Computational Aspects of L₀ Linking in the Rasch Model

Abstract

The $L_0$ linking approach replaces the $L_2$ loss function in mean–mean linking under the Rasch model with the $L_0$ loss function. Using the $L_0$ loss function offers the advantage of potential robustness against fixed differential item functioning effects. However, its nondifferentiability necessitates differentiable approximations to ensure feasible and computationally stable estimation. This article examines alternative specifications of two approximations, each controlled by a tuning parameter $\epsilon$ that determines the approximation error. Results demonstrate that the optimal $\epsilon$ value minimizing the RMSE of the linking parameter estimate depends on the magnitude of DIF effects, the number of items, and the sample size. A data-driven selection of $\epsilon$ outperformed a fixed $\epsilon$ across all conditions in both a numerical illustration and a simulation study.

1. Introduction

Item response theory (IRT) models [1,2,3] are statistical models for multivariate discrete random variables. This article focuses on dichotomous (i.e., binary) random variables and the comparison of two groups through a linking method.

Let $\mathit{X}=(X_1,\dots ,X_I)$ denote a vector of $I\in \mathbb{N}$ random variables $X_i\in \{0,1\}$, commonly referred to as items or (scored) item responses. A unidimensional IRT model [4] represents the probability distribution $\mathsf{P}(\mathit{X}=\mathit{x})$ for $\mathit{x}=(x_1,\dots ,x_I)\in {\{0,1\}}^I$,

$${\displaystyle \mathsf{P}(\mathit{X}=\mathit{x};\mathit{\delta },\mathit{\gamma })=\int \prod _{i=1}^I\left[{P_i(\theta ;{\mathit{\gamma }}_i)}^{x_i}{\left(1-P_i(\theta ;{\mathit{\gamma }}_i)\right)}^{1-x_i}\right]\varphi (\theta ;\mu ,\sigma )\mathrm{d}\theta ,}$$

(1)

where $\varphi$ denotes the density of the normal distribution with mean $\mu$ and standard deviation (SD) $\sigma$. The distribution parameters of the latent variable $\theta$, often labeled as a trait or ability variable, are collected in the vector $\mathit{\delta }=(\mu ,\sigma )$. The vector $\mathit{\gamma }=({\mathit{\gamma }}_1,\dots ,{\mathit{\gamma }}_I)$ contains the item parameters associated with the item response functions (IRFs) $P_i(\theta ;{\mathit{\gamma }}_i)=\mathsf{P}(X_i=1|\theta )$ for $i=1,\dots ,I$. The IRF of the Rasch model [5,6,7] is defined as

$$P_i(\theta ;{\mathit{\gamma }}_i)=\mathsf{\Psi }\left(\theta -b_i\right),$$

(2)

where $b_i$ denotes the item difficulty $b_i$, and $\mathsf{\Psi }\left(x\right)={(1+exp(-x))}^{-1}$ is the logistic distribution function. In this formulation, the item parameter vector is given as ${\mathit{\gamma }}_i=\left(b_i\right)$.

For a sample of N individuals with independently and identically distributed observations ${\mathit{x}}_1,\dots ,{\mathit{x}}_N$ from the distribution of the random variable $\mathit{X}$, the unknown parameters of the IRT model in (1) can be consistently estimated using maximum likelihood estimation (ML; [8,9,10]).

IRT models are commonly used to compare the performance of two groups on a test by examining differences in the latent variable $\theta$ within the framework of the IRT model in (1). This article focuses on a generalization of the mean–mean linking method [11] based on the Rasch model. The purpose of a linking method is to estimate the difference between the distributions of $\theta$ in the two groups. This difference serves as a summary measure of group performance on the multivariate vector of dichotomous items $\mathit{X}$.

The linking approach consists of two steps. First, the Rasch model is estimated separately for each group, allowing for potential differential item functioning (DIF), where items may function differently across groups [12,13,14]. Second, differences in item parameters are used to estimate group differences in the $\theta$ variable through a linking method [11,15,16].

This article investigates the performance of a generalization of mean–mean (MM; [11]) linking in the presence of fixed uniform DIF [14] in item difficulties. The traditional MM method relies on mean differences in item difficulties, which corresponds to using an $L_2$ loss function. MM linking is a widely used linking or equating method, as discussed in popular textbooks on Rasch modeling [6,7,17,18]. In this study, we investigate a generalization of the MM linking method in the Rasch model using the $L_0$ loss function [19]. Using this robust loss function can essentially remove items with DIF effects from the group comparison directly in the linking method (see also [20,21,22,23,24,25,26,27]). DIF effects can also be treated through the application of robust procedures in MM linking [19,28,29,30,31].

The $L_0$ loss function, being nondifferentiable, requires differentiable approximations to ensure feasible and computationally stable estimation. This article explores alternative specifications of these approximations, which depend on a tuning parameter $\epsilon$ that controls the approximation error. Previous research relied on fixed $\epsilon$ values based on prior knowledge or simulation studies. Here, the choice of a fixed $\epsilon$ parameter is examined and compared with a data-driven approach for determining $\epsilon$.

The rest of this article is structured as follows. Section 2 introduces $L_0$ linking in the Rasch model. Section 3 presents a numerical illustration of alternative specifications of $L_0$ linking in a simplified setting. In Section 4, findings from a simulation study are reported. An empirical example is provided in Section 5. This article closes with a discussion in Section 6 and a conclusion in Section 7.

2. ${\mathit{L}}_{\mathbf{0}}$ Linking in the Rasch Model

This section discusses the $L_0$ linking approach in the Rasch model. Section 2.1 covers item parameter identification and the ordinary MM linking method. Section 2.2 introduces the $L_0$ loss function and presents two differentiable approximations. Section 2.3 applies these approximations to define the $L_0$ linking approach in the Rasch model. Finally, Section 2.4 examines its statistical properties.

2.1. Identified Item Parameters and Mean–Mean Linking

To introduce the $L_0$ linking method as a replacement for ordinary mean–mean linking in the Rasch model, we first outline the identification of item parameters in a group-specific estimation of the Rasch model when no DIF effects are present. In this case, the identified item parameters in the first group are ${\widehat{b}}_{i1}=b_i$, where $b_i$ represents the invariant item parameters across both groups. For identification, the mean of $\theta$ in the first group is fixed at 0, while the standard deviation $\sigma$ remains identifiable within this group.

In the second group, the mean $\mu$ and the SD $\sigma$ can be identified when invariant item difficulties are assumed. Thus, $\mu$ and $\sigma$ represent group differences in $\theta$. The identified item parameters in this group, estimated separately under the assumption of a $\theta$ mean of 0 and an estimated SD $\sigma$, are given by

$${\widehat{b}}_{i2}=b_i-\mu .$$

(3)

Linking methods aim to recover the group parameter $\mu$ using the group-specific item parameters obtained from separate Rasch model estimations.

The MM linking method estimates the group mean $\widehat{\mu }$ for the second group as

$$\widehat{\mu }=-{\displaystyle \frac{1}{I}}\sum _{i=1}^I\left({\widehat{b}}_{i2}-{\widehat{b}}_{i1}\right).$$

(4)

The linking parameter $\widehat{\mu }$ in MM linking is obtained as the minimizer of the squared loss function (i.e., the $L_2$ loss function; [19])

$$\widehat{\mu }=\underset{\mu }{arg\; min}H\left(\mu \right),\mathrm{with}H\left(\mu \right)={\displaystyle \frac{1}{I}}\sum _{i=1}^I\rho \left({\widehat{b}}_{i2}-{\widehat{b}}_{i1}+\mu \right),\mathrm{where}\rho \left(x\right)={\displaystyle \frac{1}{2}}{x}^2.$$

(5)

Thus, the estimate $\widehat{\mu }$ satisfies the estimating equation

$${\displaystyle \frac{1}{I}}\sum _{i=1}^I{\rho }^{\prime }\left({\widehat{b}}_{i2}-{\widehat{b}}_{i1}+\widehat{\mu }\right)=0,\mathrm{where}{\rho }^{\prime }\left(x\right)=x$$

(6)

and ${\rho }^{\prime }$ denotes the derivative of $\rho$ with respect to x. Clearly, (6) is equivalent to (4).

This paper investigates the computational aspects of MM linking in (5) when the squared $L_2$ loss function is replaced with the $L_0$ loss function, which aims to robustify the linking method in the presence of fixed DIF effects.

2.2. $L_0$ Loss Function and Differentiable Approximations

In this section, we introduce the $L_0$ loss function [32,33], which is defined as

$$\rho \left(x\right)=\mathbf{1}(x\ne 0).$$

(7)

This loss function equals 0 for $x=0$ and 1 for $x\ne 0$. The $L_0$ loss function has been widely applied in statistical modeling, particularly for obtaining sparse solutions (see, e.g., [34,35,36,37,38,39]).

The exact $L_0$ loss function in (7) has the disadvantage of being nondifferentiable at $x=0$, making it difficult to use in numerical optimization. To address this, differentiable approximations of the $L_0$ loss function have been proposed.

The ratio loss function, as an approximation of the $L_0$ loss function, is defined as (see [40,41,42,43])

$${\rho }_{\epsilon }\left(x\right)={\displaystyle \frac{x^2}{x^2+\epsilon }},$$

(8)

where $\epsilon >0$ is a tuning parameter. We have found that $\epsilon =0.01$ performs satisfactorily in applications [19,44].

The linking parameter estimate based on the $L_2$ loss function can be obtained using the $L_0$ approximation (8) with a sufficiently large $\epsilon$ value, such as $\epsilon =1000$. In this case, ${\rho }_{\epsilon }\left(x\right)\approx x^2/\epsilon$, and the resulting estimate closely matches the $L_2$ estimate, which is the ordinary MM parameter estimate.

An alternative differentiable approximation of the $L_0$ loss function is based on the Gaussian density function, denoted as the Gaussian function hereafter, and is given by (see [45,46,47])

$${\rho }_{\epsilon }\left(x\right)=1-exp\left(-{\displaystyle \frac{x^2}{\epsilon }}\right),$$

(9)

where $\epsilon >0$ is again a tuning parameter.

Figure 1 displays the ratio and the Gaussian loss functions for $\epsilon =0.001$ and $\epsilon =0.01$. It can be seen that the two loss functions exhibit similar behavior around $x=0$, which can be explained by the fact that their first two derivatives coincide, with ${\rho }_{\epsilon }^{\prime }\left(0\right)=0$ and ${\rho }_{\epsilon }^{″}\left(0\right)=2/\epsilon$. For larger $\left|x\right|$ values, the Gaussian loss function grows faster toward the upper asymptote of 1 compared to the ratio loss function.

The ordinary MM parameter estimate can be obtained using the Gaussian $L_0$ loss function approximation, similar to the ratio function, by selecting a sufficiently large $\epsilon$ value.

2.3. $L_0$ Linking as a Robust Mean–Mean Linking in the Rasch Model

MM linking can be modified by using the $L_0$ loss function instead of the $L_2$ loss function. In the former case, the loss is defined by counting the number of non-vanishing deviations ${\widehat{b}}_{i2}$ from ${\widehat{b}}_{i1}+\mu$, which corresponds to the number of non-vanishing DIF effects. To obtain a stable linking parameter estimate in the presence of sampling errors, the $L_0$ loss function is replaced by the differentiable approximation ${\rho }_{\epsilon }$, and the estimated mean $\widehat{\mu }$ is obtained as

$$\widehat{\mu }=\underset{\mu }{arg\; min}H\left(\mu \right)\mathrm{with}H\left(\mu \right)={\displaystyle \frac{1}{I}}\sum _{i=1}^I{\rho }_{\epsilon }\left({\widehat{b}}_{i2}-{\widehat{b}}_{i1}+\mu \right).$$

(10)

This approach can also be referred to as robust MM linking or $L_0$ linking in the Rasch model, where “robust” refers to the linking parameter estimate being resilient to the presence of fixed DIF effects.

Equivalently to (10), the linking parameter estimate $\widehat{\mu }$ solves the estimating equation

$${\displaystyle \frac{\partial H}{\partial \mu }}={\displaystyle \frac{1}{I}}\sum _{i=1}^I{\rho }_{\epsilon }^{\prime }\left({\widehat{b}}_{i2}-{\widehat{b}}_{i1}+\widehat{\mu }\right)=0.$$

(11)

Equation (10) represents a one-dimensional optimization problem that can be numerically solved using standard optimizers implemented in statistical software.

2.4. Statistical Properties of the Estimated Linking Parameter in $L_0$ Linking

We now investigate the statistical properties of the linking parameter estimate $\widehat{\mu }$ obtained from (10). The difference in estimated item difficulties required in the $L_0$ linking approach can be written as

$${\widehat{b}}_{i2}-{\widehat{b}}_{i1}=-\mu +{\kappa }_i+u_i$$

(12)

with fixed DIF effects ${\kappa }_i$ and sampling errors $u_i$, where $\mathsf{E}\left(u_i\right)=0$ and $\mathsf{Var}\left(u_i\right)=V_i$. Asymptotic unbiasedness follows from the general properties of ML estimation.

The $L_0$ linking approach is analyzed by starting from the estimating equation

$${\displaystyle \frac{1}{I}}\sum _{i=1}^I{\rho }_{\epsilon }^{\prime }\left({\widehat{b}}_{i2}-{\widehat{b}}_{i1}+\widehat{\mu }\right)={\displaystyle \frac{1}{I}}\sum _{i=1}^I{\rho }_{\epsilon }^{\prime }\left(\widehat{\mu }-\mu +{\kappa }_i+u_i\right)=0.$$

(13)

Now, we apply a Taylor expansion with respect to $\mu$, as in standard M-estimation theory ([48]; see also [25,49]), and obtain

$$0={\displaystyle \frac{1}{I}}\sum _{i=1}^I{\rho }_{\epsilon }^{\prime }\left(\widehat{\mu }-\mu +{\kappa }_i+u_i\right)\simeq {\displaystyle \frac{1}{I}}\sum _{i=1}^I{\rho }_{\epsilon }^{\prime }\left({\kappa }_i+u_i\right)+{\displaystyle \frac{1}{I}}\sum _{i=1}^I{\rho }_{\epsilon }^{″}\left({\kappa }_i+u_i\right)\left(\widehat{\mu }-\mu \right)=0.$$

(14)

The bias and variance of $\widehat{\mu }$ can be derived from (14) and are given by

$$\mathsf{Bias}\left(\widehat{\mu }\right)=-{\displaystyle \frac{{\displaystyle \sum _{i=1}^I\mathsf{E}\left[{\rho }_{\epsilon }^{\prime }\left({\kappa }_i+u_i\right)\right]}}{{\displaystyle \sum _{i=1}^I\mathsf{E}\left[{\rho }_{\epsilon }^{″}\left({\kappa }_i+u_i\right)\right]}}}\mathrm{and}$$

(15)

$$\mathsf{Var}\left(\widehat{\mu }\right)={\displaystyle \frac{{\displaystyle \sum _{i=1}^I\mathsf{Var}\left[{\rho }_{\epsilon }^{\prime }\left({\kappa }_i+u_i\right)\right]}}{{\displaystyle {\left\{\sum _{i=1}^I\mathsf{E}\left[{\rho }_{\epsilon }^{″}\left({\kappa }_i+u_i\right)\right]\right\}}^2}}},$$

(16)

where approximate independence of item parameters across items is assumed in (16). As a summary precision measure of the linking parameter estimate, the mean squared error (MSE) can be determined as

$$\mathsf{MSE}\left(\widehat{\mu }\right)=\mathsf{E}{\left(\widehat{\mu }-\mu \right)}^2=\mathsf{Bias}{\left(\widehat{\mu }\right)}^2+\mathsf{Var}\left(\widehat{\mu }\right)={\displaystyle \frac{{\displaystyle \sum _{i=1}^I\left\{{\left(\mathsf{E}\left[{\rho }_{\epsilon }^{\prime }\left({\kappa }_i+u_i\right)\right]\right)}^2+\mathsf{Var}\left[{\rho }_{\epsilon }^{\prime }\left({\kappa }_i+u_i\right)\right]\right\}}}{{\displaystyle {\left\{\sum _{i=1}^I\mathsf{E}\left[{\rho }_{\epsilon }^{″}\left({\kappa }_i+u_i\right)\right]\right\}}^2}}}.$$

(17)

Although it might not be immediately evident from Equations (15)–(17), the bias, variance, and MSE depend on the choice of the tuning parameter $\epsilon$ in the $L_0$ approximation ${\rho }_{\epsilon }$.

In the next two sections, we compare the two differentiable approximations—the ratio and Gaussian loss functions—regarding their statistical properties in a numerical illustration and a simulation study. In particular, we focus on the choice of the tuning parameter $\epsilon$.

3. Numerical Illustration

3.1. Method

In this Numerical Illustration, the properties of the estimated linking parameter $\widehat{\mu }$ were studied in a simplified setting in which no item responses were simulated. It was assumed that the difference ${\widehat{b}}_{i2}-{\widehat{b}}_{i1}$ had a variance of $V/N$, where N denotes the sample size. Hence, this illustration assumed equal sampling variances of item parameter differences, which might be violated in practice. However, this assumption eases the statistical treatment and aims at yielding clearly interpretable results. Furthermore, we assumed that we had I items, and a proportion $\pi$ of the items had an unbalanced DIF effect $\kappa$, while a proportion $1-\pi$ of the items did not show DIF.

From Section 2.4, we know that the linking parameter satisfies the estimating equation

$${\displaystyle \frac{1}{I}}\sum _{i=1}^I{\rho }_{\epsilon }^{\prime }\left({\widehat{b}}_{i2}-{\widehat{b}}_{i1}+\widehat{\mu }\right)={\displaystyle \frac{1}{I}}\sum _{i=1}^I{\rho }_{\epsilon }^{\prime }\left(\widehat{\mu }-\mu +{\kappa }_i+u_i\right)=0,$$

(18)

where $\mathsf{Var}\left(u_i\right)=V/N$. Because all item parameter differences have equal sampling variances, (18) can be simplified to

$$\sum _{i=1}^{I\pi }{\rho }_{\epsilon }^{\prime }\left(\widehat{\mu }-\mu +\kappa +u_i\right)+\sum _{I\pi +1}^I{\rho }_{\epsilon }^{\prime }\left(\widehat{\mu }-\mu +u_i\right)=0,$$

(19)

assuming that $I\pi$ is an integer. By using the bias Formula (15), we obtain

$$\mathsf{Bias}\left(\widehat{\mu }\right)=-{\displaystyle \frac{\pi \mathsf{E}\left[{\rho }_{\epsilon }^{\prime }\left(\kappa +u_i\right)\right]}{\pi \mathsf{E}\left[{\rho }_{\epsilon }^{″}\left(\kappa +u_i\right)\right]+(1-\pi )\mathsf{E}\left[{\rho }_{\epsilon }^{″}\left(u_i\right)\right]}},$$

(20)

where we use $\mathsf{E}\left[{\rho }_{\epsilon }^{\prime }\left(u_i\right)\right]=0$ because $u_i$ has a symmetric distribution. The variance can be computed as (see (16))

$$\mathsf{Var}\left(\widehat{\mu }\right)={\displaystyle \frac{\pi \mathsf{Var}\left[{\rho }_{\epsilon }^{\prime }\left(\kappa +u_i\right)\right]+(1-\pi )\mathsf{Var}\left[{\rho }_{\epsilon }^{\prime }\left(u_i\right)\right]}{{\displaystyle {\left\{\pi \mathsf{E}\left[{\rho }_{\epsilon }^{″}\left(\kappa +u_i\right)\right]+(1-\pi )\mathsf{E}\left[{\rho }_{\epsilon }^{″}\left(u_i\right)\right]\right\}}^2}}},$$

(21)

The root mean square error (RMSE) as the square root of the MSE can be obtained as

$$\mathsf{RMSE}\left(\widehat{\mu }\right)=\sqrt{\mathsf{MSE}\left(\widehat{\mu }\right)}=\sqrt{\mathsf{Bias}{\left(\widehat{\mu }\right)}^2+\mathsf{Var}\left(\widehat{\mu }\right)}.$$

(22)

The expected values and variances in (20) and (21) can be numerically evaluated.

The bias, SD, and RMSE are evaluated for $\widehat{\mu }$ for sample sizes N of 250, 500, and 1000, as well as the number of items I as 10, 20, and 40. We simulated fixed DIF effects $\kappa =0.4$ and $\kappa =0.8$, representing small and large DIF. The tuning parameter $\epsilon$ in the ratio loss function and Gaussian loss function ${\rho }_{\epsilon }$ was chosen as 1, 0.95, 0.9, 0.85, 0.8, 0.75, 0.7, 0.65, 0.6, 0.55, 0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.1, 0.095, 0.09, 0.085, 0.08, 0.075, 0.07, 0.065, 0.06, 0.055, 0.05, 0.045, 0.04, 0.035, 0.03, 0.025, 0.02, 0.015, 0.01, 0.009, 0.008, 0.007, 0.006, 0.005, 0.004, 0.003, 0.002, and 0.001. In total, 46 $\epsilon$ values were evaluated in this Numerical Illustration. We only report results for the ratio loss function because the findings for the Gaussian loss function were very similar. The statistical software R (Version 4.4.1; [50]) was used for the analysis in this study. Symbolic derivatives of the two loss functions were computed with the R package Deriv (Version: 4.1.6; [51]). Replication material for this illustration can be retrieved from https://osf.io/un6q4 (accessed on 20 February 2025).

3.2. Results

Figure 2 displays the absolute bias, the SD, and the RMSE of the estimated mean $\widehat{\mu }$ for $I=20$ items as a function of the DIF effect $\kappa$ and the sample size N. The tuning parameter $\epsilon$ is displayed on a logarithmic scale on the x-axis. The bias increased with increasing values of $\epsilon$ in the loss function ${\rho }_{\epsilon }$. In contrast, the SD decreased with increasing values of $\epsilon$. The RMSE reflects the bias–variance trade-off of the linking parameter estimate. There was an optimal $\epsilon$ parameter that minimizes the RMSE. This optimal $\epsilon$ parameter decreased with increasing sample size N. Moreover, the optimal $\epsilon$ parameter was larger for $\kappa =0.8$ than for $\kappa =0.4$.

Table 1. Numerical Illustration: optimal $\epsilon$ value yielding the minimum root mean square error (RMSE) of the estimated mean $\widehat{\mu }$ using the ratio loss function as a function of the DIF effect size $\kappa$, number of items I, and sample size N.

$\mathit{I}$	$\mathit{\kappa }=0.4$, $\mathit{N}=$				$\mathit{\kappa }=0.8$, $\mathit{N}=$
	250	500	1000		250	500	1000
10	0.900	0.065	0.030		0.150	0.100	0.090
20	0.200	0.050	0.030		0.100	0.100	0.090
40	0.100	0.035	0.025		0.100	0.095	0.085

presents the optimal $\epsilon$ values that minimize the RMSE of the estimated mean $\widehat{\mu }$ under a ratio loss function. These values are reported for different DIF effect sizes $\kappa$, numbers of items I, and sample sizes N. As the sample size N increased, the optimal $\epsilon$ generally decreased. This pattern was more pronounced for small item numbers (e.g., $I=10$), where $\epsilon$ dropped sharply with increasing N. Larger item numbers I were associated with smaller optimal $\epsilon$ values, suggesting that when more items were available, the best trade-off under the ratio loss function occurred at a lower $\epsilon$ value. This effect was particularly evident for $\epsilon =0.4$, where $\epsilon$ decreased substantially as I increased from 10 to 40. When the DIF effect size $\kappa$ increased from 0.4 to 0.8, the optimal $\epsilon$ values tended to be slightly higher for the same I and N. This suggests that stronger DIF effects required a higher $\epsilon$ to achieve the lowest RMSE, likely due to greater bias introduced by DIF at the smaller effect size $\kappa =0.4$.

Overall, the results highlight a trade-off between sample size, the number of items, and DIF effect size in determining the optimal correction for minimizing the RMSE.

4. Simulation Study

4.1. Method

The Rasch model was used as the IRF in the data-generating model for two groups. For identification purposes, the mean of the latent variable $\theta$ in the first group was fixed at 0, with an SD $\sigma$ of 1. In the second group, the mean $\mu$ was set to 0.3 to represent the difference in $\theta$ between the groups, while the SD $\sigma$ was set to 1.2.

The simulation study was conducted for $I=10$, 20, and 40 items. In the $I=10$ condition, base item difficulty values $b_i$ were set to −0.314, 0.411, −1.097, −0.542, −1.854, −0.403, −0.895, 0.715, 0.841, and 0.139, resulting in a mean of $M=-0.300$ and an SD of 0.850. These item difficulties were used in the data generation for the first group. In the second group, the same values were applied, except for the first three items, where item difficulties were adjusted to $b_{i2}=b_i+\kappa$. This introduced fixed DIF in 30% of the items. The DIF effect size $\kappa$ was set to 0.4 and 0.8, representing small and large DIF conditions, respectively. For item sets larger than 10, the same 10-item parameter set was repeated accordingly. The item parameters are also available at https://osf.io/un6q4 (accessed on 20 February 2025). Note that item parameters remained fixed across all replications within each condition.

Sample sizes of $N=125$, 250, 500, and 1000 per group were selected to reflect small- to large-scale applications of the Rasch model.

A total of 5000 replications were conducted for each of the 24 conditions, corresponding to the combinations of 4 sample sizes (N) × 3 item numbers (I) × 2 DIF effect sizes ($\kappa$). For each simulated dataset, $L_0$ linking was performed using various specifications. Both the ratio and Gaussian loss functions ${\rho }_{\epsilon }$ (see Section 2.2) were applied. The ${\rho }_{\epsilon }$ loss functions were evaluated for a sequence of 11 $\epsilon$ values: 1, 0.75, 0.50, 0.25, 0.10, 0.075, 0.05, 0.025, 0.01, 0.005, and 0.001.

Additionally, as demonstrated in Section 3, the optimal $\epsilon$ value corresponding to the minimal RMSE is dependent on the sample size N. To account for this, a data-driven approach for selecting the $\epsilon$ value was explored. Let $\overline{V}$ denote the average variance of the estimated item difficulty differences ${\widehat{b}}_{i2}-{\widehat{b}}_{i1}$. For a fixed $y\in (0,1)$, ${\epsilon }_y$ is chosen such that

$${\rho }_{{\epsilon }_y}\left(\sqrt{\overline{V}}\right)=y.$$

(23)

In the simulation, the following y values were chosen: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9.

The bias, standard deviation (SD), and root mean square error (RMSE) of the estimated mean, $\widehat{\mu }$, were calculated for all specifications of the $L_0$ linking approach. The relative RMSE of $\widehat{\mu }$ was defined as the ratio of the RMSE for a given specification to the RMSE of a reference method, multiplied by 100. Across all conditions, the reference method was the ratio loss function with a fixed $\epsilon$ value of 0.01.

All analyses for this simulation study were conducted using the statistical software R (Version 4.4.1; [50]). The Rasch model was fitted with the sirt::xxirt() function from the R package sirt (Version 4.2-106; [52]). The optimization of $L_0$ linking was performed with the stats::nlminb() function. Replication materials for this simulation study can be assessed at https://osf.io/un6q4 (accessed on 20 February 2025).

4.2. Results

Table 2. Simulation Study: Absolute bias of estimated mean $\widehat{\mu }$ as a function of the DIF effect size $\kappa$, number of items I, and sample size N.

$\mathit{\kappa }$	$\mathit{I}$	$\mathit{N}$	Ratio Function, $\mathit{\epsilon }=$							Gaussian Function, $\mathit{\epsilon }=$
			0.25	0.10	0.05	0.01	${\mathit{\epsilon }}_{\mathbf{0.2}}$	${\mathit{\epsilon }}_{\mathbf{0.4}}$		0.25	0.10	0.05	0.01	${\mathit{\epsilon }}_{\mathbf{0.2}}$	${\mathit{\epsilon }}_{\mathbf{0.4}}$
0.4	10	125	0.107	0.103	0.102	0.102	0.110	0.105		0.107	0.101	0.100	0.101	0.112	0.106
		250	0.098	0.087	0.082	0.079	0.096	0.085		0.101	0.086	0.080	0.078	0.101	0.087
		500	0.088	0.066	0.052	0.041	0.068	0.049		0.097	0.067	0.046	0.033	0.075	0.048
		1000	0.082	0.050	0.031	0.015	0.034	0.019		0.096	0.057	0.026	0.010	0.035	0.013
	20	125	0.104	0.100	0.099	0.099	0.107	0.102		0.104	0.099	0.096	0.096	0.109	0.103
		250	0.098	0.086	0.080	0.075	0.097	0.084		0.102	0.084	0.075	0.069	0.102	0.086
		500	0.089	0.066	0.051	0.036	0.069	0.048		0.097	0.068	0.044	0.028	0.076	0.046
		1000	0.081	0.049	0.028	0.009	0.031	0.014		0.095	0.056	0.023	0.003	0.032	0.008
	40	125	0.105	0.101	0.098	0.096	0.108	0.103		0.106	0.099	0.096	0.096	0.110	0.105
		250	0.098	0.086	0.077	0.070	0.097	0.083		0.101	0.084	0.071	0.063	0.101	0.086
		500	0.099	0.076	0.059	0.039	0.079	0.055		0.107	0.078	0.051	0.029	0.087	0.054
		1000	0.087	0.054	0.033	0.012	0.036	0.018		0.100	0.062	0.027	0.006	0.036	0.012
0.8	10	125	0.114	0.093	0.087	0.084	0.138	0.102		0.105	0.078	0.074	0.074	0.152	0.099
		250	0.069	0.037	0.027	0.023	0.064	0.033		0.066	0.021	0.015	0.015	0.067	0.023
		500	0.048	0.017	0.008	0.003	0.019	0.006		0.047	0.004	0.000	0.001	0.009	0.000
		1000	0.039	0.011	0.004	0.001	0.005	0.001		0.040	0.002	0.000	0.000	0.000	0.000
	20	125	0.112	0.082	0.073	0.069	0.140	0.096		0.103	0.063	0.055	0.054	0.155	0.095
		250	0.069	0.033	0.020	0.014	0.063	0.029		0.065	0.015	0.007	0.005	0.065	0.017
		500	0.047	0.015	0.006	0.000	0.017	0.004		0.046	0.003	0.002	0.003	0.007	0.001
		1000	0.038	0.011	0.004	0.000	0.005	0.001		0.039	0.002	0.000	0.000	0.000	0.000
	40	125	0.109	0.075	0.061	0.054	0.140	0.092		0.099	0.052	0.041	0.036	0.156	0.090
		250	0.071	0.033	0.020	0.011	0.065	0.029		0.067	0.015	0.006	0.005	0.067	0.018
		500	0.051	0.020	0.011	0.005	0.022	0.009		0.050	0.007	0.003	0.003	0.011	0.003
		1000	0.039	0.012	0.005	0.001	0.006	0.002		0.040	0.002	0.001	0.001	0.001	0.001

Note. ${\epsilon }_{0.2},{\epsilon }_{0.4}$ = computed $\epsilon$ values in loss function ${\rho }_{\epsilon }$ such that ${\rho }_{{\epsilon }_y}\left(\sqrt{\overline{V}}\right)=y$ for $y=0.2$ and $y=0.4$, where $\sqrt{\overline{V}}$ is the square root of the average variance of the item difficulty difference ${\widehat{b}}_{i2}-{\widehat{b}}_{i1}$; absolute biases larger than 0.03 are printed in bold.

presents the absolute bias as a function of the DIF effect size $\kappa$, the number of items I, and the sample size N. Overall, the absolute bias generally decreased as $\kappa$ increased from 0.4 to 0.8. The bias in the estimated $\widehat{\mu }$ was highest for the smallest number of items ($I=10$) and slightly decreased as the number of items increased. This reduction in bias was more pronounced for smaller sample sizes. As the sample size N increased, the bias systematically decreased, with the lowest bias observed for the largest sample size ($N=1000$). For the data-driven $\epsilon$ choices ${\epsilon }_{0.2}$ and ${\epsilon }_{0.4}$, the bias also decreased as the sample size increased.

The biases for the ratio and Gaussian loss functions were relatively similar, although the bias was slightly smaller under conditions with a larger DIF effect size ($\kappa =0.8$) compared to a smaller DIF effect size ($\kappa =0.4$). Additionally, the bias decreased as $\epsilon$ values in the ${\rho }_{\epsilon }$ function were reduced. Specifically, the bias was slightly higher for $\epsilon =0.2$ compared to $\epsilon =0.4$, particularly at smaller N. These findings suggest that while the choice of the loss function influenced estimation accuracy, its effect diminished as I and N increased.

Table 3. Simulation Study: Relative root mean square error (RMSE) of estimated mean $\widehat{\mu }$ as a function of the DIF effect size $\kappa$, number of items I, and sample size N.

$\mathit{\kappa }$	$\mathit{I}$	$\mathit{N}$	Ratio Function, $\mathit{\epsilon }=$							Gaussian Function, $\mathit{\epsilon }=$
			0.25	0.10	0.05	0.01	${\mathit{\epsilon }}_{\mathbf{0.2}}$	${\mathit{\epsilon }}_{\mathbf{0.4}}$		0.25	0.10	0.05	0.01	${\mathit{\epsilon }}_{\mathbf{0.2}}$	${\mathit{\epsilon }}_{\mathbf{0.4}}$
0.4	10	125	83.5	90.5	95.0	100	80.2	86.3		83.0	92.9	99.0	105.2	79.1	84.0
		250	83.4	88.1	93.3	100	83.7	89.4		83.2	89.1	96.6	105.8	83.2	88.1
		500	92.6	89.0	91.0	100	89.1	92.0		95.6	89.8	92.3	103.8	90.5	91.6
		1000	123.1	102.5	94.8	100	95.5	96.2		134.4	108.1	94.6	102.4	97.5	94.0
	20	125	83.2	88.6	93.8	100	81.8	85.1		83.0	90.9	98.1	105.6	81.4	83.6
		250	85.9	86.9	91.3	100	85.7	87.8		86.4	87.0	93.7	104.5	86.4	86.5
		500	101.9	93.4	91.7	100	94.0	92.1		106.3	94.1	91.0	103.2	96.8	90.7
		1000	141.1	111.1	97.7	100	99.2	95.7		155.9	118.4	95.8	100.4	100.7	91.7
	40	125	85.6	88.9	93.5	100	85.2	86.5		85.4	89.8	97.1	105.9	85.2	85.6
		250	91.5	89.7	91.6	100	91.1	89.9		92.4	89.3	92.5	104.4	92.4	89.3
		500	120.2	105.7	97.9	100	107.2	96.8		126.2	107.1	94.9	101.5	112.2	95.7
		1000	169.0	127.9	106.9	100	109.1	98.3		187.6	137.6	103.0	99.7	110.8	94.2
0.8	10	125	87.2	92.6	96.0	100	85.6	89.3		87.9	95.9	100.5	106.1	86.8	88.7
		250	90.3	89.4	92.8	100	89.6	90.2		91.4	89.1	94.6	106.2	91.4	89.0
		500	94.8	87.2	89.8	100	87.1	90.7		97.7	84.0	87.5	105.3	84.6	86.6
		1000	101.5	87.9	88.4	100	88.0	93.7		106.2	85.6	86.5	102.4	87.2	91.9
	20	125	88.7	90.9	95.0	100	91.6	88.7		88.5	92.2	97.6	104.7	94.4	88.2
		250	94.6	88.3	90.9	100	92.9	88.5		94.8	85.6	90.8	104.9	94.7	85.2
		500	97.2	86.3	87.6	100	86.6	88.6		99.0	83.7	86.1	104.1	84.2	85.5
		1000	106.8	90.5	90.2	100	90.0	94.0		109.8	88.1	88.7	102.4	88.3	91.4
	40	125	92.1	90.7	94.0	100	98.3	90.2		90.2	89.4	95.3	103.8	102.7	88.8
		250	101.0	90.8	91.2	100	98.6	90.4		100.3	87.5	90.1	104.0	100.1	87.6
		500	105.4	90.3	89.7	100	90.8	90.2		106.1	87.0	87.9	104.9	87.7	87.6
		1000	116.4	94.8	93.3	100	93.3	95.4		118.6	92.0	92.2	101.8	91.9	93.7

Note. ${\epsilon }_{0.2},{\epsilon }_{0.4}$ = computed $\epsilon$ values in loss function ${\rho }_{\epsilon }$ such that ${\rho }_{{\epsilon }_y}\left(\sqrt{\overline{V}}\right)=y$ for $y=0.2$ and $y=0.4$, where $\sqrt{\overline{V}}$ is the square root of the average variance of the item difficulty difference ${\widehat{b}}_{i2}-{\widehat{b}}_{i1}$. The ratio loss function with $\epsilon =0.01$ was chosen as the reference method when computing the relative RMSE. Cells with the smallest RMSE values are printed with a gray background color.

presents the relative RMSE of the estimated mean $\widehat{\mu }$ as a function of the DIF effect size $\kappa$, the number of items I, and the sample size N. In general, larger $\epsilon$ values performed better in smaller samples and with fewer items. Thus, the optimal $\epsilon$ for minimizing the RMSE depends on $\kappa$, I, and N. The data-driven estimates ${\epsilon }_{0.2}$ and ${\epsilon }_{0.4}$ generally outperformed the fixed choice $\epsilon =0.01$ based on the loss function ${\rho }_{\epsilon }$. Overall, it can be concluded that ${\epsilon }_{0.4}$ was superior to ${\epsilon }_{0.2}$. Although ${\epsilon }_{0.2}$ produced a lower RMSE than ${\epsilon }_{0.4}$ in many simulation conditions, the differences were typically small. However, in scenarios where ${\epsilon }_{0.2}$ performed worse than ${\epsilon }_{0.4}$, its RMSE exceeded the reference value of 100 and was substantially higher than that of ${\epsilon }_{0.4}$. This supports the use of ${\epsilon }_{0.4}$ as a conservative default choice. In most conditions, the Gaussian loss function slightly outperformed the ratio loss function. In large samples ($N=1000$), smaller $\epsilon$ values, such as $\epsilon =0.05$ or $\epsilon =0.01$, proved effective.

To provide further insight into the absolute bias and relative RMSE, two additional figures are presented below.

Figure 3 presents the absolute bias, SD, and RMSE of the estimated mean $\widehat{\mu }$ using the ratio loss function for $I=20$ items as a function of the DIF effect size $\kappa$ and sample size N. The absolute bias increased with larger $\epsilon$, while the SD decreased, illustrating a bias–variance trade-off in the RMSE. This trade-off results in an optimal $\epsilon$ value that minimizes the RMSE. As shown in Figure 3, the optimal $\epsilon$ was smaller for a larger DIF effect size (i.e., $\kappa =0.8$) than for $\kappa =0.4$. Additionally, the optimal $\epsilon$ decreased with increasing sample size N. As expected, the RMSE was lower for larger sample sizes.

Figure 4 presents the relative RMSE of the estimated mean based on data-driven ${\epsilon }_y$ values for the ratio loss function ${\rho }_{\epsilon }$ with $\epsilon ={\epsilon }_y$. The results indicate that an optimal RMSE was achieved for a specific y value, depending on the DIF effect size $\kappa$, the number of items I, and the sample size N. Overall, the findings in Figure 4 support the selection of ${\epsilon }_{0.2}$ and ${\epsilon }_{0.4}$, as presented in Table 2 and Table 3.

Following the suggestion of a reviewer, bias and RMSE were regressed onto the simulation factors using a two-way analysis of variance (ANOVA). The factors included sample size N, number of items I, the size of the DIF effect $\kappa$, the type of loss function (i.e., ratio vs. Gaussian function), and the chosen $\epsilon$ value in the loss function ${\rho }_{\epsilon }$. All simulation factors were treated as categorical factors in the ANOVA. The same item numbers I (e.g., $I=10,20,40$) and $\epsilon$ values (i.e., $\epsilon =0.01,0.05,0.10,0.25$) as reported in Table 2 and Table 3 were used. The main focus of the analysis was the proportion of variance explained by the main effects and two-way interactions of the factors in the ANOVA.

In the two-way ANOVA with bias as the dependent variable, $96.0\%$ of the variance was explained by the main factors and their two-way interactions. The largest proportion was attributed to sample size N ($39.8\%$), followed by $\kappa$ ($24.8\%$) and $\epsilon$ ($24.8\%$). The number of items I ($0.1\%$) and the type of loss function ($0.2\%$) contributed minimally to the variability in bias. Among the two-way interactions, only the interaction between N and $\kappa$ ($N\times \kappa$, $2.3\%$) and $N\times \epsilon$ ($1.7\%$) accounted for non-negligible proportions of variance.

In the two-way ANOVA with RMSE as the dependent variable, $99.3\%$ of the variance was explained by the simulation factors and their two-way interactions. As expected, sample size N accounted for the largest proportion of explained variance ($89.9\%$), followed by number of items I ($3.2\%$), $\kappa$ ($1.0\%$), $\epsilon$ ($0.5\%$), and the type of loss function ($0.0\%$). Among the two-way interactions, $N\times I$ ($1.3\%$), $N\times \kappa$ ($1.0\%$), and $N\times \epsilon$ ($2.1\%$) contributed with non-negligible amounts to the variance.

5. Empirical Example

We now illustrate $L_0$ linking in the Rasch model using an empirical dataset. The dataDIF dataset from the R package equateIRT (Version 1.0.0; [53,54]) was selected for this purpose. The dataset contains 20 dichotomous items and three groups, each with 1000 subjects. For illustration, only Groups 1 and 2 were linked using the Rasch model.

As in the Simulation Study (see Section 4), the Rasch model was fitted using the sirt::xxirt() function from the R package sirt (Version 4.2-106; [52]). The optimization of $L_0$ linking was performed using the stats::nlminb() function, with R code specifically written for this paper, available at https://osf.io/un6q4 (accessed on 20 February 2025).

The average standard error of item difficulty differences was 0.126. The mean ability difference between Group 1 and Group 2, obtained from MM linking based on the $L_2$ loss function, was 0.482. Slight deviations from this estimate were observed with different specifications of the $L_0$ linking function. For the ratio loss function, the mean estimate based on $\epsilon =0.01$ was 0.535. Empirical choices of the $\epsilon$ parameter, specifically ${\epsilon }_{0.2}$ and ${\epsilon }_{0.4}$, were also explored. For the ratio loss function, the estimates for ${\epsilon }_{0.2}=0.064$ and ${\epsilon }_{0.4}=0.024$ yielded mean differences of 0.518 and 0.520, respectively. The mean estimate based on the Gaussian loss function was 0.537, with estimates for ${\epsilon }_{0.2}=0.072$ and ${\epsilon }_{0.4}=0.031$ yielding 0.519 and 0.517, respectively.

DIF effects for the 20 items were also computed based on the estimated mean difference of 0.520, obtained with ${\epsilon }_{0.4}$ and the ratio loss function. Figure 5 displays the DIF effects along with their 95% confidence intervals. Item 1 exhibited a significantly positive DIF effect (0.798, with a standard error of 0.112), while DIF effects for the remaining 19 items did not significantly differ from 0.

6. Discussion

This article examined the computational aspects of $L_0$ linking in the Rasch model. $L_0$ linking serves as a robust alternative to MM linking when fixed DIF effects are present. Since the $L_0$ loss function is not differentiable at $x=0$, the ratio loss function and the Gaussian loss function were used as differentiable approximations. Both approximations rely on a tuning parameter $\epsilon$, which controls the approximation error. A numerical illustration and a simulation study demonstrated that the optimal $\epsilon$ value minimizing the RMSE of the linking parameter estimate $\widehat{\mu }$ depends on the magnitude of DIF effects, the number of items, and the sample size. A data-driven selection of $\epsilon$ provided better performance than a fixed $\epsilon$ across all conditions. Additionally, the Gaussian loss function showed a slight advantage over the ratio loss function, though the differences are likely negligible in practice.

This study aimed to determine group differences in a robust manner, ensuring insensitivity to DIF effects. Detecting DIF items was treated as a nuisance, as the goal was not to identify deviating items but rather to obtain a consistent estimate of the mean difference in ability between the two groups. In contrast, much of the psychometric literature focuses on detecting DIF items [22,27,55,56,57], with group difference estimation being, at best, a by-product of the procedure.

Using the $L_0$ loss function effectively removes items with large DIF effects from the linking process. As a result, the estimated group difference is based solely on items with little to no DIF. This approach may pose a threat to validity, as it alters the construct being measured by treating items with DIF effects as construct-irrelevant [58,59]. Restricting the item set in this way can change the interpretation of the ability variable [60,61,62,63,64].

7. Conclusions

The $L_0$ loss function proved effective in linking two groups based on the Rasch model in the presence of differential item functioning. Two differentiable approximations (i.e., the ratio and Gaussian loss functions) of the nondifferentiable $L_0$ loss function have been proposed. The smoothness of the approximation depends on the tuning parameter $\epsilon$. A simulation study demonstrated that a data-driven choice of $\epsilon$, based on the average standard error of item difficulty differences, produced estimates with a lower RMSE compared to estimates based on a fixed $\epsilon$ value, such as $\epsilon =0.01$.