The data has 22 variables and it is supposed to load on 3 factors. The details of the problem are in the text. The solutions differ from the Mplus solution in the book a bit and am not sure why.
Let us load the problem
library(utils)
library(lavaan) ## This is lavaan 0.5-22## lavaan is BETA software! Please report any bugs.db1 <- read.fwf("https://fhssrsc.byu.edu/Documents/Textbook%20examples/Structural%20Equation%20Modeling%20with%20Mplus%20-%20Barbara%20Byrne/Ch%204/ELEMM1.DAT", c(rep(1,22)))
##The measurement model diagram can be found in the text
model1 <- '
F1 =~ V1 + V2 + V3 + V6 + V8 + V13 + V14 + V16 + V20
F2 =~ V5 + V10 + V11 + V15 + V22
F3 =~ V4 + V7 + V9 + V12 + V17 + V18 + V19 + V21'
#The first CFA. Robust (MLM) estimation is used as we are not confident about multivariate normality
fit1 <- cfa(model1, data = db1, meanstructure = TRUE, estimator = "MLM")
summary(fit1, fit.measures = TRUE)## lavaan (0.5-22) converged normally after 46 iterations
##
## Number of observations 372
##
## Estimator ML Robust
## Minimum Function Test Statistic 695.719 567.753
## Degrees of freedom 206 206
## P-value (Chi-square) 0.000 0.000
## Scaling correction factor 1.225
## for the Satorra-Bentler correction
##
## Model test baseline model:
##
## Minimum Function Test Statistic 3452.269 2911.466
## Degrees of freedom 231 231
## P-value 0.000 0.000
##
## User model versus baseline model:
##
## Comparative Fit Index (CFI) 0.848 0.865
## Tucker-Lewis Index (TLI) 0.830 0.849
##
## Robust Comparative Fit Index (CFI) 0.861
## Robust Tucker-Lewis Index (TLI) 0.844
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -12811.043 -12811.043
## Loglikelihood unrestricted model (H1) -12463.184 -12463.184
##
## Number of free parameters 69 69
## Akaike (AIC) 25760.087 25760.087
## Bayesian (BIC) 26030.490 26030.490
## Sample-size adjusted Bayesian (BIC) 25811.575 25811.575
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.080 0.069
## 90 Percent Confidence Interval 0.073 0.087 0.063 0.075
## P-value RMSEA <= 0.05 0.000 0.000
##
## Robust RMSEA 0.076
## 90 Percent Confidence Interval 0.069 0.084
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.070 0.070
##
## Parameter Estimates:
##
## Information Expected
## Standard Errors Robust.sem
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## F1 =~
## V1 1.000
## V2 0.887 0.040 22.361 0.000
## V3 1.021 0.053 19.284 0.000
## V6 0.764 0.070 10.959 0.000
## V8 1.143 0.059 19.340 0.000
## V13 1.017 0.062 16.318 0.000
## V14 0.848 0.058 14.564 0.000
## V16 0.715 0.066 10.812 0.000
## V20 0.753 0.061 12.286 0.000
## F2 =~
## V5 1.000
## V10 1.142 0.152 7.499 0.000
## V11 1.353 0.162 8.356 0.000
## V15 0.905 0.123 7.357 0.000
## V22 0.768 0.122 6.275 0.000
## F3 =~
## V4 1.000
## V7 0.970 0.128 7.553 0.000
## V9 1.780 0.322 5.521 0.000
## V12 1.499 0.241 6.223 0.000
## V17 1.348 0.200 6.748 0.000
## V18 1.918 0.298 6.426 0.000
## V19 1.716 0.287 5.970 0.000
## V21 1.356 0.227 5.976 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## F1 ~~
## F2 0.701 0.106 6.599 0.000
## F3 -0.192 0.040 -4.790 0.000
## F2 ~~
## F3 -0.172 0.036 -4.771 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .V1 4.366 0.086 50.675 0.000
## .V2 4.868 0.080 60.741 0.000
## .V3 3.527 0.090 39.243 0.000
## .V6 2.707 0.082 32.958 0.000
## .V8 3.043 0.090 33.972 0.000
## .V13 3.586 0.087 41.108 0.000
## .V14 4.027 0.090 44.989 0.000
## .V16 2.473 0.075 33.137 0.000
## .V20 2.245 0.073 30.597 0.000
## .V5 2.199 0.077 28.505 0.000
## .V10 2.204 0.075 29.388 0.000
## .V11 2.239 0.079 28.241 0.000
## .V15 1.769 0.067 26.261 0.000
## .V22 2.581 0.082 31.495 0.000
## .V4 6.298 0.052 121.663 0.000
## .V7 6.312 0.044 144.912 0.000
## .V9 6.035 0.068 88.446 0.000
## .V12 5.699 0.062 92.113 0.000
## .V17 6.406 0.044 144.908 0.000
## .V18 5.702 0.066 86.338 0.000
## .V19 5.946 0.062 96.417 0.000
## .V21 5.852 0.066 89.138 0.000
## F1 0.000
## F2 0.000
## F3 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .V1 1.128 0.093 12.160 0.000
## .V2 1.105 0.088 12.489 0.000
## .V3 1.301 0.106 12.301 0.000
## .V6 1.553 0.135 11.534 0.000
## .V8 0.852 0.082 10.436 0.000
## .V13 1.142 0.125 9.160 0.000
## .V14 1.804 0.142 12.713 0.000
## .V16 1.235 0.110 11.263 0.000
## .V20 1.075 0.137 7.849 0.000
## .V5 1.503 0.180 8.370 0.000
## .V10 1.169 0.147 7.948 0.000
## .V11 1.044 0.141 7.389 0.000
## .V15 1.106 0.153 7.210 0.000
## .V22 2.076 0.184 11.251 0.000
## .V4 0.802 0.113 7.114 0.000
## .V7 0.523 0.075 7.001 0.000
## .V9 1.117 0.149 7.477 0.000
## .V12 0.987 0.126 7.842 0.000
## .V17 0.375 0.057 6.626 0.000
## .V18 0.909 0.143 6.368 0.000
## .V19 0.844 0.111 7.612 0.000
## .V21 1.245 0.133 9.326 0.000
## F1 1.625 0.148 10.989 0.000
## F2 0.705 0.158 4.446 0.000
## F3 0.193 0.050 3.834 0.000We calculate the top Modification indices (MI) and then progressively add more contraints.
mi1 <- modindices(fit1)
head(mi1[order(mi1$mi, decreasing = TRUE),],10)## lhs op rhs mi mi.scaled epc sepc.lv sepc.all sepc.nox
## 183 V6 ~~ V16 91.282 74.492 0.733 0.733 0.322 0.322
## 120 V1 ~~ V2 82.448 67.283 0.613 0.613 0.239 0.239
## 84 F1 =~ V12 41.517 33.880 -0.313 -0.400 -0.335 -0.335
## 285 V10 ~~ V11 38.081 31.076 0.580 0.580 0.263 0.263
## 335 V7 ~~ V21 33.529 27.362 0.263 0.263 0.248 0.248
## 323 V4 ~~ V7 33.432 27.283 0.209 0.209 0.250 0.250
## 106 F3 =~ V1 28.732 23.448 0.872 0.383 0.231 0.231
## 348 V18 ~~ V19 18.607 15.184 0.250 0.250 0.165 0.165
## 185 V6 ~~ V5 17.193 14.030 0.354 0.354 0.151 0.151
## 275 V5 ~~ V15 15.584 12.718 0.313 0.313 0.162 0.162Since the MI for V6 ~~ V16 is the highest, we constrain these two items to covary. We find the fit indices, fit parameters and also the top MIs.
model2 <- '
F1 =~ V1 + V2 + V3 + V6 + V8 + V13 + V14 + V16 + V20
F2 =~ V5 + V10 + V11 + V15 + V22
F3 =~ V4 + V7 + V9 + V12 + V17 + V18 + V19 + V21
V6 ~~ V16'
#MLM CFA with V6 correlated with V16
fit2 <- cfa(model2, data = db1, estimator = "MLM")
summary(fit2, fit.measures = TRUE)## lavaan (0.5-22) converged normally after 48 iterations
##
## Number of observations 372
##
## Estimator ML Robust
## Minimum Function Test Statistic 597.731 493.398
## Degrees of freedom 205 205
## P-value (Chi-square) 0.000 0.000
## Scaling correction factor 1.211
## for the Satorra-Bentler correction
##
## Model test baseline model:
##
## Minimum Function Test Statistic 3452.269 2911.466
## Degrees of freedom 231 231
## P-value 0.000 0.000
##
## User model versus baseline model:
##
## Comparative Fit Index (CFI) 0.878 0.892
## Tucker-Lewis Index (TLI) 0.863 0.879
##
## Robust Comparative Fit Index (CFI) 0.890
## Robust Tucker-Lewis Index (TLI) 0.876
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -12762.049 -12762.049
## Loglikelihood unrestricted model (H1) -12463.184 -12463.184
##
## Number of free parameters 70 70
## Akaike (AIC) 25664.098 25664.098
## Bayesian (BIC) 25938.421 25938.421
## Sample-size adjusted Bayesian (BIC) 25716.333 25716.333
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.072 0.061
## 90 Percent Confidence Interval 0.065 0.078 0.055 0.068
## P-value RMSEA <= 0.05 0.000 0.002
##
## Robust RMSEA 0.068
## 90 Percent Confidence Interval 0.060 0.075
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.069 0.069
##
## Parameter Estimates:
##
## Information Expected
## Standard Errors Robust.sem
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## F1 =~
## V1 1.000
## V2 0.887 0.040 22.273 0.000
## V3 1.015 0.052 19.606 0.000
## V6 0.715 0.069 10.355 0.000
## V8 1.133 0.058 19.672 0.000
## V13 1.002 0.062 16.205 0.000
## V14 0.847 0.058 14.672 0.000
## V16 0.672 0.065 10.280 0.000
## V20 0.746 0.061 12.271 0.000
## F2 =~
## V5 1.000
## V10 1.151 0.154 7.463 0.000
## V11 1.363 0.164 8.317 0.000
## V15 0.909 0.124 7.341 0.000
## V22 0.771 0.123 6.243 0.000
## F3 =~
## V4 1.000
## V7 0.969 0.128 7.554 0.000
## V9 1.779 0.322 5.521 0.000
## V12 1.496 0.240 6.223 0.000
## V17 1.347 0.200 6.747 0.000
## V18 1.917 0.298 6.433 0.000
## V19 1.714 0.287 5.971 0.000
## V21 1.356 0.227 5.977 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## .V6 ~~
## .V16 0.733 0.121 6.061 0.000
## F1 ~~
## F2 0.697 0.106 6.596 0.000
## F3 -0.188 0.040 -4.663 0.000
## F2 ~~
## F3 -0.171 0.036 -4.762 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .V1 4.366 0.086 50.675 0.000
## .V2 4.868 0.080 60.741 0.000
## .V3 3.527 0.090 39.243 0.000
## .V6 2.707 0.082 32.958 0.000
## .V8 3.043 0.090 33.972 0.000
## .V13 3.586 0.087 41.108 0.000
## .V14 4.027 0.090 44.989 0.000
## .V16 2.473 0.075 33.137 0.000
## .V20 2.245 0.073 30.597 0.000
## .V5 2.199 0.077 28.505 0.000
## .V10 2.204 0.075 29.388 0.000
## .V11 2.239 0.079 28.241 0.000
## .V15 1.769 0.067 26.261 0.000
## .V22 2.581 0.082 31.495 0.000
## .V4 6.298 0.052 121.663 0.000
## .V7 6.312 0.044 144.912 0.000
## .V9 6.035 0.068 88.446 0.000
## .V12 5.699 0.062 92.113 0.000
## .V17 6.406 0.044 144.908 0.000
## .V18 5.702 0.066 86.338 0.000
## .V19 5.946 0.062 96.417 0.000
## .V21 5.852 0.066 89.138 0.000
## F1 0.000
## F2 0.000
## F3 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .V1 1.091 0.092 11.808 0.000
## .V2 1.076 0.088 12.203 0.000
## .V3 1.283 0.105 12.195 0.000
## .V6 1.654 0.141 11.695 0.000
## .V8 0.844 0.080 10.545 0.000
## .V13 1.156 0.129 8.933 0.000
## .V14 1.780 0.141 12.638 0.000
## .V16 1.317 0.116 11.397 0.000
## .V20 1.071 0.136 7.853 0.000
## .V5 1.511 0.180 8.403 0.000
## .V10 1.164 0.147 7.916 0.000
## .V11 1.038 0.141 7.354 0.000
## .V15 1.108 0.154 7.215 0.000
## .V22 2.077 0.185 11.254 0.000
## .V4 0.801 0.113 7.115 0.000
## .V7 0.523 0.075 7.002 0.000
## .V9 1.116 0.149 7.474 0.000
## .V12 0.988 0.126 7.844 0.000
## .V17 0.375 0.057 6.627 0.000
## .V18 0.909 0.143 6.368 0.000
## .V19 0.844 0.111 7.616 0.000
## .V21 1.244 0.133 9.326 0.000
## F1 1.662 0.148 11.201 0.000
## F2 0.697 0.158 4.418 0.000
## F3 0.193 0.050 3.837 0.000#Calulating MI
mi2 <- modindices(fit2)
head(mi2[order(mi2$mi, decreasing = TRUE),],10)## lhs op rhs mi mi.scaled epc sepc.lv sepc.all sepc.nox
## 121 V1 ~~ V2 78.275 64.612 0.591 0.591 0.231 0.231
## 85 F1 =~ V12 41.936 34.616 -0.310 -0.400 -0.336 -0.336
## 285 V10 ~~ V11 37.348 30.829 0.578 0.578 0.262 0.262
## 335 V7 ~~ V21 33.497 27.650 0.263 0.263 0.248 0.248
## 323 V4 ~~ V7 33.386 27.558 0.209 0.209 0.250 0.250
## 107 F3 =~ V1 28.188 23.268 0.851 0.374 0.225 0.225
## 348 V18 ~~ V19 18.617 15.367 0.250 0.250 0.165 0.165
## 275 V5 ~~ V15 16.067 13.262 0.318 0.318 0.165 0.165
## 176 V3 ~~ V12 15.294 12.625 -0.253 -0.253 -0.123 -0.123
## 112 F3 =~ V13 14.632 12.078 -0.628 -0.276 -0.164 -0.164In the next model we constrain V1 and V2 to covary. Ensure you get the detailed interpretation from the text.
model3 <- '
F1 =~ V1 + V2 + V3 + V6 + V8 + V13 + V14 + V16 + V20
F2 =~ V5 + V10 + V11 + V15 + V22
F3 =~ V4 + V7 + V9 + V12 + V17 + V18 + V19 + V21
V6 ~~ V16
V1 ~~ V2'
#MLM CFA with V1 correlated with V2
fit3 <- cfa(model3, data = db1, estimator = "MLM")
summary(fit3, fit.measures = TRUE, modindices = TRUE)## lavaan (0.5-22) converged normally after 46 iterations
##
## Number of observations 372
##
## Estimator ML Robust
## Minimum Function Test Statistic 520.481 431.496
## Degrees of freedom 204 204
## P-value (Chi-square) 0.000 0.000
## Scaling correction factor 1.206
## for the Satorra-Bentler correction
##
## Model test baseline model:
##
## Minimum Function Test Statistic 3452.269 2911.466
## Degrees of freedom 231 231
## P-value 0.000 0.000
##
## User model versus baseline model:
##
## Comparative Fit Index (CFI) 0.902 0.915
## Tucker-Lewis Index (TLI) 0.889 0.904
##
## Robust Comparative Fit Index (CFI) 0.914
## Robust Tucker-Lewis Index (TLI) 0.902
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -12723.424 -12723.424
## Loglikelihood unrestricted model (H1) -12463.184 -12463.184
##
## Number of free parameters 71 71
## Akaike (AIC) 25588.849 25588.849
## Bayesian (BIC) 25867.090 25867.090
## Sample-size adjusted Bayesian (BIC) 25641.829 25641.829
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.065 0.055
## 90 Percent Confidence Interval 0.058 0.071 0.048 0.061
## P-value RMSEA <= 0.05 0.000 0.114
##
## Robust RMSEA 0.060
## 90 Percent Confidence Interval 0.052 0.068
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.066 0.066
##
## Parameter Estimates:
##
## Information Expected
## Standard Errors Robust.sem
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## F1 =~
## V1 1.000
## V2 0.877 0.042 21.128 0.000
## V3 1.068 0.059 18.197 0.000
## V6 0.767 0.077 10.012 0.000
## V8 1.216 0.067 18.214 0.000
## V13 1.086 0.069 15.667 0.000
## V14 0.884 0.063 14.090 0.000
## V16 0.727 0.072 10.040 0.000
## V20 0.811 0.067 12.120 0.000
## F2 =~
## V5 1.000
## V10 1.151 0.154 7.468 0.000
## V11 1.363 0.164 8.335 0.000
## V15 0.910 0.124 7.353 0.000
## V22 0.769 0.123 6.256 0.000
## F3 =~
## V4 1.000
## V7 0.969 0.128 7.556 0.000
## V9 1.782 0.323 5.517 0.000
## V12 1.505 0.241 6.231 0.000
## V17 1.349 0.200 6.750 0.000
## V18 1.919 0.299 6.422 0.000
## V19 1.718 0.288 5.971 0.000
## V21 1.356 0.227 5.969 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## .V6 ~~
## .V16 0.708 0.122 5.797 0.000
## .V1 ~~
## .V2 0.596 0.087 6.882 0.000
## F1 ~~
## F2 0.672 0.103 6.516 0.000
## F3 -0.193 0.039 -4.908 0.000
## F2 ~~
## F3 -0.171 0.036 -4.758 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .V1 4.366 0.086 50.675 0.000
## .V2 4.868 0.080 60.741 0.000
## .V3 3.527 0.090 39.243 0.000
## .V6 2.707 0.082 32.958 0.000
## .V8 3.043 0.090 33.972 0.000
## .V13 3.586 0.087 41.108 0.000
## .V14 4.027 0.090 44.989 0.000
## .V16 2.473 0.075 33.137 0.000
## .V20 2.245 0.073 30.597 0.000
## .V5 2.199 0.077 28.505 0.000
## .V10 2.204 0.075 29.388 0.000
## .V11 2.239 0.079 28.241 0.000
## .V15 1.769 0.067 26.261 0.000
## .V22 2.581 0.082 31.495 0.000
## .V4 6.298 0.052 121.663 0.000
## .V7 6.312 0.044 144.912 0.000
## .V9 6.035 0.068 88.446 0.000
## .V12 5.699 0.062 92.113 0.000
## .V17 6.406 0.044 144.908 0.000
## .V18 5.702 0.066 86.338 0.000
## .V19 5.946 0.062 96.417 0.000
## .V21 5.852 0.066 89.138 0.000
## F1 0.000
## F2 0.000
## F3 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .V1 1.276 0.105 12.194 0.000
## .V2 1.246 0.099 12.652 0.000
## .V3 1.312 0.110 11.915 0.000
## .V6 1.633 0.143 11.429 0.000
## .V8 0.793 0.083 9.546 0.000
## .V13 1.081 0.124 8.700 0.000
## .V14 1.819 0.145 12.550 0.000
## .V16 1.287 0.117 11.006 0.000
## .V20 1.024 0.137 7.496 0.000
## .V5 1.511 0.179 8.417 0.000
## .V10 1.165 0.147 7.901 0.000
## .V11 1.037 0.141 7.382 0.000
## .V15 1.106 0.153 7.229 0.000
## .V22 2.079 0.184 11.277 0.000
## .V4 0.802 0.113 7.117 0.000
## .V7 0.523 0.075 7.008 0.000
## .V9 1.116 0.149 7.485 0.000
## .V12 0.985 0.126 7.842 0.000
## .V17 0.375 0.057 6.634 0.000
## .V18 0.909 0.143 6.365 0.000
## .V19 0.844 0.111 7.626 0.000
## .V21 1.245 0.134 9.322 0.000
## F1 1.477 0.150 9.855 0.000
## F2 0.697 0.158 4.422 0.000
## F3 0.192 0.050 3.831 0.000
##
## Modification Indices:
##
## lhs op rhs mi mi.scaled epc sepc.lv sepc.all sepc.nox
## 78 F1 =~ V5 0.073 0.061 0.025 0.031 0.021 0.021
## 79 F1 =~ V10 5.282 4.379 -0.209 -0.254 -0.176 -0.176
## 80 F1 =~ V11 0.167 0.138 0.041 0.049 0.032 0.032
## 81 F1 =~ V15 0.017 0.014 0.011 0.013 0.010 0.010
## 82 F1 =~ V22 6.492 5.382 0.261 0.318 0.201 0.201
## 83 F1 =~ V4 4.564 3.783 0.097 0.118 0.118 0.118
## 84 F1 =~ V7 1.985 1.646 0.053 0.064 0.076 0.076
## 85 F1 =~ V9 1.908 1.582 0.077 0.094 0.072 0.072
## 86 F1 =~ V12 41.026 34.012 -0.332 -0.404 -0.339 -0.339
## 87 F1 =~ V17 0.074 0.062 0.009 0.011 0.013 0.013
## 88 F1 =~ V18 2.003 1.660 0.074 0.090 0.071 0.071
## 89 F1 =~ V19 0.196 0.163 -0.022 -0.027 -0.023 -0.023
## 90 F1 =~ V21 0.224 0.185 0.027 0.033 0.026 0.026
## 91 F2 =~ V1 0.896 0.743 -0.103 -0.086 -0.052 -0.052
## 92 F2 =~ V2 2.151 1.783 -0.155 -0.129 -0.084 -0.084
## 93 F2 =~ V3 0.015 0.013 0.016 0.014 0.008 0.008
## 94 F2 =~ V6 0.709 0.587 0.100 0.083 0.053 0.053
## 95 F2 =~ V8 0.133 0.110 0.042 0.035 0.020 0.020
## 96 F2 =~ V13 0.895 0.742 0.115 0.096 0.057 0.057
## 97 F2 =~ V14 0.761 0.631 -0.129 -0.108 -0.062 -0.062
## 98 F2 =~ V16 4.108 3.406 0.215 0.179 0.125 0.125
## 99 F2 =~ V20 0.829 0.687 -0.103 -0.086 -0.061 -0.061
## 100 F2 =~ V4 4.293 3.559 0.160 0.133 0.134 0.134
## 101 F2 =~ V7 0.362 0.300 0.038 0.032 0.038 0.038
## 102 F2 =~ V9 1.030 0.854 0.096 0.081 0.061 0.061
## 103 F2 =~ V12 14.001 11.608 -0.329 -0.274 -0.230 -0.230
## 104 F2 =~ V17 0.035 0.029 0.011 0.009 0.011 0.011
## 105 F2 =~ V18 1.155 0.958 0.096 0.080 0.063 0.063
## 106 F2 =~ V19 0.425 0.352 -0.055 -0.046 -0.039 -0.039
## 107 F2 =~ V21 0.143 0.118 -0.036 -0.030 -0.024 -0.024
## 108 F3 =~ V1 14.440 11.971 0.560 0.246 0.148 0.148
## 109 F3 =~ V2 1.034 0.858 0.146 0.064 0.042 0.042
## 110 F3 =~ V3 0.345 0.286 0.104 0.046 0.026 0.026
## 111 F3 =~ V6 1.167 0.967 -0.175 -0.077 -0.048 -0.048
## 112 F3 =~ V8 1.460 1.211 -0.185 -0.081 -0.047 -0.047
## 113 F3 =~ V13 9.519 7.892 -0.509 -0.223 -0.133 -0.133
## 114 F3 =~ V14 5.974 4.953 0.492 0.216 0.125 0.125
## 115 F3 =~ V16 2.646 2.194 -0.235 -0.103 -0.072 -0.072
## 116 F3 =~ V20 0.332 0.275 -0.089 -0.039 -0.028 -0.028
## 117 F3 =~ V5 0.086 0.071 -0.062 -0.027 -0.018 -0.018
## 118 F3 =~ V10 0.671 0.556 -0.163 -0.072 -0.050 -0.050
## 119 F3 =~ V11 3.325 2.757 0.385 0.169 0.110 0.110
## 120 F3 =~ V15 2.948 2.444 -0.314 -0.138 -0.106 -0.106
## 121 F3 =~ V22 0.778 0.645 0.207 0.091 0.057 0.057
## 122 V1 ~~ V3 13.922 11.542 0.248 0.248 0.086 0.086
## 123 V1 ~~ V6 0.003 0.003 0.003 0.003 0.001 0.001
## 124 V1 ~~ V8 0.617 0.512 -0.046 -0.046 -0.016 -0.016
## 125 V1 ~~ V13 1.662 1.378 -0.080 -0.080 -0.029 -0.029
## 126 V1 ~~ V14 0.860 0.713 0.069 0.069 0.024 0.024
## 127 V1 ~~ V16 0.486 0.403 -0.037 -0.037 -0.016 -0.016
## 128 V1 ~~ V20 0.284 0.235 -0.031 -0.031 -0.013 -0.013
## 129 V1 ~~ V5 0.043 0.036 -0.014 -0.014 -0.006 -0.006
## 130 V1 ~~ V10 0.109 0.090 -0.021 -0.021 -0.009 -0.009
## 131 V1 ~~ V11 0.709 0.588 0.052 0.052 0.021 0.021
## 132 V1 ~~ V15 0.130 0.108 -0.021 -0.021 -0.010 -0.010
## 133 V1 ~~ V22 0.625 0.518 0.061 0.061 0.023 0.023
## 134 V1 ~~ V4 1.546 1.281 0.060 0.060 0.036 0.036
## 135 V1 ~~ V7 0.598 0.496 0.031 0.031 0.022 0.022
## 136 V1 ~~ V9 1.991 1.651 0.083 0.083 0.038 0.038
## 137 V1 ~~ V12 0.602 0.499 -0.042 -0.042 -0.021 -0.021
## 138 V1 ~~ V17 0.034 0.028 0.007 0.007 0.005 0.005
## 139 V1 ~~ V18 1.330 1.103 0.063 0.063 0.030 0.030
## 140 V1 ~~ V19 1.377 1.142 0.061 0.061 0.031 0.031
## 141 V1 ~~ V21 0.110 0.091 0.020 0.020 0.010 0.010
## 142 V2 ~~ V3 0.013 0.011 -0.007 -0.007 -0.003 -0.003
## 143 V2 ~~ V6 0.211 0.175 0.027 0.027 0.011 0.011
## 144 V2 ~~ V8 2.255 1.870 0.085 0.085 0.032 0.032
## 145 V2 ~~ V13 1.180 0.978 -0.065 -0.065 -0.025 -0.025
## 146 V2 ~~ V14 8.290 6.872 0.209 0.209 0.079 0.079
## 147 V2 ~~ V16 0.479 0.397 -0.036 -0.036 -0.016 -0.016
## 148 V2 ~~ V20 2.956 2.451 -0.096 -0.096 -0.044 -0.044
## 149 V2 ~~ V5 0.000 0.000 0.000 0.000 0.000 0.000
## 150 V2 ~~ V10 0.247 0.205 -0.030 -0.030 -0.014 -0.014
## 151 V2 ~~ V11 0.610 0.505 -0.047 -0.047 -0.020 -0.020
## 152 V2 ~~ V15 0.989 0.820 -0.057 -0.057 -0.028 -0.028
## 153 V2 ~~ V22 1.682 1.395 0.098 0.098 0.040 0.040
## 154 V2 ~~ V4 0.136 0.113 -0.017 -0.017 -0.011 -0.011
## 155 V2 ~~ V7 0.314 0.260 -0.022 -0.022 -0.017 -0.017
## 156 V2 ~~ V9 2.864 2.374 0.097 0.097 0.048 0.048
## 157 V2 ~~ V12 0.557 0.462 -0.040 -0.040 -0.022 -0.022
## 158 V2 ~~ V17 0.002 0.002 0.002 0.002 0.001 0.001
## 159 V2 ~~ V18 0.374 0.310 0.032 0.032 0.017 0.017
## 160 V2 ~~ V19 0.006 0.005 -0.004 -0.004 -0.002 -0.002
## 161 V2 ~~ V21 0.019 0.016 -0.008 -0.008 -0.004 -0.004
## 162 V3 ~~ V6 1.861 1.543 0.099 0.099 0.036 0.036
## 163 V3 ~~ V8 0.020 0.017 0.011 0.011 0.004 0.004
## 164 V3 ~~ V13 2.279 1.889 -0.115 -0.115 -0.040 -0.040
## 165 V3 ~~ V14 0.026 0.022 0.015 0.015 0.005 0.005
## 166 V3 ~~ V16 3.878 3.215 -0.127 -0.127 -0.051 -0.051
## 167 V3 ~~ V20 6.454 5.351 -0.177 -0.177 -0.072 -0.072
## 168 V3 ~~ V5 0.563 0.467 0.061 0.061 0.024 0.024
## 169 V3 ~~ V10 0.008 0.007 0.007 0.007 0.003 0.003
## 170 V3 ~~ V11 0.027 0.022 0.012 0.012 0.005 0.005
## 171 V3 ~~ V15 0.136 0.113 -0.026 -0.026 -0.012 -0.012
## 172 V3 ~~ V22 0.031 0.026 -0.016 -0.016 -0.006 -0.006
## 173 V3 ~~ V4 2.508 2.079 0.092 0.092 0.053 0.053
## 174 V3 ~~ V7 1.894 1.570 0.065 0.065 0.045 0.045
## 175 V3 ~~ V9 0.673 0.558 -0.058 -0.058 -0.025 -0.025
## 176 V3 ~~ V12 16.431 13.622 -0.265 -0.265 -0.129 -0.129
## 177 V3 ~~ V17 5.874 4.870 0.103 0.103 0.070 0.070
## 178 V3 ~~ V18 0.086 0.071 0.019 0.019 0.009 0.009
## 179 V3 ~~ V19 0.250 0.208 -0.031 -0.031 -0.015 -0.015
## 180 V3 ~~ V21 0.295 0.244 0.039 0.039 0.018 0.018
## 181 V6 ~~ V8 1.718 1.424 -0.082 -0.082 -0.030 -0.030
## 182 V6 ~~ V13 3.515 2.914 0.126 0.126 0.047 0.047
## 183 V6 ~~ V14 7.689 6.375 -0.226 -0.226 -0.083 -0.083
## 184 V6 ~~ V20 0.898 0.744 -0.059 -0.059 -0.026 -0.026
## 185 V6 ~~ V5 10.791 8.946 0.247 0.247 0.105 0.105
## 186 V6 ~~ V10 0.228 0.189 -0.033 -0.033 -0.014 -0.014
## 187 V6 ~~ V11 3.044 2.523 -0.119 -0.119 -0.049 -0.049
## 188 V6 ~~ V15 1.201 0.995 0.071 0.071 0.035 0.035
## 189 V6 ~~ V22 0.385 0.319 -0.053 -0.053 -0.021 -0.021
## 190 V6 ~~ V4 0.066 0.055 0.014 0.014 0.009 0.009
## 191 V6 ~~ V7 1.010 0.837 -0.044 -0.044 -0.033 -0.033
## 192 V6 ~~ V9 0.012 0.010 0.007 0.007 0.003 0.003
## 193 V6 ~~ V12 3.911 3.242 0.119 0.119 0.063 0.063
## 194 V6 ~~ V17 5.304 4.397 -0.090 -0.090 -0.067 -0.067
## 195 V6 ~~ V18 0.305 0.253 0.033 0.033 0.016 0.016
## 196 V6 ~~ V19 0.037 0.031 0.011 0.011 0.006 0.006
## 197 V6 ~~ V21 0.443 0.367 -0.044 -0.044 -0.022 -0.022
## 198 V8 ~~ V13 1.924 1.595 -0.097 -0.097 -0.034 -0.034
## 199 V8 ~~ V14 8.074 6.693 -0.225 -0.225 -0.076 -0.076
## 200 V8 ~~ V16 1.696 1.406 0.073 0.073 0.029 0.029
## 201 V8 ~~ V20 7.151 5.928 0.166 0.166 0.068 0.068
## 202 V8 ~~ V5 2.073 1.719 -0.100 -0.100 -0.039 -0.039
## 203 V8 ~~ V10 0.114 0.094 0.021 0.021 0.009 0.009
## 204 V8 ~~ V11 0.624 0.517 0.050 0.050 0.019 0.019
## 205 V8 ~~ V15 0.002 0.002 0.003 0.003 0.001 0.001
## 206 V8 ~~ V22 0.262 0.217 -0.040 -0.040 -0.015 -0.015
## 207 V8 ~~ V4 2.333 1.934 -0.075 -0.075 -0.043 -0.043
## 208 V8 ~~ V7 0.166 0.137 -0.016 -0.016 -0.011 -0.011
## 209 V8 ~~ V9 0.018 0.015 -0.008 -0.008 -0.004 -0.004
## 210 V8 ~~ V12 7.730 6.408 -0.154 -0.154 -0.075 -0.075
## 211 V8 ~~ V17 0.779 0.646 0.032 0.032 0.022 0.022
## 212 V8 ~~ V18 0.104 0.086 -0.018 -0.018 -0.008 -0.008
## 213 V8 ~~ V19 2.965 2.458 0.090 0.090 0.044 0.044
## 214 V8 ~~ V21 0.065 0.054 -0.016 -0.016 -0.007 -0.007
## 215 V13 ~~ V14 2.773 2.299 0.140 0.140 0.048 0.048
## 216 V13 ~~ V16 0.120 0.099 -0.021 -0.021 -0.009 -0.009
## 217 V13 ~~ V20 7.493 6.212 0.178 0.178 0.075 0.075
## 218 V13 ~~ V5 0.302 0.250 -0.041 -0.041 -0.017 -0.017
## 219 V13 ~~ V10 0.001 0.001 -0.002 -0.002 -0.001 -0.001
## 220 V13 ~~ V11 0.198 0.164 -0.031 -0.031 -0.012 -0.012
## 221 V13 ~~ V15 0.182 0.151 0.028 0.028 0.013 0.013
## 222 V13 ~~ V22 0.485 0.402 0.060 0.060 0.023 0.023
## 223 V13 ~~ V4 0.268 0.222 0.028 0.028 0.017 0.017
## 224 V13 ~~ V7 0.576 0.477 0.033 0.033 0.023 0.023
## 225 V13 ~~ V9 6.758 5.603 -0.169 -0.169 -0.077 -0.077
## 226 V13 ~~ V12 9.295 7.706 0.184 0.184 0.092 0.092
## 227 V13 ~~ V17 0.241 0.200 -0.019 -0.019 -0.013 -0.013
## 228 V13 ~~ V18 2.702 2.240 -0.099 -0.099 -0.046 -0.046
## 229 V13 ~~ V19 4.612 3.823 -0.123 -0.123 -0.062 -0.062
## 230 V13 ~~ V21 0.334 0.277 -0.039 -0.039 -0.018 -0.018
## 231 V14 ~~ V16 1.072 0.889 0.075 0.075 0.030 0.030
## 232 V14 ~~ V20 0.032 0.027 -0.014 -0.014 -0.006 -0.006
## 233 V14 ~~ V5 0.221 0.184 -0.044 -0.044 -0.017 -0.017
## 234 V14 ~~ V10 3.809 3.158 -0.166 -0.166 -0.066 -0.066
## 235 V14 ~~ V11 1.636 1.357 0.109 0.109 0.041 0.041
## 236 V14 ~~ V15 0.177 0.147 0.034 0.034 0.015 0.015
## 237 V14 ~~ V22 0.807 0.669 0.095 0.095 0.035 0.035
## 238 V14 ~~ V4 0.256 0.212 0.033 0.033 0.019 0.019
## 239 V14 ~~ V7 0.099 0.082 0.017 0.017 0.012 0.012
## 240 V14 ~~ V9 0.001 0.001 -0.003 -0.003 -0.001 -0.001
## 241 V14 ~~ V12 0.001 0.001 0.003 0.003 0.001 0.001
## 242 V14 ~~ V17 0.072 0.060 -0.013 -0.013 -0.009 -0.009
## 243 V14 ~~ V18 0.570 0.472 0.056 0.056 0.026 0.026
## 244 V14 ~~ V19 2.961 2.455 0.122 0.122 0.059 0.059
## 245 V14 ~~ V21 0.121 0.101 0.029 0.029 0.013 0.013
## 246 V16 ~~ V20 0.007 0.005 -0.005 -0.005 -0.002 -0.002
## 247 V16 ~~ V5 0.620 0.514 0.053 0.053 0.025 0.025
## 248 V16 ~~ V10 0.001 0.001 -0.002 -0.002 -0.001 -0.001
## 249 V16 ~~ V11 1.823 1.511 0.082 0.082 0.038 0.038
## 250 V16 ~~ V15 0.167 0.138 -0.024 -0.024 -0.013 -0.013
## 251 V16 ~~ V22 0.149 0.123 0.029 0.029 0.013 0.013
## 252 V16 ~~ V4 0.755 0.626 0.041 0.041 0.029 0.029
## 253 V16 ~~ V7 0.001 0.001 0.001 0.001 0.001 0.001
## 254 V16 ~~ V9 1.863 1.545 0.079 0.079 0.042 0.042
## 255 V16 ~~ V12 5.621 4.660 -0.127 -0.127 -0.074 -0.074
## 256 V16 ~~ V17 0.061 0.051 0.009 0.009 0.007 0.007
## 257 V16 ~~ V18 0.104 0.086 -0.017 -0.017 -0.009 -0.009
## 258 V16 ~~ V19 3.059 2.536 -0.089 -0.089 -0.052 -0.052
## 259 V16 ~~ V21 0.621 0.515 0.047 0.047 0.026 0.026
## 260 V20 ~~ V5 0.482 0.400 -0.049 -0.049 -0.023 -0.023
## 261 V20 ~~ V10 0.001 0.001 -0.002 -0.002 -0.001 -0.001
## 262 V20 ~~ V11 1.546 1.282 -0.081 -0.081 -0.037 -0.037
## 263 V20 ~~ V15 0.214 0.177 0.028 0.028 0.015 0.015
## 264 V20 ~~ V22 0.020 0.016 0.011 0.011 0.005 0.005
## 265 V20 ~~ V4 1.150 0.953 -0.054 -0.054 -0.038 -0.038
## 266 V20 ~~ V7 0.020 0.016 0.006 0.006 0.005 0.005
## 267 V20 ~~ V9 0.435 0.360 0.040 0.040 0.022 0.022
## 268 V20 ~~ V12 0.043 0.035 0.012 0.012 0.007 0.007
## 269 V20 ~~ V17 2.209 1.831 -0.055 -0.055 -0.046 -0.046
## 270 V20 ~~ V18 0.339 0.281 0.033 0.033 0.018 0.018
## 271 V20 ~~ V19 0.782 0.648 -0.048 -0.048 -0.028 -0.028
## 272 V20 ~~ V21 0.825 0.684 0.057 0.057 0.032 0.032
## 273 V5 ~~ V10 1.961 1.625 -0.122 -0.122 -0.057 -0.057
## 274 V5 ~~ V11 7.750 6.425 -0.258 -0.258 -0.114 -0.114
## 275 V5 ~~ V15 15.931 13.207 0.316 0.316 0.164 0.164
## 276 V5 ~~ V22 1.254 1.039 0.114 0.114 0.048 0.048
## 277 V5 ~~ V4 0.365 0.302 0.037 0.037 0.025 0.025
## 278 V5 ~~ V7 2.810 2.330 -0.084 -0.084 -0.067 -0.067
## 279 V5 ~~ V9 0.107 0.089 0.024 0.024 0.012 0.012
## 280 V5 ~~ V12 6.543 5.424 0.177 0.177 0.100 0.100
## 281 V5 ~~ V17 0.210 0.174 -0.021 -0.021 -0.016 -0.016
## 282 V5 ~~ V18 0.646 0.536 0.055 0.055 0.029 0.029
## 283 V5 ~~ V19 1.771 1.468 -0.087 -0.087 -0.050 -0.050
## 284 V5 ~~ V21 1.725 1.430 -0.101 -0.101 -0.054 -0.054
## 285 V10 ~~ V11 37.190 30.832 0.575 0.575 0.260 0.260
## 286 V10 ~~ V15 6.361 5.273 -0.191 -0.191 -0.102 -0.102
## 287 V10 ~~ V22 5.764 4.778 -0.227 -0.227 -0.100 -0.100
## 288 V10 ~~ V4 0.667 0.553 -0.046 -0.046 -0.032 -0.032
## 289 V10 ~~ V7 4.084 3.386 0.092 0.092 0.076 0.076
## 290 V10 ~~ V9 0.059 0.049 -0.017 -0.017 -0.009 -0.009
## 291 V10 ~~ V12 0.138 0.114 -0.023 -0.023 -0.014 -0.014
## 292 V10 ~~ V17 1.120 0.929 -0.044 -0.044 -0.035 -0.035
## 293 V10 ~~ V18 2.308 1.913 -0.096 -0.096 -0.052 -0.052
## 294 V10 ~~ V19 2.202 1.825 0.089 0.089 0.052 0.052
## 295 V10 ~~ V21 0.171 0.142 -0.029 -0.029 -0.016 -0.016
## 296 V11 ~~ V15 5.502 4.562 -0.191 -0.191 -0.096 -0.096
## 297 V11 ~~ V22 1.791 1.485 -0.131 -0.131 -0.054 -0.054
## 298 V11 ~~ V4 0.720 0.596 0.047 0.047 0.031 0.031
## 299 V11 ~~ V7 0.657 0.545 0.037 0.037 0.029 0.029
## 300 V11 ~~ V9 0.030 0.025 0.012 0.012 0.006 0.006
## 301 V11 ~~ V12 0.674 0.559 -0.052 -0.052 -0.029 -0.029
## 302 V11 ~~ V17 0.538 0.446 0.030 0.030 0.023 0.023
## 303 V11 ~~ V18 0.407 0.338 0.040 0.040 0.021 0.021
## 304 V11 ~~ V19 0.230 0.190 0.029 0.029 0.016 0.016
## 305 V11 ~~ V21 0.655 0.543 -0.057 -0.057 -0.029 -0.029
## 306 V15 ~~ V22 1.936 1.605 0.122 0.122 0.060 0.060
## 307 V15 ~~ V4 0.273 0.227 -0.028 -0.028 -0.021 -0.021
## 308 V15 ~~ V7 4.236 3.512 -0.089 -0.089 -0.081 -0.081
## 309 V15 ~~ V9 0.187 0.155 -0.028 -0.028 -0.016 -0.016
## 310 V15 ~~ V12 0.013 0.011 -0.007 -0.007 -0.004 -0.004
## 311 V15 ~~ V17 2.475 2.052 0.061 0.061 0.055 0.055
## 312 V15 ~~ V18 0.005 0.004 0.004 0.004 0.003 0.003
## 313 V15 ~~ V19 4.917 4.076 -0.126 -0.126 -0.081 -0.081
## 314 V15 ~~ V21 0.848 0.703 0.061 0.061 0.037 0.037
## 315 V22 ~~ V4 3.244 2.689 0.126 0.126 0.080 0.080
## 316 V22 ~~ V7 0.935 0.775 -0.055 -0.055 -0.042 -0.042
## 317 V22 ~~ V9 0.360 0.298 0.051 0.051 0.025 0.025
## 318 V22 ~~ V12 0.292 0.242 0.043 0.043 0.023 0.023
## 319 V22 ~~ V17 1.964 1.628 -0.072 -0.072 -0.054 -0.054
## 320 V22 ~~ V18 0.326 0.270 0.045 0.045 0.022 0.022
## 321 V22 ~~ V19 0.185 0.154 0.032 0.032 0.017 0.017
## 322 V22 ~~ V21 0.568 0.471 0.066 0.066 0.033 0.033
## 323 V4 ~~ V7 33.523 27.792 0.210 0.210 0.251 0.251
## 324 V4 ~~ V9 1.256 1.041 -0.061 -0.061 -0.047 -0.047
## 325 V4 ~~ V12 0.073 0.061 -0.014 -0.014 -0.011 -0.011
## 326 V4 ~~ V17 2.616 2.169 0.055 0.055 0.064 0.064
## 327 V4 ~~ V18 7.556 6.264 -0.141 -0.141 -0.111 -0.111
## 328 V4 ~~ V19 9.871 8.184 -0.152 -0.152 -0.128 -0.128
## 329 V4 ~~ V21 13.153 10.904 0.201 0.201 0.159 0.159
## 330 V7 ~~ V9 0.258 0.214 -0.023 -0.023 -0.021 -0.021
## 331 V7 ~~ V12 5.983 4.960 -0.101 -0.101 -0.101 -0.101
## 332 V7 ~~ V17 0.427 0.354 -0.018 -0.018 -0.026 -0.026
## 333 V7 ~~ V18 11.652 9.660 -0.144 -0.144 -0.135 -0.135
## 334 V7 ~~ V19 0.121 0.100 -0.014 -0.014 -0.014 -0.014
## 335 V7 ~~ V21 33.636 27.885 0.264 0.264 0.249 0.249
## 336 V9 ~~ V12 0.054 0.045 0.015 0.015 0.009 0.009
## 337 V9 ~~ V17 0.190 0.157 -0.019 -0.019 -0.017 -0.017
## 338 V9 ~~ V18 0.249 0.206 -0.032 -0.032 -0.019 -0.019
## 339 V9 ~~ V19 9.493 7.870 0.187 0.187 0.120 0.120
## 340 V9 ~~ V21 0.349 0.290 -0.041 -0.041 -0.024 -0.024
## 341 V12 ~~ V17 0.006 0.005 -0.003 -0.003 -0.003 -0.003
## 342 V12 ~~ V18 1.256 1.041 0.066 0.066 0.044 0.044
## 343 V12 ~~ V19 2.813 2.332 -0.094 -0.094 -0.066 -0.066
## 344 V12 ~~ V21 0.490 0.406 -0.044 -0.044 -0.029 -0.029
## 345 V17 ~~ V18 3.654 3.029 0.079 0.079 0.073 0.073
## 346 V17 ~~ V19 3.178 2.635 -0.069 -0.069 -0.068 -0.068
## 347 V17 ~~ V21 0.054 0.044 -0.010 -0.010 -0.009 -0.009
## 348 V18 ~~ V19 18.591 15.413 0.250 0.250 0.165 0.165
## 349 V18 ~~ V21 5.592 4.636 -0.152 -0.152 -0.095 -0.095
## 350 V19 ~~ V21 8.588 7.120 -0.178 -0.178 -0.119 -0.119#Calulating MI
mi3 <- modindices(fit3)
head(mi3[order(mi3$mi, decreasing = TRUE),],10)## lhs op rhs mi mi.scaled epc sepc.lv sepc.all sepc.nox
## 86 F1 =~ V12 41.026 34.012 -0.332 -0.404 -0.339 -0.339
## 285 V10 ~~ V11 37.190 30.832 0.575 0.575 0.260 0.260
## 335 V7 ~~ V21 33.636 27.885 0.264 0.264 0.249 0.249
## 323 V4 ~~ V7 33.523 27.792 0.210 0.210 0.251 0.251
## 348 V18 ~~ V19 18.591 15.413 0.250 0.250 0.165 0.165
## 176 V3 ~~ V12 16.431 13.622 -0.265 -0.265 -0.129 -0.129
## 275 V5 ~~ V15 15.931 13.207 0.316 0.316 0.164 0.164
## 108 F3 =~ V1 14.440 11.971 0.560 0.246 0.148 0.148
## 103 F2 =~ V12 14.001 11.608 -0.329 -0.274 -0.230 -0.230
## 122 V1 ~~ V3 13.922 11.542 0.248 0.248 0.086 0.086We could model the cross loading of V12 on F1. However, as the text says, we first model the V10 - V11 correlation as their EPC is higher (0.575)
model4 <- '
F1 =~ V1 + V2 + V3 + V6 + V8 + V13 + V14 + V16 + V20
F2 =~ V5 + V10 + V11 + V15 + V22
F3 =~ V4 + V7 + V9 + V12 + V17 + V18 + V19 + V21
V6 ~~ V16
V1 ~~ V2
V10 ~~ V11'
# MLM CFA with V10 correlated with V11
fit4 <- cfa(model4, data = db1, estimator = "MLM")
summary(fit4, fit.measures = TRUE)## lavaan (0.5-22) converged normally after 47 iterations
##
## Number of observations 372
##
## Estimator ML Robust
## Minimum Function Test Statistic 487.893 403.049
## Degrees of freedom 203 203
## P-value (Chi-square) 0.000 0.000
## Scaling correction factor 1.211
## for the Satorra-Bentler correction
##
## Model test baseline model:
##
## Minimum Function Test Statistic 3452.269 2911.466
## Degrees of freedom 231 231
## P-value 0.000 0.000
##
## User model versus baseline model:
##
## Comparative Fit Index (CFI) 0.912 0.925
## Tucker-Lewis Index (TLI) 0.899 0.915
##
## Robust Comparative Fit Index (CFI) 0.924
## Robust Tucker-Lewis Index (TLI) 0.913
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -12707.131 -12707.131
## Loglikelihood unrestricted model (H1) -12463.184 -12463.184
##
## Number of free parameters 72 72
## Akaike (AIC) 25558.261 25558.261
## Bayesian (BIC) 25840.421 25840.421
## Sample-size adjusted Bayesian (BIC) 25611.988 25611.988
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.061 0.051
## 90 Percent Confidence Interval 0.054 0.068 0.045 0.058
## P-value RMSEA <= 0.05 0.004 0.351
##
## Robust RMSEA 0.057
## 90 Percent Confidence Interval 0.049 0.065
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.065 0.065
##
## Parameter Estimates:
##
## Information Expected
## Standard Errors Robust.sem
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## F1 =~
## V1 1.000
## V2 0.878 0.042 21.120 0.000
## V3 1.068 0.059 18.245 0.000
## V6 0.770 0.077 10.056 0.000
## V8 1.214 0.067 18.149 0.000
## V13 1.086 0.069 15.656 0.000
## V14 0.885 0.063 14.070 0.000
## V16 0.727 0.072 10.057 0.000
## V20 0.811 0.067 12.105 0.000
## F2 =~
## V5 1.000
## V10 0.886 0.123 7.195 0.000
## V11 1.102 0.129 8.547 0.000
## V15 0.919 0.119 7.705 0.000
## V22 0.776 0.116 6.676 0.000
## F3 =~
## V4 1.000
## V7 0.976 0.129 7.570 0.000
## V9 1.783 0.324 5.496 0.000
## V12 1.499 0.241 6.233 0.000
## V17 1.348 0.199 6.762 0.000
## V18 1.917 0.299 6.422 0.000
## V19 1.724 0.289 5.963 0.000
## V21 1.356 0.228 5.939 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## .V6 ~~
## .V16 0.703 0.122 5.762 0.000
## .V1 ~~
## .V2 0.596 0.086 6.896 0.000
## .V10 ~~
## .V11 0.519 0.110 4.724 0.000
## F1 ~~
## F2 0.751 0.106 7.086 0.000
## F3 -0.193 0.039 -4.901 0.000
## F2 ~~
## F3 -0.190 0.039 -4.884 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .V1 4.366 0.086 50.675 0.000
## .V2 4.868 0.080 60.741 0.000
## .V3 3.527 0.090 39.243 0.000
## .V6 2.707 0.082 32.958 0.000
## .V8 3.043 0.090 33.972 0.000
## .V13 3.586 0.087 41.108 0.000
## .V14 4.027 0.090 44.989 0.000
## .V16 2.473 0.075 33.137 0.000
## .V20 2.245 0.073 30.597 0.000
## .V5 2.199 0.077 28.505 0.000
## .V10 2.204 0.075 29.388 0.000
## .V11 2.239 0.079 28.241 0.000
## .V15 1.769 0.067 26.261 0.000
## .V22 2.581 0.082 31.495 0.000
## .V4 6.298 0.052 121.663 0.000
## .V7 6.312 0.044 144.912 0.000
## .V9 6.035 0.068 88.446 0.000
## .V12 5.699 0.062 92.113 0.000
## .V17 6.406 0.044 144.908 0.000
## .V18 5.702 0.066 86.338 0.000
## .V19 5.946 0.062 96.417 0.000
## .V21 5.852 0.066 89.138 0.000
## F1 0.000
## F2 0.000
## F3 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .V1 1.276 0.105 12.165 0.000
## .V2 1.245 0.098 12.680 0.000
## .V3 1.313 0.110 11.889 0.000
## .V6 1.626 0.142 11.417 0.000
## .V8 0.799 0.083 9.622 0.000
## .V13 1.080 0.124 8.693 0.000
## .V14 1.817 0.145 12.568 0.000
## .V16 1.285 0.117 11.000 0.000
## .V20 1.024 0.137 7.500 0.000
## .V5 1.404 0.181 7.776 0.000
## .V10 1.457 0.150 9.713 0.000
## .V11 1.358 0.159 8.521 0.000
## .V15 1.005 0.142 7.101 0.000
## .V22 2.006 0.182 11.004 0.000
## .V4 0.802 0.113 7.115 0.000
## .V7 0.521 0.074 6.998 0.000
## .V9 1.116 0.149 7.480 0.000
## .V12 0.988 0.125 7.880 0.000
## .V17 0.376 0.057 6.632 0.000
## .V18 0.912 0.143 6.375 0.000
## .V19 0.840 0.111 7.596 0.000
## .V21 1.246 0.134 9.321 0.000
## F1 1.477 0.150 9.851 0.000
## F2 0.803 0.171 4.700 0.000
## F3 0.192 0.050 3.823 0.000#Calulating MI
mi4 <- modindices(fit4)
head(mi4[order(mi4$mi, decreasing = TRUE),],10)## lhs op rhs mi mi.scaled epc sepc.lv sepc.all sepc.nox
## 87 F1 =~ V12 40.621 33.557 -0.331 -0.402 -0.337 -0.337
## 335 V7 ~~ V21 33.404 27.595 0.262 0.262 0.247 0.247
## 323 V4 ~~ V7 33.318 27.524 0.209 0.209 0.250 0.250
## 348 V18 ~~ V19 18.400 15.200 0.248 0.248 0.164 0.164
## 177 V3 ~~ V12 16.749 13.837 -0.268 -0.268 -0.130 -0.130
## 109 F3 =~ V1 14.481 11.963 0.561 0.246 0.148 0.148
## 104 F2 =~ V12 14.270 11.789 -0.324 -0.290 -0.243 -0.243
## 123 V1 ~~ V3 13.974 11.544 0.249 0.249 0.087 0.087
## 329 V4 ~~ V21 13.190 10.896 0.201 0.201 0.160 0.160
## 333 V7 ~~ V18 12.056 9.960 -0.147 -0.147 -0.137 -0.137Well, if you have had the patience to last till here, this is the last model. Note that in the Model formula for F1, an additional V12 has been added at the end.
model5 <- '
F1 =~ V1 + V2 + V3 + V6 + V8 + V13 + V14 + V16 + V20 + V12
F2 =~ V5 + V10 + V11 + V15 + V22
F3 =~ V4 + V7 + V9 + V12 + V17 + V18 + V19 + V21
V6 ~~ V16
V1 ~~ V2
V10 ~~ V11'
#MLM CFA with crossloading of F1 with V12
fit5 <- cfa(model5, data = db1, estimator = "MLM")
summary(fit5, fit.measures = TRUE)## lavaan (0.5-22) converged normally after 52 iterations
##
## Number of observations 372
##
## Estimator ML Robust
## Minimum Function Test Statistic 446.419 369.998
## Degrees of freedom 202 202
## P-value (Chi-square) 0.000 0.000
## Scaling correction factor 1.207
## for the Satorra-Bentler correction
##
## Model test baseline model:
##
## Minimum Function Test Statistic 3452.269 2911.466
## Degrees of freedom 231 231
## P-value 0.000 0.000
##
## User model versus baseline model:
##
## Comparative Fit Index (CFI) 0.924 0.937
## Tucker-Lewis Index (TLI) 0.913 0.928
##
## Robust Comparative Fit Index (CFI) 0.936
## Robust Tucker-Lewis Index (TLI) 0.927
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -12686.394 -12686.394
## Loglikelihood unrestricted model (H1) -12463.184 -12463.184
##
## Number of free parameters 73 73
## Akaike (AIC) 25518.787 25518.787
## Bayesian (BIC) 25804.867 25804.867
## Sample-size adjusted Bayesian (BIC) 25573.260 25573.260
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.057 0.047
## 90 Percent Confidence Interval 0.050 0.064 0.040 0.054
## P-value RMSEA <= 0.05 0.052 0.735
##
## Robust RMSEA 0.052
## 90 Percent Confidence Interval 0.044 0.060
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.054 0.054
##
## Parameter Estimates:
##
## Information Expected
## Standard Errors Robust.sem
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## F1 =~
## V1 1.000
## V2 0.878 0.041 21.287 0.000
## V3 1.073 0.058 18.435 0.000
## V6 0.764 0.077 9.978 0.000
## V8 1.215 0.066 18.357 0.000
## V13 1.072 0.070 15.395 0.000
## V14 0.880 0.063 14.052 0.000
## V16 0.727 0.073 10.018 0.000
## V20 0.806 0.067 12.111 0.000
## V12 -0.316 0.054 -5.882 0.000
## F2 =~
## V5 1.000
## V10 0.889 0.124 7.168 0.000
## V11 1.105 0.130 8.519 0.000
## V15 0.921 0.120 7.660 0.000
## V22 0.776 0.117 6.659 0.000
## F3 =~
## V4 1.000
## V7 0.973 0.128 7.592 0.000
## V9 1.763 0.317 5.554 0.000
## V12 1.131 0.202 5.600 0.000
## V17 1.327 0.198 6.708 0.000
## V18 1.890 0.291 6.489 0.000
## V19 1.695 0.286 5.925 0.000
## V21 1.342 0.224 5.985 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## .V6 ~~
## .V16 0.706 0.122 5.765 0.000
## .V1 ~~
## .V2 0.588 0.086 6.861 0.000
## .V10 ~~
## .V11 0.517 0.110 4.713 0.000
## F1 ~~
## F2 0.747 0.106 7.029 0.000
## F3 -0.167 0.038 -4.350 0.000
## F2 ~~
## F3 -0.181 0.038 -4.781 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .V1 4.366 0.086 50.675 0.000
## .V2 4.868 0.080 60.741 0.000
## .V3 3.527 0.090 39.243 0.000
## .V6 2.707 0.082 32.958 0.000
## .V8 3.043 0.090 33.972 0.000
## .V13 3.586 0.087 41.108 0.000
## .V14 4.027 0.090 44.989 0.000
## .V16 2.473 0.075 33.137 0.000
## .V20 2.245 0.073 30.597 0.000
## .V12 5.699 0.062 92.113 0.000
## .V5 2.199 0.077 28.505 0.000
## .V10 2.204 0.075 29.388 0.000
## .V11 2.239 0.079 28.241 0.000
## .V15 1.769 0.067 26.261 0.000
## .V22 2.581 0.082 31.495 0.000
## .V4 6.298 0.052 121.663 0.000
## .V7 6.312 0.044 144.912 0.000
## .V9 6.035 0.068 88.446 0.000
## .V17 6.406 0.044 144.908 0.000
## .V18 5.702 0.066 86.338 0.000
## .V19 5.946 0.062 96.417 0.000
## .V21 5.852 0.066 89.138 0.000
## F1 0.000
## F2 0.000
## F3 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .V1 1.268 0.103 12.252 0.000
## .V2 1.238 0.098 12.614 0.000
## .V3 1.285 0.108 11.923 0.000
## .V6 1.636 0.143 11.458 0.000
## .V8 0.783 0.080 9.815 0.000
## .V13 1.115 0.128 8.681 0.000
## .V14 1.822 0.144 12.634 0.000
## .V16 1.281 0.116 11.032 0.000
## .V20 1.031 0.137 7.509 0.000
## .V12 0.898 0.105 8.545 0.000
## .V5 1.407 0.181 7.761 0.000
## .V10 1.455 0.150 9.697 0.000
## .V11 1.355 0.160 8.492 0.000
## .V15 1.004 0.142 7.084 0.000
## .V22 2.008 0.182 11.007 0.000
## .V4 0.795 0.112 7.098 0.000
## .V7 0.515 0.074 6.988 0.000
## .V9 1.108 0.150 7.397 0.000
## .V17 0.374 0.056 6.685 0.000
## .V18 0.906 0.143 6.327 0.000
## .V19 0.838 0.113 7.426 0.000
## .V21 1.240 0.133 9.354 0.000
## F1 1.486 0.150 9.919 0.000
## F2 0.800 0.171 4.678 0.000
## F3 0.199 0.051 3.891 0.000#Modification indices
mi5 <- modindices(fit5)
head(mi5[order(mi5$mi, decreasing = TRUE),],10)## lhs op rhs mi mi.scaled epc sepc.lv sepc.all sepc.nox
## 340 V7 ~~ V21 32.503 26.939 0.259 0.259 0.244 0.244
## 330 V4 ~~ V7 32.009 26.530 0.204 0.204 0.244 0.244
## 348 V18 ~~ V19 18.274 15.145 0.250 0.250 0.166 0.166
## 109 F3 =~ V1 14.649 12.141 0.541 0.241 0.145 0.145
## 338 V7 ~~ V18 14.409 11.942 -0.161 -0.161 -0.151 -0.151
## 219 V13 ~~ V12 13.063 10.826 0.212 0.212 0.106 0.106
## 123 V1 ~~ V3 12.963 10.744 0.237 0.237 0.083 0.083
## 335 V4 ~~ V21 12.460 10.327 0.195 0.195 0.155 0.155
## 334 V4 ~~ V19 11.555 9.577 -0.165 -0.165 -0.139 -0.139
## 169 V3 ~~ V12 11.417 9.463 -0.210 -0.210 -0.102 -0.102Thats all folks!