One example from Major Depression Criteria
Recently, I was asked by my friend why should we use Measurement Invariance in real research. Why not just ignore this complex and tedious process? As far I’m concerned, measurement invariance should be widely used if you have large data scale and figure out what’s going on between groups difference. In this post, I want to elaborate some problems in Measurement Invariance: 1) What is measurement invariance 2) why should we care about measurement invariance 3) how to do measurement invariance using R
LavaanPackage.
In my advisor Jonathan’s lectures slides, measurement invariance (MI) is a testing procedure in latent variable modeling to investigate “whether indicators measure the same construct in the same way in different groups or over time/condition”.
It is a neat and clear definition of MI. In my opinion, we should first know different piles of variances of indicator responses. We know that in CFA, latent trait is identified by covariances among indicators (observable features). Imaging item responses have “significant” group differences (e.g., male vs. female, international students with native speakers), then there are at least three sources of deviations between groups:
The first two points are straightforward. Taking an international math assessment for example, it may measure native speaker’s math ability while it may also measure English proficiency of international students. Or if male and female have different math ability, they might (not must) have different item responses on math assessment.
The third point suggests that even for two groups with exactly same average level of target trait, they will still have different item properties since same item have different power to measure the latent trait for male and female. For example, one item of Daily Living Ability Survey is “How often do you cook in a week?”. The item may be biased toward men, because most males may hate cooking (that is a stereotype!!!) but still have high daily living ability (such as driving, fixing), some females loving cooking but have low daily living ability. Thus, this item doesn’t account for female’s or men’s daily ability at same extent.
Actually all parameters in CFA model (e.g., factor variances, factor covariance, factor means, factor loadings, item intercepts and residential variances, co-variances) could be potentially different across groups, which leads to some problems in interpreting results. In psychometrics, the MI is splitted into multiple parts:
In previous paragraph, the first two differences are measured by Structural Invariance. The 3rd differences are measured by Measurement Invariance.
This example data is from Brown Chapter 7. Major Depression Criteria across Men and Women (n =345)
9 items rated by clinicians on a scale of 0 to 8 (0=none, 8 =very severely disturbing/disabling)
Jonathan in his Measurement Invariance Example elaborated the manual version so that learner could learn what you are doing first. I will show you how to use shortcuts.
sex item1 item2 item3 item4 item5 item6 item7 item8 item9
1 0 5 4 1 6 5 6 5 4 2
2 0 5 5 5 5 4 5 4 5 4
3 0 4 5 4 2 6 6 0 0 0
4 0 5 5 3 3 5 5 6 4 0
5 0 5 5 0 5 0 4 6 0 0
6 0 6 6 4 6 4 6 5 6 2
The sample size of female reference groups is as same as the male. The model for 2 groups should be same and check how many changes are allowed to differ.
model1.config <- "
# Constrain the factor loadings and intercepts of marker variable in ALL groups
# depress =~ c(L1F, L1M)*item1 + c(L2F, L2M)*item2 + c(L3F, L3M)*item3 +
# c(L4F, L4M)*item4 + c(L5F, L5M)*item5 + c(L6F, L6M)*item6 +
# c(L7F, L7M)*item7 + c(L8F, L8M)*item8 + c(L9F, L9M)*item9
depress =~ item1 + item2 + item3 +
item4 + item5 + item6 +
item7 + item8 + item9
#Item intercepts all freely estimated in both groups with label for each group
item1 ~ 1; item2 ~ 1; item3 ~ 1;
item4 ~ 1; item5 ~ 1; item6 ~ 1;
item7 ~ 1; item8 ~ 1; item9 ~ 1;
#Redidual variances all freely estimated with label for each group
item1 ~~ item1; item2 ~~ item2; item3 ~~ item3;
item4 ~~ item4; item5 ~~ item5; item6 ~~ item6;
item7 ~~ item7; item8 ~~ item8; item9 ~~ item9;
#Residual covariance freely estimated in both groups with label for each group
item1 ~~ item2
#==================================================
#Factor variance fixed to 1 for identification in each group
depress ~~ c(1,NA)*depress
#Factor mean fixed to zero for identification in each group
depress ~ c(0,NA)*0
"Configural Invariance Model is the first-step model which allows all estimation different for two groups except that mean and variance of factor are fixed to 0 and 1, because the model uses z-score scalling.
Compared to configural invariance, metic invariance model constrains the factor loadings for two groups equal with each other. To test metric invariance, we could use absolute model fit indices (CFI, TLI, RMSEA, SRMR) and comparable model fit indices (Log-likelihood test). It deserves noting that in metric invariance model, factor means are still constrained to be equal for two groups but the variances of factor are different. The variance of factor for reference group is fixed to 1 but that for other group is free to estimate. Since if we constrain both factor loadings and factor variances to equal, then the residual variances of items will also be equal. This is next step. Freeing one group’s factor variance will let model not too strict to Residual Variance.
Next model is Scalar Invariance Model, which constrain the intercepts of items to be equal.
fit.config <- sem(model1.config, data = mddAll,
meanstructure = T , std.lv = T,
estimator = "MLR", mimic = "mplus",
group = "sex",
group.equal = c("lv.variances", "means")) # latent variance both equal to 1
fit.metric <- sem(model1.config, data = mddAll,
meanstructure = T , std.lv = T,
estimator = "MLR", mimic = "mplus",
group = "sex",
group.equal = c("loadings", "means")) # factor mean should be equal to 0
fit.scalar <- sem(model1.config, data = mddAll,
meanstructure = T , std.lv = T,
estimator = "MLR", mimic = "mplus",
group = "sex",
group.equal = c("loadings","intercepts"))
# same: factor loadings, item intercepts
# different: reference factor mean is 1, another factor mean is 0
fit.scalar2 <- sem(model1.config, data = mddAll,
meanstructure = T , std.lv = T,
estimator = "MLR", mimic = "mplus",
group = "sex",
group.equal = c("loadings","intercepts"),
group.partial = c("item7~1"))
fit.strict <- sem(model1.config, data = mddAll,
meanstructure = T , std.lv = T,
estimator = "MLR", mimic = "mplus",
group = "sex",
group.equal = c("loadings","intercepts", "residuals"),
group.partial = c("item7~1", "item7~~item7"))
fit.strict.cov <- sem(model1.config, data = mddAll,
meanstructure = T , std.lv = T,
estimator = "MLR", mimic = "mplus",
group = "sex",
group.equal = c("loadings","intercepts", "residuals",
"residual.covariances"),
group.partial = c("item7~1", "item7~~item7"))summary(fit.config, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan 0.6.17 ended normally after 47 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 56
Number of observations per group:
Female 375
Male 375
Number of missing patterns per group:
Female 1
Male 1
Model Test User Model:
Standard Scaled
Test Statistic 98.911 94.175
Degrees of freedom 52 52
P-value (Chi-square) 0.000 0.000
Scaling correction factor 1.050
Yuan-Bentler correction (Mplus variant)
Test statistic for each group:
Female 52.954 50.418
Male 45.957 43.756
Model Test Baseline Model:
Test statistic 1343.575 1218.364
Degrees of freedom 72 72
P-value 0.000 0.000
Scaling correction factor 1.103
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.963 0.963
Tucker-Lewis Index (TLI) 0.949 0.949
Robust Comparative Fit Index (CFI) 0.965
Robust Tucker-Lewis Index (TLI) 0.952
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -13706.898 -13706.898
Scaling correction factor 0.981
for the MLR correction
Loglikelihood unrestricted model (H1) -13657.442 -13657.442
Scaling correction factor 1.014
for the MLR correction
Akaike (AIC) 27525.796 27525.796
Bayesian (BIC) 27784.520 27784.520
Sample-size adjusted Bayesian (SABIC) 27606.698 27606.698
Root Mean Square Error of Approximation:
RMSEA 0.049 0.047
90 Percent confidence interval - lower 0.034 0.031
90 Percent confidence interval - upper 0.064 0.061
P-value H_0: RMSEA <= 0.050 0.522 0.636
P-value H_0: RMSEA >= 0.080 0.000 0.000
Robust RMSEA 0.048
90 Percent confidence interval - lower 0.032
90 Percent confidence interval - upper 0.063
P-value H_0: Robust RMSEA <= 0.050 0.581
P-value H_0: Robust RMSEA >= 0.080 0.000
Standardized Root Mean Square Residual:
SRMR 0.039 0.039
Parameter Estimates:
Standard errors Sandwich
Information bread Observed
Observed information based on Hessian
Group 1 [Female]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
depress =~
item1 1.251 0.095 13.155 0.000 1.251 0.730
item2 1.385 0.103 13.426 0.000 1.385 0.688
item3 0.911 0.104 8.775 0.000 0.911 0.435
item4 1.140 0.115 9.874 0.000 1.140 0.516
item5 1.015 0.106 9.615 0.000 1.015 0.477
item6 1.155 0.103 11.238 0.000 1.155 0.577
item7 0.764 0.115 6.618 0.000 0.764 0.371
item8 1.224 0.113 10.817 0.000 1.224 0.569
item9 0.606 0.094 6.412 0.000 0.606 0.339
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 ~~
.item2 0.393 0.166 2.364 0.018 0.393 0.230
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 4.184 0.089 47.258 0.000 4.184 2.440
.item2 3.725 0.104 35.848 0.000 3.725 1.851
.item3 1.952 0.108 18.058 0.000 1.952 0.933
.item4 3.589 0.114 31.458 0.000 3.589 1.624
.item5 2.256 0.110 20.522 0.000 2.256 1.060
.item6 3.955 0.103 38.237 0.000 3.955 1.975
.item7 3.869 0.106 36.382 0.000 3.869 1.879
.item8 3.595 0.111 32.331 0.000 3.595 1.670
.item9 1.205 0.092 13.053 0.000 1.205 0.674
depress 0.000 0.000 0.000
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 1.375 0.194 7.090 0.000 1.375 0.468
.item2 2.132 0.236 9.049 0.000 2.132 0.527
.item3 3.551 0.201 17.678 0.000 3.551 0.810
.item4 3.583 0.272 13.166 0.000 3.583 0.734
.item5 3.501 0.223 15.733 0.000 3.501 0.773
.item6 2.677 0.269 9.967 0.000 2.677 0.667
.item7 3.658 0.276 13.270 0.000 3.658 0.862
.item8 3.137 0.291 10.785 0.000 3.137 0.677
.item9 2.831 0.195 14.538 0.000 2.831 0.885
depress 1.000 1.000 1.000
R-Square:
Estimate
item1 0.532
item2 0.473
item3 0.190
item4 0.266
item5 0.227
item6 0.333
item7 0.138
item8 0.323
item9 0.115
Group 2 [Male]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
depress =~
item1 1.024 0.099 10.384 0.000 1.024 0.642
item2 1.266 0.112 11.283 0.000 1.266 0.628
item3 0.805 0.115 7.011 0.000 0.805 0.385
item4 1.193 0.123 9.729 0.000 1.193 0.535
item5 0.982 0.113 8.678 0.000 0.982 0.466
item6 1.159 0.116 10.010 0.000 1.159 0.549
item7 0.784 0.131 5.994 0.000 0.784 0.343
item8 1.043 0.121 8.610 0.000 1.043 0.480
item9 0.647 0.102 6.359 0.000 0.647 0.362
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 ~~
.item2 0.920 0.205 4.499 0.000 0.920 0.479
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 4.171 0.082 50.608 0.000 4.171 2.613
.item2 3.685 0.104 35.414 0.000 3.685 1.829
.item3 1.739 0.108 16.098 0.000 1.739 0.831
.item4 3.357 0.115 29.160 0.000 3.357 1.506
.item5 2.235 0.109 20.560 0.000 2.235 1.062
.item6 3.661 0.109 33.598 0.000 3.661 1.735
.item7 3.421 0.118 29.014 0.000 3.421 1.498
.item8 3.517 0.112 31.372 0.000 3.517 1.620
.item9 1.259 0.092 13.649 0.000 1.259 0.705
depress 0.000 0.000 0.000
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 1.499 0.216 6.932 0.000 1.499 0.588
.item2 2.459 0.274 8.989 0.000 2.459 0.606
.item3 3.727 0.205 18.167 0.000 3.727 0.852
.item4 3.547 0.291 12.189 0.000 3.547 0.713
.item5 3.467 0.236 14.716 0.000 3.467 0.783
.item6 3.111 0.296 10.520 0.000 3.111 0.698
.item7 4.599 0.279 16.457 0.000 4.599 0.882
.item8 3.626 0.296 12.267 0.000 3.626 0.769
.item9 2.770 0.208 13.291 0.000 2.770 0.869
depress 1.000 1.000 1.000
R-Square:
Estimate
item1 0.412
item2 0.394
item3 0.148
item4 0.287
item5 0.217
item6 0.302
item7 0.118
item8 0.231
item9 0.131summary(fit.metric, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan 0.6.17 ended normally after 48 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 57
Number of equality constraints 9
Number of observations per group:
Female 375
Male 375
Number of missing patterns per group:
Female 1
Male 1
Model Test User Model:
Standard Scaled
Test Statistic 102.839 99.532
Degrees of freedom 60 60
P-value (Chi-square) 0.000 0.001
Scaling correction factor 1.033
Yuan-Bentler correction (Mplus variant)
Test statistic for each group:
Female 54.745 52.985
Male 48.094 46.547
Model Test Baseline Model:
Test statistic 1343.575 1218.364
Degrees of freedom 72 72
P-value 0.000 0.000
Scaling correction factor 1.103
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.966 0.966
Tucker-Lewis Index (TLI) 0.960 0.959
Robust Comparative Fit Index (CFI) 0.968
Robust Tucker-Lewis Index (TLI) 0.961
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -13708.862 -13708.862
Scaling correction factor 0.834
for the MLR correction
Loglikelihood unrestricted model (H1) -13657.442 -13657.442
Scaling correction factor 1.014
for the MLR correction
Akaike (AIC) 27513.724 27513.724
Bayesian (BIC) 27735.488 27735.488
Sample-size adjusted Bayesian (SABIC) 27583.069 27583.069
Root Mean Square Error of Approximation:
RMSEA 0.044 0.042
90 Percent confidence interval - lower 0.029 0.027
90 Percent confidence interval - upper 0.058 0.056
P-value H_0: RMSEA <= 0.050 0.758 0.818
P-value H_0: RMSEA >= 0.080 0.000 0.000
Robust RMSEA 0.043
90 Percent confidence interval - lower 0.027
90 Percent confidence interval - upper 0.057
P-value H_0: Robust RMSEA <= 0.050 0.785
P-value H_0: Robust RMSEA >= 0.080 0.000
Standardized Root Mean Square Residual:
SRMR 0.042 0.042
Parameter Estimates:
Standard errors Sandwich
Information bread Observed
Observed information based on Hessian
Group 1 [Female]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
depress =~
item1 (.p1.) 1.180 0.082 14.455 0.000 1.180 0.701
item2 (.p2.) 1.386 0.088 15.667 0.000 1.386 0.687
item3 (.p3.) 0.888 0.084 10.542 0.000 0.888 0.426
item4 (.p4.) 1.202 0.091 13.153 0.000 1.202 0.538
item5 (.p5.) 1.035 0.084 12.301 0.000 1.035 0.485
item6 (.p6.) 1.191 0.084 14.198 0.000 1.191 0.591
item7 (.p7.) 0.792 0.092 8.642 0.000 0.792 0.383
item8 (.p8.) 1.186 0.094 12.595 0.000 1.186 0.555
item9 (.p9.) 0.647 0.073 8.813 0.000 0.647 0.359
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 ~~
.item2 0.439 0.158 2.777 0.005 0.439 0.249
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 4.184 0.089 47.258 0.000 4.184 2.484
.item2 3.725 0.104 35.848 0.000 3.725 1.846
.item3 1.952 0.108 18.058 0.000 1.952 0.936
.item4 3.589 0.114 31.458 0.000 3.589 1.608
.item5 2.256 0.110 20.522 0.000 2.256 1.058
.item6 3.955 0.103 38.237 0.000 3.955 1.961
.item7 3.869 0.106 36.382 0.000 3.869 1.869
.item8 3.595 0.111 32.331 0.000 3.595 1.684
.item9 1.205 0.092 13.053 0.000 1.205 0.669
depress 0.000 0.000 0.000
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 1.444 0.189 7.646 0.000 1.444 0.509
.item2 2.151 0.220 9.794 0.000 2.151 0.528
.item3 3.556 0.190 18.738 0.000 3.556 0.818
.item4 3.540 0.261 13.543 0.000 3.540 0.710
.item5 3.479 0.206 16.850 0.000 3.479 0.765
.item6 2.648 0.261 10.140 0.000 2.648 0.651
.item7 3.656 0.271 13.482 0.000 3.656 0.853
.item8 3.153 0.275 11.465 0.000 3.153 0.692
.item9 2.827 0.195 14.492 0.000 2.827 0.871
depress 1.000 1.000 1.000
R-Square:
Estimate
item1 0.491
item2 0.472
item3 0.182
item4 0.290
item5 0.235
item6 0.349
item7 0.147
item8 0.308
item9 0.129
Group 2 [Male]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
depress =~
item1 (.p1.) 1.180 0.082 14.455 0.000 1.097 0.675
item2 (.p2.) 1.386 0.088 15.667 0.000 1.288 0.638
item3 (.p3.) 0.888 0.084 10.542 0.000 0.825 0.393
item4 (.p4.) 1.202 0.091 13.153 0.000 1.117 0.506
item5 (.p5.) 1.035 0.084 12.301 0.000 0.961 0.458
item6 (.p6.) 1.191 0.084 14.198 0.000 1.107 0.529
item7 (.p7.) 0.792 0.092 8.642 0.000 0.736 0.324
item8 (.p8.) 1.186 0.094 12.595 0.000 1.102 0.503
item9 (.p9.) 0.647 0.073 8.813 0.000 0.601 0.339
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 ~~
.item2 0.862 0.187 4.610 0.000 0.862 0.463
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 4.171 0.082 50.608 0.000 4.171 2.568
.item2 3.685 0.104 35.414 0.000 3.685 1.827
.item3 1.739 0.108 16.098 0.000 1.739 0.828
.item4 3.357 0.115 29.160 0.000 3.357 1.522
.item5 2.235 0.109 20.560 0.000 2.235 1.064
.item6 3.661 0.109 33.598 0.000 3.661 1.748
.item7 3.421 0.118 29.014 0.000 3.421 1.506
.item8 3.517 0.112 31.372 0.000 3.517 1.605
.item9 1.259 0.092 13.649 0.000 1.259 0.710
depress 0.000 0.000 0.000
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 1.436 0.203 7.060 0.000 1.436 0.544
.item2 2.412 0.245 9.854 0.000 2.412 0.593
.item3 3.731 0.196 19.064 0.000 3.731 0.846
.item4 3.617 0.258 14.027 0.000 3.617 0.744
.item5 3.488 0.216 16.176 0.000 3.488 0.790
.item6 3.161 0.270 11.688 0.000 3.161 0.721
.item7 4.619 0.260 17.798 0.000 4.619 0.895
.item8 3.587 0.276 12.998 0.000 3.587 0.747
.item9 2.781 0.208 13.395 0.000 2.781 0.885
depress 0.863 0.112 7.728 0.000 1.000 1.000
R-Square:
Estimate
item1 0.456
item2 0.407
item3 0.154
item4 0.256
item5 0.210
item6 0.279
item7 0.105
item8 0.253
item9 0.115summary(fit.scalar, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan 0.6.17 ended normally after 52 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 58
Number of equality constraints 18
Number of observations per group:
Female 375
Male 375
Number of missing patterns per group:
Female 1
Male 1
Model Test User Model:
Standard Scaled
Test Statistic 115.309 111.951
Degrees of freedom 68 68
P-value (Chi-square) 0.000 0.001
Scaling correction factor 1.030
Yuan-Bentler correction (Mplus variant)
Test statistic for each group:
Female 60.715 58.946
Male 54.594 53.004
Model Test Baseline Model:
Test statistic 1343.575 1218.364
Degrees of freedom 72 72
P-value 0.000 0.000
Scaling correction factor 1.103
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.963 0.962
Tucker-Lewis Index (TLI) 0.961 0.959
Robust Comparative Fit Index (CFI) 0.964
Robust Tucker-Lewis Index (TLI) 0.962
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -13715.097 -13715.097
Scaling correction factor 0.681
for the MLR correction
Loglikelihood unrestricted model (H1) -13657.442 -13657.442
Scaling correction factor 1.014
for the MLR correction
Akaike (AIC) 27510.194 27510.194
Bayesian (BIC) 27694.997 27694.997
Sample-size adjusted Bayesian (SABIC) 27567.981 27567.981
Root Mean Square Error of Approximation:
RMSEA 0.043 0.042
90 Percent confidence interval - lower 0.029 0.027
90 Percent confidence interval - upper 0.056 0.055
P-value H_0: RMSEA <= 0.050 0.794 0.846
P-value H_0: RMSEA >= 0.080 0.000 0.000
Robust RMSEA 0.042
90 Percent confidence interval - lower 0.028
90 Percent confidence interval - upper 0.056
P-value H_0: Robust RMSEA <= 0.050 0.817
P-value H_0: Robust RMSEA >= 0.080 0.000
Standardized Root Mean Square Residual:
SRMR 0.046 0.046
Parameter Estimates:
Standard errors Sandwich
Information bread Observed
Observed information based on Hessian
Group 1 [Female]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
depress =~
item1 (.p1.) 1.171 0.081 14.385 0.000 1.171 0.696
item2 (.p2.) 1.377 0.089 15.534 0.000 1.377 0.683
item3 (.p3.) 0.894 0.084 10.621 0.000 0.894 0.429
item4 (.p4.) 1.209 0.091 13.343 0.000 1.209 0.541
item5 (.p5.) 1.033 0.084 12.275 0.000 1.033 0.485
item6 (.p6.) 1.199 0.083 14.424 0.000 1.199 0.593
item7 (.p7.) 0.803 0.091 8.853 0.000 0.803 0.386
item8 (.p8.) 1.184 0.094 12.534 0.000 1.184 0.555
item9 (.p9.) 0.640 0.074 8.604 0.000 0.640 0.356
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 ~~
.item2 0.454 0.159 2.852 0.004 0.454 0.255
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.10.) 4.240 0.077 54.984 0.000 4.240 2.520
.item2 (.11.) 3.773 0.092 41.111 0.000 3.773 1.872
.item3 (.12.) 1.897 0.087 21.735 0.000 1.897 0.909
.item4 (.13.) 3.541 0.096 37.066 0.000 3.541 1.584
.item5 (.14.) 2.303 0.090 25.622 0.000 2.303 1.080
.item6 (.15.) 3.882 0.091 42.556 0.000 3.882 1.921
.item7 (.16.) 3.711 0.087 42.428 0.000 3.711 1.784
.item8 (.17.) 3.620 0.094 38.567 0.000 3.620 1.696
.item9 (.18.) 1.268 0.072 17.592 0.000 1.268 0.704
depress 0.000 0.000 0.000
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 1.460 0.193 7.576 0.000 1.460 0.516
.item2 2.166 0.223 9.726 0.000 2.166 0.533
.item3 3.555 0.191 18.619 0.000 3.555 0.816
.item4 3.535 0.261 13.520 0.000 3.535 0.708
.item5 3.478 0.206 16.880 0.000 3.478 0.765
.item6 2.648 0.260 10.183 0.000 2.648 0.648
.item7 3.682 0.267 13.767 0.000 3.682 0.851
.item8 3.155 0.277 11.377 0.000 3.155 0.692
.item9 2.834 0.192 14.790 0.000 2.834 0.874
depress 1.000 1.000 1.000
R-Square:
Estimate
item1 0.484
item2 0.467
item3 0.184
item4 0.292
item5 0.235
item6 0.352
item7 0.149
item8 0.308
item9 0.126
Group 2 [Male]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
depress =~
item1 (.p1.) 1.171 0.081 14.385 0.000 1.089 0.671
item2 (.p2.) 1.377 0.089 15.534 0.000 1.280 0.635
item3 (.p3.) 0.894 0.084 10.621 0.000 0.831 0.395
item4 (.p4.) 1.209 0.091 13.343 0.000 1.123 0.509
item5 (.p5.) 1.033 0.084 12.275 0.000 0.960 0.457
item6 (.p6.) 1.199 0.083 14.424 0.000 1.114 0.531
item7 (.p7.) 0.803 0.091 8.853 0.000 0.746 0.327
item8 (.p8.) 1.184 0.094 12.534 0.000 1.100 0.502
item9 (.p9.) 0.640 0.074 8.604 0.000 0.595 0.336
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 ~~
.item2 0.879 0.185 4.754 0.000 0.879 0.468
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.10.) 4.240 0.077 54.984 0.000 4.240 2.611
.item2 (.11.) 3.773 0.092 41.111 0.000 3.773 1.870
.item3 (.12.) 1.897 0.087 21.735 0.000 1.897 0.902
.item4 (.13.) 3.541 0.096 37.066 0.000 3.541 1.604
.item5 (.14.) 2.303 0.090 25.622 0.000 2.303 1.097
.item6 (.15.) 3.882 0.091 42.556 0.000 3.882 1.850
.item7 (.16.) 3.711 0.087 42.428 0.000 3.711 1.625
.item8 (.17.) 3.620 0.094 38.567 0.000 3.620 1.653
.item9 (.18.) 1.268 0.072 17.592 0.000 1.268 0.715
depress -0.112 0.083 -1.345 0.179 -0.120 -0.120
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 1.451 0.200 7.258 0.000 1.451 0.550
.item2 2.431 0.240 10.124 0.000 2.431 0.597
.item3 3.730 0.196 19.059 0.000 3.730 0.844
.item4 3.611 0.258 13.975 0.000 3.611 0.741
.item5 3.489 0.216 16.166 0.000 3.489 0.791
.item6 3.161 0.276 11.468 0.000 3.161 0.718
.item7 4.658 0.277 16.831 0.000 4.658 0.893
.item8 3.588 0.274 13.119 0.000 3.588 0.748
.item9 2.788 0.213 13.105 0.000 2.788 0.887
depress 0.864 0.112 7.720 0.000 1.000 1.000
R-Square:
Estimate
item1 0.450
item2 0.403
item3 0.156
item4 0.259
item5 0.209
item6 0.282
item7 0.107
item8 0.252
item9 0.113summary(fit.scalar2, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan 0.6.17 ended normally after 53 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 58
Number of equality constraints 17
Number of observations per group:
Female 375
Male 375
Number of missing patterns per group:
Female 1
Male 1
Model Test User Model:
Standard Scaled
Test Statistic 109.216 106.031
Degrees of freedom 67 67
P-value (Chi-square) 0.001 0.002
Scaling correction factor 1.030
Yuan-Bentler correction (Mplus variant)
Test statistic for each group:
Female 57.897 56.209
Male 51.318 49.822
Model Test Baseline Model:
Test statistic 1343.575 1218.364
Degrees of freedom 72 72
P-value 0.000 0.000
Scaling correction factor 1.103
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.967 0.966
Tucker-Lewis Index (TLI) 0.964 0.963
Robust Comparative Fit Index (CFI) 0.968
Robust Tucker-Lewis Index (TLI) 0.966
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -13712.050 -13712.050
Scaling correction factor 0.699
for the MLR correction
Loglikelihood unrestricted model (H1) -13657.442 -13657.442
Scaling correction factor 1.014
for the MLR correction
Akaike (AIC) 27506.100 27506.100
Bayesian (BIC) 27695.523 27695.523
Sample-size adjusted Bayesian (SABIC) 27565.332 27565.332
Root Mean Square Error of Approximation:
RMSEA 0.041 0.039
90 Percent confidence interval - lower 0.026 0.025
90 Percent confidence interval - upper 0.055 0.053
P-value H_0: RMSEA <= 0.050 0.855 0.896
P-value H_0: RMSEA >= 0.080 0.000 0.000
Robust RMSEA 0.040
90 Percent confidence interval - lower 0.025
90 Percent confidence interval - upper 0.054
P-value H_0: Robust RMSEA <= 0.050 0.873
P-value H_0: Robust RMSEA >= 0.080 0.000
Standardized Root Mean Square Residual:
SRMR 0.044 0.044
Parameter Estimates:
Standard errors Sandwich
Information bread Observed
Observed information based on Hessian
Group 1 [Female]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
depress =~
item1 (.p1.) 1.174 0.082 14.377 0.000 1.174 0.698
item2 (.p2.) 1.381 0.089 15.564 0.000 1.381 0.685
item3 (.p3.) 0.894 0.084 10.598 0.000 0.894 0.428
item4 (.p4.) 1.208 0.091 13.309 0.000 1.208 0.540
item5 (.p5.) 1.034 0.084 12.287 0.000 1.034 0.485
item6 (.p6.) 1.198 0.083 14.364 0.000 1.198 0.592
item7 (.p7.) 0.791 0.092 8.603 0.000 0.791 0.382
item8 (.p8.) 1.185 0.094 12.561 0.000 1.185 0.555
item9 (.p9.) 0.642 0.074 8.630 0.000 0.642 0.356
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 ~~
.item2 0.449 0.159 2.825 0.005 0.449 0.253
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.10.) 4.228 0.078 54.510 0.000 4.228 2.512
.item2 (.11.) 3.761 0.092 40.840 0.000 3.761 1.865
.item3 (.12.) 1.887 0.087 21.651 0.000 1.887 0.904
.item4 (.13.) 3.529 0.096 36.780 0.000 3.529 1.578
.item5 (.14.) 2.292 0.090 25.462 0.000 2.292 1.075
.item6 (.15.) 3.870 0.092 42.207 0.000 3.870 1.915
.item7 3.869 0.106 36.382 0.000 3.869 1.869
.item8 (.17.) 3.609 0.094 38.382 0.000 3.609 1.690
.item9 (.18.) 1.261 0.072 17.570 0.000 1.261 0.700
depress 0.000 0.000 0.000
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 1.455 0.191 7.595 0.000 1.455 0.513
.item2 2.160 0.222 9.738 0.000 2.160 0.531
.item3 3.557 0.191 18.613 0.000 3.557 0.817
.item4 3.539 0.261 13.545 0.000 3.539 0.708
.item5 3.478 0.206 16.874 0.000 3.478 0.765
.item6 2.651 0.260 10.205 0.000 2.651 0.649
.item7 3.658 0.271 13.485 0.000 3.658 0.854
.item8 3.154 0.277 11.404 0.000 3.154 0.692
.item9 2.832 0.192 14.743 0.000 2.832 0.873
depress 1.000 1.000 1.000
R-Square:
Estimate
item1 0.487
item2 0.469
item3 0.183
item4 0.292
item5 0.235
item6 0.351
item7 0.146
item8 0.308
item9 0.127
Group 2 [Male]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
depress =~
item1 (.p1.) 1.174 0.082 14.377 0.000 1.091 0.672
item2 (.p2.) 1.381 0.089 15.564 0.000 1.283 0.636
item3 (.p3.) 0.894 0.084 10.598 0.000 0.830 0.395
item4 (.p4.) 1.208 0.091 13.309 0.000 1.122 0.508
item5 (.p5.) 1.034 0.084 12.287 0.000 0.961 0.457
item6 (.p6.) 1.198 0.083 14.364 0.000 1.113 0.530
item7 (.p7.) 0.791 0.092 8.603 0.000 0.735 0.324
item8 (.p8.) 1.185 0.094 12.561 0.000 1.101 0.503
item9 (.p9.) 0.642 0.074 8.630 0.000 0.597 0.337
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 ~~
.item2 0.872 0.186 4.696 0.000 0.872 0.466
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.10.) 4.228 0.078 54.510 0.000 4.228 2.604
.item2 (.11.) 3.761 0.092 40.840 0.000 3.761 1.864
.item3 (.12.) 1.887 0.087 21.651 0.000 1.887 0.897
.item4 (.13.) 3.529 0.096 36.780 0.000 3.529 1.598
.item5 (.14.) 2.292 0.090 25.462 0.000 2.292 1.091
.item6 (.15.) 3.870 0.092 42.207 0.000 3.870 1.844
.item7 3.493 0.123 28.376 0.000 3.493 1.538
.item8 (.17.) 3.609 0.094 38.382 0.000 3.609 1.647
.item9 (.18.) 1.261 0.072 17.570 0.000 1.261 0.712
depress -0.090 0.083 -1.087 0.277 -0.097 -0.097
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 1.445 0.201 7.186 0.000 1.445 0.548
.item2 2.423 0.242 10.026 0.000 2.423 0.595
.item3 3.733 0.196 19.086 0.000 3.733 0.844
.item4 3.615 0.260 13.913 0.000 3.615 0.742
.item5 3.488 0.216 16.160 0.000 3.488 0.791
.item6 3.166 0.277 11.417 0.000 3.166 0.719
.item7 4.620 0.259 17.804 0.000 4.620 0.895
.item8 3.587 0.274 13.071 0.000 3.587 0.747
.item9 2.787 0.212 13.148 0.000 2.787 0.887
depress 0.864 0.112 7.725 0.000 1.000 1.000
R-Square:
Estimate
item1 0.452
item2 0.405
item3 0.156
item4 0.258
item5 0.209
item6 0.281
item7 0.105
item8 0.253
item9 0.113summary(fit.strict, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan 0.6.17 ended normally after 54 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 58
Number of equality constraints 25
Number of observations per group:
Female 375
Male 375
Number of missing patterns per group:
Female 1
Male 1
Model Test User Model:
Standard Scaled
Test Statistic 114.059 112.019
Degrees of freedom 75 75
P-value (Chi-square) 0.002 0.004
Scaling correction factor 1.018
Yuan-Bentler correction (Mplus variant)
Test statistic for each group:
Female 60.752 59.666
Male 53.306 52.353
Model Test Baseline Model:
Test statistic 1343.575 1218.364
Degrees of freedom 72 72
P-value 0.000 0.000
Scaling correction factor 1.103
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.969 0.968
Tucker-Lewis Index (TLI) 0.971 0.969
Robust Comparative Fit Index (CFI) 0.970
Robust Tucker-Lewis Index (TLI) 0.971
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -13714.472 -13714.472
Scaling correction factor 0.572
for the MLR correction
Loglikelihood unrestricted model (H1) -13657.442 -13657.442
Scaling correction factor 1.014
for the MLR correction
Akaike (AIC) 27494.944 27494.944
Bayesian (BIC) 27647.406 27647.406
Sample-size adjusted Bayesian (SABIC) 27542.618 27542.618
Root Mean Square Error of Approximation:
RMSEA 0.037 0.036
90 Percent confidence interval - lower 0.022 0.021
90 Percent confidence interval - upper 0.051 0.050
P-value H_0: RMSEA <= 0.050 0.942 0.956
P-value H_0: RMSEA >= 0.080 0.000 0.000
Robust RMSEA 0.037
90 Percent confidence interval - lower 0.021
90 Percent confidence interval - upper 0.050
P-value H_0: Robust RMSEA <= 0.050 0.948
P-value H_0: Robust RMSEA >= 0.080 0.000
Standardized Root Mean Square Residual:
SRMR 0.048 0.048
Parameter Estimates:
Standard errors Sandwich
Information bread Observed
Observed information based on Hessian
Group 1 [Female]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
depress =~
item1 (.p1.) 1.167 0.082 14.180 0.000 1.167 0.696
item2 (.p2.) 1.372 0.089 15.358 0.000 1.372 0.671
item3 (.p3.) 0.888 0.083 10.655 0.000 0.888 0.422
item4 (.p4.) 1.203 0.090 13.341 0.000 1.203 0.537
item5 (.p5.) 1.031 0.084 12.316 0.000 1.031 0.484
item6 (.p6.) 1.197 0.083 14.492 0.000 1.197 0.575
item7 (.p7.) 0.787 0.092 8.593 0.000 0.787 0.381
item8 (.p8.) 1.178 0.093 12.608 0.000 1.178 0.540
item9 (.p9.) 0.639 0.074 8.602 0.000 0.639 0.356
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 ~~
.item2 0.484 0.160 3.030 0.002 0.484 0.266
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.10.) 4.229 0.078 53.943 0.000 4.229 2.522
.item2 (.11.) 3.763 0.093 40.533 0.000 3.763 1.840
.item3 (.12.) 1.886 0.087 21.609 0.000 1.886 0.895
.item4 (.13.) 3.528 0.096 36.880 0.000 3.528 1.574
.item5 (.14.) 2.292 0.090 25.455 0.000 2.292 1.076
.item6 (.15.) 3.862 0.091 42.539 0.000 3.862 1.855
.item7 3.869 0.106 36.382 0.000 3.869 1.872
.item8 (.17.) 3.609 0.094 38.326 0.000 3.609 1.655
.item9 (.18.) 1.261 0.071 17.668 0.000 1.261 0.703
depress 0.000 0.000 0.000
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.19.) 1.447 0.145 9.954 0.000 1.447 0.515
.item2 (.20.) 2.300 0.177 12.965 0.000 2.300 0.550
.item3 (.21.) 3.646 0.143 25.449 0.000 3.646 0.822
.item4 (.22.) 3.574 0.197 18.123 0.000 3.574 0.712
.item5 (.23.) 3.479 0.161 21.647 0.000 3.479 0.766
.item6 (.24.) 2.903 0.199 14.558 0.000 2.903 0.670
.item7 3.653 0.271 13.462 0.000 3.653 0.855
.item8 (.26.) 3.367 0.207 16.293 0.000 3.367 0.708
.item9 (.27.) 2.809 0.143 19.650 0.000 2.809 0.873
depress 1.000 1.000 1.000
R-Square:
Estimate
item1 0.485
item2 0.450
item3 0.178
item4 0.288
item5 0.234
item6 0.330
item7 0.145
item8 0.292
item9 0.127
Group 2 [Male]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
depress =~
item1 (.p1.) 1.167 0.082 14.180 0.000 1.097 0.674
item2 (.p2.) 1.372 0.089 15.358 0.000 1.289 0.648
item3 (.p3.) 0.888 0.083 10.655 0.000 0.834 0.400
item4 (.p4.) 1.203 0.090 13.341 0.000 1.130 0.513
item5 (.p5.) 1.031 0.084 12.316 0.000 0.968 0.461
item6 (.p6.) 1.197 0.083 14.492 0.000 1.124 0.551
item7 (.p7.) 0.787 0.092 8.593 0.000 0.739 0.325
item8 (.p8.) 1.178 0.093 12.608 0.000 1.107 0.516
item9 (.p9.) 0.639 0.074 8.602 0.000 0.600 0.337
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 ~~
.item2 0.832 0.151 5.497 0.000 0.832 0.456
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.10.) 4.229 0.078 53.943 0.000 4.229 2.598
.item2 (.11.) 3.763 0.093 40.533 0.000 3.763 1.890
.item3 (.12.) 1.886 0.087 21.609 0.000 1.886 0.905
.item4 (.13.) 3.528 0.096 36.880 0.000 3.528 1.602
.item5 (.14.) 2.292 0.090 25.455 0.000 2.292 1.091
.item6 (.15.) 3.862 0.091 42.539 0.000 3.862 1.892
.item7 3.493 0.123 28.381 0.000 3.493 1.535
.item8 (.17.) 3.609 0.094 38.326 0.000 3.609 1.684
.item9 (.18.) 1.261 0.071 17.668 0.000 1.261 0.708
depress -0.091 0.084 -1.084 0.279 -0.097 -0.097
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.19.) 1.447 0.145 9.954 0.000 1.447 0.546
.item2 (.20.) 2.300 0.177 12.965 0.000 2.300 0.581
.item3 (.21.) 3.646 0.143 25.449 0.000 3.646 0.840
.item4 (.22.) 3.574 0.197 18.123 0.000 3.574 0.737
.item5 (.23.) 3.479 0.161 21.647 0.000 3.479 0.788
.item6 (.24.) 2.903 0.199 14.558 0.000 2.903 0.697
.item7 4.629 0.260 17.815 0.000 4.629 0.894
.item8 (.26.) 3.367 0.207 16.293 0.000 3.367 0.733
.item9 (.27.) 2.809 0.143 19.650 0.000 2.809 0.886
depress 0.883 0.111 7.936 0.000 1.000 1.000
R-Square:
Estimate
item1 0.454
item2 0.419
item3 0.160
item4 0.263
item5 0.212
item6 0.303
item7 0.106
item8 0.267
item9 0.114summary(fit.strict.cov, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)lavaan 0.6.17 ended normally after 55 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 58
Number of equality constraints 26
Number of observations per group:
Female 375
Male 375
Number of missing patterns per group:
Female 1
Male 1
Model Test User Model:
Standard Scaled
Test Statistic 123.351 119.281
Degrees of freedom 76 76
P-value (Chi-square) 0.000 0.001
Scaling correction factor 1.034
Yuan-Bentler correction (Mplus variant)
Test statistic for each group:
Female 65.102 62.954
Male 58.248 56.327
Model Test Baseline Model:
Test statistic 1343.575 1218.364
Degrees of freedom 72 72
P-value 0.000 0.000
Scaling correction factor 1.103
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.963 0.962
Tucker-Lewis Index (TLI) 0.965 0.964
Robust Comparative Fit Index (CFI) 0.965
Robust Tucker-Lewis Index (TLI) 0.967
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -13719.118 -13719.118
Scaling correction factor 0.534
for the MLR correction
Loglikelihood unrestricted model (H1) -13657.442 -13657.442
Scaling correction factor 1.014
for the MLR correction
Akaike (AIC) 27502.235 27502.235
Bayesian (BIC) 27650.078 27650.078
Sample-size adjusted Bayesian (SABIC) 27548.465 27548.465
Root Mean Square Error of Approximation:
RMSEA 0.041 0.039
90 Percent confidence interval - lower 0.027 0.025
90 Percent confidence interval - upper 0.054 0.052
P-value H_0: RMSEA <= 0.050 0.877 0.920
P-value H_0: RMSEA >= 0.080 0.000 0.000
Robust RMSEA 0.040
90 Percent confidence interval - lower 0.025
90 Percent confidence interval - upper 0.053
P-value H_0: Robust RMSEA <= 0.050 0.897
P-value H_0: Robust RMSEA >= 0.080 0.000
Standardized Root Mean Square Residual:
SRMR 0.048 0.048
Parameter Estimates:
Standard errors Sandwich
Information bread Observed
Observed information based on Hessian
Group 1 [Female]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
depress =~
item1 (.p1.) 1.164 0.082 14.182 0.000 1.164 0.695
item2 (.p2.) 1.355 0.088 15.395 0.000 1.355 0.666
item3 (.p3.) 0.883 0.084 10.551 0.000 0.883 0.420
item4 (.p4.) 1.200 0.091 13.260 0.000 1.200 0.536
item5 (.p5.) 1.031 0.084 12.293 0.000 1.031 0.484
item6 (.p6.) 1.191 0.083 14.394 0.000 1.191 0.573
item7 (.p7.) 0.782 0.091 8.554 0.000 0.782 0.378
item8 (.p8.) 1.173 0.094 12.528 0.000 1.173 0.539
item9 (.p9.) 0.637 0.074 8.581 0.000 0.637 0.355
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 ~~
.item2 (.28.) 0.671 0.132 5.072 0.000 0.671 0.366
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.10.) 4.231 0.079 53.848 0.000 4.231 2.524
.item2 (.11.) 3.767 0.092 40.777 0.000 3.767 1.850
.item3 (.12.) 1.886 0.087 21.608 0.000 1.886 0.896
.item4 (.13.) 3.528 0.096 36.854 0.000 3.528 1.576
.item5 (.14.) 2.292 0.090 25.422 0.000 2.292 1.077
.item6 (.15.) 3.862 0.091 42.534 0.000 3.862 1.857
.item7 3.869 0.106 36.382 0.000 3.869 1.873
.item8 (.17.) 3.610 0.094 38.318 0.000 3.610 1.657
.item9 (.18.) 1.261 0.071 17.661 0.000 1.261 0.703
depress 0.000 0.000 0.000
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.19.) 1.455 0.147 9.904 0.000 1.455 0.518
.item2 (.20.) 2.309 0.179 12.908 0.000 2.309 0.557
.item3 (.21.) 3.648 0.143 25.433 0.000 3.648 0.824
.item4 (.22.) 3.570 0.197 18.084 0.000 3.570 0.712
.item5 (.23.) 3.470 0.161 21.569 0.000 3.470 0.765
.item6 (.24.) 2.906 0.199 14.565 0.000 2.906 0.672
.item7 3.658 0.272 13.465 0.000 3.658 0.857
.item8 (.26.) 3.368 0.207 16.303 0.000 3.368 0.710
.item9 (.27.) 2.808 0.143 19.650 0.000 2.808 0.874
depress 1.000 1.000 1.000
R-Square:
Estimate
item1 0.482
item2 0.443
item3 0.176
item4 0.288
item5 0.235
item6 0.328
item7 0.143
item8 0.290
item9 0.126
Group 2 [Male]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
depress =~
item1 (.p1.) 1.164 0.082 14.182 0.000 1.103 0.675
item2 (.p2.) 1.355 0.088 15.395 0.000 1.284 0.645
item3 (.p3.) 0.883 0.084 10.551 0.000 0.836 0.401
item4 (.p4.) 1.200 0.091 13.260 0.000 1.137 0.516
item5 (.p5.) 1.031 0.084 12.293 0.000 0.977 0.464
item6 (.p6.) 1.191 0.083 14.394 0.000 1.128 0.552
item7 (.p7.) 0.782 0.091 8.554 0.000 0.741 0.325
item8 (.p8.) 1.173 0.094 12.528 0.000 1.111 0.518
item9 (.p9.) 0.637 0.074 8.581 0.000 0.603 0.339
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 ~~
.item2 (.28.) 0.671 0.132 5.072 0.000 0.671 0.366
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.10.) 4.231 0.079 53.848 0.000 4.231 2.589
.item2 (.11.) 3.767 0.092 40.777 0.000 3.767 1.894
.item3 (.12.) 1.886 0.087 21.608 0.000 1.886 0.904
.item4 (.13.) 3.528 0.096 36.854 0.000 3.528 1.600
.item5 (.14.) 2.292 0.090 25.422 0.000 2.292 1.090
.item6 (.15.) 3.862 0.091 42.534 0.000 3.862 1.890
.item7 3.493 0.123 28.383 0.000 3.493 1.535
.item8 (.17.) 3.610 0.094 38.318 0.000 3.610 1.683
.item9 (.18.) 1.261 0.071 17.661 0.000 1.261 0.708
depress -0.091 0.084 -1.086 0.278 -0.097 -0.097
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.19.) 1.455 0.147 9.904 0.000 1.455 0.545
.item2 (.20.) 2.309 0.179 12.908 0.000 2.309 0.584
.item3 (.21.) 3.648 0.143 25.433 0.000 3.648 0.839
.item4 (.22.) 3.570 0.197 18.084 0.000 3.570 0.734
.item5 (.23.) 3.470 0.161 21.569 0.000 3.470 0.784
.item6 (.24.) 2.906 0.199 14.565 0.000 2.906 0.696
.item7 4.630 0.260 17.825 0.000 4.630 0.894
.item8 (.26.) 3.368 0.207 16.303 0.000 3.368 0.732
.item9 (.27.) 2.808 0.143 19.650 0.000 2.808 0.885
depress 0.897 0.113 7.932 0.000 1.000 1.000
R-Square:
Estimate
item1 0.455
item2 0.416
item3 0.161
item4 0.266
item5 0.216
item6 0.304
item7 0.106
item8 0.268
item9 0.115model_fit <- function(lavobject) {
vars <- c("cfi", "tli", "rmsea", "rmsea.ci.lower", "rmsea.ci.upper", "rmsea.pvalue", "srmr")
return(fitmeasures(lavobject)[vars] %>% data.frame() %>% round(2) %>% t())
}
table_fit <-
list(model_fit(fit.config), model_fit(fit.metric),
model_fit(fit.scalar), model_fit(fit.scalar2),
model_fit(fit.strict), model_fit(fit.strict.cov)) %>%
reduce(rbind)
rownames(table_fit) <- c("Configural", "Metric", "Scalar", "Scalar2","Strict","Strict+Cov")
table_lik.test <-
list(anova(fit.config, fit.metric),
anova(fit.metric, fit.scalar),
anova(fit.scalar, fit.scalar2),
anova(fit.scalar2, fit.strict),
anova(fit.strict, fit.strict.cov)
) %>%
reduce(rbind) %>%
.[-c(3,5,7,9),]
rownames(table_lik.test) <- c("Configural", "Metric", "Scalar", "Scalar2","Strict","Strict+Cov")
kable(table_fit, caption = "Model Fit Indices Table")| cfi | tli | rmsea | rmsea.ci.lower | rmsea.ci.upper | rmsea.pvalue | srmr | |
|---|---|---|---|---|---|---|---|
| Configural | 0.96 | 0.95 | 0.05 | 0.03 | 0.06 | 0.52 | 0.04 |
| Metric | 0.97 | 0.96 | 0.04 | 0.03 | 0.06 | 0.76 | 0.04 |
| Scalar | 0.96 | 0.96 | 0.04 | 0.03 | 0.06 | 0.79 | 0.05 |
| Scalar2 | 0.97 | 0.96 | 0.04 | 0.03 | 0.05 | 0.85 | 0.04 |
| Strict | 0.97 | 0.97 | 0.04 | 0.02 | 0.05 | 0.94 | 0.05 |
| Strict+Cov | 0.96 | 0.96 | 0.04 | 0.03 | 0.05 | 0.88 | 0.05 |
kable(table_lik.test, caption = "Model Comparision Table")| Df | AIC | BIC | Chisq | Chisq diff | Df diff | Pr(>Chisq) | |
|---|---|---|---|---|---|---|---|
| Configural | 52 | 27525.80 | 27784.52 | 98.91085 | NA | NA | NA |
| Metric | 60 | 27513.72 | 27735.49 | 102.83941 | 4.259305 | 8 | 0.8330029 |
| Scalar | 68 | 27510.19 | 27695.00 | 115.30933 | 12.398256 | 8 | 0.1342996 |
| Scalar2 | 68 | 27510.19 | 27695.00 | 115.30933 | 5.929566 | 1 | 0.0148889 |
| Strict | 75 | 27494.94 | 27647.41 | 114.05887 | 5.269043 | 8 | 0.7284714 |
| Strict+Cov | 76 | 27502.24 | 27650.08 | 123.35057 | 4.172431 | 1 | 0.0410868 |
lavaan 0.6.17 ended normally after 54 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 57
Number of equality constraints 25
Number of observations per group:
Female 375
Male 375
Number of missing patterns per group:
Female 1
Male 1
Model Test User Model:
Standard Scaled
Test Statistic 114.904 113.113
Degrees of freedom 76 76
P-value (Chi-square) 0.003 0.004
Scaling correction factor 1.016
Yuan-Bentler correction (Mplus variant)
Test statistic for each group:
Female 61.213 60.259
Male 53.691 52.854
Model Test Baseline Model:
Test statistic 1343.575 1218.364
Degrees of freedom 72 72
P-value 0.000 0.000
Scaling correction factor 1.103
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.969 0.968
Tucker-Lewis Index (TLI) 0.971 0.969
Robust Comparative Fit Index (CFI) 0.970
Robust Tucker-Lewis Index (TLI) 0.972
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -13714.894 -13714.894
Scaling correction factor 0.567
for the MLR correction
Loglikelihood unrestricted model (H1) -13657.442 -13657.442
Scaling correction factor 1.014
for the MLR correction
Akaike (AIC) 27493.789 27493.789
Bayesian (BIC) 27641.631 27641.631
Sample-size adjusted Bayesian (SABIC) 27540.019 27540.019
Root Mean Square Error of Approximation:
RMSEA 0.037 0.036
90 Percent confidence interval - lower 0.022 0.021
90 Percent confidence interval - upper 0.050 0.049
P-value H_0: RMSEA <= 0.050 0.947 0.958
P-value H_0: RMSEA >= 0.080 0.000 0.000
Robust RMSEA 0.036
90 Percent confidence interval - lower 0.021
90 Percent confidence interval - upper 0.050
P-value H_0: Robust RMSEA <= 0.050 0.952
P-value H_0: Robust RMSEA >= 0.080 0.000
Standardized Root Mean Square Residual:
SRMR 0.050 0.050
Parameter Estimates:
Standard errors Sandwich
Information bread Observed
Observed information based on Hessian
Group 1 [Female]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
depress =~
item1 (.p1.) 1.132 0.069 16.495 0.000 1.132 0.685
item2 (.p2.) 1.332 0.076 17.634 0.000 1.332 0.660
item3 (.p3.) 0.861 0.076 11.269 0.000 0.861 0.411
item4 (.p4.) 1.169 0.083 14.123 0.000 1.169 0.526
item5 (.p5.) 1.000 0.076 13.226 0.000 1.000 0.473
item6 (.p6.) 1.162 0.077 15.167 0.000 1.162 0.564
item7 (.p7.) 0.765 0.086 8.889 0.000 0.765 0.371
item8 (.p8.) 1.142 0.082 13.922 0.000 1.142 0.528
item9 (.p9.) 0.620 0.069 8.931 0.000 0.620 0.347
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 ~~
.item2 0.490 0.159 3.077 0.002 0.490 0.268
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.10.) 4.229 0.078 53.980 0.000 4.229 2.558
.item2 (.11.) 3.763 0.093 40.555 0.000 3.763 1.864
.item3 (.12.) 1.886 0.087 21.616 0.000 1.886 0.900
.item4 (.13.) 3.528 0.096 36.887 0.000 3.528 1.588
.item5 (.14.) 2.292 0.090 25.463 0.000 2.292 1.083
.item6 (.15.) 3.862 0.091 42.543 0.000 3.862 1.873
.item7 3.869 0.106 36.382 0.000 3.869 1.879
.item8 (.17.) 3.610 0.094 38.348 0.000 3.610 1.670
.item9 (.18.) 1.261 0.071 17.671 0.000 1.261 0.706
depress 0.000 0.000 0.000
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.19.) 1.452 0.145 9.988 0.000 1.452 0.531
.item2 (.20.) 2.301 0.178 12.925 0.000 2.301 0.565
.item3 (.21.) 3.646 0.143 25.467 0.000 3.646 0.831
.item4 (.22.) 3.571 0.197 18.119 0.000 3.571 0.723
.item5 (.23.) 3.478 0.161 21.626 0.000 3.478 0.777
.item6 (.24.) 2.900 0.199 14.536 0.000 2.900 0.682
.item7 3.655 0.271 13.480 0.000 3.655 0.862
.item8 (.26.) 3.368 0.207 16.280 0.000 3.368 0.721
.item9 (.27.) 2.809 0.143 19.649 0.000 2.809 0.880
depress 1.000 1.000 1.000
R-Square:
Estimate
item1 0.469
item2 0.435
item3 0.169
item4 0.277
item5 0.223
item6 0.318
item7 0.138
item8 0.279
item9 0.120
Group 2 [Male]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
depress =~
item1 (.p1.) 1.132 0.069 16.495 0.000 1.132 0.685
item2 (.p2.) 1.332 0.076 17.634 0.000 1.332 0.660
item3 (.p3.) 0.861 0.076 11.269 0.000 0.861 0.411
item4 (.p4.) 1.169 0.083 14.123 0.000 1.169 0.526
item5 (.p5.) 1.000 0.076 13.226 0.000 1.000 0.473
item6 (.p6.) 1.162 0.077 15.167 0.000 1.162 0.564
item7 (.p7.) 0.765 0.086 8.889 0.000 0.765 0.335
item8 (.p8.) 1.142 0.082 13.922 0.000 1.142 0.528
item9 (.p9.) 0.620 0.069 8.931 0.000 0.620 0.347
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 ~~
.item2 0.834 0.152 5.483 0.000 0.834 0.456
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.10.) 4.229 0.078 53.980 0.000 4.229 2.558
.item2 (.11.) 3.763 0.093 40.555 0.000 3.763 1.864
.item3 (.12.) 1.886 0.087 21.616 0.000 1.886 0.900
.item4 (.13.) 3.528 0.096 36.887 0.000 3.528 1.588
.item5 (.14.) 2.292 0.090 25.463 0.000 2.292 1.083
.item6 (.15.) 3.862 0.091 42.543 0.000 3.862 1.873
.item7 3.493 0.123 28.386 0.000 3.493 1.530
.item8 (.17.) 3.610 0.094 38.348 0.000 3.610 1.670
.item9 (.18.) 1.261 0.071 17.671 0.000 1.261 0.706
depress -0.094 0.085 -1.098 0.272 -0.094 -0.094
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.19.) 1.452 0.145 9.988 0.000 1.452 0.531
.item2 (.20.) 2.301 0.178 12.925 0.000 2.301 0.565
.item3 (.21.) 3.646 0.143 25.467 0.000 3.646 0.831
.item4 (.22.) 3.571 0.197 18.119 0.000 3.571 0.723
.item5 (.23.) 3.478 0.161 21.626 0.000 3.478 0.777
.item6 (.24.) 2.900 0.199 14.536 0.000 2.900 0.682
.item7 4.626 0.260 17.771 0.000 4.626 0.888
.item8 (.26.) 3.368 0.207 16.280 0.000 3.368 0.721
.item9 (.27.) 2.809 0.143 19.649 0.000 2.809 0.880
depress 1.000 1.000 1.000
R-Square:
Estimate
item1 0.469
item2 0.435
item3 0.169
item4 0.277
item5 0.223
item6 0.318
item7 0.112
item8 0.279
item9 0.120
Scaled Chi-Squared Difference Test (method = "satorra.bentler.2001")
lavaan NOTE:
The "Chisq" column contains standard test statistics, not the
robust test that should be reported per model. A robust difference
test is a function of two standard (not robust) statistics.
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
fit.strict 75 27495 27647 114.06
fit.structuralVariance 76 27494 27642 114.90 1.0095 1 0.315lavaan 0.6.17 ended normally after 54 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 56
Number of equality constraints 25
Number of observations per group:
Female 375
Male 375
Number of missing patterns per group:
Female 1
Male 1
Model Test User Model:
Standard Scaled
Test Statistic 116.143 114.340
Degrees of freedom 77 77
P-value (Chi-square) 0.003 0.004
Scaling correction factor 1.016
Yuan-Bentler correction (Mplus variant)
Test statistic for each group:
Female 61.790 60.831
Male 54.353 53.509
Model Test Baseline Model:
Test statistic 1343.575 1218.364
Degrees of freedom 72 72
P-value 0.000 0.000
Scaling correction factor 1.103
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.969 0.967
Tucker-Lewis Index (TLI) 0.971 0.970
Robust Comparative Fit Index (CFI) 0.970
Robust Tucker-Lewis Index (TLI) 0.972
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -13715.514 -13715.514
Scaling correction factor 0.559
for the MLR correction
Loglikelihood unrestricted model (H1) -13657.442 -13657.442
Scaling correction factor 1.014
for the MLR correction
Akaike (AIC) 27493.027 27493.027
Bayesian (BIC) 27636.250 27636.250
Sample-size adjusted Bayesian (SABIC) 27537.813 27537.813
Root Mean Square Error of Approximation:
RMSEA 0.037 0.036
90 Percent confidence interval - lower 0.022 0.021
90 Percent confidence interval - upper 0.050 0.049
P-value H_0: RMSEA <= 0.050 0.950 0.961
P-value H_0: RMSEA >= 0.080 0.000 0.000
Robust RMSEA 0.036
90 Percent confidence interval - lower 0.021
90 Percent confidence interval - upper 0.050
P-value H_0: Robust RMSEA <= 0.050 0.954
P-value H_0: Robust RMSEA >= 0.080 0.000
Standardized Root Mean Square Residual:
SRMR 0.050 0.050
Parameter Estimates:
Standard errors Sandwich
Information bread Observed
Observed information based on Hessian
Group 1 [Female]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
depress =~
item1 (.p1.) 1.135 0.068 16.637 0.000 1.135 0.686
item2 (.p2.) 1.336 0.075 17.802 0.000 1.336 0.661
item3 (.p3.) 0.860 0.077 11.228 0.000 0.860 0.411
item4 (.p4.) 1.168 0.083 14.063 0.000 1.168 0.526
item5 (.p5.) 1.001 0.076 13.194 0.000 1.001 0.473
item6 (.p6.) 1.161 0.077 15.096 0.000 1.161 0.563
item7 (.p7.) 0.766 0.086 8.914 0.000 0.766 0.372
item8 (.p8.) 1.144 0.082 13.946 0.000 1.144 0.529
item9 (.p9.) 0.622 0.069 9.001 0.000 0.622 0.348
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 ~~
.item2 0.485 0.159 3.047 0.002 0.485 0.266
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.10.) 4.176 0.060 69.282 0.000 4.176 2.525
.item2 (.11.) 3.702 0.074 50.291 0.000 3.702 1.833
.item3 (.12.) 1.845 0.077 24.121 0.000 1.845 0.881
.item4 (.13.) 3.473 0.081 42.797 0.000 3.473 1.563
.item5 (.14.) 2.245 0.077 29.048 0.000 2.245 1.061
.item6 (.15.) 3.808 0.075 50.564 0.000 3.808 1.846
.item7 3.842 0.104 37.048 0.000 3.842 1.866
.item8 (.17.) 3.556 0.079 45.035 0.000 3.556 1.644
.item9 (.18.) 1.232 0.065 18.878 0.000 1.232 0.689
depress 0.000 0.000 0.000
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.19.) 1.447 0.145 9.949 0.000 1.447 0.529
.item2 (.20.) 2.295 0.178 12.893 0.000 2.295 0.563
.item3 (.21.) 3.649 0.143 25.557 0.000 3.649 0.831
.item4 (.22.) 3.576 0.197 18.172 0.000 3.576 0.724
.item5 (.23.) 3.478 0.161 21.617 0.000 3.478 0.776
.item6 (.24.) 2.906 0.199 14.596 0.000 2.906 0.683
.item7 3.654 0.271 13.478 0.000 3.654 0.862
.item8 (.26.) 3.368 0.207 16.275 0.000 3.368 0.720
.item9 (.27.) 2.807 0.143 19.666 0.000 2.807 0.879
depress 1.000 1.000 1.000
R-Square:
Estimate
item1 0.471
item2 0.437
item3 0.169
item4 0.276
item5 0.224
item6 0.317
item7 0.138
item8 0.280
item9 0.121
Group 2 [Male]:
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
depress =~
item1 (.p1.) 1.135 0.068 16.637 0.000 1.135 0.686
item2 (.p2.) 1.336 0.075 17.802 0.000 1.336 0.661
item3 (.p3.) 0.860 0.077 11.228 0.000 0.860 0.411
item4 (.p4.) 1.168 0.083 14.063 0.000 1.168 0.526
item5 (.p5.) 1.001 0.076 13.194 0.000 1.001 0.473
item6 (.p6.) 1.161 0.077 15.096 0.000 1.161 0.563
item7 (.p7.) 0.766 0.086 8.914 0.000 0.766 0.336
item8 (.p8.) 1.144 0.082 13.946 0.000 1.144 0.529
item9 (.p9.) 0.622 0.069 9.001 0.000 0.622 0.348
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 ~~
.item2 0.829 0.152 5.448 0.000 0.829 0.455
Intercepts:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.10.) 4.176 0.060 69.282 0.000 4.176 2.525
.item2 (.11.) 3.702 0.074 50.291 0.000 3.702 1.833
.item3 (.12.) 1.845 0.077 24.121 0.000 1.845 0.881
.item4 (.13.) 3.473 0.081 42.797 0.000 3.473 1.563
.item5 (.14.) 2.245 0.077 29.048 0.000 2.245 1.061
.item6 (.15.) 3.808 0.075 50.564 0.000 3.808 1.846
.item7 3.448 0.116 29.819 0.000 3.448 1.510
.item8 (.17.) 3.556 0.079 45.035 0.000 3.556 1.644
.item9 (.18.) 1.232 0.065 18.878 0.000 1.232 0.689
depress 0.000 0.000 0.000
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.item1 (.19.) 1.447 0.145 9.949 0.000 1.447 0.529
.item2 (.20.) 2.295 0.178 12.893 0.000 2.295 0.563
.item3 (.21.) 3.649 0.143 25.557 0.000 3.649 0.831
.item4 (.22.) 3.576 0.197 18.172 0.000 3.576 0.724
.item5 (.23.) 3.478 0.161 21.617 0.000 3.478 0.776
.item6 (.24.) 2.906 0.199 14.596 0.000 2.906 0.683
.item7 4.625 0.260 17.769 0.000 4.625 0.887
.item8 (.26.) 3.368 0.207 16.275 0.000 3.368 0.720
.item9 (.27.) 2.807 0.143 19.666 0.000 2.807 0.879
depress 1.000 1.000 1.000
R-Square:
Estimate
item1 0.471
item2 0.437
item3 0.169
item4 0.276
item5 0.224
item6 0.317
item7 0.113
item8 0.280
item9 0.121| cfi | tli | rmsea | rmsea.ci.lower | rmsea.ci.upper | rmsea.pvalue | srmr | |
|---|---|---|---|---|---|---|---|
| Configural | 0.96 | 0.95 | 0.05 | 0.03 | 0.06 | 0.52 | 0.04 |
| structuralVariance | 0.97 | 0.97 | 0.04 | 0.02 | 0.05 | 0.95 | 0.05 |
| structuralMean | 0.97 | 0.97 | 0.04 | 0.02 | 0.05 | 0.95 | 0.05 |
| Df | AIC | BIC | Chisq | Chisq diff | Df diff | Pr(>Chisq) | |
|---|---|---|---|---|---|---|---|
| Configural | 52 | 27525.80 | 27784.52 | 98.91085 | NA | NA | NA |
| structuralVariance | 76 | 27493.79 | 27641.63 | 114.90425 | 16.993054 | 24 | 0.8489582 |
| structuralMean | 77 | 27493.03 | 27636.25 | 116.14270 | 1.225188 | 1 | 0.2683450 |