, Kopf and Zeileis2015; Tutz and Berger, Reference Tutz and Berger2016; Bollmann et al., Reference Bollmann, Berger and Tutz2018). However, we note that one can always move beyond the linearity by including transformations of the raw covariates (e.g., using spline basis) into the covariate vector and increasing the dimension of the DIF parameter vector ${\gamma }_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\gamma }_j$$\end{document} simultaneously.

Ordinal response data. Finally, we note that the proposed method can be extended to IRT models for other types of response data. To elaborate, we consider the generalized partial credit model (GPCM) (Muraki, Reference Muraki1992) for ordinal response data as an example. For simplicity, we focus on the two-group setting (i.e., $x_i\in \{0,1\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_i \in \{0,1\}$$\end{document} ) and uniform DIF. Let $\{0,1,\dots ,m_j\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{0, 1, \ldots , m_j\}$$\end{document} be the ordered categories of item j. Then, the measurement model becomes:

$\begin{array}{c}\frac{P(Y_{\mathrm{ij}}=k|{\theta }_i=\theta ,x_i)}{P(Y_{\mathrm{ij}}=k-1|{\theta }_i=\theta ,x_i)}=exp(a_j\theta +d_{\mathrm{jk}}+{\gamma }_{\mathrm{jk}}{x}_i),k=1,\dots ,m_j,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{P(Y_{ij} = k\vert \theta _i = \theta , x_i)}{P(Y_{ij} = k-1\vert \theta _i = \theta , x_i)} = {\exp (a_j\theta + d_{jk} + \gamma _{jk}x_i)}, k = 1, \ldots ,m_j, \end{aligned}$$\end{document}

where the DIF parameters ${\gamma }_{\mathrm{jk}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{jk}$$\end{document} depend on both the item and the category. We keep the structural model the same as in Sect. 2.2. Under this model, an item j is DIF-free if ${\gamma }_{\mathrm{jk}}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{jk} = 0$$\end{document} for all k. The location indeterminacy under this model leads to the following ML1 condition for identifying the true model parameters ${\Xi }^{\ast }=\{{\beta }^{\ast },{\left({\sigma }^{\ast }\right)}^2,a_j^{\ast },d_j^{\ast },{\gamma }_{\mathrm{jk}}^{\ast },k=1,\dots ,m_j,j=1,\dots ,J\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi ^* = \{\beta ^*, (\sigma ^*)^2, a_j^*, d_j^*, \gamma _{jk}^*, k = 1, \ldots , m_j, j = 1, \ldots , J\}$$\end{document} :

$\begin{array}{c}\sum _{j=1}^J\sum _{k=1}^{m_j}|{\gamma }_{\mathrm{jk}}^{\ast }|<\sum _{j=1}^J\sum _{k=1}^{m_j}|{\gamma }_{\mathrm{jk}}^{\ast }-a_j^{\ast }c|,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{j=1}^{J}\sum _{k=1}^{m_j} \vert \gamma _{jk}^* \vert < \sum _{j=1}^{J}\sum _{k=1}^{m_j} \vert \gamma _{jk}^* -a_j^* c\vert , \end{aligned}$$\end{document}

for all $c\ne 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\ne 0$$\end{document} .