# Comparing Robust Linking and Regularized Estimation for Linking Two Groups in the 1PL and 2PL Models in the Presence of Sparse Uniform Differential Item Functioning

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## Abstract

**:**

## 1. Introduction

## 2. Two-Group Comparison under Sparse DIF

#### 2.1. Concurrent Calibration

#### 2.1.1. 1PL Model

#### 2.1.2. 2PL Model

#### 2.2. Regularization Approaches

#### 2.2.1. 1PL Model

#### 2.2.2. 2PL Model

#### 2.3. Robust Linking Approaches

#### 2.3.1. 1PL Model

#### Robust Linking Using the ${L}_{p}$ Loss Function

#### Robust Linking Using the MAD Statistic

#### 2.3.2. 2PL Model

#### Robust Linking Using ${L}_{p}$ Loss Function or MAD Statistic

#### Joint Haberman Linking Using Common Item Discriminations

#### Haberman Linking Based on Separate Calibration

#### 2.4. On the Relation of Robust Linking and Regularized Estimation

## 3. Simulation Study 1: DIF Effects in the 1PL Model

#### 3.1. Method

`xxirt`function in the sirt package [72]. Replication material can be found at https://osf.io/tma3f/ (accessed on 8 December 2022).

#### 3.2. Results

## 4. Focused Simulation Study 1A: Optimal Choice of two Tuning Parameters for the SCAD Penalty

#### 4.1. Method

#### 4.2. Results

## 5. Simulation Study 2: Uniform DIF Effects in the 2PL Model

#### 5.1. Method

#### 5.2. Results

## 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 |

AIC | Akaike information criterion |

BIC | Bayesian information criterion |

CC | concurrent calibration |

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 |

MAD | median absolute deviation |

PISA | programme for international student assessment |

RMSE | root mean square error |

SCAD | smoothly clipped absolute deviation |

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**Figure 1.**SCAD penalty function ${\mathcal{P}}_{\mathrm{SCAD}}$ for different values of a for $\lambda =0.2$.

**Table 1.**Simulation Study 1: Bias of estimated group means for balanced and unbalanced DIF effects as a function of the size of DIF effects $\delta $ and sample size N.

Choice of $\mathit{\lambda}$ | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\delta}$ | N | MAD | AIC | BIC | 0.05 | 0.10 | 0.15 | ${\mathit{L}}_{0.5}$ | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | CC |

Balanced DIF | |||||||||||

0.5 | 500 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

1000 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

2500 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

5000 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | −0.01 | 0.00 | 0.00 | 0.00 | 0.00 | |

1.0 | 500 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

1000 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

2500 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

5000 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

Unbalanced DIF | |||||||||||

0.5 | 500 | −0.06 | −0.02 | −0.03 | −0.04 | −0.02 | −0.06 | −0.03 | −0.05 | −0.10 | −0.09 |

1000 | −0.03 | −0.02 | −0.01 | −0.01 | −0.01 | −0.08 | −0.02 | −0.04 | −0.10 | −0.10 | |

2500 | 0.00 | −0.02 | −0.01 | −0.01 | −0.01 | −0.09 | −0.01 | −0.02 | −0.10 | −0.10 | |

5000 | 0.00 | −0.01 | −0.01 | −0.01 | −0.01 | −0.09 | −0.01 | −0.02 | −0.10 | −0.10 | |

1.0 | 500 | −0.01 | −0.02 | 0.00 | −0.05 | 0.00 | 0.00 | −0.02 | −0.05 | −0.20 | −0.18 |

1000 | 0.00 | −0.02 | 0.00 | −0.02 | 0.00 | 0.00 | −0.01 | −0.04 | −0.20 | −0.18 | |

2500 | 0.00 | −0.01 | 0.00 | −0.02 | 0.00 | 0.00 | −0.01 | −0.02 | −0.20 | −0.18 | |

5000 | 0.00 | −0.01 | 0.00 | −0.03 | 0.00 | 0.00 | −0.01 | −0.02 | −0.20 | −0.18 |

_{p}= linking employing the L

_{p}loss function with p = 0.5, 1.0, or 2.0; CC = concurrent calibration assuming invariant item parameters; Absolute biases larger than 0.03 are printed in bold.

**Table 2.**Simulation Study 1: Relative root mean sqaure error (RMSE) of estimated group means for balanced and unbalanced DIF effects as a function of the size of DIF effects $\delta $ and sample size N.

Choice of $\mathit{\lambda}$ | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\delta}$ | N | MAD | AIC | BIC | 0.05 | 0.10 | 0.15 | ${\mathit{L}}_{0.5}$ | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | CC |

Balanced DIF | |||||||||||

0.5 | 500 | 111 | 115 | 111 | 120 | 110 | 111 | 122 | 110 | 101 | 100 |

1000 | 111 | 112 | 106 | 109 | 108 | 118 | 115 | 108 | 101 | 100 | |

2500 | 104 | 133 | 103 | 122 | 118 | 135 | 114 | 108 | 101 | 100 | |

5000 | 103 | 147 | 129 | 139 | 126 | 153 | 111 | 107 | 100 | 100 | |

1.0 | 500 | 107 | 111 | 105 | 115 | 106 | 104 | 118 | 109 | 102 | 100 |

1000 | 104 | 110 | 103 | 108 | 103 | 103 | 115 | 108 | 102 | 100 | |

2500 | 105 | 113 | 104 | 112 | 103 | 103 | 114 | 109 | 102 | 100 | |

5000 | 104 | 111 | 103 | 126 | 103 | 103 | 112 | 108 | 102 | 100 | |

Unbalanced DIF | |||||||||||

0.5 | 500 | 120 | 108 | 104 | 120 | 100 | 117 | 113 | 110 | 142 | 138 |

1000 | 124 | 126 | 100 | 122 | 108 | 164 | 117 | 119 | 196 | 187 | |

2500 | 102 | 203 | 185 | 194 | 161 | 258 | 114 | 120 | 288 | 274 | |

5000 | 100 | 249 | 240 | 243 | 217 | 374 | 114 | 124 | 408 | 383 | |

1.0 | 500 | 108 | 109 | 100 | 141 | 101 | 100 | 122 | 121 | 270 | 247 |

1000 | 100 | 113 | 100 | 125 | 100 | 100 | 117 | 122 | 370 | 336 | |

2500 | 101 | 113 | 101 | 244 | 103 | 100 | 113 | 121 | 572 | 519 | |

5000 | 100 | 115 | 100 | 352 | 103 | 100 | 111 | 124 | 808 | 730 |

_{p}= linking employing the L

_{p}loss function with p = 0.5, 1.0, or 2.0; CC = concurrent calibration assuming invariant item parameters; Relative RMSE values larger than 125 are printed in bold.

**Table 3.**Focused Simulation Study 1A: Relative root mean square error (RMSE) of estimated group means for unbalanced DIF effects as a function of the size of DIF effects $\delta $ and sample size N for different values a of the SCAD penalty.

Best | Choice of $\mathit{\lambda}$ Based on AIC with $\mathit{a}=$ | Choice of $\mathit{\lambda}$ Based on BIC with $\mathit{a}=$ | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\delta}$ | N | a | $\mathit{\lambda}$ | 2.2 | 2.5 | 3 | 3.7 | 4.5 | 6 | 9 | ${\mathit{a}}_{\mathbf{opt}}$ | 2.2 | 2.5 | 3 | 3.7 | 4.5 | 6 | 9 | ${\mathit{a}}_{\mathbf{opt}}$ |

0.5 | 500 | 9 | 0.04 | 110.8 | 111.2 | 110.9 | 111.8 | 110.2 | 110.3 | 109.6 | 108.0 | 103.4 | 103.4 | 103.3 | 103.4 | 103.4 | 103.5 | 104.0 | 104.0 |

1000 | 3.7 | BIC | 128.7 | 130.1 | 129.5 | 127.7 | 129.4 | 126.1 | 122.7 | 121.4 | 100.2 | 100.0 | 100.1 | 100.0 | 100.2 | 100.1 | 100.4 | 100.2 | |

2500 | 2.2 | 0.19 | 139.2 | 138.6 | 137.7 | 136.3 | 138.1 | 137.8 | 132.3 | 129.4 | 126.0 | 126.6 | 125.2 | 122.7 | 125.1 | 124.6 | 117.8 | 111.3 | |

1 | 500 | 3.7 | 0.13 | 111.4 | 110.4 | 110.5 | 108.7 | 108.7 | 107.9 | 109.9 | 108.6 | 102.5 | 102.4 | 102.5 | 100.3 | 100.2 | 100.2 | 100.4 | 100.6 |

1000 | 3.7 | 0.13 | 113.1 | 112.5 | 112.5 | 112.9 | 112.1 | 111.2 | 108.9 | 110.3 | 100.6 | 100.6 | 100.7 | 100.6 | 100.7 | 100.6 | 100.9 | 100.8 | |

2500 | 9 | 0.08 | 126.9 | 119.8 | 120.4 | 119.5 | 120.4 | 114.4 | 116.4 | 122.8 | 111.3 | 103.4 | 103.4 | 105.8 | 103.4 | 103.4 | 103.3 | 111.2 |

_{opt}= choice of optimal a parameter based on AIC or BIC with corresponding optimal $\lambda $ parameter.

**Table 4.**Simulation Study 2: Bias of estimated group means for balanced and unbalanced DIF effects as a function of the size of DIF effects $\delta $ and sample size N.

Choice of $\mathit{\lambda}$ | JHL with $\mathit{p}=$ | HL with $\mathit{p}=$ | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\delta}$ | N | MAD | AIC | BIC | 0.05 | 0.10 | 0.15 | 0.5 | 1 | 2 | 0.5 | 1 | 2 | ${\mathit{L}}_{0.5}$ | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | CC |

Balanced DIF | |||||||||||||||||

0.5 | 500 | −0.01 | −0.01 | −0.01 | −0.02 | −0.01 | −0.04 | −0.01 | −0.01 | −0.01 | 0.00 | 0.00 | 0.01 | −0.01 | −0.01 | −0.01 | −0.04 |

1000 | 0.01 | 0.00 | 0.01 | 0.00 | 0.01 | −0.03 | 0.00 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.01 | −0.04 | |

2500 | −0.01 | −0.01 | −0.01 | −0.01 | 0.00 | −0.05 | −0.01 | −0.01 | −0.01 | 0.00 | 0.00 | 0.00 | −0.01 | −0.01 | −0.01 | −0.04 | |

5000 | 0.00 | 0.00 | 0.01 | 0.01 | 0.01 | −0.04 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | −0.04 | |

1.0 | 500 | −0.01 | −0.02 | −0.01 | −0.02 | −0.01 | −0.01 | −0.02 | −0.02 | −0.01 | 0.00 | 0.00 | 0.01 | −0.02 | −0.02 | −0.01 | −0.06 |

1000 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | −0.06 | |

2500 | −0.01 | −0.02 | −0.01 | −0.01 | −0.01 | −0.01 | −0.02 | −0.01 | −0.01 | 0.00 | 0.00 | 0.00 | −0.02 | −0.01 | −0.01 | −0.06 | |

5000 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | −0.06 | |

Unbalanced DIF | |||||||||||||||||

0.5 | 500 | −0.07 | −0.03 | −0.04 | −0.04 | −0.03 | −0.07 | −0.04 | −0.06 | −0.10 | −0.04 | −0.06 | −0.10 | −0.04 | −0.05 | −0.10 | −0.10 |

1000 | −0.03 | −0.01 | 0.00 | −0.01 | −0.01 | −0.07 | −0.02 | −0.04 | −0.10 | −0.03 | −0.05 | −0.10 | −0.02 | −0.03 | −0.10 | −0.10 | |

2500 | −0.01 | −0.03 | −0.03 | −0.03 | −0.02 | −0.10 | −0.02 | −0.04 | −0.10 | −0.02 | −0.04 | −0.10 | −0.01 | −0.03 | −0.10 | −0.10 | |

5000 | 0.01 | −0.01 | 0.00 | −0.01 | 0.00 | −0.09 | 0.00 | −0.02 | −0.10 | −0.01 | −0.03 | −0.10 | 0.00 | −0.01 | −0.10 | −0.10 | |

1.0 | 500 | −0.03 | −0.03 | −0.02 | −0.06 | −0.02 | −0.02 | −0.03 | −0.07 | −0.21 | −0.03 | −0.07 | −0.20 | −0.04 | −0.06 | −0.21 | −0.17 |

1000 | 0.00 | −0.01 | 0.00 | −0.02 | 0.00 | 0.00 | −0.01 | −0.04 | −0.20 | −0.02 | −0.05 | −0.20 | −0.01 | −0.03 | −0.20 | −0.17 | |

2500 | −0.01 | −0.02 | −0.02 | −0.05 | −0.02 | −0.02 | −0.02 | −0.04 | −0.21 | −0.01 | −0.04 | −0.20 | −0.02 | −0.03 | −0.21 | −0.17 | |

5000 | 0.00 | 0.00 | 0.00 | −0.02 | 0.00 | 0.00 | 0.00 | −0.02 | −0.20 | −0.01 | −0.03 | −0.20 | 0.00 | −0.01 | −0.20 | −0.17 |

_{p}= linking employing the unweighted L

_{p}loss function with p = 0.5, 1.0, or 2.0 using joint item discriminations; CC = concurrent calibration assuming invariant item parameters; Absolute biases larger than 0.03 are printed in bold.

**Table 5.**Simulation Study 2: Relative root mean square error (RMSE) of estimated group means for balanced and unbalanced DIF effects as a function of the size of DIF effects $\delta $ and sample size N.

Choice of $\mathit{\lambda}$ | JHL with $\mathit{p}=$ | HL with $\mathit{p}=$ | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\delta}$ | N | MAD | AIC | BIC | 0.05 | 0.10 | 0.15 | 0.5 | 1 | 2 | 0.5 | 1 | 2 | ${\mathit{L}}_{0.5}$ | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | CC |

Balanced DIF | |||||||||||||||||

0.5 | 500 | 108 | 119 | 109 | 130 | 111 | 120 | 112 | 103 | 100 | 127 | 115 | 125 | 115 | 105 | 100 | 113 |

1000 | 109 | 113 | 106 | 111 | 106 | 127 | 107 | 102 | 100 | 120 | 112 | 118 | 112 | 104 | 100 | 121 | |

2500 | 104 | 185 | 117 | 183 | 139 | 178 | 105 | 103 | 100 | 113 | 110 | 116 | 111 | 105 | 100 | 152 | |

5000 | 103 | 120 | 112 | 115 | 104 | 209 | 103 | 102 | 100 | 110 | 109 | 113 | 110 | 104 | 100 | 194 | |

1.0 | 500 | 105 | 109 | 101 | 117 | 101 | 100 | 109 | 103 | 100 | 127 | 117 | 127 | 113 | 105 | 100 | 127 |

1000 | 104 | 108 | 102 | 107 | 102 | 101 | 109 | 103 | 100 | 124 | 116 | 123 | 112 | 106 | 100 | 149 | |

2500 | 105 | 113 | 100 | 127 | 100 | 100 | 106 | 104 | 102 | 114 | 111 | 116 | 112 | 107 | 102 | 192 | |

5000 | 103 | 108 | 100 | 108 | 100 | 100 | 102 | 101 | 100 | 113 | 113 | 118 | 110 | 104 | 100 | 258 | |

Unbalanced DIF | |||||||||||||||||

0.5 | 500 | 118 | 118 | 101 | 130 | 100 | 121 | 103 | 107 | 140 | 119 | 118 | 148 | 109 | 106 | 140 | 135 |

1000 | 126 | 136 | 100 | 133 | 113 | 163 | 107 | 118 | 190 | 126 | 137 | 201 | 116 | 115 | 190 | 192 | |

2500 | 105 | 293 | 272 | 299 | 212 | 269 | 107 | 134 | 288 | 118 | 146 | 284 | 113 | 123 | 288 | 276 | |

5000 | 102 | 276 | 279 | 279 | 253 | 356 | 100 | 123 | 375 | 115 | 156 | 391 | 109 | 110 | 375 | 384 | |

1.0 | 500 | 109 | 115 | 100 | 146 | 107 | 105 | 110 | 125 | 265 | 128 | 141 | 271 | 120 | 123 | 265 | 231 |

1000 | 100 | 114 | 105 | 131 | 105 | 105 | 105 | 122 | 359 | 124 | 145 | 366 | 113 | 118 | 359 | 315 | |

2500 | 101 | 179 | 169 | 308 | 163 | 171 | 108 | 144 | 536 | 113 | 146 | 529 | 112 | 131 | 536 | 459 | |

5000 | 103 | 117 | 100 | 345 | 100 | 100 | 104 | 135 | 776 | 116 | 161 | 778 | 112 | 118 | 776 | 677 |

^{p}= linking employing the unweighted L

^{p}loss function with p = 0.5, 1.0, or 2.0 using joint item discriminations; CC = concurrent calibration assuming invariant item parameters; Relative RMSE values larger than 125 are printed in bold.

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## Share and Cite

**MDPI and ACS Style**

Robitzsch, A.
Comparing Robust Linking and Regularized Estimation for Linking Two Groups in the 1PL and 2PL Models in the Presence of Sparse Uniform Differential Item Functioning. *Stats* **2023**, *6*, 192-208.
https://doi.org/10.3390/stats6010012

**AMA Style**

Robitzsch A.
Comparing Robust Linking and Regularized Estimation for Linking Two Groups in the 1PL and 2PL Models in the Presence of Sparse Uniform Differential Item Functioning. *Stats*. 2023; 6(1):192-208.
https://doi.org/10.3390/stats6010012

**Chicago/Turabian Style**

Robitzsch, Alexander.
2023. "Comparing Robust Linking and Regularized Estimation for Linking Two Groups in the 1PL and 2PL Models in the Presence of Sparse Uniform Differential Item Functioning" *Stats* 6, no. 1: 192-208.
https://doi.org/10.3390/stats6010012