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)
- Depressed mood
- Loss of interest in usual activities
- Weight/appetite change
- Sleep disturbance
- Psychomotor agitation/retardation
- Fatigue/loss of energy
- Feelings of worthless/guilt
- Concentration difficulties
- 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
| cfi | tli | rmsea | rmsea.ci.lower | rmsea.ci.upper | rmsea.pvalue | srmr | |
|---|---|---|---|---|---|---|---|
| Configural | 0.96 | 0.95 | 0.05 | 0.03 | 0.06 | 0.52 | 0.04 |
| Metric | 0.97 | 0.96 | 0.04 | 0.03 | 0.06 | 0.76 | 0.04 |
| Scalar | 0.96 | 0.96 | 0.04 | 0.03 | 0.06 | 0.79 | 0.05 |
| Scalar2 | 0.97 | 0.96 | 0.04 | 0.03 | 0.05 | 0.85 | 0.04 |
| Strict | 0.97 | 0.97 | 0.04 | 0.02 | 0.05 | 0.94 | 0.05 |
| Strict+Cov | 0.96 | 0.96 | 0.04 | 0.03 | 0.05 | 0.88 | 0.05 |
kable(table_lik.test, caption = "Model Comparision Table")
Table: Table 1: Model Comparision Table
| Df | AIC | BIC | Chisq | Chisq diff | Df diff | Pr(>Chisq) | |
|---|---|---|---|---|---|---|---|
| Configural | 52 | 27525.80 | 27784.52 | 98.91085 | NA | NA | NA |
| Metric | 60 | 27513.72 | 27735.49 | 102.83941 | 4.259306 | 8 | 0.8330029 |
| Scalar | 68 | 27510.19 | 27695.00 | 115.30933 | 12.398255 | 8 | 0.1342996 |
| Scalar2 | 68 | 27510.19 | 27695.00 | 115.30933 | 5.929561 | 1 | 0.0148890 |
| Strict | 75 | 27494.94 | 27647.41 | 114.05887 | 5.269042 | 8 | 0.7284715 |
| Strict+Cov | 76 | 27502.24 | 27650.08 | 123.35057 | 4.172433 | 1 | 0.0410868 |
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
| cfi | tli | rmsea | rmsea.ci.lower | rmsea.ci.upper | rmsea.pvalue | srmr | |
|---|---|---|---|---|---|---|---|
| Configural | 0.96 | 0.95 | 0.05 | 0.03 | 0.06 | 0.52 | 0.04 |
| structuralVariance | 0.97 | 0.97 | 0.04 | 0.02 | 0.05 | 0.95 | 0.05 |
| structuralMean | 0.97 | 0.97 | 0.04 | 0.02 | 0.05 | 0.95 | 0.05 |
Table: Table 2: Model Comparision Table
| Df | AIC | BIC | Chisq | Chisq diff | Df diff | Pr(>Chisq) | |
|---|---|---|---|---|---|---|---|
| Configural | 52 | 27525.80 | 27784.52 | 98.91085 | NA | NA | NA |
| structuralVariance | 76 | 27493.79 | 27641.63 | 114.90425 | 16.993054 | 24 | 0.8489581 |
| structuralMean | 77 | 27493.03 | 27636.25 | 116.14270 | 1.225188 | 1 | 0.2683448 |
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.