Computational Aspects of <i>L</i><sub>0</sub> Linking in the Rasch Model
The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mn>0</mn></msub></semantics></math></inline-formula> linking approach replaces the <inline-...
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| Format: | Article |
| Language: | English |
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MDPI AG
2025-04-01
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| Series: | Algorithms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1999-4893/18/4/213 |
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| Summary: | The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mn>0</mn></msub></semantics></math></inline-formula> linking approach replaces the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mn>2</mn></msub></semantics></math></inline-formula> loss function in mean–mean linking under the Rasch model with the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mn>0</mn></msub></semantics></math></inline-formula> loss function. Using the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mn>0</mn></msub></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula> that determines the approximation error. Results demonstrate that the optimal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula> outperformed a fixed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula> across all conditions in both a numerical illustration and a simulation study. |
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| ISSN: | 1999-4893 |