One Example of Measurement Invariance

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 measuremnet invariance 3) how to do measurement invariance using R Lavaan Package.

What is Measurement Invariance (MI)?

In my advisor Jonathan’s lectures slide, MI is a testing tool investigating “whether indicators measure the same construct in the same way in different groups or over time/condition”. If so, the indicator response should depend only on latent trait scores.

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. Imaging your items responses have significant group difference (male with female, international students with native speaker), there are at least three sources of distinct:

(I start from 0 numbering because I love Python!)

(0) the measuremt measure different traits for groups

(1) the difference of true latent trait ($\theta$)

(2) the difference of effects of trait on measurements ($\lambda$).

The zero and first are easy to understant. For international math assessment, it may measure native speaker’s math ability but on the other hand measures 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 final group difference means, even if male and females have exactly same level of trait, they will still have different item responses since same item reflect efficientlty or not efficiently 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 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.

Acutally all parameters in CFA model (factor variances, factor covariance, factor means, factor loadings, item intercepts and resifial variances, covariances) could be different for different groups. Testing the difference coming from factor part is called Structual invariance. Testing the difference coming from measurement part is called Measurement Invariance. In previous paragraph, the 0st and 1st differences are measured by Structural Invariance. The 3rd differences are measurede by Measurement Invariance.

Why we should use Measurement Invariance?

How to use Measurement Invariance

Multiple Group CFA Invariance Example (data from Brown Charpter 7):

Major Deression Criteria across Men and Women (n =345)

9 items rated by clinicians on a scale iof 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 Meansurement Invariance Example elaborated the manual version so that learner could learn what you are doing first. I will show you how to use shortcuts.

Data Import

##   V1 V2 V3 V4 V5 V6 V7 V8 V9 V10
## 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.

Model Specification

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
"

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-likihood 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 euqal.

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"))

Runing Model

summary(fit.config, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)
## lavaan 0.6-10 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      Robust
##   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)                            0.951
## 
## 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 (BIC)        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 RMSEA <= 0.05                          0.522       0.636
##                                                                   
##   Robust RMSEA                                               0.048
##   90 Percent confidence interval - lower                     0.032
##   90 Percent confidence interval - upper                     0.063
## 
## 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             0.131
summary(fit.metric, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)
## lavaan 0.6-10 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      Robust
##   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 (BIC)        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 RMSEA <= 0.05                          0.758       0.818
##                                                                   
##   Robust RMSEA                                               0.043
##   90 Percent confidence interval - lower                     0.027
##   90 Percent confidence interval - upper                     0.057
## 
## 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             0.115
summary(fit.scalar, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)
## lavaan 0.6-10 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      Robust
##   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 (BIC)        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 RMSEA <= 0.05                          0.794       0.846
##                                                                   
##   Robust RMSEA                                               0.042
##   90 Percent confidence interval - lower                     0.028
##   90 Percent confidence interval - upper                     0.056
## 
## 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             0.113
summary(fit.scalar2, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)
## lavaan 0.6-10 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      Robust
##   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 (BIC)        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 RMSEA <= 0.05                          0.855       0.896
##                                                                   
##   Robust RMSEA                                               0.040
##   90 Percent confidence interval - lower                     0.025
##   90 Percent confidence interval - upper                     0.054
## 
## 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             0.113
summary(fit.strict, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)
## lavaan 0.6-10 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      Robust
##   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 (BIC)        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 RMSEA <= 0.05                          0.942       0.956
##                                                                   
##   Robust RMSEA                                               0.037
##   90 Percent confidence interval - lower                     0.021
##   90 Percent confidence interval - upper                     0.050
## 
## 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             0.114
summary(fit.strict.cov, fit.measures = TRUE, rsquare = TRUE, standardized = TRUE)
## lavaan 0.6-10 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      Robust
##   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)                            0.966
## 
## 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 (BIC)        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 RMSEA <= 0.05                          0.877       0.920
##                                                                   
##   Robust RMSEA                                               0.040
##   90 Percent confidence interval - lower                     0.025
##   90 Percent confidence interval - upper                     0.053
## 
## 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             0.115

Model Comparision

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")

Table: Table 1: Model Fit Indices Table

cfitlirmsearmsea.ci.lowerrmsea.ci.upperrmsea.pvaluesrmr
Configural0.960.950.050.030.060.520.04
Metric0.970.960.040.030.060.760.04
Scalar0.960.960.040.030.060.790.05
Scalar20.970.960.040.030.050.850.04
Strict0.970.970.040.020.050.940.05
Strict+Cov0.960.960.040.030.050.880.05
kable(table_lik.test, caption = "Model Comparision Table")

Table: Table 1: Model Comparision Table

DfAICBICChisqChisq diffDf diffPr(>Chisq)
Configural5227525.8027784.5298.91085NANANA
Metric6027513.7227735.49102.839414.25930680.8330029
Scalar6827510.1927695.00115.3093312.39825580.1342996
Scalar26827510.1927695.00115.309335.92956110.0148890
Strict7527494.9427647.41114.058875.26904280.7284715
Strict+Cov7627502.2427650.08123.350574.17243310.0410868

STRUCTUAL INVARIANCE TESTS

Factor Variance Invariance Model

## lavaan 0.6-10 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      Robust
##   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 (BIC)        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 RMSEA <= 0.05                          0.947       0.958
##                                                                   
##   Robust RMSEA                                               0.036
##   90 Percent confidence interval - lower                     0.021
##   90 Percent confidence interval - upper                     0.050
## 
## 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      0.315

Factor Mean Invariance Model

## lavaan 0.6-10 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      Robust
##   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 (BIC)        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 RMSEA <= 0.05                          0.950       0.961
##                                                                   
##   Robust RMSEA                                               0.036
##   90 Percent confidence interval - lower                     0.021
##   90 Percent confidence interval - upper                     0.050
## 
## 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

Model Comparision

Table: Table 2: Model Fit Indices Table

cfitlirmsearmsea.ci.lowerrmsea.ci.upperrmsea.pvaluesrmr
Configural0.960.950.050.030.060.520.04
structuralVariance0.970.970.040.020.050.950.05
structuralMean0.970.970.040.020.050.950.05

Table: Table 2: Model Comparision Table

DfAICBICChisqChisq diffDf diffPr(>Chisq)
Configural5227525.8027784.5298.91085NANANA
structuralVariance7627493.7927641.63114.9042516.993054240.8489581
structuralMean7727493.0327636.25116.142701.22518810.2683448
Jihong Zhang, M.S.
Jihong Zhang, M.S.
Ph.D. Candidate

My research interests mainly focus on the Bayesian Diagnostic Classification Models (DCMs) - a special kind of Item Response Model and the model checking method, as applied in the psychological, educational, and social sciences. I seek to improve the utility of advanced psychometric modeling and provide easy-to-use tools or software for researchers.