Standard Error Estimation in Invariance Alignment
Abstract
:1. Introduction
2. Invariance Alignment
3. Standard Error and Confidence Interval Estimation in Invariance Alignment
3.1. Delta Method
3.2. Bootstrap Methods
3.3. Comparison
4. Simulation Study 1: Continuous Items
4.1. Method
4.2. Results
5. Simulation Study 2: Dichotomous Items
5.1. Method
5.2. Results
6. Discussion
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
2PL | two-parameter logistic |
BAV | average bootstrap |
BBC | bias-corrected bootstrap |
BNO | bootstrap based on normal distribution |
BPE | percentile bootstrap |
CFA | confirmatory factor analysis |
CI | confidence interval |
DIF | differential item functioning |
DM | delta method |
IA | invariance alignment |
IRF | item response function |
IRT | item response theory |
MML | marginal maximum likelihood |
RMSE | root mean square error |
SD | standard deviation |
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Crit | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Bias | 250 | −0.042 | −0.007 | 0.019 | 0.022 | −0.016 | 0.005 | −0.009 | 0.005 | |||
500 | −0.027 | 0.000 | 0.007 | 0.006 | −0.013 | 0.002 | −0.007 | 0.004 | ||||
1000 | −0.020 | −0.003 | 0.004 | 0.003 | −0.009 | 0.001 | −0.007 | 0.001 | ||||
2000 | −0.012 | 0.000 | 0.003 | 0.002 | −0.007 | 0.001 | −0.005 | 0.000 | ||||
RMSE | 250 | 0.140 | 0.138 | 0.151 | 0.172 | 0.128 | 0.135 | 0.111 | 0.116 | |||
500 | 0.092 | 0.088 | 0.101 | 0.105 | 0.084 | 0.085 | 0.073 | 0.074 | ||||
1000 | 0.066 | 0.063 | 0.068 | 0.068 | 0.057 | 0.056 | 0.051 | 0.050 | ||||
2000 | 0.046 | 0.044 | 0.046 | 0.046 | 0.039 | 0.038 | 0.035 | 0.034 |
Par | DM | BNO | BPE | BBC | BAV | DM | BNO | BPE | BBC | BAV | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|
250 | 96.4 | 96.6 | 96.2 | 93.2 | 95.4 | 96.0 | 98.0 | 97.3 | 94.2 | 96.5 | ||
500 | 96.9 | 96.7 | 96.2 | 95.1 | 96.2 | 96.6 | 97.3 | 97.0 | 95.7 | 96.8 | ||
1000 | 96.0 | 95.8 | 95.4 | 94.1 | 95.1 | 95.7 | 96.2 | 95.8 | 95.4 | 96.0 | ||
2000 | 96.0 | 95.8 | 95.5 | 95.0 | 95.6 | 95.5 | 95.8 | 95.5 | 95.4 | 95.6 | ||
250 | 96.4 | 96.5 | 97.1 | 94.3 | 95.7 | 94.3 | 97.2 | 98.1 | 94.7 | 96.4 | ||
500 | 97.0 | 97.1 | 96.8 | 95.6 | 96.5 | 96.2 | 98.3 | 97.2 | 96.5 | 97.7 | ||
1000 | 96.4 | 96.6 | 96.6 | 95.2 | 95.9 | 96.3 | 97.5 | 96.7 | 96.2 | 97.0 | ||
2000 | 96.1 | 96.0 | 96.0 | 95.4 | 95.7 | 95.4 | 96.1 | 95.7 | 95.8 | 96.0 | ||
250 | 96.8 | 96.0 | 96.9 | 93.8 | 95.3 | 95.9 | 97.3 | 97.8 | 95.8 | 96.9 | ||
500 | 96.9 | 96.5 | 97.0 | 95.2 | 96.0 | 96.1 | 97.5 | 97.6 | 96.4 | 97.1 | ||
1000 | 96.4 | 96.3 | 96.4 | 94.9 | 95.8 | 95.9 | 96.5 | 96.5 | 96.1 | 96.4 | ||
2000 | 96.2 | 96.3 | 96.3 | 95.7 | 96.2 | 95.6 | 96.2 | 95.9 | 95.7 | 96.1 | ||
250 | 97.0 | 96.5 | 97.4 | 94.1 | 95.9 | 96.1 | 97.6 | 98.2 | 95.5 | 97.0 | ||
500 | 96.7 | 96.5 | 97.2 | 95.0 | 95.9 | 96.4 | 97.5 | 97.8 | 96.5 | 97.0 | ||
1000 | 96.3 | 96.1 | 96.4 | 94.9 | 95.6 | 95.8 | 96.7 | 96.7 | 96.0 | 96.4 | ||
2000 | 96.0 | 96.1 | 96.2 | 95.5 | 95.8 | 95.7 | 96.1 | 96.2 | 96.0 | 96.1 |
Crit | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Bias | 15 | 500 | 0.040 | 0.031 | −0.099 | −0.068 | 0.061 | 0.069 | 0.053 | 0.059 | |||
1000 | 0.025 | 0.011 | −0.064 | −0.018 | 0.028 | 0.032 | 0.027 | 0.029 | |||||
2000 | 0.014 | 0.003 | −0.035 | 0.000 | 0.015 | 0.016 | 0.013 | 0.014 | |||||
4000 | 0.008 | 0.001 | −0.023 | 0.000 | 0.007 | 0.007 | 0.006 | 0.006 | |||||
30 | 500 | 0.037 | 0.023 | −0.100 | −0.055 | 0.047 | 0.053 | 0.043 | 0.047 | ||||
1000 | 0.022 | 0.006 | −0.061 | −0.009 | 0.022 | 0.023 | 0.020 | 0.021 | |||||
2000 | 0.015 | 0.004 | −0.035 | 0.000 | 0.009 | 0.009 | 0.009 | 0.010 | |||||
4000 | 0.008 | 0.001 | −0.022 | 0.000 | 0.004 | 0.004 | 0.005 | 0.005 | |||||
RMSE | 15 | 500 | 0.124 | 0.146 | 0.160 | 0.180 | 0.139 | 0.170 | 0.121 | 0.149 | |||
1000 | 0.084 | 0.091 | 0.110 | 0.114 | 0.088 | 0.104 | 0.076 | 0.091 | |||||
2000 | 0.056 | 0.055 | 0.068 | 0.063 | 0.057 | 0.063 | 0.050 | 0.056 | |||||
4000 | 0.037 | 0.036 | 0.045 | 0.038 | 0.038 | 0.039 | 0.034 | 0.035 | |||||
30 | 500 | 0.109 | 0.123 | 0.145 | 0.149 | 0.105 | 0.128 | 0.093 | 0.113 | ||||
1000 | 0.072 | 0.073 | 0.094 | 0.081 | 0.067 | 0.075 | 0.058 | 0.066 | |||||
2000 | 0.049 | 0.047 | 0.058 | 0.049 | 0.044 | 0.046 | 0.039 | 0.041 | |||||
4000 | 0.033 | 0.032 | 0.039 | 0.032 | 0.030 | 0.030 | 0.026 | 0.026 |
Par | DM | BNO | BPE | BBC | BAV | DM | BNO | BPE | BBC | BAV | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
15 | 500 | 99.3 | 97.9 | 98.9 | 92.0 | 95.4 | 98.2 | 98.8 | 99.6 | 91.3 | 96.0 | ||
1000 | 99.2 | 97.9 | 98.5 | 93.6 | 96.0 | 98.5 | 99.3 | 99.6 | 95.3 | 98.2 | |||
2000 | 98.8 | 97.9 | 97.9 | 95.1 | 96.9 | 98.3 | 98.9 | 99.0 | 96.7 | 98.2 | |||
4000 | 98.1 | 97.1 | 97.4 | 95.3 | 96.6 | 97.7 | 97.8 | 97.9 | 96.7 | 97.6 | |||
30 | 500 | 99.2 | 96.8 | 97.9 | 92.1 | 94.8 | 98.6 | 98.4 | 99.4 | 92.0 | 96.2 | ||
1000 | 99.3 | 97.3 | 97.5 | 93.7 | 95.9 | 98.9 | 99.0 | 99.2 | 96.6 | 98.4 | |||
2000 | 98.6 | 97.0 | 97.0 | 94.8 | 96.4 | 98.6 | 98.5 | 98.5 | 97.0 | 97.9 | |||
4000 | 97.7 | 96.7 | 96.8 | 95.4 | 96.0 | 97.4 | 97.1 | 97.3 | 96.2 | 96.8 | |||
15 | 500 | 99.3 | 97.6 | 99.4 | 88.3 | 94.0 | 96.8 | 98.6 | 99.9 | 86.3 | 93.5 | ||
1000 | 99.2 | 97.3 | 98.8 | 89.3 | 94.5 | 97.3 | 98.3 | 99.8 | 90.8 | 95.8 | |||
2000 | 99.4 | 98.6 | 97.7 | 93.6 | 97.2 | 99.1 | 99.5 | 99.5 | 96.5 | 98.9 | |||
4000 | 99.1 | 97.9 | 98.0 | 94.1 | 96.6 | 99.0 | 98.9 | 99.3 | 96.9 | 98.2 | |||
30 | 500 | 99.6 | 96.5 | 98.3 | 89.5 | 93.8 | 98.0 | 97.9 | 99.6 | 87.2 | 94.0 | ||
1000 | 99.6 | 97.6 | 97.4 | 90.1 | 95.2 | 99.1 | 99.2 | 99.4 | 94.7 | 98.1 | |||
2000 | 99.3 | 98.0 | 96.3 | 92.9 | 96.6 | 99.2 | 99.2 | 99.3 | 97.1 | 98.4 | |||
4000 | 98.5 | 97.1 | 96.4 | 93.8 | 96.0 | 97.8 | 97.6 | 98.1 | 96.6 | 97.3 | |||
15 | 500 | 99.8 | 98.8 | 99.0 | 89.3 | 95.7 | 99.0 | 99.8 | 100.0 | 89.7 | 97.3 | ||
1000 | 99.8 | 98.5 | 99.0 | 91.2 | 96.1 | 99.5 | 99.5 | 99.9 | 92.7 | 97.7 | |||
2000 | 99.6 | 98.2 | 98.9 | 93.3 | 96.3 | 99.0 | 99.6 | 99.8 | 95.4 | 98.3 | |||
4000 | 99.2 | 97.5 | 98.4 | 94.2 | 96.2 | 98.7 | 98.9 | 99.4 | 96.7 | 98.2 | |||
30 | 500 | 99.8 | 97.9 | 97.4 | 90.9 | 95.3 | 99.3 | 99.2 | 99.9 | 90.6 | 96.7 | ||
1000 | 99.7 | 97.6 | 98.0 | 92.4 | 95.9 | 99.5 | 99.3 | 99.8 | 94.7 | 97.9 | |||
2000 | 99.3 | 97.3 | 97.6 | 93.8 | 95.7 | 99.1 | 98.9 | 99.2 | 96.1 | 97.9 | |||
4000 | 98.6 | 96.7 | 97.1 | 94.8 | 96.0 | 97.9 | 97.8 | 98.0 | 96.5 | 97.3 | |||
15 | 500 | 99.8 | 98.7 | 98.9 | 89.7 | 95.9 | 99.1 | 99.5 | 99.9 | 90.1 | 97.2 | ||
1000 | 99.7 | 98.4 | 99.2 | 91.8 | 96.1 | 99.2 | 99.4 | 99.9 | 92.4 | 97.4 | |||
2000 | 99.7 | 98.3 | 98.8 | 93.5 | 96.5 | 99.5 | 99.6 | 99.9 | 95.9 | 98.5 | |||
4000 | 99.2 | 97.6 | 98.6 | 94.3 | 96.0 | 98.8 | 99.0 | 99.6 | 96.6 | 97.9 | |||
30 | 500 | 99.9 | 97.9 | 97.4 | 90.3 | 95.5 | 99.6 | 99.5 | 99.8 | 90.5 | 96.9 | ||
1000 | 99.8 | 97.8 | 98.0 | 92.4 | 96.0 | 99.6 | 99.4 | 99.8 | 94.3 | 97.8 | |||
2000 | 99.5 | 97.4 | 97.9 | 93.3 | 96.1 | 99.2 | 99.1 | 99.5 | 96.2 | 97.9 | |||
4000 | 98.6 | 96.8 | 97.1 | 94.2 | 95.9 | 98.2 | 98.1 | 98.3 | 96.2 | 97.4 |
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Robitzsch, A. Standard Error Estimation in Invariance Alignment. Mathematics 2025, 13, 1915. https://doi.org/10.3390/math13121915
Robitzsch A. Standard Error Estimation in Invariance Alignment. Mathematics. 2025; 13(12):1915. https://doi.org/10.3390/math13121915
Chicago/Turabian StyleRobitzsch, Alexander. 2025. "Standard Error Estimation in Invariance Alignment" Mathematics 13, no. 12: 1915. https://doi.org/10.3390/math13121915
APA StyleRobitzsch, A. (2025). Standard Error Estimation in Invariance Alignment. Mathematics, 13(12), 1915. https://doi.org/10.3390/math13121915