Bias-Reduced Haebara and Stocking–Lord Linking
Abstract
:1. Introduction
2. Bias Reduction in Haebara and Stocking–Lord Linking
2.1. Group Comparisons in the 2PL Model
2.2. Bias Reduction in a Linking Method Due to DIF
2.3. Bias-Reduced Haebara Linking (brHAE)
2.4. Bias-Reduced Stocking–Lord Linking (brSL)
2.5. Mean-Geometric-Mean Linking (MGM)
3. Simulation Study
3.1. Method
3.2. Results
4. Discussion
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
2PL | two-parameter logistic |
brHAE | bias-reduced Haebara |
brSL | bias-reduced Stocking–Lord |
HAE | Haebara |
IRF | item response function |
IRT | item response theory |
MGM | mean-geometric-mean |
RMSE | root mean square error |
SD | standard deviation |
SL | Stocking–Lord |
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Bias | SD | Relative RMSE | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
HAE | brHAE | SL | brSL | MGM | HAE | brHAE | SL | brSL | MGM | HAE | brHAE | SL | brSL | MGM | ||||
No DIF (DIF SD ) | ||||||||||||||||||
10 | 500 | 0.008 | 0.012 | 0.003 | 0.010 | 0.004 | 0.094 | 0.095 | 0.091 | 0.092 | 0.103 | 100 ‡ | 101.7 | 96.7 | 98.1 | 109.9 | ||
1000 | 0.004 | 0.006 | 0.002 | 0.005 | 0.002 | 0.064 | 0.064 | 0.063 | 0.063 | 0.069 | 100 ‡ | 100.7 | 97.3 | 97.9 | 108.1 | |||
2000 | 0.002 | 0.003 | 0.001 | 0.002 | 0.001 | 0.046 | 0.046 | 0.045 | 0.045 | 0.050 | 100 ‡ | 100.3 | 97.0 | 97.3 | 107.8 | |||
4000 | 0.000 | 0.001 | 0.000 | 0.000 | 0.000 | 0.032 | 0.032 | 0.031 | 0.031 | 0.035 | 100 ‡ | 100.1 | 97.1 | 97.2 | 107.3 | |||
Inf | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 100 ‡ | 100.0 | 100.0 | 100.0 | 100.0 | |||
20 | 500 | 0.007 | 0.011 | 0.003 | 0.009 | 0.003 | 0.082 | 0.083 | 0.081 | 0.082 | 0.087 | 100 ‡ | 101.4 | 98.6 | 99.7 | 106.1 | ||
1000 | 0.004 | 0.006 | 0.002 | 0.005 | 0.002 | 0.058 | 0.058 | 0.057 | 0.057 | 0.060 | 100 ‡ | 100.7 | 98.3 | 98.9 | 104.8 | |||
2000 | 0.002 | 0.003 | 0.001 | 0.002 | 0.001 | 0.041 | 0.041 | 0.040 | 0.040 | 0.042 | 100 ‡ | 100.3 | 99.2 | 99.4 | 104.4 | |||
4000 | 0.000 | 0.000 | −0.001 | 0.000 | −0.001 | 0.029 | 0.029 | 0.028 | 0.028 | 0.030 | 100 ‡ | 100.1 | 98.7 | 98.8 | 103.4 | |||
Inf | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 100 ‡ | 100.0 | 100.0 | 100.0 | 100.0 | |||
40 | 500 | 0.007 | 0.011 | 0.003 | 0.009 | 0.004 | 0.077 | 0.078 | 0.077 | 0.077 | 0.080 | 100 ‡ | 101.4 | 99.4 | 100.6 | 102.9 | ||
1000 | 0.002 | 0.004 | 0.000 | 0.003 | 0.000 | 0.055 | 0.055 | 0.054 | 0.054 | 0.056 | 100 ‡ | 100.6 | 99.3 | 99.7 | 102.3 | |||
2000 | 0.002 | 0.003 | 0.001 | 0.002 | 0.001 | 0.039 | 0.039 | 0.039 | 0.039 | 0.040 | 100 ‡ | 100.3 | 98.9 | 99.2 | 102.3 | |||
4000 | 0.000 | 0.000 | −0.001 | 0.000 | 0.000 | 0.027 | 0.028 | 0.027 | 0.027 | 0.028 | 100 ‡ | 100.1 | 99.5 | 99.6 | 101.8 | |||
Inf | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 100 ‡ | 100.0 | 100.0 | 100.0 | 100.0 | |||
Bias | SD | Relative RMSE | ||||||||||||||||
HAE | brHAE | SL | brSL | MGM | HAE | brHAE | SL | brSL | MGM | HAE | brHAE | SL | brSL | MGM | ||||
Moderate Uniform DIF (DIF SD ) | ||||||||||||||||||
10 | 500 | 0.006 | 0.015 | 0.000 | 0.012 | 0.005 | 0.127 | 0.130 | 0.121 | 0.123 | 0.131 | 100 ‡ | 103.5 | 95.5 | 97.8 | 103.2 | ||
1000 | 0.001 | 0.007 | −0.003 | 0.006 | 0.003 | 0.107 | 0.110 | 0.102 | 0.104 | 0.107 | 100 ‡ | 102.2 | 95.4 | 96.7 | 99.9 | |||
2000 | −0.002 | 0.003 | −0.005 | 0.001 | 0.000 | 0.096 | 0.097 | 0.091 | 0.092 | 0.094 | 100 ‡ | 101.9 | 95.5 | 96.5 | 97.9 | |||
4000 | −0.003 | 0.002 | −0.006 | 0.000 | −0.001 | 0.091 | 0.092 | 0.086 | 0.087 | 0.087 | 100 ‡ | 101.7 | 95.4 | 96.2 | 96.0 | |||
Inf | −0.004 | 0.000 | −0.006 | −0.001 | 0.000 | 0.085 | 0.086 | 0.080 | 0.081 | 0.079 | 100 ‡ | 101.7 | 95.0 | 95.7 | 93.5 | |||
20 | 500 | 0.003 | 0.012 | −0.002 | 0.010 | 0.004 | 0.100 | 0.103 | 0.097 | 0.098 | 0.102 | 100 ‡ | 103.4 | 96.6 | 98.8 | 101.7 | ||
1000 | 0.000 | 0.007 | −0.003 | 0.005 | 0.003 | 0.083 | 0.085 | 0.080 | 0.081 | 0.082 | 100 ‡ | 102.3 | 96.6 | 97.8 | 99.4 | |||
2000 | −0.001 | 0.005 | −0.003 | 0.004 | 0.003 | 0.072 | 0.073 | 0.069 | 0.070 | 0.070 | 100 ‡ | 102.0 | 96.1 | 97.1 | 97.6 | |||
4000 | −0.003 | 0.002 | −0.005 | 0.001 | 0.001 | 0.066 | 0.067 | 0.063 | 0.064 | 0.063 | 100 ‡ | 101.8 | 96.0 | 96.8 | 95.5 | |||
Inf | −0.005 | 0.000 | −0.006 | 0.000 | −0.001 | 0.059 | 0.060 | 0.056 | 0.057 | 0.056 | 100 ‡ | 101.5 | 95.9 | 96.3 | 94.3 | |||
40 | 500 | 0.000 | 0.009 | −0.005 | 0.007 | 0.002 | 0.087 | 0.089 | 0.086 | 0.087 | 0.088 | 100 ‡ | 102.8 | 98.3 | 99.7 | 101.0 | ||
1000 | −0.001 | 0.006 | −0.004 | 0.004 | 0.002 | 0.068 | 0.070 | 0.067 | 0.067 | 0.068 | 100 ‡ | 102.2 | 98.0 | 99.0 | 100.1 | |||
2000 | −0.004 | 0.002 | −0.005 | 0.001 | 0.001 | 0.057 | 0.058 | 0.055 | 0.056 | 0.056 | 100 ‡ | 101.6 | 97.1 | 97.6 | 97.3 | |||
4000 | −0.003 | 0.002 | −0.005 | 0.001 | 0.000 | 0.050 | 0.051 | 0.048 | 0.048 | 0.048 | 100 ‡ | 101.7 | 96.4 | 96.9 | 95.6 | |||
Inf | −0.007 | −0.002 | −0.008 | −0.002 | −0.002 | 0.043 | 0.043 | 0.041 | 0.041 | 0.040 | 100 ‡ | 100.6 | 95.7 | 95.2 | 93.0 | |||
Large Uniform DIF (DIF SD ) | ||||||||||||||||||
10 | 500 | −0.008 | 0.014 | −0.019 | 0.007 | 0.003 | 0.188 | 0.203 | 0.177 | 0.186 | 0.189 | 100 ‡ | 107.8 | 94.5 | 98.4 | 100.2 | ||
1000 | −0.012 | 0.006 | −0.020 | 0.001 | 0.002 | 0.176 | 0.187 | 0.168 | 0.174 | 0.174 | 100 ‡ | 106.0 | 95.7 | 98.6 | 98.6 | |||
2000 | −0.017 | −0.001 | −0.024 | −0.005 | −0.004 | 0.172 | 0.182 | 0.162 | 0.168 | 0.167 | 100 ‡ | 105.5 | 95.0 | 97.5 | 96.6 | |||
4000 | −0.012 | 0.003 | −0.019 | −0.001 | 0.001 | 0.169 | 0.179 | 0.160 | 0.165 | 0.162 | 100 ‡ | 105.4 | 94.9 | 97.3 | 95.5 | |||
Inf | −0.011 | 0.005 | −0.018 | 0.000 | 0.002 | 0.165 | 0.175 | 0.156 | 0.161 | 0.156 | 100 ‡ | 105.8 | 94.6 | 97.3 | 94.4 | |||
20 | 500 | −0.008 | 0.014 | −0.017 | 0.009 | 0.007 | 0.144 | 0.154 | 0.137 | 0.142 | 0.144 | 100 ‡ | 107.0 | 95.8 | 99.0 | 100.2 | ||
1000 | −0.013 | 0.004 | −0.020 | 0.001 | 0.003 | 0.130 | 0.138 | 0.124 | 0.128 | 0.127 | 100 ‡ | 105.5 | 96.0 | 98.1 | 97.5 | |||
2000 | −0.015 | 0.001 | −0.020 | −0.001 | 0.003 | 0.121 | 0.128 | 0.116 | 0.120 | 0.118 | 100 ‡ | 105.0 | 96.2 | 98.0 | 97.0 | |||
4000 | −0.019 | −0.003 | −0.024 | −0.005 | −0.003 | 0.122 | 0.129 | 0.115 | 0.119 | 0.116 | 100 ‡ | 104.8 | 95.7 | 97.2 | 94.3 | |||
Inf | −0.015 | 0.000 | −0.020 | −0.002 | 0.002 | 0.115 | 0.122 | 0.110 | 0.113 | 0.111 | 100 ‡ | 104.8 | 95.8 | 97.5 | 95.3 | |||
40 | 500 | −0.011 | 0.009 | −0.019 | 0.006 | 0.006 | 0.112 | 0.118 | 0.108 | 0.112 | 0.113 | 100 ‡ | 105.9 | 97.5 | 99.7 | 100.7 | ||
1000 | −0.015 | 0.002 | −0.021 | 0.000 | 0.003 | 0.099 | 0.104 | 0.095 | 0.098 | 0.098 | 100 ‡ | 104.3 | 97.6 | 98.5 | 98.7 | |||
2000 | −0.018 | −0.002 | −0.023 | −0.003 | 0.000 | 0.092 | 0.097 | 0.088 | 0.091 | 0.090 | 100 ‡ | 103.8 | 96.9 | 97.2 | 95.9 | |||
4000 | −0.016 | −0.001 | −0.021 | −0.002 | 0.001 | 0.087 | 0.092 | 0.082 | 0.085 | 0.083 | 100 ‡ | 104.1 | 96.2 | 96.7 | 94.3 | |||
Inf | −0.019 | −0.004 | −0.023 | −0.005 | 0.000 | 0.082 | 0.087 | 0.077 | 0.080 | 0.078 | 100 ‡ | 103.4 | 96.0 | 95.7 | 92.8 |
Bias | SD | Relative RMSE | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
HAE | brHAE | SL | brSL | MGM | HAE | brHAE | SL | brSL | MGM | HAE | brHAE | SL | brSL | MGM | ||||
No DIF (DIF SD ) | ||||||||||||||||||
10 | 500 | 0.013 | 0.018 | 0.008 | 0.015 | 0.008 | 0.099 | 0.100 | 0.097 | 0.098 | 0.099 | 100 ‡ | 102.4 | 97.3 | 98.9 | 99.5 | ||
1000 | 0.008 | 0.011 | 0.005 | 0.008 | 0.005 | 0.068 | 0.068 | 0.067 | 0.067 | 0.068 | 100 ‡ | 101.2 | 98.0 | 98.9 | 100.4 | |||
2000 | 0.002 | 0.004 | 0.001 | 0.003 | 0.001 | 0.047 | 0.047 | 0.046 | 0.047 | 0.048 | 100 ‡ | 100.5 | 98.7 | 99.0 | 101.4 | |||
4000 | 0.001 | 0.002 | 0.000 | 0.001 | 0.000 | 0.033 | 0.033 | 0.033 | 0.033 | 0.034 | 100 ‡ | 100.2 | 99.1 | 99.2 | 101.7 | |||
Inf | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 100 ‡ | 100.0 | 100.0 | 100.0 | 100.0 | |||
20 | 500 | 0.010 | 0.015 | 0.005 | 0.012 | 0.005 | 0.077 | 0.078 | 0.076 | 0.076 | 0.077 | 100 ‡ | 102.4 | 97.5 | 99.0 | 98.8 | ||
1000 | 0.005 | 0.008 | 0.003 | 0.006 | 0.003 | 0.054 | 0.054 | 0.053 | 0.053 | 0.054 | 100 ‡ | 101.2 | 98.6 | 99.4 | 100.6 | |||
2000 | 0.003 | 0.004 | 0.001 | 0.003 | 0.002 | 0.037 | 0.038 | 0.037 | 0.037 | 0.038 | 100 ‡ | 100.6 | 99.0 | 99.4 | 100.5 | |||
4000 | 0.001 | 0.002 | 0.000 | 0.001 | 0.000 | 0.027 | 0.027 | 0.026 | 0.026 | 0.027 | 100 ‡ | 100.3 | 99.0 | 99.2 | 100.7 | |||
Inf | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 100 ‡ | 100.0 | 100.0 | 100.0 | 100.0 | |||
40 | 500 | 0.007 | 0.012 | 0.003 | 0.009 | 0.003 | 0.066 | 0.067 | 0.065 | 0.066 | 0.066 | 100 ‡ | 102.4 | 98.1 | 99.6 | 99.0 | ||
1000 | 0.004 | 0.007 | 0.002 | 0.005 | 0.002 | 0.046 | 0.046 | 0.046 | 0.046 | 0.046 | 100 ‡ | 101.2 | 98.6 | 99.4 | 99.3 | |||
2000 | 0.002 | 0.003 | 0.000 | 0.002 | 0.000 | 0.033 | 0.033 | 0.032 | 0.032 | 0.033 | 100 ‡ | 100.5 | 99.1 | 99.4 | 99.8 | |||
4000 | 0.000 | 0.001 | −0.001 | 0.000 | −0.001 | 0.022 | 0.022 | 0.022 | 0.022 | 0.022 | 100 ‡ | 100.2 | 99.4 | 99.4 | 100.5 | |||
Inf | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 100 ‡ | 100.0 | 100.0 | 100.0 | 100.0 | |||
Moderate Uniform DIF (DIF SD ) | ||||||||||||||||||
10 | 500 | 0.007 | 0.020 | −0.001 | 0.015 | 0.009 | 0.108 | 0.113 | 0.099 | 0.101 | 0.098 | 100 ‡ | 105.9 | 91.3 | 94.4 | 90.6 | ||
1000 | 0.000 | 0.010 | −0.005 | 0.007 | 0.005 | 0.081 | 0.084 | 0.072 | 0.073 | 0.069 | 100 ‡ | 104.5 | 88.5 | 90.5 | 84.8 | |||
2000 | −0.002 | 0.007 | −0.007 | 0.004 | 0.002 | 0.065 | 0.068 | 0.053 | 0.055 | 0.048 | 100 ‡ | 104.6 | 82.7 | 84.5 | 74.0 | |||
4000 | −0.005 | 0.004 | −0.008 | 0.002 | 0.001 | 0.056 | 0.058 | 0.042 | 0.044 | 0.033 | 100 ‡ | 104.4 | 76.7 | 77.8 | 59.6 | |||
Inf | −0.007 | 0.001 | −0.010 | 0.000 | 0.000 | 0.044 | 0.046 | 0.026 | 0.028 | 0.000 | 100 ‡ | 103.9 | 63.1 | 62.7 | 0.0 | |||
20 | 500 | 0.002 | 0.015 | −0.005 | 0.012 | 0.006 | 0.085 | 0.088 | 0.079 | 0.081 | 0.079 | 100 ‡ | 105.6 | 93.4 | 96.2 | 92.9 | ||
1000 | −0.003 | 0.008 | −0.008 | 0.005 | 0.003 | 0.062 | 0.064 | 0.056 | 0.057 | 0.053 | 100 ‡ | 104.1 | 90.9 | 92.0 | 86.6 | |||
2000 | −0.006 | 0.004 | −0.009 | 0.003 | 0.002 | 0.050 | 0.052 | 0.042 | 0.043 | 0.038 | 100 ‡ | 103.7 | 85.9 | 86.5 | 76.6 | |||
4000 | −0.009 | 0.000 | −0.011 | 0.000 | 0.000 | 0.041 | 0.043 | 0.032 | 0.033 | 0.026 | 100 ‡ | 102.0 | 80.7 | 78.6 | 62.8 | |||
Inf | −0.008 | 0.001 | −0.010 | 0.000 | 0.000 | 0.031 | 0.032 | 0.018 | 0.020 | 0.000 | 100 ‡ | 102.3 | 66.3 | 61.9 | 0.0 | |||
40 | 500 | −0.001 | 0.012 | −0.007 | 0.009 | 0.004 | 0.070 | 0.073 | 0.066 | 0.068 | 0.065 | 100 ‡ | 105.2 | 94.9 | 97.0 | 93.3 | ||
1000 | −0.006 | 0.005 | −0.010 | 0.004 | 0.001 | 0.051 | 0.053 | 0.047 | 0.048 | 0.046 | 100 ‡ | 103.2 | 94.0 | 94.0 | 90.2 | |||
2000 | −0.008 | 0.001 | −0.011 | 0.001 | 0.000 | 0.039 | 0.041 | 0.035 | 0.035 | 0.032 | 100 ‡ | 101.4 | 90.3 | 87.9 | 80.3 | |||
4000 | −0.009 | 0.000 | −0.012 | −0.001 | −0.002 | 0.032 | 0.033 | 0.026 | 0.027 | 0.023 | 100 ‡ | 100.1 | 86.4 | 80.8 | 68.8 | |||
Inf | −0.010 | −0.001 | −0.011 | −0.001 | 0.000 | 0.022 | 0.023 | 0.013 | 0.014 | 0.000 | 100 ‡ | 96.8 | 72.1 | 58.7 | 0.0 | |||
Bias | SD | Relative RMSE | ||||||||||||||||
HAE | brHAE | SL | brSL | MGM | HAE | brHAE | SL | brSL | MGM | HAE | brHAE | SL | brSL | MGM | ||||
Large Uniform DIF (DIF SD ) | ||||||||||||||||||
10 | 500 | −0.017 | 0.014 | −0.032 | 0.006 | 0.006 | 0.130 | 0.149 | 0.106 | 0.115 | 0.098 | 100 ‡ | 114.2 | 84.3 | 87.6 | 74.6 | ||
1000 | −0.022 | 0.007 | −0.033 | 0.001 | 0.005 | 0.108 | 0.124 | 0.082 | 0.090 | 0.069 | 100 ‡ | 112.5 | 80.5 | 82.0 | 62.9 | |||
2000 | −0.025 | 0.003 | −0.036 | −0.003 | 0.002 | 0.097 | 0.113 | 0.067 | 0.076 | 0.048 | 100 ‡ | 112.7 | 75.9 | 76.2 | 47.6 | |||
4000 | −0.025 | 0.002 | −0.036 | −0.004 | 0.000 | 0.092 | 0.108 | 0.060 | 0.070 | 0.033 | 100 ‡ | 112.9 | 73.6 | 73.2 | 35.0 | |||
Inf | −0.026 | 0.001 | −0.036 | −0.005 | 0.000 | 0.085 | 0.100 | 0.050 | 0.060 | 0.000 | 100 ‡ | 113.0 | 70.0 | 68.2 | 0.0 | |||
20 | 500 | −0.023 | 0.008 | −0.035 | 0.003 | 0.005 | 0.096 | 0.109 | 0.081 | 0.087 | 0.077 | 100 ‡ | 110.9 | 89.3 | 88.4 | 77.9 | ||
1000 | −0.029 | −0.001 | −0.039 | −0.004 | 0.002 | 0.080 | 0.091 | 0.063 | 0.068 | 0.054 | 100 ‡ | 107.1 | 86.3 | 80.2 | 62.7 | |||
2000 | −0.033 | −0.005 | −0.041 | −0.007 | 0.000 | 0.070 | 0.080 | 0.051 | 0.057 | 0.038 | 100 ‡ | 104.5 | 84.3 | 74.2 | 48.9 | |||
4000 | −0.030 | −0.002 | −0.039 | −0.005 | 0.001 | 0.064 | 0.075 | 0.043 | 0.050 | 0.027 | 100 ‡ | 105.7 | 82.1 | 70.7 | 37.9 | |||
Inf | −0.032 | −0.005 | −0.040 | −0.007 | 0.000 | 0.059 | 0.070 | 0.036 | 0.043 | 0.000 | 100 ‡ | 104.0 | 79.4 | 64.6 | 0.0 | |||
40 | 500 | −0.026 | 0.004 | −0.037 | 0.002 | 0.005 | 0.080 | 0.089 | 0.070 | 0.075 | 0.068 | 100 ‡ | 106.4 | 94.5 | 89.1 | 81.0 | ||
1000 | −0.033 | −0.005 | −0.041 | −0.006 | 0.001 | 0.062 | 0.070 | 0.051 | 0.056 | 0.046 | 100 ‡ | 100.1 | 93.5 | 79.2 | 65.2 | |||
2000 | −0.034 | −0.006 | −0.042 | −0.007 | −0.001 | 0.053 | 0.060 | 0.041 | 0.045 | 0.033 | 100 ‡ | 96.3 | 92.9 | 72.0 | 52.3 | |||
4000 | −0.034 | −0.007 | −0.041 | −0.007 | 0.000 | 0.048 | 0.055 | 0.033 | 0.038 | 0.023 | 100 ‡ | 95.0 | 90.9 | 66.2 | 39.0 | |||
Inf | −0.035 | −0.008 | −0.042 | −0.008 | 0.000 | 0.042 | 0.050 | 0.026 | 0.031 | 0.000 | 100 ‡ | 91.9 | 89.0 | 58.1 | 0.0 |
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Robitzsch, A. Bias-Reduced Haebara and Stocking–Lord Linking. J 2024, 7, 373-384. https://doi.org/10.3390/j7030021
Robitzsch A. Bias-Reduced Haebara and Stocking–Lord Linking. J. 2024; 7(3):373-384. https://doi.org/10.3390/j7030021
Chicago/Turabian StyleRobitzsch, Alexander. 2024. "Bias-Reduced Haebara and Stocking–Lord Linking" J 7, no. 3: 373-384. https://doi.org/10.3390/j7030021
APA StyleRobitzsch, A. (2024). Bias-Reduced Haebara and Stocking–Lord Linking. J, 7(3), 373-384. https://doi.org/10.3390/j7030021