CFA Measurement Invariance

One example from Major Depression Criteria

R
SEM
lavaan
Author

Jihong Zhang

Published

November 12, 2017

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 Lavaan Package.

1 What and Why?

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:

1. the scale measures varied trait(s) for different groups (e.g., for group A, the scale actually measures trait \alpha while the scale measures trait \beta for group B)
2. the difference of true latent trait (\theta; the scale scores for group A has varied location and scale with the scale scores for group B)
3. the difference of item effects on trait (\lambda; the scale items have varied difficulties/discrimination between group A and group B).

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:

1. Testing the difference coming from factor part is called Structural invariance.
2. Testing the difference coming from measurement part is called Measurement invariance.

In previous paragraph, the first two differences are measured by Structural Invariance. The 3rd differences are measured by Measurement Invariance.

1.1 An example of Multiple Group CFA 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)

1. Depressed mood
2. Loss of interest in usual activities
3. Weight/appetite change
4. Sleep disturbance
5. Psychomotor agitation/retardation
6. Fatigue/loss of energy
7. Feelings of worthless/guilt
8. Concentration difficulties
9. Thoughts of death/suicidality

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.

1.1.1 Data Import

  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.

1.1.2 Model Specification

⌘+C
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
"

1.1.3 Model Options

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.

⌘+C
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",
# 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.partial = c("item7~1"))

fit.strict <- sem(model1.config, data = mddAll,
meanstructure = T , std.lv = T,
estimator = "MLR", mimic = "mplus",
group = "sex",
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",
"residual.covariances"),
group.partial = c("item7~1", "item7~~item7"))

1.1.4 Runing Model

⌘+C
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
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             0.131
⌘+C
summary(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
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             0.115
⌘+C
summary(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
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             0.113
⌘+C
summary(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
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             0.113
⌘+C
summary(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
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             0.114
⌘+C
summary(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
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             0.115

1.1.5 Model Comparison

⌘+C
model_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")
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
⌘+C
kable(table_lik.test, caption = "Model Comparision Table")
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

1.2 STRUCTUAL INVARIANCE TESTS

1.2.1 Factor Variance Invariance Model

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
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      0.315

1.2.2 Factor Mean Invariance Model

lavaan 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
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

1.2.3 Model Comparison

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
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
Model Comparision Table
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