Implementation Aspects in Simulation Extrapolation-Based Stocking–Lord Linking
Abstract
:1. Introduction
2. Stocking–Lord Linking
3. Proposed Computational Shortcuts in SIMEX-Based Linking
3.1. A Glimpse of the SIMEX Technique
3.2. Applying the SIMEX Approach to Linking Methods
3.3. Replication Method to SIMEX-Based Linking
3.4. Approximate Estimation in SIMEX-Based Linking
4. Simulation Study
4.1. Method
4.2. Results
5. Discussion
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
2PL | Two-parameter logistic |
DIF | Differential item functioning |
IRF | Item response function |
IRT | Item response theory |
RMSE | Root-mean-square error |
SD | Standard deviation |
SIMEX | Simulation extrapolation |
SL | Stocking–Lord |
TCF | Test characteristic function |
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SL | RE | RA | SE | SA | SL | RE | RA | SE | SA | |||
No DIF (DIF SD ) | ||||||||||||
10 | 500 | 0.005 | 0.005 | 0.005 | 0.004 | 0.004 | 0.007 | 0.008 | 0.008 | 0.008 | 0.008 | |
1000 | 0.001 | 0.001 | 0.001 | 0.000 | 0.000 | 0.003 | 0.004 | 0.004 | 0.004 | 0.004 | ||
2000 | −0.001 | −0.001 | −0.001 | −0.001 | −0.001 | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 | ||
4000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | −0.001 | −0.001 | −0.001 | −0.001 | −0.001 | ||
20 | 500 | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 | 0.005 | 0.006 | 0.006 | 0.006 | 0.006 | |
1000 | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | ||
2000 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 | ||
4000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | ||
30 | 500 | −0.001 | −0.001 | −0.001 | −0.002 | −0.002 | 0.007 | 0.008 | 0.008 | 0.008 | 0.008 | |
1000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 | ||
2000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.002 | 0.002 | 0.002 | 0.002 | ||
4000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | ||
Small DIF (DIF SD ) | ||||||||||||
10 | 500 | 0.001 | 0.003 | 0.003 | −0.004 | −0.004 | −0.002 | 0.006 | 0.006 | 0.004 | 0.004 | |
1000 | 0.002 | 0.004 | 0.004 | −0.004 | −0.004 | −0.005 | 0.003 | 0.003 | 0.001 | 0.001 | ||
2000 | −0.001 | 0.001 | 0.001 | −0.006 | −0.007 | −0.008 | 0.000 | 0.000 | −0.001 | −0.001 | ||
4000 | −0.001 | 0.001 | 0.001 | −0.006 | −0.007 | −0.008 | 0.000 | 0.000 | −0.001 | −0.001 | ||
20 | 500 | −0.001 | 0.001 | 0.001 | 0.003 | 0.003 | −0.004 | 0.004 | 0.004 | 0.004 | 0.004 | |
1000 | −0.001 | 0.001 | 0.001 | 0.002 | 0.003 | −0.006 | 0.003 | 0.003 | 0.003 | 0.003 | ||
2000 | −0.001 | 0.001 | 0.001 | 0.003 | 0.003 | −0.009 | 0.001 | 0.001 | 0.000 | 0.000 | ||
4000 | −0.003 | 0.000 | 0.000 | 0.001 | 0.001 | −0.009 | 0.000 | 0.000 | 0.000 | 0.000 | ||
30 | 500 | −0.003 | 0.000 | 0.000 | −0.002 | −0.002 | −0.007 | 0.002 | 0.002 | 0.002 | 0.002 | |
1000 | −0.003 | −0.001 | −0.001 | −0.002 | −0.002 | −0.009 | 0.001 | 0.001 | 0.000 | 0.000 | ||
2000 | −0.003 | 0.000 | 0.000 | −0.002 | −0.002 | −0.009 | 0.001 | 0.001 | 0.001 | 0.001 | ||
4000 | −0.004 | −0.002 | −0.002 | −0.003 | −0.003 | −0.009 | 0.001 | 0.001 | 0.000 | 0.000 | ||
Large DIF (DIF SD ) | ||||||||||||
10 | 500 | −0.005 | 0.003 | 0.003 | −0.012 | −0.013 | −0.027 | 0.005 | 0.005 | −0.001 | 0.000 | |
1000 | −0.007 | 0.001 | 0.001 | −0.014 | −0.015 | −0.031 | 0.001 | 0.001 | −0.005 | −0.004 | ||
2000 | −0.007 | 0.001 | 0.001 | −0.015 | −0.015 | −0.033 | −0.001 | −0.001 | −0.007 | −0.006 | ||
4000 | −0.010 | −0.002 | −0.002 | −0.018 | −0.019 | −0.035 | −0.002 | −0.003 | −0.009 | −0.008 | ||
20 | 500 | −0.008 | 0.001 | 0.000 | 0.002 | 0.003 | −0.036 | 0.000 | −0.001 | −0.003 | −0.003 | |
1000 | −0.011 | −0.002 | −0.003 | −0.001 | 0.000 | −0.036 | −0.001 | −0.002 | −0.003 | −0.003 | ||
2000 | −0.010 | −0.001 | −0.001 | 0.001 | 0.001 | −0.037 | −0.001 | −0.002 | −0.004 | −0.004 | ||
4000 | −0.009 | 0.000 | 0.000 | 0.001 | 0.002 | −0.037 | −0.001 | −0.002 | −0.004 | −0.004 | ||
30 | 500 | −0.011 | −0.002 | −0.002 | −0.004 | −0.005 | −0.034 | 0.003 | 0.002 | 0.002 | 0.001 | |
1000 | −0.010 | 0.000 | −0.001 | −0.003 | −0.003 | −0.034 | 0.002 | 0.001 | 0.001 | 0.001 | ||
2000 | −0.010 | 0.000 | −0.001 | −0.003 | −0.004 | −0.039 | −0.002 | −0.003 | −0.004 | −0.004 | ||
4000 | −0.011 | −0.002 | −0.002 | −0.005 | −0.005 | −0.038 | −0.001 | −0.002 | −0.003 | −0.003 |
SL | RE | RA | SE | SA | SL | RE | RA | SE | SA | |||
No DIF (DIF SD ) | ||||||||||||
10 | 500 | 100.0 | 100.0 | 100.0 | 100.0 | 100 | 99.7 | 100.0 | 100.0 | 100.0 | 100 | |
1000 | 100.0 | 100.0 | 100.0 | 100.0 | 100 | 99.9 | 100.0 | 100.0 | 100.0 | 100 | ||
2000 | 100.0 | 100.0 | 100.0 | 100.0 | 100 | 99.9 | 100.0 | 100.0 | 100.0 | 100 | ||
4000 | 100.0 | 100.0 | 100.0 | 100.0 | 100 | 100.0 | 100.0 | 100.0 | 100.0 | 100 | ||
20 | 500 | 99.9 | 100.0 | 100.0 | 100.0 | 100 | 99.7 | 100.0 | 100.0 | 100.0 | 100 | |
1000 | 99.9 | 100.0 | 100.0 | 100.0 | 100 | 99.9 | 100.0 | 100.0 | 100.0 | 100 | ||
2000 | 100.0 | 100.0 | 100.0 | 100.0 | 100 | 100.0 | 100.0 | 100.0 | 100.0 | 100 | ||
4000 | 100.0 | 100.0 | 100.0 | 100.0 | 100 | 100.0 | 100.0 | 100.0 | 100.0 | 100 | ||
30 | 500 | 100.0 | 100.0 | 100.0 | 100.0 | 100 | 99.8 | 100.0 | 100.0 | 100.0 | 100 | |
1000 | 100.0 | 100.0 | 100.0 | 100.0 | 100 | 99.9 | 100.0 | 100.0 | 100.0 | 100 | ||
2000 | 100.0 | 100.0 | 100.0 | 100.0 | 100 | 100.0 | 100.0 | 100.0 | 100.0 | 100 | ||
4000 | 100.0 | 100.0 | 100.0 | 100.0 | 100 | 100.0 | 100.0 | 100.0 | 100.0 | 100 | ||
Small DIF (DIF SD ) | ||||||||||||
10 | 500 | 99.4 | 99.9 | 99.9 | 100.0 | 100 | 99.3 | 100.3 | 100.2 | 100.0 | 100 | |
1000 | 99.4 | 100.1 | 100.1 | 100.0 | 100 | 99.4 | 100.3 | 100.3 | 100.0 | 100 | ||
2000 | 99.2 | 99.8 | 99.8 | 100.0 | 100 | 100.4 | 100.3 | 100.2 | 100.1 | 100 | ||
4000 | 99.1 | 99.8 | 99.8 | 99.9 | 100 | 100.8 | 100.3 | 100.3 | 100.1 | 100 | ||
20 | 500 | 99.4 | 100.0 | 100.0 | 100.0 | 100 | 99.4 | 100.1 | 100.1 | 100.0 | 100 | |
1000 | 99.3 | 100.0 | 100.0 | 100.0 | 100 | 99.5 | 100.1 | 100.1 | 100.0 | 100 | ||
2000 | 99.2 | 100.0 | 100.0 | 100.0 | 100 | 101.1 | 100.1 | 100.1 | 100.0 | 100 | ||
4000 | 99.4 | 100.0 | 100.0 | 100.0 | 100 | 102.3 | 100.1 | 100.0 | 100.0 | 100 | ||
30 | 500 | 99.5 | 100.0 | 100.0 | 100.0 | 100 | 99.4 | 100.1 | 100.1 | 100.0 | 100 | |
1000 | 99.5 | 100.0 | 100.0 | 100.0 | 100 | 100.5 | 100.0 | 100.0 | 100.0 | 100 | ||
2000 | 99.4 | 100.0 | 100.0 | 100.0 | 100 | 101.6 | 100.1 | 100.1 | 100.0 | 100 | ||
4000 | 99.4 | 99.9 | 99.9 | 100.0 | 100 | 103.5 | 100.1 | 100.1 | 100.1 | 100 | ||
Large DIF (DIF SD ) | ||||||||||||
10 | 500 | 97.8 | 100.3 | 100.2 | 99.9 | 100 | 99.8 | 101.0 | 100.9 | 100.0 | 100 | |
1000 | 97.6 | 100.1 | 100.1 | 99.9 | 100 | 102.4 | 100.8 | 100.6 | 100.0 | 100 | ||
2000 | 97.4 | 100.0 | 99.9 | 99.9 | 100 | 104.7 | 101.0 | 100.8 | 100.3 | 100 | ||
4000 | 97.2 | 99.8 | 99.8 | 99.8 | 100 | 108.5 | 100.8 | 100.6 | 100.6 | 100 | ||
20 | 500 | 97.8 | 100.2 | 100.1 | 100.1 | 100 | 105.2 | 100.4 | 100.2 | 100.1 | 100 | |
1000 | 97.9 | 100.2 | 100.1 | 100.0 | 100 | 111.1 | 100.3 | 100.1 | 100.2 | 100 | ||
2000 | 97.6 | 100.2 | 100.2 | 100.0 | 100 | 117.5 | 100.4 | 100.3 | 100.4 | 100 | ||
4000 | 97.5 | 100.2 | 100.2 | 100.0 | 100 | 125.0 | 100.3 | 100.1 | 100.3 | 100 | ||
30 | 500 | 98.0 | 100.0 | 100.0 | 100.0 | 100 | 105.6 | 100.3 | 100.1 | 100.1 | 100 | |
1000 | 97.7 | 100.1 | 100.0 | 100.0 | 100 | 111.9 | 100.3 | 100.0 | 100.1 | 100 | ||
2000 | 97.6 | 100.1 | 100.0 | 100.0 | 100 | 126.3 | 100.1 | 100.0 | 100.2 | 100 | ||
4000 | 97.7 | 100.0 | 99.9 | 100.0 | 100 | 132.4 | 100.2 | 100.0 | 100.3 | 100 |
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Robitzsch, A. Implementation Aspects in Simulation Extrapolation-Based Stocking–Lord Linking. Appl. Sci. 2025, 15, 901. https://doi.org/10.3390/app15020901
Robitzsch A. Implementation Aspects in Simulation Extrapolation-Based Stocking–Lord Linking. Applied Sciences. 2025; 15(2):901. https://doi.org/10.3390/app15020901
Chicago/Turabian StyleRobitzsch, Alexander. 2025. "Implementation Aspects in Simulation Extrapolation-Based Stocking–Lord Linking" Applied Sciences 15, no. 2: 901. https://doi.org/10.3390/app15020901
APA StyleRobitzsch, A. (2025). Implementation Aspects in Simulation Extrapolation-Based Stocking–Lord Linking. Applied Sciences, 15(2), 901. https://doi.org/10.3390/app15020901