Model-Robust Estimation of Multiple-Group Structural Equation Models
Abstract
:1. Introduction
2. Model-Robust Estimation in Structural Equation Modeling
2.1. Multiple-Group Structural Equation Modeling
2.2. Robust Moment Estimation Using Robust Loss Functions
2.2.1. Bias Derivation in the Presence of Model Errors
2.2.2. Standard Error Estimation
2.3. Regularized Maximum Likelihood Estimation
3. Simulation Study 1: Unmodeled Residual Error Correlation
3.1. Method
3.2. Results
4. Focused Simulation Study 1A: Computation of Standard Errors
4.1. Method
4.2. Results
5. Simulation Study 2: Noninvariant Item Intercepts (DIF)
5.1. Method
5.2. Results
6. Empirical Example: ESS 2005 Data
6.1. Method
6.2. Results
7. Discussion
Funding
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
BIC | Bayesian information criterion |
BS | bootstrap |
CFA | confirmatory factor analysis |
DF | delta formula |
DIF | differential item functioning |
DWLS | diagonally weighted least squares |
ESS | European social survey |
JK | jackknife |
MAD | mean absolute deviation |
ML | maximum likelihood |
MVN | multivariate normal |
OI | observed information |
RegML | regularized maximum likelihood |
RME | robust moment estimation |
RMSE | root mean square error |
SCAD | smoothly clipped absolute deviation |
SEM | structural equation model |
ULS | unweighted least squares |
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Bias | Relative RMSE | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
RME with p = | RME with p = | ||||||||||||||
Par | N | RegML | 0.25 | 0.5 | 1 | ULS | ML | RegML | 0.25 | 0.5 | 1 | ULS | ML | ||
0 | 500 | −0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 102 | 102 | 101 | 101 | 100 | 101 | ||
1000 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 102 | 101 | 101 | 100 | 100 | 100 | |||
2500 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 103 | 100 | 100 | 100 | 100 | 100 | |||
0.3 | 500 | 0.02 | 0.02 | 0.02 | 0.03 | 0.07 | 0.07 | 100 | 101 | 103 | 113 | 162 | 166 | ||
1000 | 0.01 | 0.01 | 0.01 | 0.03 | 0.07 | 0.07 | 100 | 101 | 105 | 122 | 213 | 218 | |||
2500 | 0.01 | 0.01 | 0.01 | 0.03 | 0.07 | 0.07 | 100 | 101 | 108 | 144 | 318 | 328 | |||
0.6 | 500 | 0.01 | 0.01 | 0.01 | 0.03 | 0.15 | 0.16 | 100 | 101 | 102 | 119 | 312 | 336 | ||
1000 | 0.00 | 0.01 | 0.01 | 0.03 | 0.15 | 0.16 | 100 | 100 | 103 | 129 | 429 | 462 | |||
2500 | 0.00 | 0.01 | 0.01 | 0.03 | 0.15 | 0.16 | 100 | 100 | 105 | 152 | 672 | 722 | |||
0 | 500 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 101 | 108 | 106 | 104 | 104 | 100 | ||
1000 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 101 | 105 | 104 | 103 | 104 | 100 | |||
2500 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 101 | 104 | 104 | 104 | 104 | 100 | |||
0.3 | 500 | 0.00 | 0.00 | 0.00 | 0.01 | 0.02 | 0.01 | 102 | 104 | 103 | 101 | 109 | 100 | ||
1000 | 0.00 | 0.00 | 0.00 | 0.01 | 0.02 | 0.01 | 100 | 101 | 101 | 102 | 117 | 102 | |||
2500 | 0.00 | 0.00 | 0.00 | 0.01 | 0.02 | 0.01 | 100 | 100 | 101 | 105 | 137 | 108 | |||
0.6 | 500 | 0.00 | 0.00 | 0.00 | 0.01 | 0.04 | 0.02 | 100 | 108 | 107 | 107 | 139 | 138 | ||
1000 | 0.00 | 0.00 | 0.00 | 0.01 | 0.04 | 0.02 | 100 | 107 | 107 | 110 | 166 | 146 | |||
2500 | 0.00 | 0.00 | 0.00 | 0.01 | 0.04 | 0.02 | 100 | 104 | 105 | 111 | 219 | 160 | |||
0 | 500 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 100 | 117 | 113 | 109 | 109 | 100 | ||
1000 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 100 | 109 | 107 | 106 | 107 | 100 | |||
2500 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 101 | 107 | 107 | 107 | 108 | 100 | |||
0.3 | 500 | −0.02 | −0.01 | −0.01 | −0.01 | −0.03 | −0.02 | 107 | 110 | 107 | 105 | 116 | 100 | ||
1000 | −0.01 | −0.01 | −0.01 | −0.01 | −0.03 | −0.02 | 104 | 100 | 100 | 103 | 127 | 102 | |||
2500 | −0.01 | 0.00 | −0.01 | −0.01 | −0.03 | −0.01 | 107 | 100 | 101 | 108 | 155 | 111 | |||
0.6 | 500 | −0.02 | 0.00 | −0.01 | −0.01 | −0.06 | −0.03 | 105 | 102 | 100 | 100 | 139 | 136 | ||
1000 | −0.01 | 0.00 | 0.00 | −0.01 | −0.06 | −0.03 | 102 | 101 | 100 | 104 | 176 | 145 | |||
2500 | 0.00 | 0.00 | 0.00 | −0.01 | −0.06 | −0.03 | 100 | 101 | 102 | 112 | 260 | 177 |
RME with p = | ||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0.25 | 0.5 | 1 | ULS | ML | ||||||||||||||||||
Par | N | DF | JK | BS | DF | JK | BS | DF | JK | BS | DF | JK | BS | OI | DF | JK | BS | |||||
0 | 500 | 95.8 | 95.1 | 95.5 | 95.5 | 94.8 | 94.9 | 95.3 | 94.4 | 95.0 | 94.9 | 94.2 | 94.6 | 95.0 | 95.1 | 94.2 | 94.6 | |||||
1000 | 95.2 | 94.7 | 94.9 | 95.1 | 94.6 | 94.7 | 94.8 | 94.4 | 94.7 | 94.6 | 94.2 | 94.3 | 94.7 | 94.7 | 94.4 | 94.5 | ||||||
2500 | 95.4 | 94.7 | 95.3 | 95.3 | 94.7 | 94.9 | 95.3 | 94.6 | 94.9 | 95.2 | 94.7 | 94.7 | 95.2 | 95.2 | 94.7 | 94.9 | ||||||
0.6 | 500 | 94.8 | 94.5 | 94.6 | 94.5 | 94.1 | 94.5 | 93.7 | 93.0 | 93.3 | 94.7 | 94.0 | 94.3 | 93.5 | 94.4 | 93.7 | 94.4 | |||||
1000 | 94.3 | 94.0 | 94.7 | 94.3 | 93.7 | 94.1 | 93.7 | 93.3 | 93.1 | 94.7 | 94.0 | 94.4 | 93.6 | 94.6 | 94.0 | 94.1 | ||||||
2500 | 95.1 | 94.5 | 95.0 | 95.0 | 94.3 | 94.7 | 95.0 | 94.2 | 94.1 | 95.2 | 94.5 | 94.8 | 94.3 | 95.2 | 94.3 | 94.6 | ||||||
0 | 500 | 95.8 | 95.4 | 96.0 | 95.6 | 95.1 | 95.9 | 95.2 | 94.5 | 95.1 | 94.7 | 94.0 | 94.3 | 95.0 | 95.1 | 93.9 | 94.7 | |||||
1000 | 95.2 | 94.3 | 95.6 | 95.1 | 93.9 | 95.1 | 94.6 | 93.7 | 94.4 | 94.4 | 93.4 | 93.7 | 94.6 | 94.7 | 93.8 | 93.8 | ||||||
2500 | 94.5 | 93.8 | 94.6 | 94.4 | 93.8 | 94.1 | 94.2 | 93.9 | 94.3 | 94.2 | 93.8 | 94.0 | 94.6 | 94.6 | 93.8 | 94.1 | ||||||
0.6 | 500 | 95.5 | 94.9 | 96.0 | 95.6 | 94.7 | 96.2 | 95.6 | 94.5 | 95.3 | 94.9 | 93.8 | 94.5 | 88.5 | 96.2 | 95.7 | 96.4 | |||||
1000 | 95.1 | 94.5 | 95.5 | 94.9 | 94.5 | 95.0 | 94.7 | 94.4 | 94.8 | 94.6 | 93.8 | 94.4 | 88.5 | 95.6 | 94.7 | 95.9 | ||||||
2500 | 95.1 | 94.7 | 95.2 | 95.0 | 94.7 | 94.9 | 95.3 | 94.5 | 94.9 | 95.2 | 94.3 | 94.9 | 89.0 | 95.4 | 94.5 | 95.4 | ||||||
0 | 500 | 96.6 | 96.2 | 97.6 | 96.4 | 95.8 | 96.9 | 95.8 | 94.8 | 95.4 | 95.1 | 94.3 | 94.4 | 95.0 | 95.0 | 94.4 | 94.9 | |||||
1000 | 96.1 | 95.3 | 96.7 | 95.9 | 95.1 | 95.9 | 95.5 | 94.8 | 95.3 | 95.3 | 94.6 | 94.9 | 95.1 | 95.1 | 94.7 | 94.9 | ||||||
2500 | 95.0 | 94.7 | 95.3 | 95.1 | 94.6 | 95.1 | 95.1 | 94.6 | 94.9 | 95.0 | 94.5 | 94.7 | 95.0 | 95.0 | 94.3 | 94.4 | ||||||
0.6 | 500 | 96.1 | 95.7 | 97.1 | 96.1 | 95.6 | 96.9 | 95.7 | 95.2 | 95.8 | 94.9 | 94.5 | 94.6 | 88.7 | 96.7 | 96.5 | 96.8 | |||||
1000 | 95.3 | 95.0 | 95.9 | 95.4 | 95.0 | 95.8 | 95.3 | 94.7 | 95.3 | 94.9 | 94.2 | 94.7 | 88.6 | 95.8 | 95.1 | 96.4 | ||||||
2500 | 95.1 | 94.7 | 95.0 | 95.1 | 94.6 | 95.0 | 94.9 | 94.5 | 94.8 | 95.1 | 94.5 | 94.7 | 88.8 | 95.2 | 94.8 | 95.6 |
Bias | Relative RMSE | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
RME with p = | RME with p = | ||||||||||||||
Par | N | RegML | 0.25 | 0.5 | 1 | ULS | ML | RegML | 0.25 | 0.5 | 1 | ULS | ML | ||
0 | 500 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 100 | 103 | 102 | 100 | 100 | 100 | ||
1000 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 100 | 102 | 101 | 100 | 100 | 100 | |||
2500 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 100 | 100 | 100 | 100 | 100 | 100 | |||
0.3 | 500 | −0.02 | −0.03 | −0.03 | −0.07 | −0.12 | −0.12 | 101 | 100 | 100 | 113 | 152 | 152 | ||
1000 | 0.00 | −0.01 | −0.02 | −0.06 | −0.12 | −0.12 | 100 | 103 | 106 | 140 | 231 | 230 | |||
2500 | 0.00 | −0.01 | −0.01 | −0.05 | −0.12 | −0.12 | 100 | 103 | 107 | 169 | 356 | 355 | |||
0.6 | 500 | 0.00 | −0.01 | −0.02 | −0.07 | −0.24 | −0.24 | 100 | 104 | 104 | 129 | 303 | 302 | ||
1000 | 0.00 | −0.01 | −0.01 | −0.06 | −0.24 | −0.24 | 100 | 102 | 103 | 144 | 435 | 433 | |||
2500 | 0.00 | 0.00 | −0.01 | −0.05 | −0.24 | −0.24 | 100 | 101 | 104 | 171 | 698 | 692 | |||
0 | 500 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 100 | 103 | 102 | 101 | 100 | 100 | ||
1000 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 100 | 102 | 101 | 100 | 100 | 100 | |||
2500 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 100 | 101 | 101 | 100 | 100 | 100 | |||
0.3 | 500 | −0.02 | −0.03 | −0.03 | −0.07 | −0.12 | −0.11 | 100 | 103 | 102 | 115 | 153 | 149 | ||
1000 | 0.00 | −0.01 | −0.02 | −0.06 | −0.12 | −0.11 | 100 | 104 | 107 | 139 | 223 | 216 | |||
2500 | 0.00 | −0.01 | −0.01 | −0.05 | −0.12 | −0.12 | 100 | 103 | 108 | 167 | 344 | 332 | |||
0.6 | 500 | 0.00 | −0.01 | −0.01 | −0.06 | −0.23 | −0.21 | 100 | 103 | 102 | 124 | 286 | 267 | ||
1000 | 0.00 | 0.00 | −0.01 | −0.05 | −0.23 | −0.22 | 100 | 103 | 103 | 139 | 421 | 390 | |||
2500 | 0.00 | −0.01 | −0.01 | −0.05 | −0.24 | −0.22 | 100 | 101 | 104 | 174 | 693 | 641 |
RME with p = | |||||||
---|---|---|---|---|---|---|---|
Rank | CNT | RegML | 0.25 | 0.5 | 1 | ULS | ML |
1 | C10 | 0.37 | 0.37 | 0.37 | 0.34 | 0.34 | 0.35 |
2 | C21 | 0.01 | −0.01 | 0.01 | 0.18 | 0.27 | 0.33 |
3 | C06 | 0.06 | 0.17 | 0.19 | 0.26 | 0.25 | 0.26 |
4 | C03 | 0.33 | 0.33 | 0.32 | 0.25 | 0.21 | 0.21 |
5 | C08 | 0.13 | 0.15 | 0.16 | 0.23 | 0.23 | 0.20 |
6 | C12 | 0.12 | 0.14 | 0.15 | 0.20 | 0.19 | 0.16 |
7 | C05 | 0.15 | 0.17 | 0.17 | 0.15 | 0.11 | 0.13 |
8 | C16 | 0.14 | 0.05 | 0.06 | 0.09 | 0.06 | 0.07 |
9 | C01 | 0.12 | 0.11 | 0.10 | 0.07 | 0.08 | 0.07 |
10 | C14 | −0.11 | −0.10 | −0.10 | −0.09 | −0.05 | −0.06 |
11 | C22 | 0.02 | −0.01 | −0.02 | −0.06 | −0.08 | −0.07 |
12 | C15 | −0.18 | −0.17 | −0.17 | −0.16 | −0.01 | −0.07 |
13 | C09 | −0.21 | −0.21 | −0.21 | −0.21 | −0.19 | −0.23 |
14 | C13 | −0.30 | −0.30 | −0.30 | −0.32 | −0.28 | −0.30 |
15 | C17 | −0.20 | −0.19 | −0.21 | −0.29 | −0.36 | −0.33 |
16 | C25 | −0.17 | −0.18 | −0.19 | −0.29 | −0.39 | −0.34 |
17 | C24 | −0.29 | −0.32 | −0.34 | −0.37 | −0.38 | −0.39 |
CNT | TR09 | TR20 | CO07 | CO16 |
---|---|---|---|---|
C01 | · | −0.17 | · | −0.17 |
C03 | · | −0.32 | · | −0.26 |
C05 | −0.50 | 0.29 | · | · |
C06 | 0.09 | · | 0.24 | · |
C08 | · | 0.12 | · | · |
C09 | · | −0.24 | · | 0.09 |
C10 | −0.47 | 0.25 | · | · |
C12 | · | 0.05 | · | · |
C13 | 0.29 | · | · | −0.47 |
C14 | 0.10 | 0.31 | · | −0.42 |
C15 | 0.42 | 0.07 | · | −0.22 |
C16 | −0.26 | · | · | −0.13 |
C17 | −0.52 | −0.12 | · | · |
C21 | · | · | 0.48 | · |
C22 | −0.14 | · | · | −0.36 |
C24 | · | −0.32 | · | −0.17 |
C25 | −0.52 | · | · | −0.31 |
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Robitzsch, A. Model-Robust Estimation of Multiple-Group Structural Equation Models. Algorithms 2023, 16, 210. https://doi.org/10.3390/a16040210
Robitzsch A. Model-Robust Estimation of Multiple-Group Structural Equation Models. Algorithms. 2023; 16(4):210. https://doi.org/10.3390/a16040210
Chicago/Turabian StyleRobitzsch, Alexander. 2023. "Model-Robust Estimation of Multiple-Group Structural Equation Models" Algorithms 16, no. 4: 210. https://doi.org/10.3390/a16040210
APA StyleRobitzsch, A. (2023). Model-Robust Estimation of Multiple-Group Structural Equation Models. Algorithms, 16(4), 210. https://doi.org/10.3390/a16040210