Computational Aspects of L0 Linking in the Rasch Model
Abstract
:1. Introduction
2. Linking in the Rasch Model
2.1. Identified Item Parameters and Mean–Mean Linking
2.2. Loss Function and Differentiable Approximations
2.3. Linking as a Robust Mean–Mean Linking in the Rasch Model
2.4. Statistical Properties of the Estimated Linking Parameter in Linking
3. Numerical Illustration
3.1. Method
3.2. Results
4. Simulation Study
4.1. Method
4.2. Results
5. Empirical Example
6. Discussion
7. Conclusions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
ANOVA | analysis of variance |
DIF | differential item functioning |
IRF | item response function |
IRT | item response theory |
ML | maximum likelihood |
MM | mean–mean |
MSE | mean square error |
RMSE | root mean square error |
SD | standard deviation |
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, | , | |||||||
---|---|---|---|---|---|---|---|---|
250 | 500 | 1000 | 250 | 500 | 1000 | |||
10 | 0.900 | 0.065 | 0.030 | 0.150 | 0.100 | 0.090 | ||
20 | 0.200 | 0.050 | 0.030 | 0.100 | 0.100 | 0.090 | ||
40 | 0.100 | 0.035 | 0.025 | 0.100 | 0.095 | 0.085 |
Ratio Function, | Gaussian Function, | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0.25 | 0.10 | 0.05 | 0.01 | 0.25 | 0.10 | 0.05 | 0.01 | |||||||||
0.4 | 10 | 125 | 0.107 | 0.103 | 0.102 | 0.102 | 0.110 | 0.105 | 0.107 | 0.101 | 0.100 | 0.101 | 0.112 | 0.106 | ||
250 | 0.098 | 0.087 | 0.082 | 0.079 | 0.096 | 0.085 | 0.101 | 0.086 | 0.080 | 0.078 | 0.101 | 0.087 | ||||
500 | 0.088 | 0.066 | 0.052 | 0.041 | 0.068 | 0.049 | 0.097 | 0.067 | 0.046 | 0.033 | 0.075 | 0.048 | ||||
1000 | 0.082 | 0.050 | 0.031 | 0.015 | 0.034 | 0.019 | 0.096 | 0.057 | 0.026 | 0.010 | 0.035 | 0.013 | ||||
20 | 125 | 0.104 | 0.100 | 0.099 | 0.099 | 0.107 | 0.102 | 0.104 | 0.099 | 0.096 | 0.096 | 0.109 | 0.103 | |||
250 | 0.098 | 0.086 | 0.080 | 0.075 | 0.097 | 0.084 | 0.102 | 0.084 | 0.075 | 0.069 | 0.102 | 0.086 | ||||
500 | 0.089 | 0.066 | 0.051 | 0.036 | 0.069 | 0.048 | 0.097 | 0.068 | 0.044 | 0.028 | 0.076 | 0.046 | ||||
1000 | 0.081 | 0.049 | 0.028 | 0.009 | 0.031 | 0.014 | 0.095 | 0.056 | 0.023 | 0.003 | 0.032 | 0.008 | ||||
40 | 125 | 0.105 | 0.101 | 0.098 | 0.096 | 0.108 | 0.103 | 0.106 | 0.099 | 0.096 | 0.096 | 0.110 | 0.105 | |||
250 | 0.098 | 0.086 | 0.077 | 0.070 | 0.097 | 0.083 | 0.101 | 0.084 | 0.071 | 0.063 | 0.101 | 0.086 | ||||
500 | 0.099 | 0.076 | 0.059 | 0.039 | 0.079 | 0.055 | 0.107 | 0.078 | 0.051 | 0.029 | 0.087 | 0.054 | ||||
1000 | 0.087 | 0.054 | 0.033 | 0.012 | 0.036 | 0.018 | 0.100 | 0.062 | 0.027 | 0.006 | 0.036 | 0.012 | ||||
0.8 | 10 | 125 | 0.114 | 0.093 | 0.087 | 0.084 | 0.138 | 0.102 | 0.105 | 0.078 | 0.074 | 0.074 | 0.152 | 0.099 | ||
250 | 0.069 | 0.037 | 0.027 | 0.023 | 0.064 | 0.033 | 0.066 | 0.021 | 0.015 | 0.015 | 0.067 | 0.023 | ||||
500 | 0.048 | 0.017 | 0.008 | 0.003 | 0.019 | 0.006 | 0.047 | 0.004 | 0.000 | 0.001 | 0.009 | 0.000 | ||||
1000 | 0.039 | 0.011 | 0.004 | 0.001 | 0.005 | 0.001 | 0.040 | 0.002 | 0.000 | 0.000 | 0.000 | 0.000 | ||||
20 | 125 | 0.112 | 0.082 | 0.073 | 0.069 | 0.140 | 0.096 | 0.103 | 0.063 | 0.055 | 0.054 | 0.155 | 0.095 | |||
250 | 0.069 | 0.033 | 0.020 | 0.014 | 0.063 | 0.029 | 0.065 | 0.015 | 0.007 | 0.005 | 0.065 | 0.017 | ||||
500 | 0.047 | 0.015 | 0.006 | 0.000 | 0.017 | 0.004 | 0.046 | 0.003 | 0.002 | 0.003 | 0.007 | 0.001 | ||||
1000 | 0.038 | 0.011 | 0.004 | 0.000 | 0.005 | 0.001 | 0.039 | 0.002 | 0.000 | 0.000 | 0.000 | 0.000 | ||||
40 | 125 | 0.109 | 0.075 | 0.061 | 0.054 | 0.140 | 0.092 | 0.099 | 0.052 | 0.041 | 0.036 | 0.156 | 0.090 | |||
250 | 0.071 | 0.033 | 0.020 | 0.011 | 0.065 | 0.029 | 0.067 | 0.015 | 0.006 | 0.005 | 0.067 | 0.018 | ||||
500 | 0.051 | 0.020 | 0.011 | 0.005 | 0.022 | 0.009 | 0.050 | 0.007 | 0.003 | 0.003 | 0.011 | 0.003 | ||||
1000 | 0.039 | 0.012 | 0.005 | 0.001 | 0.006 | 0.002 | 0.040 | 0.002 | 0.001 | 0.001 | 0.001 | 0.001 |
Ratio Function, | Gaussian Function, | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0.25 | 0.10 | 0.05 | 0.01 | 0.25 | 0.10 | 0.05 | 0.01 | |||||||||
0.4 | 10 | 125 | 83.5 | 90.5 | 95.0 | 100 | 80.2 | 86.3 | 83.0 | 92.9 | 99.0 | 105.2 | 79.1 | 84.0 | ||
250 | 83.4 | 88.1 | 93.3 | 100 | 83.7 | 89.4 | 83.2 | 89.1 | 96.6 | 105.8 | 83.2 | 88.1 | ||||
500 | 92.6 | 89.0 | 91.0 | 100 | 89.1 | 92.0 | 95.6 | 89.8 | 92.3 | 103.8 | 90.5 | 91.6 | ||||
1000 | 123.1 | 102.5 | 94.8 | 100 | 95.5 | 96.2 | 134.4 | 108.1 | 94.6 | 102.4 | 97.5 | 94.0 | ||||
20 | 125 | 83.2 | 88.6 | 93.8 | 100 | 81.8 | 85.1 | 83.0 | 90.9 | 98.1 | 105.6 | 81.4 | 83.6 | |||
250 | 85.9 | 86.9 | 91.3 | 100 | 85.7 | 87.8 | 86.4 | 87.0 | 93.7 | 104.5 | 86.4 | 86.5 | ||||
500 | 101.9 | 93.4 | 91.7 | 100 | 94.0 | 92.1 | 106.3 | 94.1 | 91.0 | 103.2 | 96.8 | 90.7 | ||||
1000 | 141.1 | 111.1 | 97.7 | 100 | 99.2 | 95.7 | 155.9 | 118.4 | 95.8 | 100.4 | 100.7 | 91.7 | ||||
40 | 125 | 85.6 | 88.9 | 93.5 | 100 | 85.2 | 86.5 | 85.4 | 89.8 | 97.1 | 105.9 | 85.2 | 85.6 | |||
250 | 91.5 | 89.7 | 91.6 | 100 | 91.1 | 89.9 | 92.4 | 89.3 | 92.5 | 104.4 | 92.4 | 89.3 | ||||
500 | 120.2 | 105.7 | 97.9 | 100 | 107.2 | 96.8 | 126.2 | 107.1 | 94.9 | 101.5 | 112.2 | 95.7 | ||||
1000 | 169.0 | 127.9 | 106.9 | 100 | 109.1 | 98.3 | 187.6 | 137.6 | 103.0 | 99.7 | 110.8 | 94.2 | ||||
0.8 | 10 | 125 | 87.2 | 92.6 | 96.0 | 100 | 85.6 | 89.3 | 87.9 | 95.9 | 100.5 | 106.1 | 86.8 | 88.7 | ||
250 | 90.3 | 89.4 | 92.8 | 100 | 89.6 | 90.2 | 91.4 | 89.1 | 94.6 | 106.2 | 91.4 | 89.0 | ||||
500 | 94.8 | 87.2 | 89.8 | 100 | 87.1 | 90.7 | 97.7 | 84.0 | 87.5 | 105.3 | 84.6 | 86.6 | ||||
1000 | 101.5 | 87.9 | 88.4 | 100 | 88.0 | 93.7 | 106.2 | 85.6 | 86.5 | 102.4 | 87.2 | 91.9 | ||||
20 | 125 | 88.7 | 90.9 | 95.0 | 100 | 91.6 | 88.7 | 88.5 | 92.2 | 97.6 | 104.7 | 94.4 | 88.2 | |||
250 | 94.6 | 88.3 | 90.9 | 100 | 92.9 | 88.5 | 94.8 | 85.6 | 90.8 | 104.9 | 94.7 | 85.2 | ||||
500 | 97.2 | 86.3 | 87.6 | 100 | 86.6 | 88.6 | 99.0 | 83.7 | 86.1 | 104.1 | 84.2 | 85.5 | ||||
1000 | 106.8 | 90.5 | 90.2 | 100 | 90.0 | 94.0 | 109.8 | 88.1 | 88.7 | 102.4 | 88.3 | 91.4 | ||||
40 | 125 | 92.1 | 90.7 | 94.0 | 100 | 98.3 | 90.2 | 90.2 | 89.4 | 95.3 | 103.8 | 102.7 | 88.8 | |||
250 | 101.0 | 90.8 | 91.2 | 100 | 98.6 | 90.4 | 100.3 | 87.5 | 90.1 | 104.0 | 100.1 | 87.6 | ||||
500 | 105.4 | 90.3 | 89.7 | 100 | 90.8 | 90.2 | 106.1 | 87.0 | 87.9 | 104.9 | 87.7 | 87.6 | ||||
1000 | 116.4 | 94.8 | 93.3 | 100 | 93.3 | 95.4 | 118.6 | 92.0 | 92.2 | 101.8 | 91.9 | 93.7 |
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Robitzsch, A. Computational Aspects of L0 Linking in the Rasch Model. Algorithms 2025, 18, 213. https://doi.org/10.3390/a18040213
Robitzsch A. Computational Aspects of L0 Linking in the Rasch Model. Algorithms. 2025; 18(4):213. https://doi.org/10.3390/a18040213
Chicago/Turabian StyleRobitzsch, Alexander. 2025. "Computational Aspects of L0 Linking in the Rasch Model" Algorithms 18, no. 4: 213. https://doi.org/10.3390/a18040213
APA StyleRobitzsch, A. (2025). Computational Aspects of L0 Linking in the Rasch Model. Algorithms, 18(4), 213. https://doi.org/10.3390/a18040213