# Linking Error in the 2PL Model

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Linking Error and M-Estimation

## 3. Linking Error of Log-Mean-Mean Linking

## 4. Simulation Study

#### 4.1. Method

`qmixnorm`function from the R package KScorrect [37] for determining quantiles in the data simulation of DIF effects. Because analytical solutions are not available to compute a quantile function for the normal mixture model, the

`qmixnorm`function approximates the quantile function using a spline function calculated from cumulative density functions for the specified mixture distribution [37]. Quantiles for probabilities near zero or one are approximated by taking a randomly generated sample.

#### 4.2. Results

## 5. Further Applications of the Linking Error in the 2PL Model

#### 5.1. Different Linking Methods

#### 5.1.1. Robust Log-Mean-Mean Linking

#### 5.1.2. Haebara Linking

#### 5.2. Linking Error with Testlets

#### 5.3. Linking Error in Chain Linking

#### 5.4. Linking Error for Trend Estimates in Educational Large-Scale Assessment Studies

#### 5.5. Linking Error in Fixed Item Parameter Calibration

#### 5.6. Linking Error in Concurrent Calibration

#### 5.7. Linking Error for Derived Parameters

#### 5.7.1. Proportions

#### 5.7.2. Percentiles

#### 5.8. Computation of Total Error and Sampling Error Correction for Linking Error Estimates

## 6. Discussion

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

1PL | one-parameter logistic |

2PL | two-parameter logistic |

DIF | differential item functioning |

DWLS | diagonally weighted least squares |

FIPC | fixed item parameter calibration |

IPD | item parameter drift |

IRT | item response theory |

JK | jackknife |

LE | linking error |

LSA | large-scale assessment studies |

PIRLS | progress in international reading literacy study |

PISA | programme for international student assessment |

ULS | unweighted least squares |

## References

- Chen, Y.; Li, X.; Liu, J.; Ying, Z. Item response theory—A statistical framework for educational and psychological measurement. arXiv
**2021**, arXiv:2108.08604. [Google Scholar] [CrossRef] - van der Linden, W.J. Unidimensional logistic response models. In Handbook of Item Response Theory, Volume 1: Models; van der Linden, W.J., Ed.; CRC Press: Boca Raton, FL, USA, 2016; pp. 11–30. [Google Scholar]
- Rutkowski, L.; von Davier, M.; Rutkowski, D. (Eds.) A Handbook of International Large-Scale Assessment: Background, Technical Issues, and Methods of Data Analysis; Chapman Hall: London, UK; CRC Press: Boca Raton, FL, USA, 2013. [Google Scholar] [CrossRef]
- OECD. PISA 2018. Technical Report; OECD: Paris, France, 2020. [Google Scholar]
- Yen, W.M.; Fitzpatrick, A.R. Item response theory. In Educational Measurement; Brennan, R.L., Ed.; Praeger Publishers: Westport, CT, USA, 2006; pp. 111–154. [Google Scholar]
- Rasch, G. Probabilistic Models for Some Intelligence and Attainment Tests; Danish Institute for Educational Research: Copenhagen, Denmark, 1960. [Google Scholar]
- Birnbaum, A. Some latent trait models and their use in inferring an examinee’s ability. In Statistical Theories of Mental Test Scores; Lord, F.M., Novick, M.R., Eds.; MIT Press: Reading, MA, USA, 1968; pp. 397–479. [Google Scholar]
- Aitkin, M. Expectation maximization algorithm and extensions. In Handbook of Item Response Theory, Vol. 2: Statistical Tools; van der Linden, W.J., Ed.; CRC Press: Boca Raton, FL, USA, 2016; pp. 217–236. [Google Scholar] [CrossRef]
- Bock, R.D.; Aitkin, M. Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika
**1981**, 46, 443–459. [Google Scholar] [CrossRef] - Holland, P.W.; Wainer, H. (Eds.) Differential Item Functioning: Theory and Practice; Lawrence Erlbaum: Hillsdale, NJ, USA, 1993. [Google Scholar] [CrossRef]
- Penfield, R.D.; Camilli, G. Differential item functioning and item bias. In Handbook of Statistics, Vol. 26: Psychometrics; Rao, C.R., Sinharay, S., Eds.; Elsevier: Amsterdam, The Netherlands, 2007; pp. 125–167. [Google Scholar] [CrossRef]
- Joo, S.; Ali, U.; Robin, F.; Shin, H.J. Impact of differential item functioning on group score reporting in the context of large-scale assessments. Large-Scale Assess. Educ.
**2022**, 10, 18. [Google Scholar] [CrossRef] - Sachse, K.A.; Roppelt, A.; Haag, N. A comparison of linking methods for estimating national trends in international comparative large-scale assessments in the presence of cross-national DIF. J. Educ. Meas.
**2016**, 53, 152–171. [Google Scholar] [CrossRef] - Battauz, M. Multiple equating of separate IRT calibrations. Psychometrika
**2017**, 82, 610–636. [Google Scholar] [CrossRef] - Monseur, C.; Berezner, A. The computation of equating errors in international surveys in education. J. Appl. Meas.
**2007**, 8, 323–335. [Google Scholar] - OECD. PISA 2012. Technical Report; OECD: Paris, France, 2014; Available online: https://bit.ly/2YLG24g (accessed on 3 December 2022).
- Robitzsch, A.; Lüdtke, O. Linking errors in international large-scale assessments: Calculation of standard errors for trend estimation. Assess. Educ.
**2019**, 26, 444–465. [Google Scholar] [CrossRef] - Robitzsch, A. Robust and nonrobust linking of two groups for the Rasch model with balanced and unbalanced random DIF: A comparative simulation study and the simultaneous assessment of standard errors and linking errors with resampling techniques. Symmetry
**2021**, 13, 2198. [Google Scholar] [CrossRef] - Wu, M. Measurement, sampling, and equating errors in large-scale assessments. Educ. Meas.
**2010**, 29, 15–27. [Google Scholar] [CrossRef] - Efron, B.; Tibshirani, R.J. An Introduction to the Bootstrap; CRC Press: Boca Raton, FL, USA, 1994. [Google Scholar] [CrossRef]
- Kolenikov, S. Resampling variance estimation for complex survey data. Stata J.
**2010**, 10, 165–199. [Google Scholar] [CrossRef] [Green Version] - Boos, D.D.; Stefanski, L.A. Essential Statistical Inference; Springer: New York, NY, USA, 2013. [Google Scholar] [CrossRef]
- Stefanski, L.A.; Boos, D.D. The calculus of M-estimation. Am. Stat.
**2002**, 56, 29–38. [Google Scholar] [CrossRef] - Zeileis, A. Object-oriented computation of sandwich estimators. J. Stat. Softw.
**2006**, 16, 1–16. [Google Scholar] [CrossRef] [Green Version] - Fay, M.P.; Graubard, B.I. Small-sample adjustments for Wald-type tests using sandwich estimators. Biometrics
**2001**, 57, 1198–1206. [Google Scholar] [CrossRef] - Li, P.; Redden, D.T. Small sample performance of bias-corrected sandwich estimators for cluster-randomized trials with binary outcomes. Stat. Med.
**2015**, 34, 281–296. [Google Scholar] [CrossRef] [Green Version] - Zeileis, A.; Köll, S.; Graham, N. Various versatile variances: An object-oriented implementation of clustered covariances in R. J. Stat. Softw.
**2020**, 95, 1–36. [Google Scholar] [CrossRef] - Chen, Y.; Li, C.; Xu, G. DIF statistical inference and detection without knowing anchoring items. arXiv
**2021**, arXiv:2110.11112. [Google Scholar] [CrossRef] - Halpin, P.F. Differential item functioning via robust scaling. arXiv
**2022**, arXiv:2207.04598. [Google Scholar] [CrossRef] - Wang, W.; Liu, Y.; Liu, H. Testing differential item functioning without predefined anchor items using robust regression. J. Educ. Behav. Stat.
**2022**, 47, 666–692. [Google Scholar] [CrossRef] - Robitzsch, A. L
_{p}loss functions in invariance alignment and Haberman linking with few or many groups. Stats**2020**, 3, 246–283. [Google Scholar] [CrossRef] - Hunter, J.E. Probabilistic foundations for coefficients of generalizability. Psychometrika
**1968**, 33, 1–18. [Google Scholar] [CrossRef] - Kolen, M.J.; Brennan, R.L. Test Equating, Scaling, and Linking; Springer: New York, NY, USA, 2014. [Google Scholar] [CrossRef]
- Robitzsch, A. A comparison of linking methods for two groups for the two-parameter logistic item response model in the presence and absence of random differential item functioning. Foundations
**2021**, 1, 116–144. [Google Scholar] [CrossRef] - Maronna, R.A.; Martin, R.D.; Yohai, V.J. Robust Statistics: Theory and Methods; Wiley: New York, NY, USA, 2006. [Google Scholar] [CrossRef]
- R Core Team. R: A Language and Environment for Statistical Computing; R Core Team: Vienna, Austria, 2022; Available online: https://www.R-project.org/ (accessed on 11 January 2022).
- Novack-Gottshall, P.; Wang, S.C. KScorrect: Lilliefors-Corrected Kolmogorov-Smirnov Goodness-of-Fit Tests; R Package Version 1.4-0. 2019. Available online: https://CRAN.R-project.org/package=KScorrect (accessed on 3 July 2019).
- Haebara, T. Equating logistic ability scales by a weighted least squares method. Jpn. Psychol. Res.
**1980**, 22, 144–149. [Google Scholar] [CrossRef] [Green Version] - Bradlow, E.T.; Wainer, H.; Wang, X. A Bayesian random effects model for testlets. Psychometrika
**1999**, 64, 153–168. [Google Scholar] [CrossRef] - Sireci, S.G.; Thissen, D.; Wainer, H. On the reliability of testlet-based tests. J. Educ. Meas.
**1991**, 28, 237–247. [Google Scholar] [CrossRef] - Wainer, H.; Bradlow, E.T.; Wang, X. Testlet Response Theory and Its Applications; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar] [CrossRef]
- Monseur, C.; Sibberns, H.; Hastedt, D. Linking errors in trend estimation for international surveys in education. IERI Monogr. Ser.
**2008**, 1, 113–122. [Google Scholar] - Battauz, M. IRT test equating in complex linkage plans. Psychometrika
**2013**, 78, 464–480. [Google Scholar] [CrossRef] - Battauz, M. Factors affecting the variability of IRT equating coefficients. Stat. Neerl.
**2015**, 69, 85–101. [Google Scholar] [CrossRef] - Arce-Ferrer, A.J.; Bulut, O. Investigating separate and concurrent approaches for item parameter drift in 3PL item response theory equating. Int. J. Test.
**2017**, 17, 1–22. [Google Scholar] [CrossRef] - Taherbhai, H.; Seo, D. The philosophical aspects of IRT equating: Modeling drift to evaluate cohort growth in large-scale assessments. Educ. Meas.
**2013**, 32, 2–14. [Google Scholar] [CrossRef] - Grothendieck, G. rSymPy: R Interface to SymPy Computer Algebra System. R Package Version 0.2-1.2. 2010. Available online: https://CRAN.R-project.org/package=rSymPy (accessed on 31 July 2010).
- Meurer, A.; Smith, C.P.; Paprocki, M.; Čertík, O.; Kirpichev, S.B.; Rocklin, M.; Kumar, A.; Ivanov, S.; Moore, J.K.; Singh, S.; et al. SymPy: Symbolic computing in Python. PeerJ Comput. Sci.
**2017**, 3, e103. [Google Scholar] [CrossRef] [Green Version] - Fischer, L.; Gnambs, T.; Rohm, T.; Carstensen, C.H. Longitudinal linking of Rasch-model-scaled competence tests in large-scale assessments: A comparison and evaluation of different linking methods and anchoring designs based on two tests on mathematical competence administered in grades 5 and 7. Psych. Test Assess. Model.
**2019**, 61, 37–64. [Google Scholar] - Sachse, K.A.; Haag, N. Standard errors for national trends in international large-scale assessments in the case of cross-national differential item functioning. Appl. Meas. Educ.
**2017**, 30, 102–116. [Google Scholar] [CrossRef] - Sachse, K.A.; Mahler, N.; Pohl, S. When nonresponse mechanisms change: Effects on trends and group comparisons in international large-scale assessments. Educ. Psychol. Meas.
**2019**, 79, 699–726. [Google Scholar] [CrossRef] - OECD. PISA 2015. Technical Report; OECD: Paris, France, 2017; Available online: https://bit.ly/32buWnZ (accessed on 3 December 2022).
- Weeks, J.; von Davier, M.; Yamamoto, K. Design considerations for the program for international student assessment. In A Handbook of International Large-Scale Assessment: Background, Technical Issues, and Methods of Data Analysis; Rutkowski, L., von Davier, M., Rutkowski, D., Eds.; Chapman Hall: London, UK; CRC Press: Boca Raton, FL, USA, 2013; pp. 259–276. [Google Scholar] [CrossRef]
- Kang, H.A.; Lu, Y.; Chang, H.H. IRT item parameter scaling for developing new item pools. Appl. Meas. Educ.
**2017**, 30, 1–15. [Google Scholar] [CrossRef] - König, C.; Khorramdel, L.; Yamamoto, K.; Frey, A. The benefits of fixed item parameter calibration for parameter accuracy in small sample situations in large-scale assessments. Educ. Meas.
**2021**, 40, 17–27. [Google Scholar] [CrossRef] - Cai, L.; Moustaki, I. Estimation methods in latent variable models for categorical outcome variables. In The Wiley Handbook of Psychometric Testing: A Multidisciplinary Reference on Survey, Scale and Test; Irwing, P., Booth, T., Hughes, D.J., Eds.; Wiley: New York, NY, USA, 2018; pp. 253–277. [Google Scholar] [CrossRef]
- Yuan, K.H.; Cheng, Y.; Patton, J. Information matrices and standard errors for MLEs of item parameters in IRT. Psychometrika
**2014**, 79, 232–254. [Google Scholar] [CrossRef] - González, J.; Wiberg, M. Applying Test Equating Methods. Using R; Springer: New York, NY, USA, 2017. [Google Scholar] [CrossRef]
- Jewsbury, P.A. Error Variance in Common Population Linking Bridge Studies; (Research Report No. RR-19-42); Educational Testing Service: Princeton, NJ, USA, 2019. [Google Scholar] [CrossRef] [Green Version]
- Martin, M.O.; Mullis, I.V.S.; Foy, P.; Brossman, B.; Stanco, G.M. Estimating linking error in PIRLS. IERI Monogr. Ser.
**2012**, 5, 35–47. Available online: https://bit.ly/2Vx3el8 (accessed on 3 December 2022). - Frey, A.; Hartig, J.; Rupp, A.A. An NCME instructional module on booklet designs in large-scale assessments of student achievement: Theory and practice. Educ. Meas.
**2009**, 28, 39–53. [Google Scholar] [CrossRef] - Chen, Y.; Li, X.; Zhang, S. Joint maximum likelihood estimation for high-dimensional exploratory item factor analysis. Psychometrika
**2019**, 84, 124–146. [Google Scholar] [CrossRef] [Green Version] - Chen, Y.; Li, X.; Zhang, S. Structured latent factor analysis for large-scale data: Identifiability, estimability, and their implications. J. Am. Stat. Assoc.
**2020**, 115, 1756–1770. [Google Scholar] [CrossRef] [Green Version] - Haberman, S.J. Maximum likelihood estimates in exponential response models. Ann. Stat.
**1977**, 5, 815–841. [Google Scholar] [CrossRef]

**Figure 2.**Trend estimation for country means and standard deviations at two time points in an international large-scale assessment study.

Item | ${\mathit{a}}_{\mathit{i}}$ | ${\mathit{b}}_{\mathit{i}}$ |
---|---|---|

1 | 0.73 | −1.31 |

2 | 1.25 | $\phantom{-}$1.44 |

3 | 1.20 | −1.20 |

4 | 1.47 | $\phantom{-}$0.10 |

5 | 0.97 | $\phantom{-}$0.10 |

6 | 1.38 | −0.74 |

7 | 1.05 | $\phantom{-}$1.48 |

8 | 1.14 | −0.61 |

9 | 1.15 | $\phantom{-}$0.82 |

10 | 0.67 | −0.07 |

_{i}= item discrimination; b

_{i}= item difficulty.

**Table 2.**Simulation Study: Coverage rates for estimated mean $\widehat{\mu}$ as a function of the standard deviation of DIF effects for a (${\tau}_{a}$) and b (${\tau}_{b}$), number of items (I), and the type of distribution for DIF effects.

Normal | ${\mathit{t}}_{4}$ | Normal Mixture | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\tau}}_{\mathit{a}}$ | ${\mathit{\tau}}_{\mathit{b}}$ | I | JK | ESW | OSW | BOSW | JK | ESW | OSW | BOSW | JK | ESW | OSW | BOSW |

0.01 | 0.25 | 10 | 92.2 | 92.2 | 90.8 | 92.1 | 92.9 | 92.9 | 91.5 | 92.9 | 92.7 | 92.7 | 91.3 | 92.7 |

20 | 93.8 | 93.8 | 93.2 | 93.8 | 94.5 | 94.5 | 93.8 | 94.5 | 94.4 | 94.4 | 93.8 | 94.4 | ||

40 | 94.6 | 94.6 | 94.3 | 94.6 | 94.9 | 94.9 | 94.5 | 94.8 | 94.9 | 94.9 | 94.6 | 94.9 | ||

80 | 95.2 | 95.2 | 95.0 | 95.1 | 95.2 | 95.2 | 95.0 | 95.1 | 95.2 | 95.2 | 95.1 | 95.2 | ||

0.01 | 0.50 | 10 | 92.5 | 92.5 | 91.2 | 92.5 | 92.9 | 92.9 | 91.5 | 92.9 | 93.1 | 93.1 | 91.7 | 93.0 |

20 | 94.1 | 94.1 | 93.5 | 94.1 | 94.6 | 94.6 | 94.0 | 94.6 | 94.4 | 94.4 | 93.8 | 94.4 | ||

40 | 94.7 | 94.7 | 94.4 | 94.7 | 95.0 | 95.0 | 94.7 | 95.0 | 94.8 | 94.8 | 94.5 | 94.8 | ||

80 | 95.1 | 95.1 | 94.9 | 95.0 | 95.3 | 95.3 | 95.1 | 95.3 | 95.4 | 95.4 | 95.2 | 95.3 | ||

0.25 | 0.25 | 10 | 93.9 | 93.8 | 91.1 | 92.6 | 94.6 | 94.4 | 92.0 | 93.4 | 94.4 | 94.3 | 91.8 | 93.1 |

20 | 94.8 | 94.8 | 93.1 | 93.8 | 94.9 | 94.9 | 93.2 | 93.9 | 95.1 | 95.1 | 93.4 | 94.0 | ||

40 | 95.1 | 95.1 | 93.7 | 94.0 | 95.5 | 95.5 | 94.0 | 94.4 | 95.2 | 95.2 | 93.6 | 94.0 | ||

80 | 95.2 | 95.2 | 93.9 | 94.0 | 95.4 | 95.4 | 94.2 | 94.4 | 95.5 | 95.5 | 94.3 | 94.4 | ||

0.25 | 0.50 | 10 | 93.0 | 92.9 | 90.9 | 92.3 | 93.7 | 93.6 | 91.6 | 93.0 | 93.5 | 93.5 | 91.3 | 92.7 |

20 | 94.3 | 94.3 | 93.1 | 93.8 | 94.4 | 94.4 | 93.2 | 93.8 | 94.5 | 94.5 | 93.3 | 94.0 | ||

40 | 95.0 | 95.0 | 94.2 | 94.6 | 95.1 | 95.1 | 94.3 | 94.7 | 95.1 | 95.1 | 94.3 | 94.6 | ||

80 | 95.2 | 95.2 | 94.6 | 94.8 | 95.2 | 95.1 | 94.6 | 94.7 | 95.1 | 95.1 | 94.4 | 94.6 |

_{4}= DIF effects distributed according to scaled t

_{4}distribution; Normal Mixture = DIF effects distributed according to contaminated mixture model; JK = linking error (LE) estimated by jackknife; ESW = LE estimated by expected sandwich estimator; OSW = LE estimated by observed sandwich estimator; BOSW = LE estimated by bias-corrected observed sandwich estimator; coverage rates smaller than 92.5% or larger than 97.5% are printed in bold.

**Table 3.**Simulation Study: Coverage rates for estimated standard deviation $\widehat{\sigma}$ as a function of the standard deviation of DIF effects for a (${\tau}_{a}$) and b (${\tau}_{b}$), number of items (I), and the type of distribution for DIF effects.

Normal | ${\mathit{t}}_{4}$ | Normal Mixture | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\tau}}_{\mathit{a}}$ | ${\mathit{\tau}}_{\mathit{b}}$ | I | JK | ESW | OSW | BOSW | JK | ESW | OSW | BOSW | JK | ESW | OSW | BOSW |

0.01 | 0.25 | 10 | 92.6 | 92.6 | 86.1 | 87.3 | 92.9 | 92.9 | 84.9 | 86.2 | 92.9 | 92.9 | 85.3 | 86.7 |

20 | 93.8 | 93.8 | 83.0 | 83.8 | 94.5 | 94.5 | 82.1 | 82.8 | 94.4 | 94.4 | 82.5 | 83.3 | ||

40 | 94.6 | 94.6 | 80.2 | 80.6 | 95.0 | 95.0 | 79.1 | 79.5 | 94.9 | 94.9 | 79.2 | 79.5 | ||

80 | 95.2 | 95.2 | 77.9 | 78.1 | 95.1 | 95.1 | 76.2 | 76.4 | 95.0 | 95.0 | 76.3 | 76.6 | ||

0.01 | 0.50 | 10 | 92.1 | 92.2 | 92.3 | 93.0 | 93.2 | 93.2 | 91.8 | 92.6 | 93.0 | 93.0 | 92.2 | 92.9 |

20 | 94.1 | 94.1 | 91.3 | 91.7 | 94.4 | 94.4 | 90.7 | 91.2 | 94.2 | 94.2 | 90.8 | 91.3 | ||

40 | 94.5 | 94.5 | 91.3 | 91.6 | 94.8 | 94.8 | 90.6 | 90.8 | 94.9 | 94.9 | 90.7 | 90.9 | ||

80 | 95.1 | 95.1 | 93.7 | 93.7 | 95.0 | 95.0 | 92.5 | 92.6 | 95.3 | 95.3 | 92.9 | 93.0 | ||

0.25 | 0.25 | 10 | 92.3 | 92.3 | 89.8 | 91.3 | 93.1 | 93.1 | 90.5 | 92.0 | 92.6 | 92.6 | 90.0 | 91.4 |

20 | 93.8 | 93.8 | 92.4 | 93.0 | 94.5 | 94.5 | 93.0 | 93.6 | 94.2 | 94.2 | 92.7 | 93.3 | ||

40 | 94.7 | 94.7 | 93.6 | 93.9 | 95.1 | 95.1 | 93.9 | 94.3 | 94.7 | 94.7 | 93.7 | 94.0 | ||

80 | 94.9 | 94.9 | 94.0 | 94.2 | 95.2 | 95.2 | 94.3 | 94.5 | 95.2 | 95.2 | 94.3 | 94.5 | ||

0.25 | 0.50 | 10 | 92.3 | 92.3 | 87.7 | 89.3 | 93.0 | 92.9 | 88.2 | 89.7 | 92.7 | 92.6 | 88.0 | 89.5 |

20 | 94.1 | 94.1 | 91.2 | 91.8 | 94.3 | 94.3 | 91.2 | 91.9 | 94.1 | 94.1 | 90.9 | 91.6 | ||

40 | 94.8 | 94.8 | 92.7 | 93.0 | 94.9 | 94.9 | 92.7 | 93.0 | 94.9 | 94.9 | 92.7 | 93.1 | ||

80 | 95.1 | 95.1 | 93.3 | 93.4 | 95.4 | 95.4 | 93.5 | 93.7 | 95.1 | 95.1 | 93.3 | 93.4 |

_{4}= DIF effects distributed according to scaled t

_{4}distribution; Normal Mixture = DIF effects distributed according to contaminated mixture model; JK = linking error (LE) estimated by jackknife; ESW = LE estimated by expected sandwich estimator; OSW = LE estimated by observed sandwich estimator; BOSW = LE estimated by bias-corrected observed sandwich estimator; coverage rates smaller than 92.5% or larger than 97.5% are printed in bold.

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**MDPI and ACS Style**

Robitzsch, A.
Linking Error in the 2PL Model. *J* **2023**, *6*, 58-84.
https://doi.org/10.3390/j6010005

**AMA Style**

Robitzsch A.
Linking Error in the 2PL Model. *J*. 2023; 6(1):58-84.
https://doi.org/10.3390/j6010005

**Chicago/Turabian Style**

Robitzsch, Alexander.
2023. "Linking Error in the 2PL Model" *J* 6, no. 1: 58-84.
https://doi.org/10.3390/j6010005