# Linking Error in the 2PL Model

^{1}

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## 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 |

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**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