if (!require(lavaan)) install.packages("lavaan")Loading required package: lavaan
This is lavaan 0.5-23.1097
lavaan is BETA software! Please report any bugs.library(lavaan)lavaan (N = 151; 6 items over 3 occasions)These data measuring a latent trait of social functioning were collected at a Psychiatric Rehabilitation center, in which time 1 was admittance, and times 2 and 3 were collected at six-month intervals. There were six subscales that were completed by the hospital staff for each patient, including positively-oriented measures of Social Competence, Social Interest, and Personal Neatness, and negatively-oriented measures of Psychoticism, Motor Retardation, and Irritability. The negatively-oriented subscales were reflected (*−1) prior to analysis. Initial models examined the fit of one-factor versus two-factor models given the two valences of the subscales, but the fit of the two-factor model was not a significant improvement, and thus a one-factor model with all six items was used here.
#read in data
cafData = read.table(file = "CAF.dat", header = FALSE, na.strings = "9999")
names(cafData) = c("ID", "v1T1", "v1T2", "v1T3", "v2T1", "v2T2", "v2T3", "v3T1", "v3T2", "v3T3", "v4T1", "v4T2", "v4T3", "v5T1", "v5T2", "v5T3", "v6T1", "v6T2", "v6T3")
#recode negatively worded items
cafData$v4T1 = cafData$v4T1*(-1); cafData$v4T2 = cafData$v4T2*(-1); cafData$v4T3 = cafData$v4T3*(-1);
cafData$v5T1 = cafData$v5T1*(-1); cafData$v5T2 = cafData$v5T2*(-1); cafData$v5T3 = cafData$v5T3*(-1);
cafData$v6T1 = cafData$v6T1*(-1); cafData$v6T2 = cafData$v6T2*(-1); cafData$v6T3 = cafData$v6T3*(-1);Note: as groups are not defined in our data, we will not be using the the multiple group syntax in lavaan. Instead, we will build our code where the three factors are the three groups.
configuralSyntax = "
#===================================================================================================
#Factor loadings all freely estimated in each group with labels
Time1 =~ L1T1*v1T1 + L2T1*v2T1 + L3T1*v3T1 + L4T1*v4T1 + L5T1*v5T1 + L6T1*v6T1
Time2 =~ L1T2*v1T2 + L2T2*v2T2 + L3T2*v3T2 + L4T2*v4T2 + L5T2*v5T2 + L6T2*v6T2
Time3 =~ L1T3*v1T3 + L2T3*v2T3 + L3T3*v3T3 + L4T3*v4T3 + L5T3*v5T3 + L6T3*v6T3
#===================================================================================================
#Item intercepts all freely estimated in each group with labels
v1T1 ~ I1T1*1; v2T1 ~ I2T1*1; v3T1 ~ I3T1*1; v4T1 ~ I4T1*1; v5T1 ~ I5T1*1; v6T1 ~ I6T1*1;
v1T2 ~ I1T2*1; v2T2 ~ I2T2*1; v3T2 ~ I3T2*1; v4T2 ~ I4T2*1; v5T2 ~ I5T2*1; v6T2 ~ I6T2*1;
v1T3 ~ I1T3*1; v2T3 ~ I2T3*1; v3T3 ~ I3T3*1; v4T3 ~ I4T3*1; v5T3 ~ I5T3*1; v6T3 ~ I6T3*1;
#===================================================================================================
#Redidual variances all freely estimated in each group with labels
v1T1 ~~ E1T1*v1T1; v2T1 ~~ E2T1*v2T1; v3T1 ~~ E3T1*v3T1; v4T1 ~~ E4T1*v4T1; v5T1 ~~ E5T1*v5T1; v6T1 ~~ E6T1*v6T1;
v1T2 ~~ E1T2*v1T2; v2T2 ~~ E2T2*v2T2; v3T2 ~~ E3T2*v3T2; v4T2 ~~ E4T2*v4T2; v5T2 ~~ E5T2*v5T2; v6T2 ~~ E6T2*v6T2;
v1T3 ~~ E1T3*v1T3; v2T3 ~~ E2T3*v2T3; v3T3 ~~ E3T3*v3T3; v4T3 ~~ E4T3*v4T3; v5T3 ~~ E5T3*v5T3; v6T3 ~~ E6T3*v6T3;
#===================================================================================================
#Residual covariances freely estimated in each group with labels
v1T1 ~~ C1T12*v1T2 + C1T13*v1T3; v1T2 ~~ C1T23*v1T3;
v2T1 ~~ C2T12*v2T2 + C2T13*v2T3; v2T2 ~~ C2T23*v2T3;
v3T1 ~~ C3T12*v3T2 + C3T13*v3T3; v3T2 ~~ C3T23*v3T3;
v4T1 ~~ C4T12*v4T2 + C4T13*v4T3; v4T2 ~~ C4T23*v4T3;
v5T1 ~~ C5T12*v5T2 + C5T13*v5T3; v5T2 ~~ C5T23*v5T3;
v6T1 ~~ C6T12*v6T2 + C6T13*v6T3; v6T2 ~~ C6T23*v6T3;
#===================================================================================================
#Factor variance fixed to 1 for identification in each group
Time1 ~~ 1*Time1
Time2 ~~ 1*Time2
Time3 ~~ 1*Time3
#===================================================================================================
#Factor mean fixed to zero for identification in each group
Time1 ~ 0
Time2 ~ 0
Time3 ~ 0
#===================================================================================================
#Factor covariances all freely estimated
Time1 ~~ Time2 + Time3
Time2 ~~ Time3
#===================================================================================================
"
configuralEstimates = lavaan(model = configuralSyntax, data = cafData, estimator = "MLR", mimic = "mplus")lavaan WARNING: some cases are empty and will be ignored:
16 58 64 75 76 133 153 156summary(configuralEstimates, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan (0.5-23.1097) converged normally after 118 iterations
Used Total
Number of observations 151 159
Number of missing patterns 7
Estimator ML Robust
Minimum Function Test Statistic 292.514 283.247
Degrees of freedom 114 114
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.033
for the Yuan-Bentler correction (Mplus variant)
Model test baseline model:
Minimum Function Test Statistic 2192.158 1896.786
Degrees of freedom 153 153
P-value 0.000 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.912 0.903
Tucker-Lewis Index (TLI) 0.883 0.870
Robust Comparative Fit Index (CFI) 0.913
Robust Tucker-Lewis Index (TLI) 0.884
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -4430.302 -4430.302
Scaling correction factor 1.462
for the MLR correction
Loglikelihood unrestricted model (H1) -4284.045 -4284.045
Scaling correction factor 1.203
for the MLR correction
Number of free parameters 75 75
Akaike (AIC) 9010.604 9010.604
Bayesian (BIC) 9236.900 9236.900
Sample-size adjusted Bayesian (BIC) 8999.533 8999.533
Root Mean Square Error of Approximation:
RMSEA 0.102 0.099
90 Percent Confidence Interval 0.088 0.116 0.085 0.113
P-value RMSEA <= 0.05 0.000 0.000
Robust RMSEA 0.101
90 Percent Confidence Interval 0.086 0.116
Standardized Root Mean Square Residual:
SRMR 0.089 0.089
Parameter Estimates:
Information Observed
Standard Errors Robust.huber.white
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
Time1 =~
v1T1 (L1T1) 3.222 0.267 12.063 0.000 3.222 0.989
v2T1 (L2T1) 1.915 0.274 6.997 0.000 1.915 0.545
v3T1 (L3T1) 2.080 0.209 9.956 0.000 2.080 0.801
v4T1 (L4T1) 1.975 0.271 7.298 0.000 1.975 0.593
v5T1 (L5T1) 0.931 0.148 6.281 0.000 0.931 0.568
v6T1 (L6T1) 1.441 0.119 12.101 0.000 1.441 0.742
Time2 =~
v1T2 (L1T2) 2.863 0.305 9.372 0.000 2.863 0.970
v2T2 (L2T2) 2.072 0.197 10.490 0.000 2.072 0.649
v3T2 (L3T2) 2.133 0.185 11.509 0.000 2.133 0.821
v4T2 (L4T2) 2.098 0.322 6.514 0.000 2.098 0.628
v5T2 (L5T2) 1.175 0.239 4.921 0.000 1.175 0.477
v6T2 (L6T2) 1.512 0.129 11.749 0.000 1.512 0.821
Time3 =~
v1T3 (L1T3) 2.550 0.288 8.865 0.000 2.550 0.962
v2T3 (L2T3) 1.961 0.230 8.539 0.000 1.961 0.654
v3T3 (L3T3) 1.751 0.210 8.323 0.000 1.751 0.750
v4T3 (L4T3) 1.678 0.260 6.448 0.000 1.678 0.551
v5T3 (L5T3) 1.021 0.170 6.012 0.000 1.021 0.511
v6T3 (L6T3) 1.523 0.159 9.574 0.000 1.523 0.869
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 ~~
.v1T2 (C1T12) -0.214 0.250 -0.855 0.393 -0.214 -0.609
.v1T3 (C1T13) -0.004 0.247 -0.016 0.987 -0.004 -0.011
.v1T2 ~~
.v1T3 (C1T2) 0.113 0.231 0.488 0.626 0.113 0.218
.v2T1 ~~
.v2T2 (C2T12) 3.949 0.682 5.787 0.000 3.949 0.552
.v2T3 (C2T13) 2.256 0.805 2.803 0.005 2.256 0.338
.v2T2 ~~
.v2T3 (C2T2) 2.154 0.580 3.718 0.000 2.154 0.391
.v3T1 ~~
.v3T2 (C3T12) 0.965 0.289 3.340 0.001 0.965 0.419
.v3T3 (C3T13) 1.172 0.284 4.135 0.000 1.172 0.489
.v3T2 ~~
.v3T3 (C3T2) 1.241 0.316 3.931 0.000 1.241 0.542
.v4T1 ~~
.v4T2 (C4T12) 4.853 0.863 5.623 0.000 4.853 0.695
.v4T3 (C4T13) 3.801 1.033 3.679 0.000 3.801 0.557
.v4T2 ~~
.v4T3 (C4T2) 4.588 0.821 5.592 0.000 4.588 0.694
.v5T1 ~~
.v5T2 (C5T12) 2.349 0.767 3.064 0.002 2.349 0.804
.v5T3 (C5T13) 1.429 0.441 3.243 0.001 1.429 0.617
.v5T2 ~~
.v5T3 (C5T2) 2.887 0.777 3.717 0.000 2.887 0.778
.v6T1 ~~
.v6T2 (C6T12) 0.824 0.162 5.096 0.000 0.824 0.603
.v6T3 (C6T13) 0.592 0.182 3.251 0.001 0.592 0.525
.v6T2 ~~
.v6T3 (C6T2) 0.547 0.140 3.910 0.000 0.547 0.601
Time1 ~~
Time2 0.786 0.042 18.827 0.000 0.786 0.786
Time3 0.707 0.084 8.456 0.000 0.707 0.707
Time2 ~~
Time3 0.671 0.089 7.532 0.000 0.671 0.671
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (I1T1) 16.077 0.276 58.220 0.000 16.077 4.933
.v2T1 (I2T1) 8.672 0.298 29.132 0.000 8.672 2.469
.v3T1 (I3T1) 11.970 0.225 53.108 0.000 11.970 4.611
.v4T1 (I4T1) -3.037 0.271 -11.216 0.000 -3.037 -0.912
.v5T1 (I5T1) -1.283 0.138 -9.293 0.000 -1.283 -0.782
.v6T1 (I6T1) -2.871 0.164 -17.508 0.000 -2.871 -1.478
.v1T2 (I1T2) 17.226 0.245 70.294 0.000 17.226 5.837
.v2T2 (I2T2) 9.981 0.263 37.921 0.000 9.981 3.124
.v3T2 (I3T2) 12.467 0.218 57.265 0.000 12.467 4.799
.v4T2 (I4T2) -3.211 0.260 -12.349 0.000 -3.211 -0.961
.v5T2 (I5T2) -1.664 0.200 -8.338 0.000 -1.664 -0.676
.v6T2 (I6T2) -2.413 0.158 -15.316 0.000 -2.413 -1.311
.v1T3 (I1T3) 17.756 0.220 80.621 0.000 17.756 6.699
.v2T3 (I2T3) 10.442 0.281 37.204 0.000 10.442 3.483
.v3T3 (I3T3) 13.029 0.213 61.157 0.000 13.029 5.583
.v4T3 (I4T3) -2.738 0.249 -11.014 0.000 -2.738 -0.899
.v5T3 (I5T3) -1.247 0.166 -7.511 0.000 -1.247 -0.625
.v6T3 (I6T3) -2.075 0.152 -13.618 0.000 -2.075 -1.184
Time1 0.000 0.000 0.000
Time2 0.000 0.000 0.000
Time3 0.000 0.000 0.000
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (E1T1) 0.241 0.395 0.610 0.542 0.241 0.023
.v2T1 (E2T1) 8.672 1.022 8.484 0.000 8.672 0.703
.v3T1 (E3T1) 2.413 0.398 6.067 0.000 2.413 0.358
.v4T1 (E4T1) 7.199 1.036 6.950 0.000 7.199 0.649
.v5T1 (E5T1) 1.824 0.446 4.093 0.000 1.824 0.678
.v6T1 (E6T1) 1.694 0.243 6.974 0.000 1.694 0.449
.v1T2 (E1T2) 0.511 0.268 1.907 0.056 0.511 0.059
.v2T2 (E2T2) 5.913 0.617 9.581 0.000 5.913 0.579
.v3T2 (E3T2) 2.202 0.369 5.972 0.000 2.202 0.326
.v4T2 (E4T2) 6.765 0.990 6.834 0.000 6.765 0.606
.v5T2 (E5T2) 4.676 1.439 3.251 0.001 4.676 0.772
.v6T2 (E6T2) 1.103 0.166 6.643 0.000 1.103 0.326
.v1T3 (E1T3) 0.523 0.349 1.497 0.134 0.523 0.074
.v2T3 (E2T3) 5.142 0.806 6.379 0.000 5.142 0.572
.v3T3 (E3T3) 2.381 0.430 5.542 0.000 2.381 0.437
.v4T3 (E4T3) 6.456 1.078 5.988 0.000 6.456 0.696
.v5T3 (E5T3) 2.944 0.752 3.913 0.000 2.944 0.739
.v6T3 (E6T3) 0.751 0.162 4.630 0.000 0.751 0.244
Time1 1.000 1.000 1.000
Time2 1.000 1.000 1.000
Time3 1.000 1.000 1.000
R-Square:
Estimate
v1T1 0.977
v2T1 0.297
v3T1 0.642
v4T1 0.351
v5T1 0.322
v6T1 0.551
v1T2 0.941
v2T2 0.421
v3T2 0.674
v4T2 0.394
v5T2 0.228
v6T2 0.674
v1T3 0.926
v2T3 0.428
v3T3 0.563
v4T3 0.304
v5T3 0.261
v6T3 0.756Although the fit is not great, attempts to improve it logically were unsuccessful, so we proceed from here with this as the configural invariance model. This foreshadows the metric invariance section up next.
metricSyntax1 = "
#===================================================================================================
#Factor loadings all constrained across groups with labels *****
Time1 =~ L1*v1T1 + L2*v2T1 + L3*v3T1 + L4*v4T1 + L5*v5T1 + L6*v6T1
Time2 =~ L1*v1T2 + L2*v2T2 + L3*v3T2 + L4*v4T2 + L5*v5T2 + L6*v6T2
Time3 =~ L1*v1T3 + L2*v2T3 + L3*v3T3 + L4*v4T3 + L5*v5T3 + L6*v6T3
#===================================================================================================
#Item intercepts all freely estimated in each group with labels
v1T1 ~ I1T1*1; v2T1 ~ I2T1*1; v3T1 ~ I3T1*1; v4T1 ~ I4T1*1; v5T1 ~ I5T1*1; v6T1 ~ I6T1*1;
v1T2 ~ I1T2*1; v2T2 ~ I2T2*1; v3T2 ~ I3T2*1; v4T2 ~ I4T2*1; v5T2 ~ I5T2*1; v6T2 ~ I6T2*1;
v1T3 ~ I1T3*1; v2T3 ~ I2T3*1; v3T3 ~ I3T3*1; v4T3 ~ I4T3*1; v5T3 ~ I5T3*1; v6T3 ~ I6T3*1;
#===================================================================================================
#Redidual variances all freely estimated in each group with labels
v1T1 ~~ E1T1*v1T1; v2T1 ~~ E2T1*v2T1; v3T1 ~~ E3T1*v3T1; v4T1 ~~ E4T1*v4T1; v5T1 ~~ E5T1*v5T1; v6T1 ~~ E6T1*v6T1;
v1T2 ~~ E1T2*v1T2; v2T2 ~~ E2T2*v2T2; v3T2 ~~ E3T2*v3T2; v4T2 ~~ E4T2*v4T2; v5T2 ~~ E5T2*v5T2; v6T2 ~~ E6T2*v6T2;
v1T3 ~~ E1T3*v1T3; v2T3 ~~ E2T3*v2T3; v3T3 ~~ E3T3*v3T3; v4T3 ~~ E4T3*v4T3; v5T3 ~~ E5T3*v5T3; v6T3 ~~ E6T3*v6T3;
#===================================================================================================
#Residual covariances freely estimated in each group with labels
v1T1 ~~ C1T12*v1T2 + C1T13*v1T3; v1T2 ~~ C1T23*v1T3;
v2T1 ~~ C2T12*v2T2 + C2T13*v2T3; v2T2 ~~ C2T23*v2T3;
v3T1 ~~ C3T12*v3T2 + C3T13*v3T3; v3T2 ~~ C3T23*v3T3;
v4T1 ~~ C4T12*v4T2 + C4T13*v4T3; v4T2 ~~ C4T23*v4T3;
v5T1 ~~ C5T12*v5T2 + C5T13*v5T3; v5T2 ~~ C5T23*v5T3;
v6T1 ~~ C6T12*v6T2 + C6T13*v6T3; v6T2 ~~ C6T23*v6T3;
#===================================================================================================
#Factor variance fixed to 1 for identification in first group; estimated in others *****
Time1 ~~ 1*Time1
Time2 ~~ Time2
Time3 ~~ Time3
#===================================================================================================
#Factor mean fixed to zero for identification in each group
Time1 ~ 0
Time2 ~ 0
Time3 ~ 0
#===================================================================================================
#Factor covariances all freely estimated
Time1 ~~ Time2 + Time3
Time2 ~~ Time3
#===================================================================================================
"
metricEstimates1 = lavaan(model = metricSyntax1, data = cafData, estimator = "MLR", mimic = "mplus")lavaan WARNING: some cases are empty and will be ignored:
16 58 64 75 76 133 153 156summary(metricEstimates1, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan (0.5-23.1097) converged normally after 120 iterations
Used Total
Number of observations 151 159
Number of missing patterns 7
Estimator ML Robust
Minimum Function Test Statistic 316.712 301.234
Degrees of freedom 124 124
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.051
for the Yuan-Bentler correction (Mplus variant)
Model test baseline model:
Minimum Function Test Statistic 2192.158 1896.786
Degrees of freedom 153 153
P-value 0.000 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.905 0.898
Tucker-Lewis Index (TLI) 0.883 0.875
Robust Comparative Fit Index (CFI) 0.908
Robust Tucker-Lewis Index (TLI) 0.886
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -4442.401 -4442.401
Scaling correction factor 1.260
for the MLR correction
Loglikelihood unrestricted model (H1) -4284.045 -4284.045
Scaling correction factor 1.203
for the MLR correction
Number of free parameters 65 65
Akaike (AIC) 9014.803 9014.803
Bayesian (BIC) 9210.926 9210.926
Sample-size adjusted Bayesian (BIC) 9005.208 9005.208
Root Mean Square Error of Approximation:
RMSEA 0.101 0.097
90 Percent Confidence Interval 0.088 0.115 0.084 0.111
P-value RMSEA <= 0.05 0.000 0.000
Robust RMSEA 0.100
90 Percent Confidence Interval 0.085 0.114
Standardized Root Mean Square Residual:
SRMR 0.094 0.094
Parameter Estimates:
Information Observed
Standard Errors Robust.huber.white
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
Time1 =~
v1T1 (L1) 3.004 0.249 12.073 0.000 3.004 0.962
v2T1 (L2) 2.117 0.204 10.365 0.000 2.117 0.588
v3T1 (L3) 2.139 0.200 10.699 0.000 2.139 0.822
v4T1 (L4) 2.018 0.256 7.873 0.000 2.018 0.599
v5T1 (L5) 0.981 0.147 6.659 0.000 0.981 0.588
v6T1 (L6) 1.611 0.125 12.865 0.000 1.611 0.783
Time2 =~
v1T2 (L1) 3.004 0.249 12.073 0.000 2.938 0.975
v2T2 (L2) 2.117 0.204 10.365 0.000 2.070 0.647
v3T2 (L3) 2.139 0.200 10.699 0.000 2.091 0.814
v4T2 (L4) 2.018 0.256 7.873 0.000 1.973 0.605
v5T2 (L5) 0.981 0.147 6.659 0.000 0.959 0.405
v6T2 (L6) 1.611 0.125 12.865 0.000 1.575 0.834
Time3 =~
v1T3 (L1) 3.004 0.249 12.073 0.000 2.621 0.969
v2T3 (L2) 2.117 0.204 10.365 0.000 1.847 0.630
v3T3 (L3) 2.139 0.200 10.699 0.000 1.866 0.772
v4T3 (L4) 2.018 0.256 7.873 0.000 1.760 0.568
v5T3 (L5) 0.981 0.147 6.659 0.000 0.856 0.446
v6T3 (L6) 1.611 0.125 12.865 0.000 1.406 0.844
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 ~~
.v1T2 (C1T12) -0.208 0.255 -0.815 0.415 -0.208 -0.361
.v1T3 (C1T13) 0.035 0.235 0.150 0.880 0.035 0.062
.v1T2 ~~
.v1T3 (C1T2) 0.096 0.230 0.417 0.677 0.096 0.212
.v2T1 ~~
.v2T2 (C2T12) 3.930 0.678 5.796 0.000 3.930 0.552
.v2T3 (C2T13) 2.107 0.792 2.659 0.008 2.107 0.318
.v2T2 ~~
.v2T3 (C2T2) 2.131 0.583 3.655 0.000 2.131 0.383
.v3T1 ~~
.v3T2 (C3T12) 0.953 0.281 3.388 0.001 0.953 0.431
.v3T3 (C3T13) 1.090 0.279 3.911 0.000 1.090 0.479
.v3T2 ~~
.v3T3 (C3T2) 1.222 0.325 3.763 0.000 1.222 0.532
.v4T1 ~~
.v4T2 (C4T12) 4.821 0.874 5.515 0.000 4.821 0.688
.v4T3 (C4T13) 3.895 1.031 3.778 0.000 3.895 0.565
.v4T2 ~~
.v4T3 (C4T2) 4.590 0.810 5.668 0.000 4.590 0.692
.v5T1 ~~
.v5T2 (C5T12) 2.325 0.762 3.052 0.002 2.325 0.796
.v5T3 (C5T13) 1.403 0.435 3.223 0.001 1.403 0.605
.v5T2 ~~
.v5T3 (C5T2) 2.863 0.783 3.656 0.000 2.863 0.768
.v6T1 ~~
.v6T2 (C6T12) 0.809 0.158 5.133 0.000 0.809 0.609
.v6T3 (C6T13) 0.559 0.176 3.172 0.002 0.559 0.490
.v6T2 ~~
.v6T3 (C6T2) 0.531 0.133 3.981 0.000 0.531 0.573
Time1 ~~
Time2 0.769 0.072 10.625 0.000 0.787 0.787
Time3 0.626 0.115 5.431 0.000 0.717 0.717
Time2 ~~
Time3 0.573 0.116 4.935 0.000 0.671 0.671
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (I1T1) 16.086 0.274 58.644 0.000 16.086 5.150
.v2T1 (I2T1) 8.662 0.299 28.948 0.000 8.662 2.406
.v3T1 (I3T1) 11.965 0.226 52.891 0.000 11.965 4.600
.v4T1 (I4T1) -3.042 0.269 -11.325 0.000 -3.042 -0.903
.v5T1 (I5T1) -1.292 0.138 -9.363 0.000 -1.292 -0.775
.v6T1 (I6T1) -2.877 0.164 -17.490 0.000 -2.877 -1.399
.v1T2 (I1T2) 17.230 0.245 70.337 0.000 17.230 5.716
.v2T2 (I2T2) 9.983 0.264 37.869 0.000 9.983 3.119
.v3T2 (I3T2) 12.472 0.217 57.359 0.000 12.472 4.851
.v4T2 (I4T2) -3.207 0.259 -12.367 0.000 -3.207 -0.983
.v5T2 (I5T2) -1.664 0.200 -8.334 0.000 -1.664 -0.702
.v6T2 (I6T2) -2.412 0.158 -15.294 0.000 -2.412 -1.278
.v1T3 (I1T3) 17.768 0.221 80.464 0.000 17.768 6.567
.v2T3 (I2T3) 10.431 0.280 37.301 0.000 10.431 3.557
.v3T3 (I3T3) 13.041 0.212 61.633 0.000 13.041 5.395
.v4T3 (I4T3) -2.723 0.254 -10.709 0.000 -2.723 -0.878
.v5T3 (I5T3) -1.246 0.169 -7.382 0.000 -1.246 -0.649
.v6T3 (I6T3) -2.086 0.153 -13.615 0.000 -2.086 -1.253
Time1 0.000 0.000 0.000
Time2 0.000 0.000 0.000
Time3 0.000 0.000 0.000
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (E1T1) 0.729 0.353 2.064 0.039 0.729 0.075
.v2T1 (E2T1) 8.485 1.026 8.270 0.000 8.485 0.654
.v3T1 (E3T1) 2.190 0.334 6.552 0.000 2.190 0.324
.v4T1 (E4T1) 7.281 1.045 6.968 0.000 7.281 0.641
.v5T1 (E5T1) 1.819 0.444 4.096 0.000 1.819 0.654
.v6T1 (E6T1) 1.635 0.237 6.906 0.000 1.635 0.386
.v1T2 (E1T2) 0.455 0.263 1.730 0.084 0.455 0.050
.v2T2 (E2T2) 5.963 0.612 9.738 0.000 5.963 0.582
.v3T2 (E3T2) 2.235 0.361 6.191 0.000 2.235 0.338
.v4T2 (E4T2) 6.745 0.964 6.996 0.000 6.745 0.634
.v5T2 (E5T2) 4.694 1.429 3.286 0.001 4.694 0.836
.v6T2 (E6T2) 1.082 0.162 6.674 0.000 1.082 0.304
.v1T3 (E1T3) 0.452 0.325 1.391 0.164 0.452 0.062
.v2T3 (E2T3) 5.187 0.793 6.538 0.000 5.187 0.603
.v3T3 (E3T3) 2.361 0.427 5.527 0.000 2.361 0.404
.v4T3 (E4T3) 6.522 1.101 5.925 0.000 6.522 0.678
.v5T3 (E5T3) 2.958 0.772 3.834 0.000 2.958 0.802
.v6T3 (E6T3) 0.796 0.169 4.699 0.000 0.796 0.287
Time1 1.000 1.000 1.000
Time2 0.956 0.153 6.233 0.000 1.000 1.000
Time3 0.761 0.134 5.663 0.000 1.000 1.000
R-Square:
Estimate
v1T1 0.925
v2T1 0.346
v3T1 0.676
v4T1 0.359
v5T1 0.346
v6T1 0.614
v1T2 0.950
v2T2 0.418
v3T2 0.662
v4T2 0.366
v5T2 0.164
v6T2 0.696
v1T3 0.938
v2T3 0.397
v3T3 0.596
v4T3 0.322
v5T3 0.198
v6T3 0.713Does the metric model (metricEstimates1) fit worse than the configural model (configuralEstimates)? Yes, −2∆LL(df=10) = 19.14, p =.04.
anova(metricEstimates1, configuralEstimates)Scaled Chi Square Difference Test (method = "satorra.bentler.2001")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
configuralEstimates 114 9010.6 9236.9 292.51
metricEstimates1 124 9014.8 9210.9 316.71 19.142 10 0.0385 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1To investigate which parameter to free, we will look at the modification indices:
#get modification indices for all parameters:
metricMI1 = modificationindices(metricEstimates1, free.remove = FALSE, maximum.number = 10000)
#restrict output to only factor loading parameters
metricMI1a = metricMI1[which(metricMI1$op == "=~"),]
#restrict output to only factors and items with same time point
metricMI1a$factorTime = metricMI1a$lhs
#extract time characters from items
metricMI1a$itemTime = substr(metricMI1a$rhs, 3, 4)
#convert to same name as factors
metricMI1a$itemTime[which(metricMI1a$itemTime == "T1")] = "Time1"
metricMI1a$itemTime[which(metricMI1a$itemTime == "T2")] = "Time2"
metricMI1a$itemTime[which(metricMI1a$itemTime == "T3")] = "Time3"
#show only MIs for same time
metricMI1b = metricMI1a[which(metricMI1a$itemTime == metricMI1a$factorTime),]
metricMI1b[order(-metricMI1b$mi),] lhs op rhs mi mi.scaled epc sepc.lv sepc.all sepc.nox factorTime itemTime
18 Time3 =~ v6T3 5.684 5.406 0.210 0.183 0.110 0.110 Time3 Time3
1 Time1 =~ v1T1 5.678 5.400 0.271 0.271 0.087 0.087 Time1 Time1
11 Time2 =~ v5T2 3.302 3.140 0.186 0.182 0.077 0.077 Time2 Time2
5 Time1 =~ v5T1 3.297 3.136 -0.138 -0.138 -0.083 -0.083 Time1 Time1
6 Time1 =~ v6T1 2.694 2.562 -0.156 -0.156 -0.076 -0.076 Time1 Time1
13 Time3 =~ v1T3 1.817 1.729 -0.155 -0.135 -0.050 -0.050 Time3 Time3
10 Time2 =~ v4T2 0.946 0.899 0.150 0.147 0.045 0.045 Time2 Time2
15 Time3 =~ v3T3 0.924 0.879 -0.142 -0.124 -0.051 -0.051 Time3 Time3
7 Time2 =~ v1T2 0.826 0.786 -0.089 -0.087 -0.029 -0.029 Time2 Time2
17 Time3 =~ v5T3 0.633 0.602 0.102 0.089 0.046 0.046 Time3 Time3
2 Time1 =~ v2T1 0.553 0.526 -0.163 -0.163 -0.045 -0.045 Time1 Time1
12 Time2 =~ v6T2 0.542 0.515 -0.055 -0.054 -0.028 -0.028 Time2 Time2
9 Time2 =~ v3T2 0.423 0.402 0.076 0.074 0.029 0.029 Time2 Time2
16 Time3 =~ v4T3 0.410 0.390 -0.136 -0.119 -0.038 -0.038 Time3 Time3
14 Time3 =~ v2T3 0.396 0.377 0.151 0.132 0.045 0.045 Time3 Time3
4 Time1 =~ v4T1 0.339 0.323 -0.104 -0.104 -0.031 -0.031 Time1 Time1
8 Time2 =~ v2T2 0.020 0.019 0.025 0.025 0.008 0.008 Time2 Time2
3 Time1 =~ v3T1 0.013 0.012 0.014 0.014 0.005 0.005 Time1 Time1From these MIs there are two that are large-ish, the loading of item1 at time1 and the loading of item3 at time3, which have almost the same values. For consistency with the Mplus example, we will select the loading of item1 at time1 to free and continue. In practice, we would free one of these two parameters, not necessarily the one we have chosen.
metricSyntax2 = "
#===================================================================================================
#Factor loadings constrained across groups except non-invariant ones *****
Time1 =~ L1T1*v1T1 + L2*v2T1 + L3*v3T1 + L4*v4T1 + L5*v5T1 + L6*v6T1
Time2 =~ L1*v1T2 + L2*v2T2 + L3*v3T2 + L4*v4T2 + L5*v5T2 + L6*v6T2
Time3 =~ L1*v1T3 + L2*v2T3 + L3*v3T3 + L4*v4T3 + L5*v5T3 + L6*v6T3
#===================================================================================================
#Item intercepts all freely estimated in each group with labels
v1T1 ~ I1T1*1; v2T1 ~ I2T1*1; v3T1 ~ I3T1*1; v4T1 ~ I4T1*1; v5T1 ~ I5T1*1; v6T1 ~ I6T1*1;
v1T2 ~ I1T2*1; v2T2 ~ I2T2*1; v3T2 ~ I3T2*1; v4T2 ~ I4T2*1; v5T2 ~ I5T2*1; v6T2 ~ I6T2*1;
v1T3 ~ I1T3*1; v2T3 ~ I2T3*1; v3T3 ~ I3T3*1; v4T3 ~ I4T3*1; v5T3 ~ I5T3*1; v6T3 ~ I6T3*1;
#===================================================================================================
#Redidual variances all freely estimated in each group with labels
v1T1 ~~ E1T1*v1T1; v2T1 ~~ E2T1*v2T1; v3T1 ~~ E3T1*v3T1; v4T1 ~~ E4T1*v4T1; v5T1 ~~ E5T1*v5T1; v6T1 ~~ E6T1*v6T1;
v1T2 ~~ E1T2*v1T2; v2T2 ~~ E2T2*v2T2; v3T2 ~~ E3T2*v3T2; v4T2 ~~ E4T2*v4T2; v5T2 ~~ E5T2*v5T2; v6T2 ~~ E6T2*v6T2;
v1T3 ~~ E1T3*v1T3; v2T3 ~~ E2T3*v2T3; v3T3 ~~ E3T3*v3T3; v4T3 ~~ E4T3*v4T3; v5T3 ~~ E5T3*v5T3; v6T3 ~~ E6T3*v6T3;
#===================================================================================================
#Residual covariances freely estimated in each group with labels
v1T1 ~~ C1T12*v1T2 + C1T13*v1T3; v1T2 ~~ C1T23*v1T3;
v2T1 ~~ C2T12*v2T2 + C2T13*v2T3; v2T2 ~~ C2T23*v2T3;
v3T1 ~~ C3T12*v3T2 + C3T13*v3T3; v3T2 ~~ C3T23*v3T3;
v4T1 ~~ C4T12*v4T2 + C4T13*v4T3; v4T2 ~~ C4T23*v4T3;
v5T1 ~~ C5T12*v5T2 + C5T13*v5T3; v5T2 ~~ C5T23*v5T3;
v6T1 ~~ C6T12*v6T2 + C6T13*v6T3; v6T2 ~~ C6T23*v6T3;
#===================================================================================================
#Factor variance fixed to 1 for identification in first group; estimated in others *****
Time1 ~~ 1*Time1
Time2 ~~ Time2
Time3 ~~ Time3
#===================================================================================================
#Factor mean fixed to zero for identification in each group
Time1 ~ 0
Time2 ~ 0
Time3 ~ 0
#===================================================================================================
#Factor covariances all freely estimated
Time1 ~~ Time2 + Time3
Time2 ~~ Time3
#===================================================================================================
"
metricEstimates2 = lavaan(model = metricSyntax2, data = cafData, estimator = "MLR", mimic = "mplus")lavaan WARNING: some cases are empty and will be ignored:
16 58 64 75 76 133 153 156summary(metricEstimates2, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan (0.5-23.1097) converged normally after 116 iterations
Used Total
Number of observations 151 159
Number of missing patterns 7
Estimator ML Robust
Minimum Function Test Statistic 303.247 290.300
Degrees of freedom 123 123
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.045
for the Yuan-Bentler correction (Mplus variant)
Model test baseline model:
Minimum Function Test Statistic 2192.158 1896.786
Degrees of freedom 153 153
P-value 0.000 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.912 0.904
Tucker-Lewis Index (TLI) 0.890 0.881
Robust Comparative Fit Index (CFI) 0.913
Robust Tucker-Lewis Index (TLI) 0.892
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -4435.669 -4435.669
Scaling correction factor 1.284
for the MLR correction
Loglikelihood unrestricted model (H1) -4284.045 -4284.045
Scaling correction factor 1.203
for the MLR correction
Number of free parameters 66 66
Akaike (AIC) 9003.337 9003.337
Bayesian (BIC) 9202.478 9202.478
Sample-size adjusted Bayesian (BIC) 8993.595 8993.595
Root Mean Square Error of Approximation:
RMSEA 0.099 0.095
90 Percent Confidence Interval 0.085 0.113 0.081 0.109
P-value RMSEA <= 0.05 0.000 0.000
Robust RMSEA 0.097
90 Percent Confidence Interval 0.083 0.111
Standardized Root Mean Square Residual:
SRMR 0.091 0.091
Parameter Estimates:
Information Observed
Standard Errors Robust.huber.white
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
Time1 =~
v1T1 (L1T1) 3.233 0.261 12.362 0.000 3.233 0.992
v2T1 (L2) 1.950 0.201 9.707 0.000 1.950 0.551
v3T1 (L3) 1.967 0.198 9.910 0.000 1.967 0.779
v4T1 (L4) 1.899 0.224 8.481 0.000 1.899 0.578
v5T1 (L5) 0.968 0.137 7.055 0.000 0.968 0.582
v6T1 (L6) 1.476 0.131 11.247 0.000 1.476 0.749
Time2 =~
v1T2 (L1) 2.644 0.234 11.315 0.000 2.849 0.968
v2T2 (L2) 1.950 0.201 9.707 0.000 2.102 0.655
v3T2 (L3) 1.967 0.198 9.910 0.000 2.120 0.821
v4T2 (L4) 1.899 0.224 8.481 0.000 2.047 0.619
v5T2 (L5) 0.968 0.137 7.055 0.000 1.043 0.434
v6T2 (L6) 1.476 0.131 11.247 0.000 1.591 0.836
Time3 =~
v1T3 (L1) 2.644 0.234 11.315 0.000 2.564 0.963
v2T3 (L2) 1.950 0.201 9.707 0.000 1.892 0.639
v3T3 (L3) 1.967 0.198 9.910 0.000 1.908 0.782
v4T3 (L4) 1.899 0.224 8.481 0.000 1.842 0.582
v5T3 (L5) 0.968 0.137 7.055 0.000 0.939 0.480
v6T3 (L6) 1.476 0.131 11.247 0.000 1.432 0.850
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 ~~
.v1T2 (C1T12) -0.225 0.249 -0.904 0.366 -0.225 -0.737
.v1T3 (C1T13) -0.012 0.236 -0.049 0.961 -0.012 -0.039
.v1T2 ~~
.v1T3 (C1T2) 0.132 0.230 0.574 0.566 0.132 0.250
.v2T1 ~~
.v2T2 (C2T12) 3.963 0.679 5.834 0.000 3.963 0.553
.v2T3 (C2T13) 2.251 0.806 2.791 0.005 2.251 0.335
.v2T2 ~~
.v2T3 (C2T2) 2.142 0.580 3.691 0.000 2.142 0.388
.v3T1 ~~
.v3T2 (C3T12) 1.008 0.296 3.411 0.001 1.008 0.432
.v3T3 (C3T13) 1.148 0.285 4.023 0.000 1.148 0.478
.v3T2 ~~
.v3T3 (C3T2) 1.175 0.316 3.717 0.000 1.175 0.524
.v4T1 ~~
.v4T2 (C4T12) 4.811 0.848 5.672 0.000 4.811 0.691
.v4T3 (C4T13) 3.890 1.027 3.787 0.000 3.890 0.565
.v4T2 ~~
.v4T3 (C4T2) 4.685 0.827 5.667 0.000 4.685 0.701
.v5T1 ~~
.v5T2 (C5T12) 2.347 0.762 3.079 0.002 2.347 0.802
.v5T3 (C5T13) 1.423 0.438 3.248 0.001 1.423 0.613
.v5T2 ~~
.v5T3 (C5T2) 2.868 0.779 3.684 0.000 2.868 0.773
.v6T1 ~~
.v6T2 (C6T12) 0.833 0.164 5.090 0.000 0.833 0.610
.v6T3 (C6T13) 0.592 0.177 3.350 0.001 0.592 0.511
.v6T2 ~~
.v6T3 (C6T2) 0.533 0.135 3.951 0.000 0.533 0.576
Time1 ~~
Time2 0.847 0.078 10.837 0.000 0.786 0.786
Time3 0.682 0.124 5.508 0.000 0.703 0.703
Time2 ~~
Time3 0.699 0.128 5.473 0.000 0.669 0.669
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (I1T1) 16.078 0.276 58.267 0.000 16.078 4.934
.v2T1 (I2T1) 8.672 0.298 29.071 0.000 8.672 2.452
.v3T1 (I3T1) 11.978 0.225 53.192 0.000 11.978 4.745
.v4T1 (I4T1) -3.034 0.267 -11.343 0.000 -3.034 -0.924
.v5T1 (I5T1) -1.288 0.137 -9.377 0.000 -1.288 -0.775
.v6T1 (I6T1) -2.871 0.164 -17.506 0.000 -2.871 -1.457
.v1T2 (I1T2) 17.225 0.245 70.282 0.000 17.225 5.851
.v2T2 (I2T2) 9.980 0.264 37.872 0.000 9.980 3.108
.v3T2 (I3T2) 12.468 0.217 57.325 0.000 12.468 4.826
.v4T2 (I4T2) -3.210 0.260 -12.365 0.000 -3.210 -0.970
.v5T2 (I5T2) -1.663 0.199 -8.340 0.000 -1.663 -0.692
.v6T2 (I6T2) -2.414 0.158 -15.319 0.000 -2.414 -1.269
.v1T3 (I1T3) 17.756 0.222 80.036 0.000 17.756 6.671
.v2T3 (I2T3) 10.434 0.280 37.245 0.000 10.434 3.526
.v3T3 (I3T3) 13.041 0.212 61.441 0.000 13.041 5.347
.v4T3 (I4T3) -2.720 0.254 -10.720 0.000 -2.720 -0.860
.v5T3 (I5T3) -1.246 0.169 -7.373 0.000 -1.246 -0.637
.v6T3 (I6T3) -2.087 0.154 -13.571 0.000 -2.087 -1.240
Time1 0.000 0.000 0.000
Time2 0.000 0.000 0.000
Time3 0.000 0.000 0.000
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (E1T1) 0.170 0.374 0.454 0.650 0.170 0.016
.v2T1 (E2T1) 8.702 1.026 8.483 0.000 8.702 0.696
.v3T1 (E3T1) 2.502 0.386 6.484 0.000 2.502 0.393
.v4T1 (E4T1) 7.172 1.021 7.021 0.000 7.172 0.665
.v5T1 (E5T1) 1.829 0.443 4.131 0.000 1.829 0.661
.v6T1 (E6T1) 1.707 0.242 7.059 0.000 1.707 0.439
.v1T2 (E1T2) 0.548 0.265 2.070 0.038 0.548 0.063
.v2T2 (E2T2) 5.895 0.605 9.746 0.000 5.895 0.572
.v3T2 (E3T2) 2.178 0.352 6.183 0.000 2.178 0.326
.v4T2 (E4T2) 6.759 0.967 6.990 0.000 6.759 0.617
.v5T2 (E5T2) 4.678 1.430 3.272 0.001 4.678 0.811
.v6T2 (E6T2) 1.090 0.165 6.599 0.000 1.090 0.301
.v1T3 (E1T3) 0.509 0.314 1.618 0.106 0.509 0.072
.v2T3 (E2T3) 5.177 0.795 6.514 0.000 5.177 0.591
.v3T3 (E3T3) 2.309 0.416 5.548 0.000 2.309 0.388
.v4T3 (E4T3) 6.613 1.129 5.860 0.000 6.613 0.661
.v5T3 (E5T3) 2.944 0.760 3.872 0.000 2.944 0.770
.v6T3 (E6T3) 0.784 0.170 4.618 0.000 0.784 0.277
Time1 1.000 1.000 1.000
Time2 1.162 0.185 6.270 0.000 1.000 1.000
Time3 0.941 0.157 5.999 0.000 1.000 1.000
R-Square:
Estimate
v1T1 0.984
v2T1 0.304
v3T1 0.607
v4T1 0.335
v5T1 0.339
v6T1 0.561
v1T2 0.937
v2T2 0.428
v3T2 0.674
v4T2 0.383
v5T2 0.189
v6T2 0.699
v1T3 0.928
v2T3 0.409
v3T3 0.612
v4T3 0.339
v5T3 0.230
v6T3 0.723anova(metricEstimates2, configuralEstimates)Scaled Chi Square Difference Test (method = "satorra.bentler.2001")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
configuralEstimates 114 9010.6 9236.9 292.51
metricEstimates2 123 9003.3 9202.5 303.25 8.981 9 0.439The likelihood ratio test suggests that the partial metric invariance model fits as well as the configural model.
#get modification indices for all parameters:
metricMI2 = modificationindices(metricEstimates2, free.remove = FALSE, maximum.number = 10000)
#restrict output to only factor loading parameters
metricMI2a = metricMI2[which(metricMI2$op == "=~"),]
#restrict output to only factors and items with same time point
metricMI2a$factorTime = metricMI2a$lhs
#extract time characters from items
metricMI2a$itemTime = substr(metricMI2a$rhs, 3, 4)
#convert to same name as factors
metricMI2a$itemTime[which(metricMI2a$itemTime == "T1")] = "Time1"
metricMI2a$itemTime[which(metricMI2a$itemTime == "T2")] = "Time2"
metricMI2a$itemTime[which(metricMI2a$itemTime == "T3")] = "Time3"
#show only MIs for same time
metricMI2b = metricMI2a[which(metricMI2a$itemTime == metricMI2a$factorTime),]
metricMI2b[order(-metricMI2b$mi),] lhs op rhs mi mi.scaled epc sepc.lv sepc.all sepc.nox factorTime itemTime
18 Time3 =~ v6T3 3.279 3.139 0.143 0.138 0.082 0.082 Time3 Time3
15 Time3 =~ v3T3 2.363 2.262 -0.204 -0.197 -0.081 -0.081 Time3 Time3
12 Time2 =~ v6T2 2.020 1.934 -0.097 -0.105 -0.055 -0.055 Time2 Time2
11 Time2 =~ v5T2 1.506 1.442 0.112 0.121 0.050 0.050 Time2 Time2
5 Time1 =~ v5T1 1.393 1.333 -0.086 -0.086 -0.052 -0.052 Time1 Time1
16 Time3 =~ v4T3 1.030 0.986 -0.194 -0.188 -0.059 -0.059 Time3 Time3
3 Time1 =~ v3T1 1.024 0.980 0.124 0.124 0.049 0.049 Time1 Time1
10 Time2 =~ v4T2 0.192 0.184 0.061 0.066 0.020 0.020 Time2 Time2
4 Time1 =~ v4T1 0.139 0.133 0.064 0.064 0.020 0.020 Time1 Time1
14 Time3 =~ v2T3 0.127 0.122 0.077 0.075 0.025 0.025 Time3 Time3
9 Time2 =~ v3T2 0.125 0.120 0.037 0.040 0.016 0.016 Time2 Time2
17 Time3 =~ v5T3 0.100 0.095 0.036 0.035 0.018 0.018 Time3 Time3
13 Time3 =~ v1T3 0.072 0.069 -0.028 -0.027 -0.010 -0.010 Time3 Time3
7 Time2 =~ v1T2 0.054 0.052 0.021 0.022 0.008 0.008 Time2 Time2
6 Time1 =~ v6T1 0.041 0.040 -0.019 -0.019 -0.010 -0.010 Time1 Time1
2 Time1 =~ v2T1 0.025 0.024 -0.034 -0.034 -0.010 -0.010 Time1 Time1
8 Time2 =~ v2T2 0.022 0.021 -0.024 -0.026 -0.008 -0.008 Time2 Time2
1 Time1 =~ v1T1 0.000 0.000 0.000 0.000 0.000 0.000 Time1 Time1The modification indices indicate no other changes are needed, so we can continue.
scalarSyntax1 = "
#===================================================================================================
#Factor loadings constrained across groups except non-invariant ones
Time1 =~ L1T1*v1T1 + L2*v2T1 + L3*v3T1 + L4*v4T1 + L5*v5T1 + L6*v6T1
Time2 =~ L1*v1T2 + L2*v2T2 + L3*v3T2 + L4*v4T2 + L5*v5T2 + L6*v6T2
Time3 =~ L1*v1T3 + L2*v2T3 + L3*v3T3 + L4*v4T3 + L5*v5T3 + L6*v6T3
#===================================================================================================
#Item intercepts all constrained across groups except non-invariant ones ****
v1T1 ~ I1T1*1; v2T1 ~ I2*1; v3T1 ~ I3*1; v4T1 ~ I4*1; v5T1 ~ I5*1; v6T1 ~ I6*1;
v1T2 ~ I1*1; v2T2 ~ I2*1; v3T2 ~ I3*1; v4T2 ~ I4*1; v5T2 ~ I5*1; v6T2 ~ I6*1;
v1T3 ~ I1*1; v2T3 ~ I2*1; v3T3 ~ I3*1; v4T3 ~ I4*1; v5T3 ~ I5*1; v6T3 ~ I6*1;
#===================================================================================================
#Redidual variances all freely estimated in each group with labels
v1T1 ~~ E1T1*v1T1; v2T1 ~~ E2T1*v2T1; v3T1 ~~ E3T1*v3T1; v4T1 ~~ E4T1*v4T1; v5T1 ~~ E5T1*v5T1; v6T1 ~~ E6T1*v6T1;
v1T2 ~~ E1T2*v1T2; v2T2 ~~ E2T2*v2T2; v3T2 ~~ E3T2*v3T2; v4T2 ~~ E4T2*v4T2; v5T2 ~~ E5T2*v5T2; v6T2 ~~ E6T2*v6T2;
v1T3 ~~ E1T3*v1T3; v2T3 ~~ E2T3*v2T3; v3T3 ~~ E3T3*v3T3; v4T3 ~~ E4T3*v4T3; v5T3 ~~ E5T3*v5T3; v6T3 ~~ E6T3*v6T3;
#===================================================================================================
#Residual covariances freely estimated in each group with labels
v1T1 ~~ C1T12*v1T2 + C1T13*v1T3; v1T2 ~~ C1T23*v1T3;
v2T1 ~~ C2T12*v2T2 + C2T13*v2T3; v2T2 ~~ C2T23*v2T3;
v3T1 ~~ C3T12*v3T2 + C3T13*v3T3; v3T2 ~~ C3T23*v3T3;
v4T1 ~~ C4T12*v4T2 + C4T13*v4T3; v4T2 ~~ C4T23*v4T3;
v5T1 ~~ C5T12*v5T2 + C5T13*v5T3; v5T2 ~~ C5T23*v5T3;
v6T1 ~~ C6T12*v6T2 + C6T13*v6T3; v6T2 ~~ C6T23*v6T3;
#===================================================================================================
#Factor variance fixed to 1 for identification in first group; estimated in others
Time1 ~~ 1*Time1
Time2 ~~ Time2
Time3 ~~ Time3
#===================================================================================================
#Factor mean fixed to zero for identification first group, estimated in others ****
Time1 ~ 0
Time2 ~ 1
Time3 ~ 1
#===================================================================================================
#Factor covariances all freely estimated
Time1 ~~ Time2 + Time3
Time2 ~~ Time3
"
scalarEstimates1 = lavaan(model = scalarSyntax1, data = cafData, estimator = "MLR", mimic = "mplus")lavaan WARNING: some cases are empty and will be ignored:
16 58 64 75 76 133 153 156summary(scalarEstimates1, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan (0.5-23.1097) converged normally after 116 iterations
Used Total
Number of observations 151 159
Number of missing patterns 7
Estimator ML Robust
Minimum Function Test Statistic 355.594 342.529
Degrees of freedom 132 132
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.038
for the Yuan-Bentler correction (Mplus variant)
Model test baseline model:
Minimum Function Test Statistic 2192.158 1896.786
Degrees of freedom 153 153
P-value 0.000 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.890 0.879
Tucker-Lewis Index (TLI) 0.873 0.860
Robust Comparative Fit Index (CFI) 0.892
Robust Tucker-Lewis Index (TLI) 0.874
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -4461.842 -4461.842
Scaling correction factor 1.143
for the MLR correction
Loglikelihood unrestricted model (H1) -4284.045 -4284.045
Scaling correction factor 1.203
for the MLR correction
Number of free parameters 57 57
Akaike (AIC) 9037.685 9037.685
Bayesian (BIC) 9209.670 9209.670
Sample-size adjusted Bayesian (BIC) 9029.271 9029.271
Root Mean Square Error of Approximation:
RMSEA 0.106 0.103
90 Percent Confidence Interval 0.093 0.119 0.090 0.116
P-value RMSEA <= 0.05 0.000 0.000
Robust RMSEA 0.105
90 Percent Confidence Interval 0.091 0.118
Standardized Root Mean Square Residual:
SRMR 0.093 0.093
Parameter Estimates:
Information Observed
Standard Errors Robust.huber.white
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
Time1 =~
v1T1 (L1T1) 3.223 0.264 12.216 0.000 3.223 0.988
v2T1 (L2) 2.072 0.198 10.454 0.000 2.072 0.563
v3T1 (L3) 2.023 0.193 10.496 0.000 2.023 0.791
v4T1 (L4) 1.794 0.234 7.660 0.000 1.794 0.553
v5T1 (L5) 0.907 0.152 5.962 0.000 0.907 0.555
v6T1 (L6) 1.516 0.127 11.977 0.000 1.516 0.759
Time2 =~
v1T2 (L1) 2.662 0.230 11.597 0.000 2.834 0.964
v2T2 (L2) 2.072 0.198 10.454 0.000 2.206 0.673
v3T2 (L3) 2.023 0.193 10.496 0.000 2.153 0.827
v4T2 (L4) 1.794 0.234 7.660 0.000 1.910 0.593
v5T2 (L5) 0.907 0.152 5.962 0.000 0.966 0.390
v6T2 (L6) 1.516 0.127 11.977 0.000 1.613 0.841
Time3 =~
v1T3 (L1) 2.662 0.230 11.597 0.000 2.557 0.960
v2T3 (L2) 2.072 0.198 10.454 0.000 1.991 0.663
v3T3 (L3) 2.023 0.193 10.496 0.000 1.943 0.789
v4T3 (L4) 1.794 0.234 7.660 0.000 1.724 0.560
v5T3 (L5) 0.907 0.152 5.962 0.000 0.872 0.438
v6T3 (L6) 1.516 0.127 11.977 0.000 1.456 0.858
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 ~~
.v1T2 (C1T12) -0.187 0.251 -0.746 0.456 -0.187 -0.484
.v1T3 (C1T13) 0.026 0.240 0.107 0.914 0.026 0.069
.v1T2 ~~
.v1T3 (C1T2) 0.140 0.230 0.610 0.542 0.140 0.242
.v2T1 ~~
.v2T2 (C2T12) 3.926 0.712 5.517 0.000 3.926 0.532
.v2T3 (C2T13) 2.047 0.812 2.519 0.012 2.047 0.299
.v2T2 ~~
.v2T3 (C2T2) 2.079 0.560 3.715 0.000 2.079 0.381
.v3T1 ~~
.v3T2 (C3T12) 0.985 0.291 3.378 0.001 0.985 0.430
.v3T3 (C3T13) 1.090 0.289 3.770 0.000 1.090 0.462
.v3T2 ~~
.v3T3 (C3T2) 1.166 0.317 3.675 0.000 1.166 0.527
.v4T1 ~~
.v4T2 (C4T12) 4.606 0.837 5.501 0.000 4.606 0.658
.v4T3 (C4T13) 3.681 1.016 3.624 0.000 3.681 0.534
.v4T2 ~~
.v4T3 (C4T2) 4.515 0.796 5.674 0.000 4.515 0.683
.v5T1 ~~
.v5T2 (C5T12) 2.430 0.835 2.910 0.004 2.430 0.785
.v5T3 (C5T13) 1.487 0.489 3.041 0.002 1.487 0.613
.v5T2 ~~
.v5T3 (C5T2) 3.203 0.935 3.425 0.001 3.203 0.786
.v6T1 ~~
.v6T2 (C6T12) 0.823 0.166 4.959 0.000 0.823 0.611
.v6T3 (C6T13) 0.571 0.177 3.231 0.001 0.571 0.505
.v6T2 ~~
.v6T3 (C6T2) 0.519 0.134 3.884 0.000 0.519 0.576
Time1 ~~
Time2 0.837 0.077 10.811 0.000 0.787 0.787
Time3 0.678 0.121 5.588 0.000 0.706 0.706
Time2 ~~
Time3 0.689 0.124 5.553 0.000 0.674 0.674
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (I1T1) 16.089 0.274 58.671 0.000 16.089 4.933
.v2T1 (I2) 9.368 0.263 35.672 0.000 9.368 2.545
.v3T1 (I3) 12.026 0.227 53.055 0.000 12.026 4.704
.v4T1 (I4) -3.388 0.261 -12.988 0.000 -3.388 -1.045
.v5T1 (I5) -1.189 0.125 -9.520 0.000 -1.189 -0.728
.v6T1 (I6) -2.775 0.169 -16.456 0.000 -2.775 -1.390
.v1T2 (I1) 16.589 0.281 59.112 0.000 16.589 5.646
.v2T2 (I2) 9.368 0.263 35.672 0.000 9.368 2.859
.v3T2 (I3) 12.026 0.227 53.055 0.000 12.026 4.618
.v4T2 (I4) -3.388 0.261 -12.988 0.000 -3.388 -1.052
.v5T2 (I5) -1.189 0.125 -9.520 0.000 -1.189 -0.480
.v6T2 (I6) -2.775 0.169 -16.456 0.000 -2.775 -1.447
.v1T3 (I1) 16.589 0.281 59.112 0.000 16.589 6.226
.v2T3 (I2) 9.368 0.263 35.672 0.000 9.368 3.119
.v3T3 (I3) 12.026 0.227 53.055 0.000 12.026 4.885
.v4T3 (I4) -3.388 0.261 -12.988 0.000 -3.388 -1.100
.v5T3 (I5) -1.189 0.125 -9.520 0.000 -1.189 -0.598
.v6T3 (I6) -2.775 0.169 -16.456 0.000 -2.775 -1.636
Time1 0.000 0.000 0.000
Time2 0.228 0.081 2.831 0.005 0.214 0.214
Time3 0.452 0.083 5.456 0.000 0.471 0.471
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (E1T1) 0.247 0.390 0.635 0.526 0.247 0.023
.v2T1 (E2T1) 9.256 1.165 7.947 0.000 9.256 0.683
.v3T1 (E3T1) 2.443 0.391 6.243 0.000 2.443 0.374
.v4T1 (E4T1) 7.291 1.014 7.189 0.000 7.291 0.694
.v5T1 (E5T1) 1.846 0.458 4.027 0.000 1.846 0.692
.v6T1 (E6T1) 1.692 0.247 6.849 0.000 1.692 0.424
.v1T2 (E1T2) 0.603 0.269 2.247 0.025 0.603 0.070
.v2T2 (E2T2) 5.874 0.599 9.812 0.000 5.874 0.547
.v3T2 (E3T2) 2.145 0.351 6.115 0.000 2.145 0.316
.v4T2 (E4T2) 6.714 0.962 6.978 0.000 6.714 0.648
.v5T2 (E5T2) 5.196 1.648 3.153 0.002 5.196 0.848
.v6T2 (E6T2) 1.073 0.164 6.560 0.000 1.073 0.292
.v1T3 (E1T3) 0.560 0.312 1.794 0.073 0.560 0.079
.v2T3 (E2T3) 5.060 0.769 6.577 0.000 5.060 0.561
.v3T3 (E3T3) 2.284 0.416 5.494 0.000 2.284 0.377
.v4T3 (E4T3) 6.512 1.117 5.832 0.000 6.512 0.687
.v5T3 (E5T3) 3.193 0.882 3.620 0.000 3.193 0.808
.v6T3 (E6T3) 0.757 0.169 4.484 0.000 0.757 0.263
Time1 1.000 1.000 1.000
Time2 1.133 0.179 6.315 0.000 1.000 1.000
Time3 0.923 0.152 6.082 0.000 1.000 1.000
R-Square:
Estimate
v1T1 0.977
v2T1 0.317
v3T1 0.626
v4T1 0.306
v5T1 0.308
v6T1 0.576
v1T2 0.930
v2T2 0.453
v3T2 0.684
v4T2 0.352
v5T2 0.152
v6T2 0.708
v1T3 0.921
v2T3 0.439
v3T3 0.623
v4T3 0.313
v5T3 0.192
v6T3 0.737anova(metricEstimates2, scalarEstimates1)Scaled Chi Square Difference Test (method = "satorra.bentler.2001")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
metricEstimates2 123 9003.3 9202.5 303.25
scalarEstimates1 132 9037.7 9209.7 355.59 55.106 9 1.162e-08 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1anova(configuralEstimates, scalarEstimates1)Scaled Chi Square Difference Test (method = "satorra.bentler.2001")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
configuralEstimates 114 9010.6 9236.9 292.51
scalarEstimates1 132 9037.7 9209.7 355.59 58.816 18 3.174e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Does the full scalar model (3a) fit worse than the partial metric model (2b)? Yes, −2∆LL(df=9) = 55.13, p <.01. A look at the modification indices reveals a number of items with large values. We will pick the intercept from item2 at time1 to free.
#get modification indices for all parameters:
scalarMI1 = modificationindices(scalarEstimates1, free.remove = FALSE, maximum.number = 10000)
#restrict output to only factor loading parameters
scalarMI1a = scalarMI1[which(scalarMI1$op == "~1"),]
scalarMI1a[order(-scalarMI1a$mi),] lhs op rhs mi mi.scaled epc sepc.lv sepc.all sepc.nox
20 v2T1 ~1 12.968 12.491 -0.814 -0.814 -0.221 -0.221
29 v5T2 ~1 12.432 11.975 -0.364 -0.364 -0.147 -0.147
23 v5T1 ~1 8.446 8.136 0.219 0.219 0.134 0.134
22 v4T1 ~1 8.406 8.097 0.519 0.519 0.160 0.160
28 v4T2 ~1 4.843 4.665 -0.335 -0.335 -0.104 -0.104
26 v2T2 ~1 4.608 4.438 0.376 0.376 0.115 0.115
24 v6T1 ~1 1.667 1.606 -0.120 -0.120 -0.060 -0.060
33 v3T3 ~1 1.131 1.089 0.134 0.134 0.055 0.055
31 v1T3 ~1 0.702 0.676 -0.085 -0.085 -0.032 -0.032
25 v1T2 ~1 0.657 0.633 0.079 0.079 0.027 0.027
30 v6T2 ~1 0.631 0.607 0.058 0.058 0.030 0.030
32 v2T3 ~1 0.563 0.542 0.153 0.153 0.051 0.051
21 v3T1 ~1 0.362 0.349 -0.073 -0.073 -0.029 -0.029
35 v5T3 ~1 0.230 0.221 -0.053 -0.053 -0.026 -0.026
27 v3T2 ~1 0.151 0.146 -0.043 -0.043 -0.017 -0.017
34 v4T3 ~1 0.104 0.100 -0.059 -0.059 -0.019 -0.019
36 v6T3 ~1 0.049 0.047 0.017 0.017 0.010 0.010
78 Time3 ~1 0.000 0.000 0.000 0.000 0.000 0.000
77 Time2 ~1 0.000 0.000 0.000 0.000 0.000 0.000
19 v1T1 ~1 0.000 0.000 0.000 0.000 0.000 0.000scalarSyntax2 = "
#===================================================================================================
#Factor loadings constrained across groups except non-invariant ones
Time1 =~ L1T1*v1T1 + L2*v2T1 + L3*v3T1 + L4*v4T1 + L5*v5T1 + L6*v6T1
Time2 =~ L1*v1T2 + L2*v2T2 + L3*v3T2 + L4*v4T2 + L5*v5T2 + L6*v6T2
Time3 =~ L1*v1T3 + L2*v2T3 + L3*v3T3 + L4*v4T3 + L5*v5T3 + L6*v6T3
#===================================================================================================
#Item intercepts all constrained across groups except non-invariant ones ****
v1T1 ~ I1T1*1; v2T1 ~ I2T1*1; v3T1 ~ I3*1; v4T1 ~ I4*1; v5T1 ~ I5*1; v6T1 ~ I6*1;
v1T2 ~ I1*1; v2T2 ~ I2*1; v3T2 ~ I3*1; v4T2 ~ I4*1; v5T2 ~ I5*1; v6T2 ~ I6*1;
v1T3 ~ I1*1; v2T3 ~ I2*1; v3T3 ~ I3*1; v4T3 ~ I4*1; v5T3 ~ I5*1; v6T3 ~ I6*1;
#===================================================================================================
#Redidual variances all freely estimated in each group with labels
v1T1 ~~ E1T1*v1T1; v2T1 ~~ E2T1*v2T1; v3T1 ~~ E3T1*v3T1; v4T1 ~~ E4T1*v4T1; v5T1 ~~ E5T1*v5T1; v6T1 ~~ E6T1*v6T1;
v1T2 ~~ E1T2*v1T2; v2T2 ~~ E2T2*v2T2; v3T2 ~~ E3T2*v3T2; v4T2 ~~ E4T2*v4T2; v5T2 ~~ E5T2*v5T2; v6T2 ~~ E6T2*v6T2;
v1T3 ~~ E1T3*v1T3; v2T3 ~~ E2T3*v2T3; v3T3 ~~ E3T3*v3T3; v4T3 ~~ E4T3*v4T3; v5T3 ~~ E5T3*v5T3; v6T3 ~~ E6T3*v6T3;
#===================================================================================================
#Residual covariances freely estimated in each group with labels
v1T1 ~~ C1T12*v1T2 + C1T13*v1T3; v1T2 ~~ C1T23*v1T3;
v2T1 ~~ C2T12*v2T2 + C2T13*v2T3; v2T2 ~~ C2T23*v2T3;
v3T1 ~~ C3T12*v3T2 + C3T13*v3T3; v3T2 ~~ C3T23*v3T3;
v4T1 ~~ C4T12*v4T2 + C4T13*v4T3; v4T2 ~~ C4T23*v4T3;
v5T1 ~~ C5T12*v5T2 + C5T13*v5T3; v5T2 ~~ C5T23*v5T3;
v6T1 ~~ C6T12*v6T2 + C6T13*v6T3; v6T2 ~~ C6T23*v6T3;
#===================================================================================================
#Factor variance fixed to 1 for identification in first group; estimated in others
Time1 ~~ 1*Time1
Time2 ~~ Time2
Time3 ~~ Time3
#===================================================================================================
#Factor mean fixed to zero for identification first group, estimated in others ****
Time1 ~ 0
Time2 ~ 1
Time3 ~ 1
#===================================================================================================
#Factor covariances all freely estimated
Time1 ~~ Time2 + Time3
Time2 ~~ Time3
"
scalarEstimates2 = lavaan(model = scalarSyntax2, data = cafData, estimator = "MLR", mimic = "mplus")lavaan WARNING: some cases are empty and will be ignored:
16 58 64 75 76 133 153 156summary(scalarEstimates2, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan (0.5-23.1097) converged normally after 116 iterations
Used Total
Number of observations 151 159
Number of missing patterns 7
Estimator ML Robust
Minimum Function Test Statistic 339.321 327.268
Degrees of freedom 131 131
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.037
for the Yuan-Bentler correction (Mplus variant)
Model test baseline model:
Minimum Function Test Statistic 2192.158 1896.786
Degrees of freedom 153 153
P-value 0.000 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.898 0.887
Tucker-Lewis Index (TLI) 0.881 0.869
Robust Comparative Fit Index (CFI) 0.899
Robust Tucker-Lewis Index (TLI) 0.882
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -4453.706 -4453.706
Scaling correction factor 1.159
for the MLR correction
Loglikelihood unrestricted model (H1) -4284.045 -4284.045
Scaling correction factor 1.203
for the MLR correction
Number of free parameters 58 58
Akaike (AIC) 9023.412 9023.412
Bayesian (BIC) 9198.414 9198.414
Sample-size adjusted Bayesian (BIC) 9014.850 9014.850
Root Mean Square Error of Approximation:
RMSEA 0.103 0.100
90 Percent Confidence Interval 0.089 0.116 0.086 0.113
P-value RMSEA <= 0.05 0.000 0.000
Robust RMSEA 0.101
90 Percent Confidence Interval 0.088 0.115
Standardized Root Mean Square Residual:
SRMR 0.092 0.092
Parameter Estimates:
Information Observed
Standard Errors Robust.huber.white
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
Time1 =~
v1T1 (L1T1) 3.227 0.263 12.260 0.000 3.227 0.990
v2T1 (L2) 1.977 0.201 9.826 0.000 1.977 0.558
v3T1 (L3) 2.023 0.195 10.359 0.000 2.023 0.789
v4T1 (L4) 1.829 0.232 7.889 0.000 1.829 0.562
v5T1 (L5) 0.927 0.150 6.165 0.000 0.927 0.564
v6T1 (L6) 1.511 0.130 11.634 0.000 1.511 0.756
Time2 =~
v1T2 (L1) 2.652 0.230 11.505 0.000 2.836 0.965
v2T2 (L2) 1.977 0.201 9.826 0.000 2.114 0.658
v3T2 (L3) 2.023 0.195 10.359 0.000 2.163 0.828
v4T2 (L4) 1.829 0.232 7.889 0.000 1.956 0.602
v5T2 (L5) 0.927 0.150 6.165 0.000 0.991 0.401
v6T2 (L6) 1.511 0.130 11.634 0.000 1.616 0.841
Time3 =~
v1T3 (L1) 2.652 0.230 11.505 0.000 2.559 0.961
v2T3 (L2) 1.977 0.201 9.826 0.000 1.907 0.644
v3T3 (L3) 2.023 0.195 10.359 0.000 1.952 0.790
v4T3 (L4) 1.829 0.232 7.889 0.000 1.765 0.568
v5T3 (L5) 0.927 0.150 6.165 0.000 0.894 0.450
v6T3 (L6) 1.511 0.130 11.634 0.000 1.458 0.858
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 ~~
.v1T2 (C1T12) -0.196 0.250 -0.783 0.434 -0.196 -0.543
.v1T3 (C1T13) 0.009 0.239 0.037 0.971 0.009 0.026
.v1T2 ~~
.v1T3 (C1T2) 0.133 0.231 0.574 0.566 0.133 0.234
.v2T1 ~~
.v2T2 (C2T12) 3.948 0.679 5.818 0.000 3.948 0.554
.v2T3 (C2T13) 2.234 0.799 2.794 0.005 2.234 0.335
.v2T2 ~~
.v2T3 (C2T2) 2.144 0.563 3.804 0.000 2.144 0.391
.v3T1 ~~
.v3T2 (C3T12) 0.989 0.295 3.349 0.001 0.989 0.429
.v3T3 (C3T13) 1.091 0.292 3.732 0.000 1.091 0.457
.v3T2 ~~
.v3T3 (C3T2) 1.171 0.319 3.673 0.000 1.171 0.527
.v4T1 ~~
.v4T2 (C4T12) 4.665 0.840 5.550 0.000 4.665 0.668
.v4T3 (C4T13) 3.742 1.019 3.674 0.000 3.742 0.543
.v4T2 ~~
.v4T3 (C4T2) 4.561 0.805 5.667 0.000 4.561 0.688
.v5T1 ~~
.v5T2 (C5T12) 2.417 0.826 2.927 0.003 2.417 0.788
.v5T3 (C5T13) 1.475 0.480 3.071 0.002 1.475 0.613
.v5T2 ~~
.v5T3 (C5T2) 3.147 0.914 3.443 0.001 3.147 0.785
.v6T1 ~~
.v6T2 (C6T12) 0.813 0.168 4.838 0.000 0.813 0.598
.v6T3 (C6T13) 0.571 0.177 3.232 0.001 0.571 0.499
.v6T2 ~~
.v6T3 (C6T2) 0.522 0.134 3.902 0.000 0.522 0.575
Time1 ~~
Time2 0.841 0.078 10.747 0.000 0.787 0.787
Time3 0.681 0.122 5.582 0.000 0.706 0.706
Time2 ~~
Time3 0.695 0.125 5.554 0.000 0.674 0.674
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (I1T1) 16.090 0.274 58.695 0.000 16.090 4.935
.v2T1 (I2T1) 8.672 0.294 29.525 0.000 8.672 2.446
.v3T1 (I3) 12.100 0.226 53.558 0.000 12.100 4.719
.v4T1 (I4) -3.329 0.254 -13.103 0.000 -3.329 -1.022
.v5T1 (I5) -1.193 0.124 -9.630 0.000 -1.193 -0.726
.v6T1 (I6) -2.699 0.166 -16.286 0.000 -2.699 -1.350
.v1T2 (I1) 16.738 0.275 60.784 0.000 16.738 5.696
.v2T2 (I2) 9.645 0.261 36.936 0.000 9.645 3.000
.v3T2 (I3) 12.100 0.226 53.558 0.000 12.100 4.633
.v4T2 (I4) -3.329 0.254 -13.103 0.000 -3.329 -1.025
.v5T2 (I5) -1.193 0.124 -9.630 0.000 -1.193 -0.483
.v6T2 (I6) -2.699 0.166 -16.286 0.000 -2.699 -1.405
.v1T3 (I1) 16.738 0.275 60.784 0.000 16.738 6.287
.v2T3 (I2) 9.645 0.261 36.936 0.000 9.645 3.257
.v3T3 (I3) 12.100 0.226 53.558 0.000 12.100 4.895
.v4T3 (I4) -3.329 0.254 -13.103 0.000 -3.329 -1.071
.v5T3 (I5) -1.193 0.124 -9.630 0.000 -1.193 -0.600
.v6T3 (I6) -2.699 0.166 -16.286 0.000 -2.699 -1.589
Time1 0.000 0.000 0.000
Time2 0.171 0.079 2.162 0.031 0.160 0.160
Time3 0.399 0.080 4.963 0.000 0.413 0.413
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (E1T1) 0.220 0.387 0.568 0.570 0.220 0.021
.v2T1 (E2T1) 8.663 1.028 8.430 0.000 8.663 0.689
.v3T1 (E3T1) 2.484 0.406 6.123 0.000 2.484 0.378
.v4T1 (E4T1) 7.258 1.013 7.163 0.000 7.258 0.684
.v5T1 (E5T1) 1.841 0.456 4.041 0.000 1.841 0.682
.v6T1 (E6T1) 1.713 0.253 6.769 0.000 1.713 0.429
.v1T2 (E1T2) 0.594 0.270 2.203 0.028 0.594 0.069
.v2T2 (E2T2) 5.864 0.600 9.771 0.000 5.864 0.568
.v3T2 (E3T2) 2.143 0.352 6.093 0.000 2.143 0.314
.v4T2 (E4T2) 6.714 0.963 6.972 0.000 6.714 0.637
.v5T2 (E5T2) 5.115 1.625 3.147 0.002 5.115 0.839
.v6T2 (E6T2) 1.080 0.164 6.593 0.000 1.080 0.293
.v1T3 (E1T3) 0.541 0.310 1.743 0.081 0.541 0.076
.v2T3 (E2T3) 5.132 0.789 6.504 0.000 5.132 0.585
.v3T3 (E3T3) 2.301 0.419 5.497 0.000 2.301 0.377
.v4T3 (E4T3) 6.544 1.123 5.828 0.000 6.544 0.678
.v5T3 (E5T3) 3.145 0.863 3.644 0.000 3.145 0.797
.v6T3 (E6T3) 0.762 0.169 4.498 0.000 0.762 0.264
Time1 1.000 1.000 1.000
Time2 1.143 0.183 6.261 0.000 1.000 1.000
Time3 0.931 0.153 6.089 0.000 1.000 1.000
R-Square:
Estimate
v1T1 0.979
v2T1 0.311
v3T1 0.622
v4T1 0.316
v5T1 0.318
v6T1 0.571
v1T2 0.931
v2T2 0.432
v3T2 0.686
v4T2 0.363
v5T2 0.161
v6T2 0.707
v1T3 0.924
v2T3 0.415
v3T3 0.623
v4T3 0.322
v5T3 0.203
v6T3 0.736anova(metricEstimates2, scalarEstimates2)Scaled Chi Square Difference Test (method = "satorra.bentler.2001")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
metricEstimates2 123 9003.3 9202.5 303.25
scalarEstimates2 131 9023.4 9198.4 339.32 39.321 8 4.285e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1anova(configuralEstimates, scalarEstimates2)Scaled Chi Square Difference Test (method = "satorra.bentler.2001")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
configuralEstimates 114 9010.6 9236.9 292.51
scalarEstimates2 131 9023.4 9198.4 339.32 43.975 17 0.0003454 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Does the partial scalar model fit worse than the partial metric model? Yes, −2∆LL(df=8) = 39.321, p <.01. A look at the modification indices reveals a number of items with large values. We will pick the intercept from item5 at time2 to free.
#get modification indices for all parameters:
scalarMI2 = modificationindices(scalarEstimates2, free.remove = FALSE, maximum.number = 10000)
#restrict output to only item intercept parameters:
scalarMI2a = scalarMI2[which(scalarMI2$op == "~1"),]
scalarMI2a[order(-scalarMI2a$mi),] lhs op rhs mi mi.scaled epc sepc.lv sepc.all sepc.nox
29 v5T2 ~1 11.097 10.703 -0.340 -0.340 -0.138 -0.138
23 v5T1 ~1 7.254 6.997 0.202 0.202 0.123 0.123
22 v4T1 ~1 6.089 5.873 0.436 0.436 0.134 0.134
24 v6T1 ~1 4.979 4.802 -0.210 -0.210 -0.105 -0.105
28 v4T2 ~1 3.566 3.440 -0.285 -0.285 -0.088 -0.088
21 v3T1 ~1 1.977 1.907 -0.173 -0.173 -0.068 -0.068
33 v3T3 ~1 1.817 1.753 0.171 0.171 0.069 0.069
30 v6T2 ~1 1.711 1.650 0.096 0.096 0.050 0.050
31 v1T3 ~1 0.886 0.854 -0.095 -0.095 -0.036 -0.036
25 v1T2 ~1 0.833 0.803 0.089 0.089 0.030 0.030
36 v6T3 ~1 0.191 0.184 0.033 0.033 0.019 0.019
35 v5T3 ~1 0.135 0.130 -0.040 -0.040 -0.020 -0.020
34 v4T3 ~1 0.070 0.068 -0.049 -0.049 -0.016 -0.016
27 v3T2 ~1 0.008 0.008 0.010 0.010 0.004 0.004
32 v2T3 ~1 0.000 0.000 0.004 0.004 0.001 0.001
26 v2T2 ~1 0.000 0.000 -0.003 -0.003 -0.001 -0.001
77 Time2 ~1 0.000 0.000 0.000 0.000 0.000 0.000
78 Time3 ~1 0.000 0.000 0.000 0.000 0.000 0.000
19 v1T1 ~1 0.000 0.000 0.000 0.000 0.000 0.000
20 v2T1 ~1 0.000 0.000 0.000 0.000 0.000 0.000scalarSyntax3 = "
#===================================================================================================
#Factor loadings constrained across groups except non-invariant ones
Time1 =~ L1T1*v1T1 + L2*v2T1 + L3*v3T1 + L4*v4T1 + L5*v5T1 + L6*v6T1
Time2 =~ L1*v1T2 + L2*v2T2 + L3*v3T2 + L4*v4T2 + L5*v5T2 + L6*v6T2
Time3 =~ L1*v1T3 + L2*v2T3 + L3*v3T3 + L4*v4T3 + L5*v5T3 + L6*v6T3
#===================================================================================================
#Item intercepts all constrained across groups except non-invariant ones ****
v1T1 ~ I1T1*1; v2T1 ~ I2T1*1; v3T1 ~ I3*1; v4T1 ~ I4*1; v5T1 ~ I5*1; v6T1 ~ I6*1;
v1T2 ~ I1*1; v2T2 ~ I2*1; v3T2 ~ I3*1; v4T2 ~ I4*1; v5T2 ~ I5T2*1; v6T2 ~ I6*1;
v1T3 ~ I1*1; v2T3 ~ I2*1; v3T3 ~ I3*1; v4T3 ~ I4*1; v5T3 ~ I5*1; v6T3 ~ I6*1;
#===================================================================================================
#Redidual variances all freely estimated in each group with labels
v1T1 ~~ E1T1*v1T1; v2T1 ~~ E2T1*v2T1; v3T1 ~~ E3T1*v3T1; v4T1 ~~ E4T1*v4T1; v5T1 ~~ E5T1*v5T1; v6T1 ~~ E6T1*v6T1;
v1T2 ~~ E1T2*v1T2; v2T2 ~~ E2T2*v2T2; v3T2 ~~ E3T2*v3T2; v4T2 ~~ E4T2*v4T2; v5T2 ~~ E5T2*v5T2; v6T2 ~~ E6T2*v6T2;
v1T3 ~~ E1T3*v1T3; v2T3 ~~ E2T3*v2T3; v3T3 ~~ E3T3*v3T3; v4T3 ~~ E4T3*v4T3; v5T3 ~~ E5T3*v5T3; v6T3 ~~ E6T3*v6T3;
#===================================================================================================
#Residual covariances freely estimated in each group with labels
v1T1 ~~ C1T12*v1T2 + C1T13*v1T3; v1T2 ~~ C1T23*v1T3;
v2T1 ~~ C2T12*v2T2 + C2T13*v2T3; v2T2 ~~ C2T23*v2T3;
v3T1 ~~ C3T12*v3T2 + C3T13*v3T3; v3T2 ~~ C3T23*v3T3;
v4T1 ~~ C4T12*v4T2 + C4T13*v4T3; v4T2 ~~ C4T23*v4T3;
v5T1 ~~ C5T12*v5T2 + C5T13*v5T3; v5T2 ~~ C5T23*v5T3;
v6T1 ~~ C6T12*v6T2 + C6T13*v6T3; v6T2 ~~ C6T23*v6T3;
#===================================================================================================
#Factor variance fixed to 1 for identification in first group; estimated in others
Time1 ~~ 1*Time1
Time2 ~~ Time2
Time3 ~~ Time3
#===================================================================================================
#Factor mean fixed to zero for identification first group, estimated in others ****
Time1 ~ 0
Time2 ~ 1
Time3 ~ 1
#===================================================================================================
#Factor covariances all freely estimated
Time1 ~~ Time2 + Time3
Time2 ~~ Time3
"
scalarEstimates3 = lavaan(model = scalarSyntax3, data = cafData, estimator = "MLR", mimic = "mplus")lavaan WARNING: some cases are empty and will be ignored:
16 58 64 75 76 133 153 156summary(scalarEstimates3, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan (0.5-23.1097) converged normally after 121 iterations
Used Total
Number of observations 151 159
Number of missing patterns 7
Estimator ML Robust
Minimum Function Test Statistic 324.707 311.362
Degrees of freedom 130 130
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.043
for the Yuan-Bentler correction (Mplus variant)
Model test baseline model:
Minimum Function Test Statistic 2192.158 1896.786
Degrees of freedom 153 153
P-value 0.000 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.905 0.896
Tucker-Lewis Index (TLI) 0.888 0.878
Robust Comparative Fit Index (CFI) 0.906
Robust Tucker-Lewis Index (TLI) 0.890
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -4446.399 -4446.399
Scaling correction factor 1.162
for the MLR correction
Loglikelihood unrestricted model (H1) -4284.045 -4284.045
Scaling correction factor 1.203
for the MLR correction
Number of free parameters 59 59
Akaike (AIC) 9010.798 9010.798
Bayesian (BIC) 9188.817 9188.817
Sample-size adjusted Bayesian (BIC) 9002.088 9002.088
Root Mean Square Error of Approximation:
RMSEA 0.100 0.096
90 Percent Confidence Interval 0.086 0.113 0.083 0.110
P-value RMSEA <= 0.05 0.000 0.000
Robust RMSEA 0.098
90 Percent Confidence Interval 0.084 0.112
Standardized Root Mean Square Residual:
SRMR 0.088 0.088
Parameter Estimates:
Information Observed
Standard Errors Robust.huber.white
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
Time1 =~
v1T1 (L1T1) 3.226 0.263 12.275 0.000 3.226 0.989
v2T1 (L2) 1.980 0.200 9.888 0.000 1.980 0.558
v3T1 (L3) 2.021 0.194 10.423 0.000 2.021 0.790
v4T1 (L4) 1.814 0.231 7.849 0.000 1.814 0.558
v5T1 (L5) 0.916 0.142 6.456 0.000 0.916 0.561
v6T1 (L6) 1.513 0.128 11.821 0.000 1.513 0.757
Time2 =~
v1T2 (L1) 2.660 0.231 11.533 0.000 2.840 0.966
v2T2 (L2) 1.980 0.200 9.888 0.000 2.114 0.657
v3T2 (L3) 2.021 0.194 10.423 0.000 2.158 0.827
v4T2 (L4) 1.814 0.231 7.849 0.000 1.937 0.599
v5T2 (L5) 0.916 0.142 6.456 0.000 0.978 0.412
v6T2 (L6) 1.513 0.128 11.821 0.000 1.615 0.841
Time3 =~
v1T3 (L1) 2.660 0.231 11.533 0.000 2.561 0.961
v2T3 (L2) 1.980 0.200 9.888 0.000 1.906 0.644
v3T3 (L3) 2.021 0.194 10.423 0.000 1.946 0.789
v4T3 (L4) 1.814 0.231 7.849 0.000 1.747 0.565
v5T3 (L5) 0.916 0.142 6.456 0.000 0.882 0.449
v6T3 (L6) 1.513 0.128 11.821 0.000 1.457 0.858
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 ~~
.v1T2 (C1T12) -0.201 0.252 -0.796 0.426 -0.201 -0.556
.v1T3 (C1T13) 0.014 0.241 0.056 0.955 0.014 0.039
.v1T2 ~~
.v1T3 (C1T2) 0.134 0.231 0.579 0.562 0.134 0.239
.v2T1 ~~
.v2T2 (C2T12) 3.951 0.679 5.815 0.000 3.951 0.554
.v2T3 (C2T13) 2.223 0.799 2.781 0.005 2.223 0.334
.v2T2 ~~
.v2T3 (C2T2) 2.135 0.563 3.790 0.000 2.135 0.389
.v3T1 ~~
.v3T2 (C3T12) 0.992 0.296 3.358 0.001 0.992 0.430
.v3T3 (C3T13) 1.092 0.292 3.743 0.000 1.092 0.458
.v3T2 ~~
.v3T3 (C3T2) 1.171 0.319 3.671 0.000 1.171 0.526
.v4T1 ~~
.v4T2 (C4T12) 4.635 0.838 5.529 0.000 4.635 0.663
.v4T3 (C4T13) 3.720 1.019 3.650 0.000 3.720 0.540
.v4T2 ~~
.v4T3 (C4T2) 4.538 0.800 5.675 0.000 4.538 0.686
.v5T1 ~~
.v5T2 (C5T12) 2.342 0.761 3.079 0.002 2.342 0.799
.v5T3 (C5T13) 1.423 0.446 3.188 0.001 1.423 0.599
.v5T2 ~~
.v5T3 (C5T2) 2.909 0.802 3.629 0.000 2.909 0.765
.v6T1 ~~
.v6T2 (C6T12) 0.820 0.167 4.912 0.000 0.820 0.604
.v6T3 (C6T13) 0.572 0.176 3.246 0.001 0.572 0.503
.v6T2 ~~
.v6T3 (C6T2) 0.522 0.134 3.902 0.000 0.522 0.575
Time1 ~~
Time2 0.840 0.077 10.902 0.000 0.786 0.786
Time3 0.679 0.122 5.564 0.000 0.705 0.705
Time2 ~~
Time3 0.692 0.125 5.533 0.000 0.673 0.673
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (I1T1) 16.085 0.275 58.539 0.000 16.085 4.934
.v2T1 (I2T1) 8.677 0.293 29.587 0.000 8.677 2.446
.v3T1 (I3) 12.062 0.225 53.554 0.000 12.062 4.712
.v4T1 (I4) -3.359 0.259 -12.963 0.000 -3.359 -1.033
.v5T1 (I5) -1.353 0.141 -9.604 0.000 -1.353 -0.828
.v6T1 (I6) -2.735 0.165 -16.560 0.000 -2.735 -1.369
.v1T2 (I1) 16.664 0.276 60.330 0.000 16.664 5.667
.v2T2 (I2) 9.589 0.261 36.774 0.000 9.589 2.982
.v3T2 (I3) 12.062 0.225 53.554 0.000 12.062 4.622
.v4T2 (I4) -3.359 0.259 -12.963 0.000 -3.359 -1.039
.v5T2 (I5T2) -1.779 0.214 -8.299 0.000 -1.779 -0.748
.v6T2 (I6) -2.735 0.165 -16.560 0.000 -2.735 -1.423
.v1T3 (I1) 16.664 0.276 60.330 0.000 16.664 6.254
.v2T3 (I2) 9.589 0.261 36.774 0.000 9.589 3.240
.v3T3 (I3) 12.062 0.225 53.554 0.000 12.062 4.887
.v4T3 (I4) -3.359 0.259 -12.963 0.000 -3.359 -1.086
.v5T3 (I5) -1.353 0.141 -9.604 0.000 -1.353 -0.689
.v6T3 (I6) -2.735 0.165 -16.560 0.000 -2.735 -1.610
Time1 0.000 0.000 0.000
Time2 0.205 0.076 2.691 0.007 0.192 0.192
Time3 0.419 0.080 5.223 0.000 0.435 0.435
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (E1T1) 0.225 0.386 0.583 0.560 0.225 0.021
.v2T1 (E2T1) 8.663 1.029 8.419 0.000 8.663 0.688
.v3T1 (E3T1) 2.468 0.398 6.200 0.000 2.468 0.377
.v4T1 (E4T1) 7.281 1.016 7.169 0.000 7.281 0.689
.v5T1 (E5T1) 1.831 0.442 4.140 0.000 1.831 0.686
.v6T1 (E6T1) 1.703 0.249 6.832 0.000 1.703 0.427
.v1T2 (E1T2) 0.580 0.269 2.158 0.031 0.580 0.067
.v2T2 (E2T2) 5.872 0.601 9.773 0.000 5.872 0.568
.v3T2 (E3T2) 2.154 0.352 6.111 0.000 2.154 0.316
.v4T2 (E4T2) 6.706 0.962 6.973 0.000 6.706 0.641
.v5T2 (E5T2) 4.691 1.426 3.290 0.001 4.691 0.831
.v6T2 (E6T2) 1.082 0.164 6.582 0.000 1.082 0.293
.v1T3 (E1T3) 0.538 0.314 1.714 0.087 0.538 0.076
.v2T3 (E2T3) 5.126 0.788 6.506 0.000 5.126 0.585
.v3T3 (E3T3) 2.304 0.418 5.507 0.000 2.304 0.378
.v4T3 (E4T3) 6.524 1.117 5.839 0.000 6.524 0.681
.v5T3 (E5T3) 3.080 0.833 3.699 0.000 3.080 0.798
.v6T3 (E6T3) 0.762 0.168 4.522 0.000 0.762 0.264
Time1 1.000 1.000 1.000
Time2 1.140 0.179 6.374 0.000 1.000 1.000
Time3 0.927 0.154 6.028 0.000 1.000 1.000
R-Square:
Estimate
v1T1 0.979
v2T1 0.312
v3T1 0.623
v4T1 0.311
v5T1 0.314
v6T1 0.573
v1T2 0.933
v2T2 0.432
v3T2 0.684
v4T2 0.359
v5T2 0.169
v6T2 0.707
v1T3 0.924
v2T3 0.415
v3T3 0.622
v4T3 0.319
v5T3 0.202
v6T3 0.736anova(metricEstimates2, scalarEstimates3)Scaled Chi Square Difference Test (method = "satorra.bentler.2001")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
metricEstimates2 123 9003.3 9202.5 303.25
scalarEstimates3 130 9010.8 9188.8 324.71 21.199 7 0.003486 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1anova(configuralEstimates, scalarEstimates3)Scaled Chi Square Difference Test (method = "satorra.bentler.2001")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
configuralEstimates 114 9010.6 9236.9 292.51
scalarEstimates3 130 9010.8 9188.8 324.71 28.87 16 0.02483 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Does the partial scalar model fit worse than the partial metric model? Yes, −2∆LL(df=7) = 21.199, p <.01. A look at the modification indices reveals several items with large values. We will pick the intercept from item4 at time1 to free.
#get modification indices for all parameters:
scalarMI3 = modificationindices(scalarEstimates3, free.remove = FALSE, maximum.number = 10000)
#restrict output to only item intercept parameters:
scalarMI3a = scalarMI3[which(scalarMI3$op == "~1"),]
scalarMI3a[order(-scalarMI3a$mi),] lhs op rhs mi mi.scaled epc sepc.lv sepc.all sepc.nox
22 v4T1 ~1 7.360 7.057 0.483 0.483 0.148 0.148
28 v4T2 ~1 4.582 4.394 -0.324 -0.324 -0.100 -0.100
35 v5T3 ~1 4.389 4.209 -0.235 -0.235 -0.120 -0.120
24 v6T1 ~1 3.049 2.923 -0.163 -0.163 -0.082 -0.082
23 v5T1 ~1 1.858 1.782 0.099 0.099 0.061 0.061
33 v3T3 ~1 1.838 1.762 0.172 0.172 0.070 0.070
21 v3T1 ~1 1.004 0.963 -0.123 -0.123 -0.048 -0.048
30 v6T2 ~1 0.749 0.718 0.063 0.063 0.033 0.033
31 v1T3 ~1 0.273 0.262 -0.053 -0.053 -0.020 -0.020
36 v6T3 ~1 0.260 0.250 0.038 0.038 0.022 0.022
25 v1T2 ~1 0.255 0.245 0.049 0.049 0.017 0.017
27 v3T2 ~1 0.078 0.075 -0.031 -0.031 -0.012 -0.012
34 v4T3 ~1 0.042 0.040 -0.038 -0.038 -0.012 -0.012
32 v2T3 ~1 0.011 0.011 0.021 0.021 0.007 0.007
26 v2T2 ~1 0.008 0.008 -0.015 -0.015 -0.005 -0.005
29 v5T2 ~1 0.000 0.000 0.000 0.000 0.000 0.000
19 v1T1 ~1 0.000 0.000 0.000 0.000 0.000 0.000
77 Time2 ~1 0.000 0.000 0.000 0.000 0.000 0.000
78 Time3 ~1 0.000 0.000 0.000 0.000 0.000 0.000
20 v2T1 ~1 0.000 0.000 0.000 0.000 0.000 0.000scalarSyntax4 = "
#===================================================================================================
#Factor loadings constrained across groups except non-invariant ones
Time1 =~ L1T1*v1T1 + L2*v2T1 + L3*v3T1 + L4*v4T1 + L5*v5T1 + L6*v6T1
Time2 =~ L1*v1T2 + L2*v2T2 + L3*v3T2 + L4*v4T2 + L5*v5T2 + L6*v6T2
Time3 =~ L1*v1T3 + L2*v2T3 + L3*v3T3 + L4*v4T3 + L5*v5T3 + L6*v6T3
#===================================================================================================
#Item intercepts all constrained across groups except non-invariant ones ****
v1T1 ~ I1T1*1; v2T1 ~ I2T1*1; v3T1 ~ I3*1; v4T1 ~ I4T1*1; v5T1 ~ I5*1; v6T1 ~ I6*1;
v1T2 ~ I1*1; v2T2 ~ I2*1; v3T2 ~ I3*1; v4T2 ~ I4*1; v5T2 ~ I5T2*1; v6T2 ~ I6*1;
v1T3 ~ I1*1; v2T3 ~ I2*1; v3T3 ~ I3*1; v4T3 ~ I4*1; v5T3 ~ I5*1; v6T3 ~ I6*1;
#===================================================================================================
#Redidual variances all freely estimated in each group with labels
v1T1 ~~ E1T1*v1T1; v2T1 ~~ E2T1*v2T1; v3T1 ~~ E3T1*v3T1; v4T1 ~~ E4T1*v4T1; v5T1 ~~ E5T1*v5T1; v6T1 ~~ E6T1*v6T1;
v1T2 ~~ E1T2*v1T2; v2T2 ~~ E2T2*v2T2; v3T2 ~~ E3T2*v3T2; v4T2 ~~ E4T2*v4T2; v5T2 ~~ E5T2*v5T2; v6T2 ~~ E6T2*v6T2;
v1T3 ~~ E1T3*v1T3; v2T3 ~~ E2T3*v2T3; v3T3 ~~ E3T3*v3T3; v4T3 ~~ E4T3*v4T3; v5T3 ~~ E5T3*v5T3; v6T3 ~~ E6T3*v6T3;
#===================================================================================================
#Residual covariances freely estimated in each group with labels
v1T1 ~~ C1T12*v1T2 + C1T13*v1T3; v1T2 ~~ C1T23*v1T3;
v2T1 ~~ C2T12*v2T2 + C2T13*v2T3; v2T2 ~~ C2T23*v2T3;
v3T1 ~~ C3T12*v3T2 + C3T13*v3T3; v3T2 ~~ C3T23*v3T3;
v4T1 ~~ C4T12*v4T2 + C4T13*v4T3; v4T2 ~~ C4T23*v4T3;
v5T1 ~~ C5T12*v5T2 + C5T13*v5T3; v5T2 ~~ C5T23*v5T3;
v6T1 ~~ C6T12*v6T2 + C6T13*v6T3; v6T2 ~~ C6T23*v6T3;
#===================================================================================================
#Factor variance fixed to 1 for identification in first group; estimated in others
Time1 ~~ 1*Time1
Time2 ~~ Time2
Time3 ~~ Time3
#===================================================================================================
#Factor mean fixed to zero for identification first group, estimated in others ****
Time1 ~ 0
Time2 ~ 1
Time3 ~ 1
#===================================================================================================
#Factor covariances all freely estimated
Time1 ~~ Time2 + Time3
Time2 ~~ Time3
"
scalarEstimates4 = lavaan(model = scalarSyntax4, data = cafData, estimator = "MLR", mimic = "mplus")lavaan WARNING: some cases are empty and will be ignored:
16 58 64 75 76 133 153 156summary(scalarEstimates4, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan (0.5-23.1097) converged normally after 110 iterations
Used Total
Number of observations 151 159
Number of missing patterns 7
Estimator ML Robust
Minimum Function Test Statistic 314.564 302.389
Degrees of freedom 129 129
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.040
for the Yuan-Bentler correction (Mplus variant)
Model test baseline model:
Minimum Function Test Statistic 2192.158 1896.786
Degrees of freedom 153 153
P-value 0.000 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.909 0.901
Tucker-Lewis Index (TLI) 0.892 0.882
Robust Comparative Fit Index (CFI) 0.911
Robust Tucker-Lewis Index (TLI) 0.894
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -4441.327 -4441.327
Scaling correction factor 1.179
for the MLR correction
Loglikelihood unrestricted model (H1) -4284.045 -4284.045
Scaling correction factor 1.203
for the MLR correction
Number of free parameters 60 60
Akaike (AIC) 9002.655 9002.655
Bayesian (BIC) 9183.692 9183.692
Sample-size adjusted Bayesian (BIC) 8993.798 8993.798
Root Mean Square Error of Approximation:
RMSEA 0.098 0.094
90 Percent Confidence Interval 0.084 0.111 0.081 0.108
P-value RMSEA <= 0.05 0.000 0.000
Robust RMSEA 0.096
90 Percent Confidence Interval 0.082 0.110
Standardized Root Mean Square Residual:
SRMR 0.092 0.092
Parameter Estimates:
Information Observed
Standard Errors Robust.huber.white
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
Time1 =~
v1T1 (L1T1) 3.226 0.262 12.322 0.000 3.226 0.990
v2T1 (L2) 1.977 0.200 9.905 0.000 1.977 0.558
v3T1 (L3) 2.009 0.194 10.340 0.000 2.009 0.788
v4T1 (L4) 1.916 0.226 8.490 0.000 1.916 0.582
v5T1 (L5) 0.906 0.141 6.424 0.000 0.906 0.556
v6T1 (L6) 1.505 0.127 11.899 0.000 1.505 0.756
Time2 =~
v1T2 (L1) 2.664 0.231 11.557 0.000 2.843 0.966
v2T2 (L2) 1.977 0.200 9.905 0.000 2.110 0.656
v3T2 (L3) 2.009 0.194 10.340 0.000 2.143 0.824
v4T2 (L4) 1.916 0.226 8.490 0.000 2.045 0.618
v5T2 (L5) 0.906 0.141 6.424 0.000 0.967 0.408
v6T2 (L6) 1.505 0.127 11.899 0.000 1.606 0.839
Time3 =~
v1T3 (L1) 2.664 0.231 11.557 0.000 2.562 0.962
v2T3 (L2) 1.977 0.200 9.905 0.000 1.902 0.643
v3T3 (L3) 2.009 0.194 10.340 0.000 1.932 0.786
v4T3 (L4) 1.916 0.226 8.490 0.000 1.843 0.583
v5T3 (L5) 0.906 0.141 6.424 0.000 0.871 0.443
v6T3 (L6) 1.505 0.127 11.899 0.000 1.448 0.855
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 ~~
.v1T2 (C1T12) -0.204 0.251 -0.814 0.416 -0.204 -0.578
.v1T3 (C1T13) 0.015 0.240 0.061 0.951 0.015 0.043
.v1T2 ~~
.v1T3 (C1T2) 0.134 0.231 0.580 0.562 0.134 0.243
.v2T1 ~~
.v2T2 (C2T12) 3.954 0.679 5.820 0.000 3.954 0.554
.v2T3 (C2T13) 2.222 0.798 2.784 0.005 2.222 0.333
.v2T2 ~~
.v2T3 (C2T2) 2.133 0.564 3.782 0.000 2.133 0.388
.v3T1 ~~
.v3T2 (C3T12) 0.994 0.294 3.382 0.001 0.994 0.431
.v3T3 (C3T13) 1.105 0.290 3.813 0.000 1.105 0.464
.v3T2 ~~
.v3T3 (C3T2) 1.168 0.319 3.663 0.000 1.168 0.523
.v4T1 ~~
.v4T2 (C4T12) 4.809 0.851 5.649 0.000 4.809 0.691
.v4T3 (C4T13) 3.881 1.032 3.762 0.000 3.881 0.564
.v4T2 ~~
.v4T3 (C4T2) 4.657 0.831 5.602 0.000 4.657 0.697
.v5T1 ~~
.v5T2 (C5T12) 2.344 0.761 3.081 0.002 2.344 0.799
.v5T3 (C5T13) 1.423 0.448 3.174 0.002 1.423 0.596
.v5T2 ~~
.v5T3 (C5T2) 2.918 0.805 3.627 0.000 2.918 0.764
.v6T1 ~~
.v6T2 (C6T12) 0.826 0.165 5.019 0.000 0.826 0.609
.v6T3 (C6T13) 0.579 0.176 3.293 0.001 0.579 0.507
.v6T2 ~~
.v6T3 (C6T2) 0.525 0.134 3.933 0.000 0.525 0.576
Time1 ~~
Time2 0.838 0.077 10.916 0.000 0.786 0.786
Time3 0.677 0.123 5.525 0.000 0.704 0.704
Time2 ~~
Time3 0.689 0.126 5.491 0.000 0.672 0.672
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (I1T1) 16.084 0.275 58.538 0.000 16.084 4.933
.v2T1 (I2T1) 8.678 0.293 29.586 0.000 8.678 2.447
.v3T1 (I3) 11.996 0.222 53.986 0.000 11.996 4.707
.v4T1 (I4T1) -3.020 0.267 -11.307 0.000 -3.020 -0.917
.v5T1 (I5) -1.360 0.142 -9.574 0.000 -1.360 -0.835
.v6T1 (I6) -2.802 0.161 -17.432 0.000 -2.802 -1.408
.v1T2 (I1) 16.526 0.273 60.493 0.000 16.526 5.618
.v2T2 (I2) 9.489 0.257 36.850 0.000 9.489 2.952
.v3T2 (I3) 11.996 0.222 53.986 0.000 11.996 4.615
.v4T2 (I4) -3.668 0.284 -12.896 0.000 -3.668 -1.109
.v5T2 (I5T2) -1.816 0.219 -8.277 0.000 -1.816 -0.766
.v6T2 (I6) -2.802 0.161 -17.432 0.000 -2.802 -1.464
.v1T3 (I1) 16.526 0.273 60.493 0.000 16.526 6.203
.v2T3 (I2) 9.489 0.257 36.850 0.000 9.489 3.208
.v3T3 (I3) 11.996 0.222 53.986 0.000 11.996 4.884
.v4T3 (I4) -3.668 0.284 -12.896 0.000 -3.668 -1.159
.v5T3 (I5) -1.360 0.142 -9.574 0.000 -1.360 -0.692
.v6T3 (I6) -2.802 0.161 -17.432 0.000 -2.802 -1.656
Time1 0.000 0.000 0.000
Time2 0.257 0.075 3.432 0.001 0.241 0.241
Time3 0.468 0.085 5.506 0.000 0.487 0.487
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (E1T1) 0.219 0.377 0.580 0.562 0.219 0.021
.v2T1 (E2T1) 8.669 1.027 8.437 0.000 8.669 0.689
.v3T1 (E3T1) 2.462 0.387 6.362 0.000 2.462 0.379
.v4T1 (E4T1) 7.173 1.023 7.014 0.000 7.173 0.661
.v5T1 (E5T1) 1.833 0.443 4.142 0.000 1.833 0.691
.v6T1 (E6T1) 1.696 0.244 6.960 0.000 1.696 0.428
.v1T2 (E1T2) 0.571 0.267 2.139 0.032 0.571 0.066
.v2T2 (E2T2) 5.882 0.603 9.751 0.000 5.882 0.569
.v3T2 (E3T2) 2.163 0.352 6.145 0.000 2.163 0.320
.v4T2 (E4T2) 6.761 0.970 6.967 0.000 6.761 0.618
.v5T2 (E5T2) 4.693 1.424 3.296 0.001 4.693 0.834
.v6T2 (E6T2) 1.084 0.164 6.600 0.000 1.084 0.296
.v1T3 (E1T3) 0.532 0.316 1.685 0.092 0.532 0.075
.v2T3 (E2T3) 5.134 0.788 6.513 0.000 5.134 0.587
.v3T3 (E3T3) 2.303 0.416 5.534 0.000 2.303 0.382
.v4T3 (E4T3) 6.610 1.131 5.844 0.000 6.610 0.661
.v5T3 (E5T3) 3.106 0.841 3.692 0.000 3.106 0.804
.v6T3 (E6T3) 0.768 0.169 4.540 0.000 0.768 0.268
Time1 1.000 1.000 1.000
Time2 1.138 0.178 6.395 0.000 1.000 1.000
Time3 0.925 0.154 6.024 0.000 1.000 1.000
R-Square:
Estimate
v1T1 0.979
v2T1 0.311
v3T1 0.621
v4T1 0.339
v5T1 0.309
v6T1 0.572
v1T2 0.934
v2T2 0.431
v3T2 0.680
v4T2 0.382
v5T2 0.166
v6T2 0.704
v1T3 0.925
v2T3 0.413
v3T3 0.618
v4T3 0.339
v5T3 0.196
v6T3 0.732anova(metricEstimates2, scalarEstimates4)Scaled Chi Square Difference Test (method = "satorra.bentler.2001")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
metricEstimates2 123 9003.3 9202.5 303.25
scalarEstimates4 129 9002.7 9183.7 314.56 11.895 6 0.06435 .
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1anova(configuralEstimates, scalarEstimates4)Scaled Chi Square Difference Test (method = "satorra.bentler.2001")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
configuralEstimates 114 9010.6 9236.9 292.51
scalarEstimates4 129 9002.7 9183.7 314.56 20.089 15 0.1685We finally see our likelihood ratio test cooperate and consider this to be the model to move forward with. Next is testing residual invariance.
residVarSyntax1 = "
#===================================================================================================
#Factor loadings constrained across groups except non-invariant ones
Time1 =~ L1T1*v1T1 + L2*v2T1 + L3*v3T1 + L4*v4T1 + L5*v5T1 + L6*v6T1
Time2 =~ L1*v1T2 + L2*v2T2 + L3*v3T2 + L4*v4T2 + L5*v5T2 + L6*v6T2
Time3 =~ L1*v1T3 + L2*v2T3 + L3*v3T3 + L4*v4T3 + L5*v5T3 + L6*v6T3
#===================================================================================================
#Item intercepts all constrained across groups except non-invariant ones
v1T1 ~ I1T1*1; v2T1 ~ I2T1*1; v3T1 ~ I3*1; v4T1 ~ I4T1*1; v5T1 ~ I5*1; v6T1 ~ I6*1;
v1T2 ~ I1*1; v2T2 ~ I2*1; v3T2 ~ I3*1; v4T2 ~ I4*1; v5T2 ~ I5T2*1; v6T2 ~ I6*1;
v1T3 ~ I1*1; v2T3 ~ I2*1; v3T3 ~ I3*1; v4T3 ~ I4*1; v5T3 ~ I5*1; v6T3 ~ I6*1;
#===================================================================================================
#Redidual variances all constrained in each group except non-invariant ones ****
v1T1 ~~ E1T1*v1T1; v2T1 ~~ E2T1*v2T1; v3T1 ~~ E3*v3T1; v4T1 ~~ E4T1*v4T1; v5T1 ~~ E5*v5T1; v6T1 ~~ E6*v6T1;
v1T2 ~~ E1*v1T2; v2T2 ~~ E2*v2T2; v3T2 ~~ E3*v3T2; v4T2 ~~ E4*v4T2; v5T2 ~~ E5T2*v5T2; v6T2 ~~ E6*v6T2;
v1T3 ~~ E1*v1T3; v2T3 ~~ E2*v2T3; v3T3 ~~ E3*v3T3; v4T3 ~~ E4*v4T3; v5T3 ~~ E5*v5T3; v6T3 ~~ E6*v6T3;
#===================================================================================================
#Residual covariances freely estimated in each group with labels
v1T1 ~~ C1T12*v1T2 + C1T13*v1T3; v1T2 ~~ C1T23*v1T3;
v2T1 ~~ C2T12*v2T2 + C2T13*v2T3; v2T2 ~~ C2T23*v2T3;
v3T1 ~~ C3T12*v3T2 + C3T13*v3T3; v3T2 ~~ C3T23*v3T3;
v4T1 ~~ C4T12*v4T2 + C4T13*v4T3; v4T2 ~~ C4T23*v4T3;
v5T1 ~~ C5T12*v5T2 + C5T13*v5T3; v5T2 ~~ C5T23*v5T3;
v6T1 ~~ C6T12*v6T2 + C6T13*v6T3; v6T2 ~~ C6T23*v6T3;
#===================================================================================================
#Factor variance fixed to 1 for identification in first group; estimated in others
Time1 ~~ 1*Time1
Time2 ~~ Time2
Time3 ~~ Time3
#===================================================================================================
#Factor mean fixed to zero for identification first group, estimated in others
Time1 ~ 0
Time2 ~ 1
Time3 ~ 1
#===================================================================================================
#Factor covariances all freely estimated
Time1 ~~ Time2 + Time3
Time2 ~~ Time3
"
residVarEstimates1 = lavaan(model = residVarSyntax1, data = cafData, estimator = "MLR", mimic = "mplus")lavaan WARNING: some cases are empty and will be ignored:
16 58 64 75 76 133 153 156summary(residVarEstimates1, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan (0.5-23.1097) converged normally after 114 iterations
Used Total
Number of observations 151 159
Number of missing patterns 7
Estimator ML Robust
Minimum Function Test Statistic 344.917 322.637
Degrees of freedom 137 137
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.069
for the Yuan-Bentler correction (Mplus variant)
Model test baseline model:
Minimum Function Test Statistic 2192.158 1896.786
Degrees of freedom 153 153
P-value 0.000 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.898 0.894
Tucker-Lewis Index (TLI) 0.886 0.881
Robust Comparative Fit Index (CFI) 0.902
Robust Tucker-Lewis Index (TLI) 0.890
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -4456.504 -4456.504
Scaling correction factor 1.024
for the MLR correction
Loglikelihood unrestricted model (H1) -4284.045 -4284.045
Scaling correction factor 1.203
for the MLR correction
Number of free parameters 52 52
Akaike (AIC) 9017.008 9017.008
Bayesian (BIC) 9173.907 9173.907
Sample-size adjusted Bayesian (BIC) 9009.332 9009.332
Root Mean Square Error of Approximation:
RMSEA 0.100 0.095
90 Percent Confidence Interval 0.087 0.114 0.082 0.108
P-value RMSEA <= 0.05 0.000 0.000
Robust RMSEA 0.098
90 Percent Confidence Interval 0.084 0.112
Standardized Root Mean Square Residual:
SRMR 0.095 0.095
Parameter Estimates:
Information Observed
Standard Errors Robust.huber.white
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
Time1 =~
v1T1 (L1T1) 3.167 0.262 12.087 0.000 3.167 0.970
v2T1 (L2) 2.013 0.203 9.935 0.000 2.013 0.573
v3T1 (L3) 2.058 0.194 10.599 0.000 2.058 0.811
v4T1 (L4) 1.943 0.232 8.370 0.000 1.943 0.586
v5T1 (L5) 0.931 0.144 6.473 0.000 0.931 0.516
v6T1 (L6) 1.490 0.119 12.540 0.000 1.490 0.806
Time2 =~
v1T2 (L1) 2.697 0.229 11.777 0.000 2.834 0.963
v2T2 (L2) 2.013 0.203 9.935 0.000 2.115 0.669
v3T2 (L3) 2.058 0.194 10.599 0.000 2.163 0.825
v4T2 (L4) 1.943 0.232 8.370 0.000 2.042 0.623
v5T2 (L5) 0.931 0.144 6.473 0.000 0.978 0.407
v6T2 (L6) 1.490 0.119 12.540 0.000 1.566 0.819
Time3 =~
v1T3 (L1) 2.697 0.229 11.777 0.000 2.552 0.955
v2T3 (L2) 2.013 0.203 9.935 0.000 1.905 0.630
v3T3 (L3) 2.058 0.194 10.599 0.000 1.948 0.796
v4T3 (L4) 1.943 0.232 8.370 0.000 1.838 0.582
v5T3 (L5) 0.931 0.144 6.473 0.000 0.881 0.495
v6T3 (L6) 1.490 0.119 12.540 0.000 1.410 0.790
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 ~~
.v1T2 (C1T12) -0.068 0.248 -0.272 0.785 -0.068 -0.108
.v1T3 (C1T13) 0.197 0.231 0.854 0.393 0.197 0.315
.v1T2 ~~
.v1T3 (C1T2) 0.261 0.259 1.010 0.312 0.261 0.417
.v2T1 ~~
.v2T2 (C2T12) 3.678 0.628 5.858 0.000 3.678 0.543
.v2T3 (C2T13) 2.130 0.767 2.775 0.006 2.130 0.315
.v2T2 ~~
.v2T3 (C2T2) 2.047 0.558 3.670 0.000 2.047 0.371
.v3T1 ~~
.v3T2 (C3T12) 0.914 0.285 3.204 0.001 0.914 0.416
.v3T3 (C3T13) 0.946 0.265 3.568 0.000 0.946 0.430
.v3T2 ~~
.v3T3 (C3T2) 1.102 0.298 3.700 0.000 1.102 0.501
.v4T1 ~~
.v4T2 (C4T12) 4.732 0.819 5.775 0.000 4.732 0.687
.v4T3 (C4T13) 3.908 1.000 3.910 0.000 3.908 0.567
.v4T2 ~~
.v4T3 (C4T2) 4.585 0.822 5.582 0.000 4.585 0.697
.v5T1 ~~
.v5T2 (C5T12) 2.760 0.861 3.205 0.001 2.760 0.813
.v5T3 (C5T13) 1.339 0.423 3.168 0.002 1.339 0.561
.v5T2 ~~
.v5T3 (C5T2) 2.514 0.707 3.556 0.000 2.514 0.741
.v6T1 ~~
.v6T2 (C6T12) 0.673 0.139 4.851 0.000 0.673 0.560
.v6T3 (C6T13) 0.549 0.164 3.357 0.001 0.549 0.458
.v6T2 ~~
.v6T3 (C6T2) 0.741 0.176 4.204 0.000 0.741 0.617
Time1 ~~
Time2 0.825 0.074 11.091 0.000 0.785 0.785
Time3 0.670 0.124 5.390 0.000 0.708 0.708
Time2 ~~
Time3 0.655 0.125 5.240 0.000 0.659 0.659
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (I1T1) 16.087 0.274 58.687 0.000 16.087 4.928
.v2T1 (I2T1) 8.678 0.294 29.535 0.000 8.678 2.469
.v3T1 (I3) 11.994 0.223 53.861 0.000 11.994 4.728
.v4T1 (I4T1) -3.021 0.267 -11.332 0.000 -3.021 -0.911
.v5T1 (I5) -1.459 0.150 -9.756 0.000 -1.459 -0.809
.v6T1 (I6) -2.810 0.159 -17.717 0.000 -2.810 -1.519
.v1T2 (I1) 16.533 0.275 60.022 0.000 16.533 5.619
.v2T2 (I2) 9.489 0.259 36.566 0.000 9.489 3.001
.v3T2 (I3) 11.994 0.223 53.861 0.000 11.994 4.573
.v4T2 (I4) -3.665 0.284 -12.904 0.000 -3.665 -1.118
.v5T2 (I5T2) -1.935 0.233 -8.302 0.000 -1.935 -0.805
.v6T2 (I6) -2.810 0.159 -17.717 0.000 -2.810 -1.470
.v1T3 (I1) 16.533 0.275 60.022 0.000 16.533 6.188
.v2T3 (I2) 9.489 0.259 36.566 0.000 9.489 3.136
.v3T3 (I3) 11.994 0.223 53.861 0.000 11.994 4.900
.v4T3 (I4) -3.665 0.284 -12.904 0.000 -3.665 -1.161
.v5T3 (I5) -1.459 0.150 -9.756 0.000 -1.459 -0.821
.v6T3 (I6) -2.810 0.159 -17.717 0.000 -2.810 -1.573
Time1 0.000 0.000 0.000
Time2 0.254 0.075 3.382 0.001 0.242 0.242
Time3 0.459 0.083 5.493 0.000 0.485 0.485
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (E1T1) 0.625 0.396 1.581 0.114 0.625 0.059
.v2T1 (E2T1) 8.302 0.997 8.325 0.000 8.302 0.672
.v3T1 (E3) 2.198 0.278 7.894 0.000 2.198 0.342
.v4T1 (E4T1) 7.216 1.055 6.837 0.000 7.216 0.657
.v5T1 (E5) 2.386 0.530 4.501 0.000 2.386 0.734
.v6T1 (E6) 1.201 0.153 7.843 0.000 1.201 0.351
.v1T2 (E1) 0.626 0.259 2.417 0.016 0.626 0.072
.v2T2 (E2) 5.524 0.508 10.882 0.000 5.524 0.552
.v3T2 (E3) 2.198 0.278 7.894 0.000 2.198 0.320
.v4T2 (E4) 6.581 0.858 7.674 0.000 6.581 0.612
.v5T2 (E5T2) 4.825 1.547 3.118 0.002 4.825 0.835
.v6T2 (E6) 1.201 0.153 7.843 0.000 1.201 0.329
.v1T3 (E1) 0.626 0.259 2.417 0.016 0.626 0.088
.v2T3 (E2) 5.524 0.508 10.882 0.000 5.524 0.604
.v3T3 (E3) 2.198 0.278 7.894 0.000 2.198 0.367
.v4T3 (E4) 6.581 0.858 7.674 0.000 6.581 0.661
.v5T3 (E5) 2.386 0.530 4.501 0.000 2.386 0.755
.v6T3 (E6) 1.201 0.153 7.843 0.000 1.201 0.376
Time1 1.000 1.000 1.000
Time2 1.104 0.169 6.551 0.000 1.000 1.000
Time3 0.895 0.153 5.844 0.000 1.000 1.000
R-Square:
Estimate
v1T1 0.941
v2T1 0.328
v3T1 0.658
v4T1 0.343
v5T1 0.266
v6T1 0.649
v1T2 0.928
v2T2 0.448
v3T2 0.680
v4T2 0.388
v5T2 0.165
v6T2 0.671
v1T3 0.912
v2T3 0.396
v3T3 0.633
v4T3 0.339
v5T3 0.245
v6T3 0.624anova(scalarEstimates4, residVarEstimates1)Scaled Chi Square Difference Test (method = "satorra.bentler.2001")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
scalarEstimates4 129 9002.7 9183.7 314.56
residVarEstimates1 137 9017.0 9173.9 344.92 19.796 8 0.01114 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1anova(configuralEstimates, residVarEstimates1)Scaled Chi Square Difference Test (method = "satorra.bentler.2001")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
configuralEstimates 114 9010.6 9236.9 292.51
residVarEstimates1 137 9017.0 9173.9 344.92 41.951 23 0.009199 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Does the partial residual variance model fit worse than the partial scalar model? Yes, −2∆LL(df=14) = 32.455, p <.01. A look at the modification indices reveals several items with large values. We will pick the residual variance from item6 at time1 to free.
#get modification indices for all parameters:
residVarMI1 = modificationindices(residVarEstimates1, free.remove = FALSE, maximum.number = 10000)
#restrict output to only residual variance parameters:
residVarMI1a = residVarMI1[which(residVarMI1$op == "~~" & residVarMI1$lhs == residVarMI1$rhs),]
residVarMI1a[order(-residVarMI1a$mi),] lhs op rhs mi mi.scaled epc sepc.lv sepc.all sepc.nox
42 v6T1 ~~ v6T1 5.060 4.733 0.253 0.253 0.074 0.074
54 v6T3 ~~ v6T3 2.649 2.478 -0.185 -0.185 -0.058 -0.058
53 v5T3 ~~ v5T3 1.922 1.798 0.213 0.213 0.067 0.067
41 v5T1 ~~ v5T1 0.909 0.850 -0.101 -0.101 -0.031 -0.031
50 v2T3 ~~ v2T3 0.297 0.278 -0.363 -0.363 -0.040 -0.040
48 v6T2 ~~ v6T2 0.283 0.265 -0.050 -0.050 -0.014 -0.014
44 v2T2 ~~ v2T2 0.152 0.142 0.186 0.186 0.019 0.019
45 v3T2 ~~ v3T2 0.053 0.050 -0.050 -0.050 -0.007 -0.007
43 v1T2 ~~ v1T2 0.049 0.046 0.034 0.034 0.004 0.004
49 v1T3 ~~ v1T3 0.048 0.045 -0.034 -0.034 -0.005 -0.005
39 v3T1 ~~ v3T1 0.029 0.027 0.040 0.040 0.006 0.006
52 v4T3 ~~ v4T3 0.014 0.013 -0.057 -0.057 -0.006 -0.006
46 v4T2 ~~ v4T2 0.008 0.007 0.032 0.032 0.003 0.003
51 v3T3 ~~ v3T3 0.006 0.006 0.019 0.019 0.003 0.003
75 Time3 ~~ Time3 0.000 0.000 0.000 0.000 0.000 0.000
38 v2T1 ~~ v2T1 0.000 0.000 0.000 0.000 0.000 0.000
74 Time2 ~~ Time2 0.000 0.000 0.000 0.000 0.000 0.000
37 v1T1 ~~ v1T1 0.000 0.000 0.000 0.000 0.000 0.000
40 v4T1 ~~ v4T1 0.000 0.000 0.000 0.000 0.000 0.000
47 v5T2 ~~ v5T2 0.000 0.000 0.000 0.000 0.000 0.000residVarSyntax2 = "
#===================================================================================================
#Factor loadings constrained across groups except non-invariant ones
Time1 =~ L1T1*v1T1 + L2*v2T1 + L3*v3T1 + L4*v4T1 + L5*v5T1 + L6*v6T1
Time2 =~ L1*v1T2 + L2*v2T2 + L3*v3T2 + L4*v4T2 + L5*v5T2 + L6*v6T2
Time3 =~ L1*v1T3 + L2*v2T3 + L3*v3T3 + L4*v4T3 + L5*v5T3 + L6*v6T3
#===================================================================================================
#Item intercepts all constrained across groups except non-invariant ones
v1T1 ~ I1T1*1; v2T1 ~ I2T1*1; v3T1 ~ I3*1; v4T1 ~ I4T1*1; v5T1 ~ I5*1; v6T1 ~ I6*1;
v1T2 ~ I1*1; v2T2 ~ I2*1; v3T2 ~ I3*1; v4T2 ~ I4*1; v5T2 ~ I5T2*1; v6T2 ~ I6*1;
v1T3 ~ I1*1; v2T3 ~ I2*1; v3T3 ~ I3*1; v4T3 ~ I4*1; v5T3 ~ I5*1; v6T3 ~ I6*1;
#===================================================================================================
#Redidual variances all constrained in each group except non-invariant ones
v1T1 ~~ E1T1*v1T1; v2T1 ~~ E2T1*v2T1; v3T1 ~~ E3*v3T1; v4T1 ~~ E4T1*v4T1; v5T1 ~~ E5*v5T1; v6T1 ~~ E6T1*v6T1;
v1T2 ~~ E1*v1T2; v2T2 ~~ E2*v2T2; v3T2 ~~ E3*v3T2; v4T2 ~~ E4*v4T2; v5T2 ~~ E5T2*v5T2; v6T2 ~~ E6*v6T2;
v1T3 ~~ E1*v1T3; v2T3 ~~ E2*v2T3; v3T3 ~~ E3*v3T3; v4T3 ~~ E4*v4T3; v5T3 ~~ E5*v5T3; v6T3 ~~ E6*v6T3;
#===================================================================================================
#Residual covariances freely estimated in each group with labels
v1T1 ~~ C1T12*v1T2 + C1T13*v1T3; v1T2 ~~ C1T23*v1T3;
v2T1 ~~ C2T12*v2T2 + C2T13*v2T3; v2T2 ~~ C2T23*v2T3;
v3T1 ~~ C3T12*v3T2 + C3T13*v3T3; v3T2 ~~ C3T23*v3T3;
v4T1 ~~ C4T12*v4T2 + C4T13*v4T3; v4T2 ~~ C4T23*v4T3;
v5T1 ~~ C5T12*v5T2 + C5T13*v5T3; v5T2 ~~ C5T23*v5T3;
v6T1 ~~ C6T12*v6T2 + C6T13*v6T3; v6T2 ~~ C6T23*v6T3;
#===================================================================================================
#Factor variance fixed to 1 for identification in first group; estimated in others
Time1 ~~ 1*Time1
Time2 ~~ Time2
Time3 ~~ Time3
#===================================================================================================
#Factor mean fixed to zero for identification first group, estimated in others
Time1 ~ 0
Time2 ~ 1
Time3 ~ 1
#===================================================================================================
#Factor covariances all freely estimated
Time1 ~~ Time2 + Time3
Time2 ~~ Time3
"
residVarEstimates2 = lavaan(model = residVarSyntax2, data = cafData, estimator = "MLR", mimic = "mplus")lavaan WARNING: some cases are empty and will be ignored:
16 58 64 75 76 133 153 156summary(residVarEstimates2, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan (0.5-23.1097) converged normally after 114 iterations
Used Total
Number of observations 151 159
Number of missing patterns 7
Estimator ML Robust
Minimum Function Test Statistic 331.477 310.378
Degrees of freedom 136 136
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.068
for the Yuan-Bentler correction (Mplus variant)
Model test baseline model:
Minimum Function Test Statistic 2192.158 1896.786
Degrees of freedom 153 153
P-value 0.000 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.904 0.900
Tucker-Lewis Index (TLI) 0.892 0.888
Robust Comparative Fit Index (CFI) 0.908
Robust Tucker-Lewis Index (TLI) 0.896
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -4449.784 -4449.784
Scaling correction factor 1.039
for the MLR correction
Loglikelihood unrestricted model (H1) -4284.045 -4284.045
Scaling correction factor 1.203
for the MLR correction
Number of free parameters 53 53
Akaike (AIC) 9005.568 9005.568
Bayesian (BIC) 9165.484 9165.484
Sample-size adjusted Bayesian (BIC) 8997.744 8997.744
Root Mean Square Error of Approximation:
RMSEA 0.098 0.092
90 Percent Confidence Interval 0.084 0.111 0.079 0.105
P-value RMSEA <= 0.05 0.000 0.000
Robust RMSEA 0.095
90 Percent Confidence Interval 0.081 0.109
Standardized Root Mean Square Residual:
SRMR 0.096 0.096
Parameter Estimates:
Information Observed
Standard Errors Robust.huber.white
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
Time1 =~
v1T1 (L1T1) 3.207 0.259 12.371 0.000 3.207 0.984
v2T1 (L2) 1.968 0.198 9.931 0.000 1.968 0.559
v3T1 (L3) 2.018 0.193 10.453 0.000 2.018 0.800
v4T1 (L4) 1.925 0.226 8.525 0.000 1.925 0.584
v5T1 (L5) 0.925 0.140 6.591 0.000 0.925 0.514
v6T1 (L6) 1.494 0.119 12.526 0.000 1.494 0.759
Time2 =~
v1T2 (L1) 2.676 0.230 11.657 0.000 2.846 0.968
v2T2 (L2) 1.968 0.198 9.931 0.000 2.093 0.663
v3T2 (L3) 2.018 0.193 10.453 0.000 2.146 0.817
v4T2 (L4) 1.925 0.226 8.525 0.000 2.047 0.621
v5T2 (L5) 0.925 0.140 6.591 0.000 0.984 0.407
v6T2 (L6) 1.494 0.119 12.526 0.000 1.589 0.852
Time3 =~
v1T3 (L1) 2.676 0.230 11.657 0.000 2.562 0.960
v2T3 (L2) 1.968 0.198 9.931 0.000 1.884 0.624
v3T3 (L3) 2.018 0.193 10.453 0.000 1.931 0.787
v4T3 (L4) 1.925 0.226 8.525 0.000 1.842 0.581
v5T3 (L5) 0.925 0.140 6.591 0.000 0.886 0.497
v6T3 (L6) 1.494 0.119 12.526 0.000 1.430 0.826
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 ~~
.v1T2 (C1T12) -0.148 0.244 -0.606 0.545 -0.148 -0.340
.v1T3 (C1T13) 0.084 0.224 0.375 0.708 0.084 0.193
.v1T2 ~~
.v1T3 (C1T2) 0.186 0.235 0.789 0.430 0.186 0.336
.v2T1 ~~
.v2T2 (C2T12) 3.726 0.643 5.797 0.000 3.726 0.541
.v2T3 (C2T13) 2.252 0.781 2.883 0.004 2.252 0.327
.v2T2 ~~
.v2T3 (C2T2) 2.112 0.559 3.781 0.000 2.112 0.379
.v3T1 ~~
.v3T2 (C3T12) 0.962 0.285 3.378 0.001 0.962 0.420
.v3T3 (C3T13) 1.028 0.261 3.939 0.000 1.028 0.449
.v3T2 ~~
.v3T3 (C3T2) 1.178 0.298 3.953 0.000 1.178 0.515
.v4T1 ~~
.v4T2 (C4T12) 4.747 0.816 5.816 0.000 4.747 0.688
.v4T3 (C4T13) 3.896 0.996 3.913 0.000 3.896 0.565
.v4T2 ~~
.v4T3 (C4T2) 4.663 0.833 5.595 0.000 4.663 0.700
.v5T1 ~~
.v5T2 (C5T12) 2.787 0.868 3.211 0.001 2.787 0.817
.v5T3 (C5T13) 1.344 0.425 3.162 0.002 1.344 0.563
.v5T2 ~~
.v5T3 (C5T2) 2.523 0.710 3.557 0.000 2.523 0.740
.v6T1 ~~
.v6T2 (C6T12) 0.743 0.146 5.075 0.000 0.743 0.593
.v6T3 (C6T13) 0.628 0.189 3.316 0.001 0.628 0.502
.v6T2 ~~
.v6T3 (C6T2) 0.539 0.133 4.046 0.000 0.539 0.565
Time1 ~~
Time2 0.832 0.075 11.061 0.000 0.782 0.782
Time3 0.673 0.125 5.395 0.000 0.703 0.703
Time2 ~~
Time3 0.674 0.125 5.395 0.000 0.662 0.662
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (I1T1) 16.084 0.275 58.554 0.000 16.084 4.934
.v2T1 (I2T1) 8.679 0.294 29.565 0.000 8.679 2.467
.v3T1 (I3) 11.998 0.222 53.971 0.000 11.998 4.758
.v4T1 (I4T1) -3.020 0.266 -11.335 0.000 -3.020 -0.917
.v5T1 (I5) -1.460 0.149 -9.769 0.000 -1.460 -0.811
.v6T1 (I6) -2.797 0.160 -17.520 0.000 -2.797 -1.421
.v1T2 (I1) 16.529 0.276 59.867 0.000 16.529 5.619
.v2T2 (I2) 9.497 0.258 36.784 0.000 9.497 3.010
.v3T2 (I3) 11.998 0.222 53.971 0.000 11.998 4.570
.v4T2 (I4) -3.666 0.284 -12.899 0.000 -3.666 -1.113
.v5T2 (I5T2) -1.938 0.234 -8.294 0.000 -1.938 -0.802
.v6T2 (I6) -2.797 0.160 -17.520 0.000 -2.797 -1.500
.v1T3 (I1) 16.529 0.276 59.867 0.000 16.529 6.197
.v2T3 (I2) 9.497 0.258 36.784 0.000 9.497 3.144
.v3T3 (I3) 11.998 0.222 53.971 0.000 11.998 4.891
.v4T3 (I4) -3.666 0.284 -12.899 0.000 -3.666 -1.156
.v5T3 (I5) -1.460 0.149 -9.769 0.000 -1.460 -0.820
.v6T3 (I6) -2.797 0.160 -17.520 0.000 -2.797 -1.615
Time1 0.000 0.000 0.000
Time2 0.256 0.075 3.393 0.001 0.241 0.241
Time3 0.464 0.085 5.461 0.000 0.484 0.484
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (E1T1) 0.341 0.372 0.916 0.360 0.341 0.032
.v2T1 (E2T1) 8.509 1.009 8.434 0.000 8.509 0.687
.v3T1 (E3) 2.288 0.277 8.276 0.000 2.288 0.360
.v4T1 (E4T1) 7.140 1.044 6.838 0.000 7.140 0.658
.v5T1 (E5) 2.389 0.531 4.496 0.000 2.389 0.736
.v6T1 (E6T1) 1.641 0.238 6.902 0.000 1.641 0.424
.v1T2 (E1) 0.553 0.238 2.327 0.020 0.553 0.064
.v2T2 (E2) 5.575 0.503 11.093 0.000 5.575 0.560
.v3T2 (E3) 2.288 0.277 8.276 0.000 2.288 0.332
.v4T2 (E4) 6.665 0.866 7.697 0.000 6.665 0.614
.v5T2 (E5T2) 4.870 1.560 3.123 0.002 4.870 0.834
.v6T2 (E6) 0.956 0.136 7.003 0.000 0.956 0.275
.v1T3 (E1) 0.553 0.238 2.327 0.020 0.553 0.078
.v2T3 (E2) 5.575 0.503 11.093 0.000 5.575 0.611
.v3T3 (E3) 2.288 0.277 8.276 0.000 2.288 0.380
.v4T3 (E4) 6.665 0.866 7.697 0.000 6.665 0.663
.v5T3 (E5) 2.389 0.531 4.496 0.000 2.389 0.753
.v6T3 (E6) 0.956 0.136 7.003 0.000 0.956 0.319
Time1 1.000 1.000 1.000
Time2 1.131 0.174 6.487 0.000 1.000 1.000
Time3 0.916 0.151 6.050 0.000 1.000 1.000
R-Square:
Estimate
v1T1 0.968
v2T1 0.313
v3T1 0.640
v4T1 0.342
v5T1 0.264
v6T1 0.576
v1T2 0.936
v2T2 0.440
v3T2 0.668
v4T2 0.386
v5T2 0.166
v6T2 0.725
v1T3 0.922
v2T3 0.389
v3T3 0.620
v4T3 0.337
v5T3 0.247
v6T3 0.681anova(scalarEstimates4, residVarEstimates2)Scaled Chi Square Difference Test (method = "satorra.bentler.2001")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
scalarEstimates4 129 9002.7 9183.7 314.56
residVarEstimates2 136 9005.6 9165.5 331.48 10.713 7 0.1516anova(configuralEstimates, residVarEstimates2)Scaled Chi Square Difference Test (method = "satorra.bentler.2001")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
configuralEstimates 114 9010.6 9236.9 292.51
residVarEstimates2 136 9005.6 9165.5 331.48 31.154 22 0.09302 .
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Does the partial residual variance model fit worse than the partial scalar model? No, −2∆LL(df=13) = 21.898, p = .06. The residual covariances are not tested in this model as we allow/expect them to be non-zero due to the same items appearing across time. Up next is the structural model.
factorVarCovSyntax1 = "
#===================================================================================================
#Factor loadings constrained across groups except non-invariant ones
Time1 =~ L1T1*v1T1 + L2*v2T1 + L3*v3T1 + L4*v4T1 + L5*v5T1 + L6*v6T1
Time2 =~ L1*v1T2 + L2*v2T2 + L3*v3T2 + L4*v4T2 + L5*v5T2 + L6*v6T2
Time3 =~ L1*v1T3 + L2*v2T3 + L3*v3T3 + L4*v4T3 + L5*v5T3 + L6*v6T3
#===================================================================================================
#Item intercepts all constrained across groups except non-invariant ones
v1T1 ~ I1T1*1; v2T1 ~ I2T1*1; v3T1 ~ I3*1; v4T1 ~ I4T1*1; v5T1 ~ I5*1; v6T1 ~ I6*1;
v1T2 ~ I1*1; v2T2 ~ I2*1; v3T2 ~ I3*1; v4T2 ~ I4*1; v5T2 ~ I5T2*1; v6T2 ~ I6*1;
v1T3 ~ I1*1; v2T3 ~ I2*1; v3T3 ~ I3*1; v4T3 ~ I4*1; v5T3 ~ I5*1; v6T3 ~ I6*1;
#===================================================================================================
#Redidual variances all constrained in each group except non-invariant ones
v1T1 ~~ E1T1*v1T1; v2T1 ~~ E2T1*v2T1; v3T1 ~~ E3*v3T1; v4T1 ~~ E4T1*v4T1; v5T1 ~~ E5*v5T1; v6T1 ~~ E6T1*v6T1;
v1T2 ~~ E1*v1T2; v2T2 ~~ E2*v2T2; v3T2 ~~ E3*v3T2; v4T2 ~~ E4*v4T2; v5T2 ~~ E5T2*v5T2; v6T2 ~~ E6*v6T2;
v1T3 ~~ E1*v1T3; v2T3 ~~ E2*v2T3; v3T3 ~~ E3*v3T3; v4T3 ~~ E4*v4T3; v5T3 ~~ E5*v5T3; v6T3 ~~ E6*v6T3;
#===================================================================================================
#Residual covariances freely estimated in each group with labels
v1T1 ~~ C1T12*v1T2 + C1T13*v1T3; v1T2 ~~ C1T23*v1T3;
v2T1 ~~ C2T12*v2T2 + C2T13*v2T3; v2T2 ~~ C2T23*v2T3;
v3T1 ~~ C3T12*v3T2 + C3T13*v3T3; v3T2 ~~ C3T23*v3T3;
v4T1 ~~ C4T12*v4T2 + C4T13*v4T3; v4T2 ~~ C4T23*v4T3;
v5T1 ~~ C5T12*v5T2 + C5T13*v5T3; v5T2 ~~ C5T23*v5T3;
v6T1 ~~ C6T12*v6T2 + C6T13*v6T3; v6T2 ~~ C6T23*v6T3;
#===================================================================================================
#Factor variance fixed to 1 for identification in first group; estimated in others
Time1 ~~ 1*Time1
Time2 ~~ 1*Time2
Time3 ~~ 1*Time3
#===================================================================================================
#Factor mean fixed to zero for identification first group, estimated in others
Time1 ~ 0
Time2 ~ Fmean2*1
Time3 ~ Fmean3*1
#===================================================================================================
#Factor covariances all freely estimated
Time1 ~~ Fcov*Time2 + Fcov*Time3
Time2 ~~ Fcov*Time3
#===================================================================================================
#Wald Tests for Factor Means:
DF1F2 := -Fmean2; DF1F3 := -Fmean3; DF2F3 := Fmean2-Fmean3
"
factorVarCovEstimates2 = lavaan(model = factorVarCovSyntax1, data = cafData, estimator = "MLR", mimic = "mplus")lavaan WARNING: some cases are empty and will be ignored:
16 58 64 75 76 133 153 156summary(factorVarCovEstimates2, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan (0.5-23.1097) converged normally after 110 iterations
Used Total
Number of observations 151 159
Number of missing patterns 7
Estimator ML Robust
Minimum Function Test Statistic 338.526 311.664
Degrees of freedom 140 140
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.086
for the Yuan-Bentler correction (Mplus variant)
Model test baseline model:
Minimum Function Test Statistic 2192.158 1896.786
Degrees of freedom 153 153
P-value 0.000 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.903 0.902
Tucker-Lewis Index (TLI) 0.894 0.892
Robust Comparative Fit Index (CFI) 0.907
Robust Tucker-Lewis Index (TLI) 0.899
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -4453.308 -4453.308
Scaling correction factor 0.978
for the MLR correction
Loglikelihood unrestricted model (H1) -4284.045 -4284.045
Scaling correction factor 1.203
for the MLR correction
Number of free parameters 49 49
Akaike (AIC) 9004.617 9004.617
Bayesian (BIC) 9152.463 9152.463
Sample-size adjusted Bayesian (BIC) 8997.383 8997.383
Root Mean Square Error of Approximation:
RMSEA 0.097 0.090
90 Percent Confidence Interval 0.084 0.110 0.077 0.103
P-value RMSEA <= 0.05 0.000 0.000
Robust RMSEA 0.094
90 Percent Confidence Interval 0.080 0.108
Standardized Root Mean Square Residual:
SRMR 0.104 0.104
Parameter Estimates:
Information Observed
Standard Errors Robust.huber.white
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
Time1 =~
v1T1 (L1T1) 3.207 0.242 13.227 0.000 3.207 0.980
v2T1 (L2) 2.001 0.170 11.787 0.000 2.001 0.567
v3T1 (L3) 2.048 0.169 12.143 0.000 2.048 0.806
v4T1 (L4) 1.935 0.215 8.994 0.000 1.935 0.587
v5T1 (L5) 0.918 0.147 6.232 0.000 0.918 0.510
v6T1 (L6) 1.519 0.108 14.072 0.000 1.519 0.766
Time2 =~
v1T2 (L1) 2.712 0.232 11.712 0.000 2.712 0.963
v2T2 (L2) 2.001 0.170 11.787 0.000 2.001 0.648
v3T2 (L3) 2.048 0.169 12.143 0.000 2.048 0.806
v4T2 (L4) 1.935 0.215 8.994 0.000 1.935 0.599
v5T2 (L5) 0.918 0.147 6.232 0.000 0.918 0.384
v6T2 (L6) 1.519 0.108 14.072 0.000 1.519 0.843
Time3 =~
v1T3 (L1) 2.712 0.232 11.712 0.000 2.712 0.963
v2T3 (L2) 2.001 0.170 11.787 0.000 2.001 0.648
v3T3 (L3) 2.048 0.169 12.143 0.000 2.048 0.806
v4T3 (L4) 1.935 0.215 8.994 0.000 1.935 0.599
v5T3 (L5) 0.918 0.147 6.232 0.000 0.918 0.510
v6T3 (L6) 1.519 0.108 14.072 0.000 1.519 0.843
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 ~~
.v1T2 (C1T12) -0.085 0.236 -0.360 0.719 -0.085 -0.173
.v1T3 (C1T13) 0.108 0.227 0.477 0.634 0.108 0.220
.v1T2 ~~
.v1T3 (C1T2) 0.160 0.247 0.649 0.517 0.160 0.277
.v2T1 ~~
.v2T2 (C2T12) 3.683 0.631 5.833 0.000 3.683 0.538
.v2T3 (C2T13) 2.247 0.775 2.897 0.004 2.247 0.328
.v2T2 ~~
.v2T3 (C2T2) 2.129 0.558 3.813 0.000 2.129 0.384
.v3T1 ~~
.v3T2 (C3T12) 0.930 0.279 3.332 0.001 0.930 0.411
.v3T3 (C3T13) 1.014 0.259 3.910 0.000 1.014 0.448
.v3T2 ~~
.v3T3 (C3T2) 1.174 0.297 3.952 0.000 1.174 0.519
.v4T1 ~~
.v4T2 (C4T12) 4.756 0.815 5.832 0.000 4.756 0.689
.v4T3 (C4T13) 3.879 0.997 3.890 0.000 3.879 0.562
.v4T2 ~~
.v4T3 (C4T2) 4.633 0.835 5.552 0.000 4.633 0.694
.v5T1 ~~
.v5T2 (C5T12) 2.792 0.870 3.209 0.001 2.792 0.817
.v5T3 (C5T13) 1.343 0.426 3.152 0.002 1.343 0.561
.v5T2 ~~
.v5T3 (C5T2) 2.519 0.711 3.544 0.000 2.519 0.737
.v6T1 ~~
.v6T2 (C6T12) 0.736 0.145 5.085 0.000 0.736 0.595
.v6T3 (C6T13) 0.622 0.189 3.296 0.001 0.622 0.503
.v6T2 ~~
.v6T3 (C6T2) 0.538 0.132 4.072 0.000 0.538 0.573
Time1 ~~
Time2 (Fcov) 0.722 0.053 13.697 0.000 0.722 0.722
Time3 (Fcov) 0.722 0.053 13.697 0.000 0.722 0.722
Time2 ~~
Time3 (Fcov) 0.722 0.053 13.697 0.000 0.722 0.722
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (I1T1) 16.097 0.271 59.488 0.000 16.097 4.920
.v2T1 (I2T1) 8.686 0.292 29.752 0.000 8.686 2.462
.v3T1 (I3) 12.003 0.220 54.487 0.000 12.003 4.724
.v4T1 (I4T1) -3.012 0.263 -11.451 0.000 -3.012 -0.913
.v5T1 (I5) -1.454 0.148 -9.792 0.000 -1.454 -0.808
.v6T1 (I6) -2.794 0.159 -17.598 0.000 -2.794 -1.408
.v1T2 (I1) 16.537 0.274 60.252 0.000 16.537 5.871
.v2T2 (I2) 9.501 0.256 37.133 0.000 9.501 3.075
.v3T2 (I3) 12.003 0.220 54.487 0.000 12.003 4.724
.v4T2 (I4) -3.656 0.279 -13.089 0.000 -3.656 -1.133
.v5T2 (I5T2) -1.931 0.232 -8.319 0.000 -1.931 -0.807
.v6T2 (I6) -2.794 0.159 -17.598 0.000 -2.794 -1.551
.v1T3 (I1) 16.537 0.274 60.252 0.000 16.537 5.871
.v2T3 (I2) 9.501 0.256 37.133 0.000 9.501 3.075
.v3T3 (I3) 12.003 0.220 54.487 0.000 12.003 4.724
.v4T3 (I4) -3.656 0.279 -13.089 0.000 -3.656 -1.133
.v5T3 (I5) -1.454 0.148 -9.792 0.000 -1.454 -0.808
.v6T3 (I6) -2.794 0.159 -17.598 0.000 -2.794 -1.551
Time1 0.000 0.000 0.000
Time2 (Fmn2) 0.250 0.075 3.325 0.001 0.250 0.250
Time3 (Fmn3) 0.470 0.088 5.319 0.000 0.470 0.470
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.v1T1 (E1T1) 0.418 0.360 1.161 0.246 0.418 0.039
.v2T1 (E2T1) 8.443 1.001 8.435 0.000 8.443 0.678
.v3T1 (E3) 2.262 0.273 8.274 0.000 2.262 0.350
.v4T1 (E4T1) 7.139 1.042 6.852 0.000 7.139 0.656
.v5T1 (E5) 2.395 0.532 4.506 0.000 2.395 0.740
.v6T1 (E6T1) 1.629 0.239 6.818 0.000 1.629 0.414
.v1T2 (E1) 0.579 0.235 2.463 0.014 0.579 0.073
.v2T2 (E2) 5.542 0.493 11.243 0.000 5.542 0.581
.v3T2 (E3) 2.262 0.273 8.274 0.000 2.262 0.350
.v4T2 (E4) 6.676 0.870 7.677 0.000 6.676 0.641
.v5T2 (E5T2) 4.882 1.563 3.124 0.002 4.882 0.853
.v6T2 (E6) 0.940 0.131 7.163 0.000 0.940 0.289
.v1T3 (E1) 0.579 0.235 2.463 0.014 0.579 0.073
.v2T3 (E2) 5.542 0.493 11.243 0.000 5.542 0.581
.v3T3 (E3) 2.262 0.273 8.274 0.000 2.262 0.350
.v4T3 (E4) 6.676 0.870 7.677 0.000 6.676 0.641
.v5T3 (E5) 2.395 0.532 4.506 0.000 2.395 0.740
.v6T3 (E6) 0.940 0.131 7.163 0.000 0.940 0.289
Time1 1.000 1.000 1.000
Time2 1.000 1.000 1.000
Time3 1.000 1.000 1.000
R-Square:
Estimate
v1T1 0.961
v2T1 0.322
v3T1 0.650
v4T1 0.344
v5T1 0.260
v6T1 0.586
v1T2 0.927
v2T2 0.419
v3T2 0.650
v4T2 0.359
v5T2 0.147
v6T2 0.711
v1T3 0.927
v2T3 0.419
v3T3 0.650
v4T3 0.359
v5T3 0.260
v6T3 0.711
Defined Parameters:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
DF1F2 -0.250 0.075 -3.325 0.001 -0.250 -0.250
DF1F3 -0.470 0.088 -5.319 0.000 -0.470 -0.470
DF2F3 -0.220 0.077 -2.837 0.005 -0.220 -0.220anova(residVarEstimates2, factorVarCovEstimates2)Scaled Chi Square Difference Test (method = "satorra.bentler.2001")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
residVarEstimates2 136 9005.6 9165.5 331.48
factorVarCovEstimates2 140 9004.6 9152.5 338.53 4.1332 4 0.3883anova(configuralEstimates, factorVarCovEstimates2)Scaled Chi Square Difference Test (method = "satorra.bentler.2001")
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
configuralEstimates 114 9010.6 9236.9 292.51
factorVarCovEstimates2 140 9004.6 9152.5 338.53 34.841 26 0.1151Does constraining the factor variances and covariances cause the model to fit worse? No,-2∆LL = 4.1332 with 4 df, for a p-value of .3883. We will stop with the models here and use Wald tests to test the difference between factor means. From the defined parameters, we wee that all three differences in factor means are significant. The negative signs of the estimates tell us the factor is decreasing across time.
The extent to which a confirmatory factor model measuring social functioning (with six observed indicators) exhibited measurement invariance and structural invariance over time (i.e., across three occasions taken at 6-month intervals) was examined using the lavaan package (Rosseel, 2012) in R (R Core Team, 2017). Robust maximum likelihood (MLR) estimation was used for all analyses; accordingly, nested model comparisons were conducted using the rescaled difference in the model −2LL values as a function of the difference in model degrees of freedom. A configural invariance model was initially specified in which three correlated factors (i.e., the factor at three occasions) were estimated simultaneously; all factor means were fixed to 0 and all factor variances were fixed to 1 for identification. Residual covariances between the same indicators across occasions were estimated as well. As shown in Table 1, although the configural invariance model had marginal fit, reasonable attempts to improve the fit were unsuccessful. Thus, the analysis proceeded by applying parameter constraints in successive models to examine potential decreases in fit resulting from measurement or structural non-invariance over the three occasions.
Equality of the unstandardized indicator factor loadings across occasions was then examined in a metric invariance model. The factor variance was fixed to 1 at time 1 but was freely estimated at times 2 and 3. All factor loadings were constrained equal across time; all intercepts and residual variances were still permitted to vary across time. Factor covariances and residual covariances were estimated as described previously. The metric invariance model fit significantly worse than the configural invariance model −2ΔLL(10) = 19.14, p = .04. The modification indices suggested that the loading of indicator 1 at time 1 was a source of misfit and should be freed. After doing so, the partial metric invariance model fit significantly better than the full metric invariance model, −2ΔLL(1) = 7.16, p < .001, and the partial metric invariance model did not fit worse than the configural invariance model−2ΔLL(9) = 8.98, p = .44. The fact that partial metric invariance (i.e., “weak invariance”) held indicates that the indicators were related to the latent factor equivalently across time, or more simply, that the same latent factor was being measured at each of occasion (with the exception of indicator 1, which was more related to the factor at time 1 than at times 2 or 3).
Equality of the unstandardized indicator intercepts across time was then examined in a scalar invariance model. The factor mean and variance were fixed to 0 and 1, respectively, at time 1 for identification, but the factor mean and variance were then estimated at times 2 and 3. All factor loadings and indicator intercepts were constrained equal across time (except for indicator 1 at time 1); all residual variances were still permitted to differ across time. Factor covariances and residual covariances were estimated as described previously. The scalar invariance model fit significantly worse than the partial metric invariance model, −2ΔLL(9) = 55.11, p < .01. The modification indices suggested that the intercept of indicator 2 at time 1 was the largest source of the misfit and should be freed. After doing so, although the partial scalar invariance model had significantly better fit than the full scalar invariance model, it still fit worse than the partial metric invariance model, −2ΔLL(8) = 39.321, p < 001. The modification indices suggested that the intercept of indicator 5 at time 2 was the largest remaining source of the misfit and should be freed. After doing so, although the new partial scalar invariance model (with the intercepts for indicators 1 and 2 at time 1 and 5 freed at time 2) fit significantly better than the previous partial scalar invariance model but it still fit marginally worse than the partial metric invariance model, −2ΔLL(7) = 21.99, p < .01. The modification indices suggested that the intercept of indicator 4 at time 1 was the largest remaining source of the misfit and should be freed. After doing so, the new partial scalar invariance model (with the intercepts for indicators 1, 2, and 4 at time 1 and 5 freed at time 2) fit significantly better than the previous partial scalar invariance model (without the intercept for indicator 2 freed at time 1), and it did not fit significantly worse than the partial metric invariance model, −2ΔLL(6) = 11.90, p = .06.
Equality of the unstandardized residual variances across time was then examined in a residual variance invariance model. As in the partial scalar invariance model, the factor mean and variance were fixed to 0 and 1, respectively, for identification at time 1, but the factor mean and variance were still estimated at times 2 and 3. All factor loadings (except for indicator 1 at time 1), item intercepts (except for indicators 1, 2, and 4 at time1 and 5 at time 2), and all residual variances (except for indicators 1, 2, and 4 at time1 and 5 at time 2) were constrained to be equal across groups. Factor covariances and residual covariances were estimated as described previously. The residual variance invariance model fit significantly worse than the last partial scalar invariance model, −2ΔLL(8) = 32.46, p < .01. The modification indices suggested that the residual variance of indicator 6 at time 1 was the largest remaining source of the misfit and should be freed. After doing so, the partial residual variance invariance model fit significantly better than the residual invariance model, −2ΔLL(1) = 21.898, p = 0.06.
After achieving partial measurement invariance as was just described, structural invariance of factor variances and covariances was then tested by constraining the factor variances to one and the factor covariances to be equal across all time points, resulting in a nonsignificant decrease in fit relative to the last partial residual invariance model, −2ΔLL(4) = 4.13, p = .39. The difference between factor means at each time point was tested using Wald tests using the previous model with constrained factor variances and covariances. The difference between the mean at time 1 and time 2 was -.250 (p < .01), the difference between the factor means at times 1 and 3 was -.47, (p < .01), and the difference between the factor means at times 2 and 3 was -.22, (p < .01).
In conclusion, these analyses showed that partial measurement invariance was obtained over time-that is, the relationships of the indicators to the latent factor of social functioning were equivalent at times 2 and 3, although primarily not equivalent at time 1, as previous described. These analyses also showed that partial structural invariance was obtained over time, such that the same amount of individual differences variance in social functioning was observed with equal covariance over time across occasions (i.e., compound symmetry of the latent factor), although the amount of social functioning on average increased significantly over time. Model parameters from the final model are given in Table 2.
(Table 1 would have likelihood ratio tests; Table 2 would have unstandardized and standardized estimates and their SEs)