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Table 3 Model fit and parameter estimates of the bivariate “butterfly” models

From: Psychological determinants of test motivation in low-stakes test situations: A longitudinal study of singletrait–multistate models in accounting

# Model n Par χ2 df χ2/df CFI TLI RMSEA SRMR β_t (VALUE) p β_s11 (VALUE) p β_s12 (VALUE) p
Multiconstruct bifactor approach (butterfly models)
1 INTRI 632 85 522.08 320 1.63 0.965 0.962 0.032 0.076 0.370 0.000 0.195 0.013 0.297 0.003
2 IDENT 632 79 356.13 245 1.45 0.971 0.968 0.027 0.064 0.344 0.005 0.053 0.723 0.238 0.201
3 INTRO 632 79 438.62 245 1.79 0.928 0.918 0.035 0.073 0.334 0.000 0.512 0.000 0.295 0.017
4 EXTRI 632 79 362.82 245 1.48 0.944 0.937 0.028 0.063 − 0.096 0.279 0.186 0.107 0.074 0.701
5 ACMOT 796 78 333.37 246 1.36 0.967 0.963 0.021 0.059 0.385 0.000 0.439 0.000 NA NA
6 ASELF 796 99 981.11 495 1.98 0.912 0.906 0.035 0.083 0.122 0.297 0.241 0.046 NA NA
Bifactor models using the residual approach
7 GRITGt 506 63 284.04 126 2.25 0.900 0.879 0.050 0.057 0.600 0.205
8 GRITPt 506 63 197.45 126 1.57 0.958 0.949 0.033 0.062 0.181 0.771
9 GRITG s 506 60 290.39 129 2.25 0.898 0.879 0.050 0.062 − 0.543 0.025 − 0.545 0.251
10 GRITPs 506 63 195.21 126 1.55 0.960 0.951 0.033 0.063 0.430 0.284 0.590 0.238
# Model n Par χ2 df χ2/df CFI TLI RMSEA SRMR β_t (EFFOR) p β_s11 (EFFOR) p β_s12 (EFFOR) p
Multiconstruct bifactor approach (butterfly models)
11 INTRI 632 102 867.95 492 1.76 0.949 0.945 0.035 0.072 0.296 0.000 − 0.148 0.187 − 0.026 0.776
12 IDENT 632 97 645.05 398 1.62 0.954 0.949 0.031 0.064 0.386 0.001 − 0.074 0.491 0.066 0.716
13 INTRO 632 97 682.05 398 1.71 0.923 0.916 0.034 0.068 0.342 0.000 0.019 0.895 − 0.173 0.277
14 EXTRI 632 97 641.69 398 1.61 0.928 0.921 0.031 0.065 0.030 0.786 − 0.041 0.758 − 0.115 0.515
15 ACMOT 796 96 647.61 399 1.62 0.939 0.933 0.028 0.068 0.256 0.025 0.137 0.326 NA NA
16 ASELF 796 113 1408.30 706 1.99 0.896 0.891 0.035 0.077 0.103 0.314 0.048 0.720 NA NA
Bifactor models using the residual approach
17 GRITGt 506 81 465.48 243 1.92 0.922 0.911 0.043 0.060 − 0.046 0.851
18 GRITP t 506 81 375.03 243 1.54 0.954 0.948 0.033 0.057 0.481 0.002
19 GRITGs 506 79 515.09 245 2.10 0.905 0.893 0.047 0.063 − 0.005 0.989 -0.292 0.470
20 GRITPs 506 81 379.13 243 1.56 0.953 0.947 0.033 0.057 − 0.016 0.973 0.009 0.988
  1. Italic values indicate p value < 0.05
  2. β_t, association between traits; β_s11, association between states at grade 11; β_s12, association between states at grade 12; VALUE, Subjective test value; EFFOR, Effort; INTRI, Intrinsic regulation; IDENT, Identified regulation; INTRO, Introjected regulation; EXTRI, External regulation; ACMOT, Achievement motivation; ASELF, Academic self-concept; GRITG, Passion for long-term goals; GRITP, Perseverance. Subscript t, prediction of the trait factor; s, prediction of the state factors. N, observations; #Par, number of parameters; χ2, Chi square value; df, degrees of freedom; CFI, Comparative Fit Index; TLI, Tucker Lewis Index; RMSEA, Root Mean Square Error of Approximation; SRMR, Standardized Root Mean Square Residual. NA, not applicable as the covariate was not measured at that time