The Higher-Order Model Imposes a Proportionality Constraint: That is Why the Bifactor Model Tends to Fit Better
Supplementary Material
Table S1
Simulated Correlation Matrix: Proportionality Constraint Satisfied 1
|
a1 |
a2 |
a3 |
b1 |
b2 |
b3 |
c1 |
c2 |
c3 |
a1 |
1.000 |
|
|
|
|
|
|
|
|
a2 |
.524 |
1.000 |
|
|
|
|
|
|
|
a3 |
.500 |
.475 |
1.000 |
|
|
|
|
|
|
b1 |
.342 |
.325 |
.310 |
1.000 |
|
|
|
|
|
b2 |
.342 |
.325 |
.310 |
.500 |
1.000 |
|
|
|
|
b3 |
.342 |
.325 |
.310 |
.500 |
.500 |
1.000 |
|
|
|
c1 |
.342 |
.325 |
.310 |
.350 |
.350 |
.350 |
1.000 |
|
|
c2 |
.342 |
.325 |
.310 |
.350 |
.350 |
.350 |
.500 |
1.000 |
|
c3 |
.342 |
.325 |
.310 |
.350 |
.350 |
.350 |
.500 |
.500 |
1.000 |
Table S2
Simulated Correlation Matrix: Proportionality Constraint Violated 1
|
a1 |
a2 |
a3 |
b1 |
b2 |
b3 |
c1 |
c2 |
c3 |
a1 |
1.000 |
|
|
|
|
|
|
|
|
a2 |
.550 |
1.000 |
|
|
|
|
|
|
|
a3 |
.500 |
.451 |
1.000 |
|
|
|
|
|
|
b1 |
.342 |
.325 |
.310 |
1.000 |
|
|
|
|
|
b2 |
.342 |
.325 |
.310 |
.500 |
1.000 |
|
|
|
|
b3 |
.342 |
.325 |
.310 |
.500 |
.500 |
1.000 |
|
|
|
c1 |
.342 |
.325 |
.310 |
.350 |
.350 |
.350 |
1.000 |
|
|
c2 |
.342 |
.325 |
.310 |
.350 |
.350 |
.350 |
.500 |
1.000 |
|
c3 |
.342 |
.325 |
.310 |
.350 |
.350 |
.350 |
.500 |
.500 |
1.000 |
Table S3
Simulated Correlation Matrix: Proportionality Constraint Violated 2
|
a1 |
a2 |
a3 |
b1 |
b2 |
b3 |
c1 |
c2 |
c3 |
a1 |
1.000 |
|
|
|
|
|
|
|
|
a2 |
.550 |
1.000 |
|
|
|
|
|
|
|
a3 |
.500 |
.451 |
1.000 |
|
|
|
|
|
|
b1 |
.324 |
.325 |
.325 |
1.000 |
|
|
|
|
|
b2 |
.324 |
.325 |
.325 |
.500 |
1.000 |
|
|
|
|
b3 |
.324 |
.325 |
.325 |
.500 |
.500 |
1.000 |
|
|
|
c1 |
.324 |
.325 |
.325 |
.350 |
.350 |
.350 |
1.000 |
|
|
c2 |
.324 |
.325 |
.325 |
.350 |
.350 |
.350 |
.500 |
1.000 |
|
c3 |
.324 |
.325 |
.325 |
.350 |
.350 |
.350 |
.500 |
.500 |
1.000 |
Table S4
Simulated Correlation Matrix: Proportionality Constraint Violated 3
|
a1 |
a2 |
a3 |
b1 |
b2 |
b3 |
c1 |
c2 |
c3 |
a1 |
1.000 |
|
|
|
|
|
|
|
|
a2 |
.576 |
1.000 |
|
|
|
|
|
|
|
a3 |
.500 |
.428 |
1.000 |
|
|
|
|
|
|
b1 |
.324 |
.325 |
.325 |
1.000 |
|
|
|
|
|
b2 |
.324 |
.325 |
.325 |
.500 |
1.000 |
|
|
|
|
b3 |
.324 |
.325 |
.325 |
.500 |
.500 |
1.000 |
|
|
|
c1 |
.324 |
.325 |
.325 |
.350 |
.350 |
.350 |
1.000 |
|
|
c2 |
.324 |
.325 |
.325 |
.350 |
.350 |
.350 |
.500 |
1.000 |
|
c3 |
.324 |
.325 |
.325 |
.350 |
.350 |
.350 |
.500 |
.500 |
1.000 |
Table S5
Simulated Correlation Matrix: Proportionality Constraint Violated 4
|
a1 |
a2 |
a3 |
b1 |
b2 |
b3 |
c1 |
c2 |
c3 |
a1 |
1.000 |
|
|
|
|
|
|
|
|
a2 |
.576 |
1.000 |
|
|
|
|
|
|
|
a3 |
.500 |
.428 |
1.000 |
|
|
|
|
|
|
b1 |
.308 |
.325 |
.341 |
1.000 |
|
|
|
|
|
b2 |
.308 |
.325 |
.341 |
.500 |
1.000 |
|
|
|
|
b3 |
.308 |
.325 |
.341 |
.500 |
.500 |
1.000 |
|
|
|
c1 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
1.000 |
|
|
c2 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
.500 |
1.000 |
|
c3 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
.500 |
.500 |
1.000 |
Table S6
Simulated Correlation Matrix: Proportionality Constraint Violated 5
|
a1 |
a2 |
a3 |
b1 |
b2 |
b3 |
c1 |
c2 |
c3 |
a1 |
1.000 |
|
|
|
|
|
|
|
|
a2 |
.605 |
1.000 |
|
|
|
|
|
|
|
a3 |
.500 |
.406 |
1.000 |
|
|
|
|
|
|
b1 |
.308 |
.325 |
.341 |
1.000 |
|
|
|
|
|
b2 |
.308 |
.325 |
.341 |
.500 |
1.000 |
|
|
|
|
b3 |
.308 |
.325 |
.341 |
.500 |
.500 |
1.000 |
|
|
|
c1 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
1.000 |
|
|
c2 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
.500 |
1.000 |
|
c3 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
.500 |
.500 |
1.000 |
Table S7
Simulated Correlation Matrix: Proportionality Constraint Violated 6
|
a1 |
a2 |
a3 |
b1 |
b2 |
b3 |
c1 |
c2 |
c3 |
a1 |
1.000 |
|
|
|
|
|
|
|
|
a2 |
.605 |
1.000 |
|
|
|
|
|
|
|
a3 |
.500 |
.406 |
1.000 |
|
|
|
|
|
|
b1 |
.308 |
.325 |
.341 |
1.000 |
|
|
|
|
|
b2 |
.308 |
.325 |
.341 |
.550 |
1.000 |
|
|
|
|
b3 |
.308 |
.325 |
.341 |
.500 |
.451 |
1.000 |
|
|
|
c1 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
1.000 |
|
|
c2 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
.500 |
1.000 |
|
c3 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
.500 |
.500 |
1.000 |
Table S8
Simulated Correlation Matrix: Proportionality Constraint Violated 7
|
a1 |
a2 |
a3 |
b1 |
b2 |
b3 |
c1 |
c2 |
c3 |
a1 |
1.000 |
|
|
|
|
|
|
|
|
a2 |
.605 |
1.000 |
|
|
|
|
|
|
|
a3 |
.500 |
.406 |
1.000 |
|
|
|
|
|
|
b1 |
.308 |
.325 |
.341 |
1.000 |
|
|
|
|
|
b2 |
.308 |
.325 |
.341 |
.576 |
1.000 |
|
|
|
|
b3 |
.308 |
.325 |
.341 |
.500 |
.428 |
1.000 |
|
|
|
c1 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
1.000 |
|
|
c2 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
.500 |
1.000 |
|
c3 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
.500 |
.500 |
1.000 |
Table S9
Simulated Correlation Matrix: Proportionality Constraint Violated 8
|
a1 |
a2 |
a3 |
b1 |
b2 |
b3 |
c1 |
c2 |
c3 |
a1 |
1.000 |
|
|
|
|
|
|
|
|
a2 |
.605 |
1.000 |
|
|
|
|
|
|
|
a3 |
.500 |
.406 |
1.000 |
|
|
|
|
|
|
b1 |
.308 |
.325 |
.341 |
1.000 |
|
|
|
|
|
b2 |
.308 |
.325 |
.341 |
.605 |
1.000 |
|
|
|
|
b3 |
.308 |
.325 |
.341 |
.500 |
.406 |
1.000 |
|
|
|
c1 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
1.000 |
|
|
c2 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
.500 |
1.000 |
|
c3 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
.500 |
.500 |
1.000 |
Table S10
Simulated Correlation Matrix: Proportionality Constraint Violated 9
|
a1 |
a2 |
a3 |
b1 |
b2 |
b3 |
c1 |
c2 |
c3 |
a1 |
1.000 |
|
|
|
|
|
|
|
|
a2 |
.605 |
1.000 |
|
|
|
|
|
|
|
a3 |
.500 |
.406 |
1.000 |
|
|
|
|
|
|
b1 |
.308 |
.325 |
.341 |
1.000 |
|
|
|
|
|
b2 |
.308 |
.325 |
.341 |
.605 |
1.000 |
|
|
|
|
b3 |
.308 |
.325 |
.341 |
.500 |
.406 |
1.000 |
|
|
|
c1 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
1.000 |
|
|
c2 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
.550 |
1.000 |
|
c3 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
.500 |
.451 |
1.000 |
Table S11
Simulated Correlation Matrix: Proportionality Constraint Violated 10
|
a1 |
a2 |
a3 |
b1 |
b2 |
b3 |
c1 |
c2 |
c3 |
a1 |
1.000 |
|
|
|
|
|
|
|
|
a2 |
.605 |
1.000 |
|
|
|
|
|
|
|
a3 |
.500 |
.406 |
1.000 |
|
|
|
|
|
|
b1 |
.308 |
.325 |
.341 |
1.000 |
|
|
|
|
|
b2 |
.308 |
.325 |
.341 |
.605 |
1.000 |
|
|
|
|
b3 |
.308 |
.325 |
.341 |
.500 |
.406 |
1.000 |
|
|
|
c1 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
1.000 |
|
|
c2 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
.576 |
1.000 |
|
c3 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
.500 |
.428 |
1.000 |
Table S12
Simulated Correlation Matrix: Proportionality Constraint Violated 11
|
a1 |
a2 |
a3 |
b1 |
b2 |
b3 |
c1 |
c2 |
c3 |
a1 |
1.000 |
|
|
|
|
|
|
|
|
a2 |
.605 |
1.000 |
|
|
|
|
|
|
|
a3 |
.500 |
.406 |
1.000 |
|
|
|
|
|
|
b1 |
.308 |
.325 |
.341 |
1.000 |
|
|
|
|
|
b2 |
.308 |
.325 |
.341 |
.605 |
1.000 |
|
|
|
|
b3 |
.308 |
.325 |
.341 |
.500 |
.406 |
1.000 |
|
|
|
c1 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
1.000 |
|
|
c2 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
.605 |
1.000 |
|
c3 |
.308 |
.325 |
.341 |
.350 |
.350 |
.350 |
.500 |
.406 |
1.000 |