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