Advanced Factor Analysis
Rongting Niu
2025-10-28
library(tidyverse)
library(summarytools)
library(psych)
library(lavaan)
library(semTools)
library(MVN)
library(EFAtools)
library(EGAnet)
library(haven)
library(lavaan.mi)
options(scipen=999)
options(digits=3)Mydata <- read_sav("/Users/niurongting/Desktop/JPPR/Crisis Emotion - Main Study-Final.sav")
head(Mydata)## # A tibble: 6 × 201
## StartDate EndDate Status IPAddress Progress
## <dttm> <dttm> <dbl+lbl> <chr> <dbl>
## 1 2025-03-03 09:50:07 2025-03-03 09:54:14 0 [IP Address] 100.12.196.35 100
## 2 2025-03-03 09:50:10 2025-03-03 09:55:02 0 [IP Address] 72.0.134.210 100
## 3 2025-03-03 09:50:05 2025-03-03 09:55:41 0 [IP Address] 73.188.88.156 100
## 4 2025-03-03 09:50:03 2025-03-03 09:55:46 0 [IP Address] 47.39.239.232 100
## 5 2025-03-03 09:50:10 2025-03-03 09:55:53 0 [IP Address] 173.91.121.220 100
## 6 2025-03-03 09:50:07 2025-03-03 09:55:57 0 [IP Address] 73.85.98.93 100
## # ℹ 196 more variables: Duration__in_seconds_ <dbl>, Finished <dbl+lbl>,
## # RecordedDate <dttm>, ResponseId <chr>, RecipientLastName <chr>,
## # RecipientFirstName <chr>, RecipientEmail <chr>, ExternalReference <chr>,
## # LocationLatitude <chr>, LocationLongitude <chr>, DistributionChannel <chr>,
## # UserLanguage <chr>, S0Q1 <dbl+lbl>, Q79_First_Click <dbl>,
## # Q79_Last_Click <dbl>, Q79_Page_Submit <dbl>, Q79_Click_Count <dbl>,
## # Q80_First_Click <dbl>, Q80_Last_Click <dbl>, Q80_Page_Submit <dbl>, …Mydata_ea <- select(Mydata, EmpatheticAnger__1:EmpatheticAnger__6)
Mydata_mo <- select(Mydata, Moraloutrage__1:Moraloutrage__6)
head(Mydata_ea)## # A tibble: 6 × 6
## EmpatheticAnger__1 EmpatheticAnger__2 EmpatheticAnger__3 EmpatheticAnger__4
## <dbl+lbl> <dbl+lbl> <dbl+lbl> <dbl+lbl>
## 1 6 [Agree] 7 [Strongly agree] 7 [Strongly agree] 6 [Agree]
## 2 5 [Somewhat Agree] 5 [Somewhat Agree] 5 [Somewhat Agree] 5 [Somewhat Agree]
## 3 5 [Somewhat Agree] 5 [Somewhat Agree] 5 [Somewhat Agree] 5 [Somewhat Agree]
## 4 7 [Strongly agree] 7 [Strongly agree] 7 [Strongly agree] 7 [Strongly agree]
## 5 7 [Strongly agree] 7 [Strongly agree] 7 [Strongly agree] 7 [Strongly agree]
## 6 6 [Agree] 6 [Agree] 6 [Agree] 6 [Agree]
## # ℹ 2 more variables: EmpatheticAnger__5 <dbl+lbl>,
## # EmpatheticAnger__6 <dbl+lbl>head(Mydata_mo)## # A tibble: 6 × 6
## Moraloutrage__1 Moraloutrage__2 Moraloutrage__3 Moraloutrage__4
## <dbl+lbl> <dbl+lbl> <dbl+lbl> <dbl+lbl>
## 1 4 [Moderately] 4 [Moderately] 3 [Somewhat] 4 [Moderately]
## 2 4 [Moderately] 4 [Moderately] 4 [Moderately] 4 [Moderately]
## 3 4 [Moderately] 3 [Somewhat] 4 [Moderately] 4 [Moderately]
## 4 1 [Not at all] 1 [Not at all] 1 [Not at all] 1 [Not at all]
## 5 7 [Extremely] 7 [Extremely] 7 [Extremely] 7 [Extremely]
## 6 4 [Moderately] 1 [Not at all] 3 [Somewhat] 4 [Moderately]
## # ℹ 2 more variables: Moraloutrage__5 <dbl+lbl>, Moraloutrage__6 <dbl+lbl>Mydata_two <- Mydata %>%
select(
EmpatheticAnger__1:EmpatheticAnger__6,
Moraloutrage__2, Moraloutrage__3, Moraloutrage__5, Moraloutrage__6
) %>%
mutate(across(everything(), ~as.numeric(as.character(.))))
head(Mydata_two)## # A tibble: 6 × 10
## EmpatheticAnger__1 EmpatheticAnger__2 EmpatheticAnger__3 EmpatheticAnger__4
## <dbl> <dbl> <dbl> <dbl>
## 1 6 7 7 6
## 2 5 5 5 5
## 3 5 5 5 5
## 4 7 7 7 7
## 5 7 7 7 7
## 6 6 6 6 6
## # ℹ 6 more variables: EmpatheticAnger__5 <dbl>, EmpatheticAnger__6 <dbl>,
## # Moraloutrage__2 <dbl>, Moraloutrage__3 <dbl>, Moraloutrage__5 <dbl>,
## # Moraloutrage__6 <dbl>We always need to look at our data before running any analyses. We are looking for 1) any out of range values (coding errors), 2) non-normal univariate distributions, 3) high skewness and kurtosis values, 4) nonlinear relations between variables, 5) very high item intercorrelations, 6) item redundancy.
multi.hist(Mydata_ea,freq=F)
describe(Mydata_ea)## vars n mean sd median trimmed mad min max range skew
## EmpatheticAnger__1 1 482 5.06 1.50 5 5.22 1.48 1 7 6 -0.82
## EmpatheticAnger__2 2 482 5.09 1.48 5 5.26 1.48 1 7 6 -0.91
## EmpatheticAnger__3 3 482 5.25 1.48 5 5.44 1.48 1 7 6 -1.01
## EmpatheticAnger__4 4 482 4.94 1.59 5 5.11 1.48 1 7 6 -0.80
## EmpatheticAnger__5 5 482 5.22 1.51 5 5.42 1.48 1 7 6 -1.02
## EmpatheticAnger__6 6 482 5.07 1.56 5 5.26 1.48 1 7 6 -0.93
## kurtosis se
## EmpatheticAnger__1 0.22 0.07
## EmpatheticAnger__2 0.52 0.07
## EmpatheticAnger__3 0.82 0.07
## EmpatheticAnger__4 0.12 0.07
## EmpatheticAnger__5 0.77 0.07
## EmpatheticAnger__6 0.40 0.07pairs.panels(Mydata_two,
method = "pearson",
density = TRUE,
gap=0) # show density plots
corPlot(Mydata_ea,scale=F)
uva <- UVA(Mydata_ea)
uva## Variable pairs with wTO > 0.30 (large-to-very large redundancy)
##
## ----
##
## Variable pairs with wTO > 0.25 (moderate-to-large redundancy)
##
## ----
##
## Variable pairs with wTO > 0.20 (small-to-moderate redundancy)
##
## node_i node_j wto
## EmpatheticAnger__3 EmpatheticAnger__5 0.245
## EmpatheticAnger__1 EmpatheticAnger__2 0.228
## EmpatheticAnger__5 EmpatheticAnger__6 0.228
## EmpatheticAnger__2 EmpatheticAnger__6 0.226
## EmpatheticAnger__3 EmpatheticAnger__4 0.214
## EmpatheticAnger__3 EmpatheticAnger__6 0.214
## EmpatheticAnger__1 EmpatheticAnger__6 0.205uva$keep_remove## NULLmulti.hist(Mydata_mo,freq=F)
describe(Mydata_mo)## vars n mean sd median trimmed mad min max range skew
## Moraloutrage__1 1 482 4.32 1.91 4 4.40 2.97 1 7 6 -0.24
## Moraloutrage__2 2 482 3.72 2.07 4 3.66 2.97 1 7 6 0.10
## Moraloutrage__3 3 482 4.03 2.02 4 4.04 2.97 1 7 6 -0.14
## Moraloutrage__4 4 482 4.18 1.97 4 4.22 2.97 1 7 6 -0.19
## Moraloutrage__5 5 482 4.18 1.91 4 4.22 2.97 1 7 6 -0.15
## Moraloutrage__6 6 482 3.77 1.97 4 3.72 2.97 1 7 6 0.05
## kurtosis se
## Moraloutrage__1 -1.06 0.09
## Moraloutrage__2 -1.31 0.09
## Moraloutrage__3 -1.23 0.09
## Moraloutrage__4 -1.14 0.09
## Moraloutrage__5 -1.07 0.09
## Moraloutrage__6 -1.16 0.09pairs.panels(Mydata_mo,
method = "pearson",
density = TRUE,
gap=0) # show density plots
corPlot(Mydata_mo,scale=F)
uva <- UVA(Mydata_mo)
uva## Variable pairs with wTO > 0.30 (large-to-very large redundancy)
##
## node_i node_j wto
## Moraloutrage__1 Moraloutrage__4 0.31
##
## ----
##
## Variable pairs with wTO > 0.25 (moderate-to-large redundancy)
##
## node_i node_j wto
## Moraloutrage__3 Moraloutrage__4 0.292
## Moraloutrage__1 Moraloutrage__3 0.277
##
## ----
##
## Variable pairs with wTO > 0.20 (small-to-moderate redundancy)
##
## node_i node_j wto
## Moraloutrage__5 Moraloutrage__6 0.234
## Moraloutrage__2 Moraloutrage__3 0.225uva$keep_remove## $keep
## [1] "Moraloutrage__3" "Moraloutrage__4"
##
## $remove
## [1] "Moraloutrage__1" "Moraloutrage__1" "Moraloutrage__3"multi.hist(Mydata_two,freq=F)
describe(Mydata_two)## vars n mean sd median trimmed mad min max range skew
## EmpatheticAnger__1 1 482 5.06 1.50 5 5.22 1.48 1 7 6 -0.82
## EmpatheticAnger__2 2 482 5.09 1.48 5 5.26 1.48 1 7 6 -0.91
## EmpatheticAnger__3 3 482 5.25 1.48 5 5.44 1.48 1 7 6 -1.01
## EmpatheticAnger__4 4 482 4.94 1.59 5 5.11 1.48 1 7 6 -0.80
## EmpatheticAnger__5 5 482 5.22 1.51 5 5.42 1.48 1 7 6 -1.02
## EmpatheticAnger__6 6 482 5.07 1.56 5 5.26 1.48 1 7 6 -0.93
## Moraloutrage__2 7 482 3.72 2.07 4 3.66 2.97 1 7 6 0.10
## Moraloutrage__3 8 482 4.03 2.02 4 4.04 2.97 1 7 6 -0.14
## Moraloutrage__5 9 482 4.18 1.91 4 4.22 2.97 1 7 6 -0.15
## Moraloutrage__6 10 482 3.77 1.97 4 3.72 2.97 1 7 6 0.05
## kurtosis se
## EmpatheticAnger__1 0.22 0.07
## EmpatheticAnger__2 0.52 0.07
## EmpatheticAnger__3 0.82 0.07
## EmpatheticAnger__4 0.12 0.07
## EmpatheticAnger__5 0.77 0.07
## EmpatheticAnger__6 0.40 0.07
## Moraloutrage__2 -1.31 0.09
## Moraloutrage__3 -1.23 0.09
## Moraloutrage__5 -1.07 0.09
## Moraloutrage__6 -1.16 0.09pairs.panels(Mydata_two,
method = "pearson",
density = TRUE,
gap=0) # show density plots
corPlot(Mydata_two,scale=F)
uva <- UVA(Mydata_two)
uva## Variable pairs with wTO > 0.30 (large-to-very large redundancy)
##
## node_i node_j wto
## Moraloutrage__2 Moraloutrage__3 0.335
## Moraloutrage__5 Moraloutrage__6 0.310
##
## ----
##
## Variable pairs with wTO > 0.25 (moderate-to-large redundancy)
##
## node_i node_j wto
## Moraloutrage__3 Moraloutrage__6 0.283
## Moraloutrage__3 Moraloutrage__5 0.262
##
## ----
##
## Variable pairs with wTO > 0.20 (small-to-moderate redundancy)
##
## node_i node_j wto
## Moraloutrage__2 Moraloutrage__6 0.233
## EmpatheticAnger__2 EmpatheticAnger__6 0.217
## EmpatheticAnger__3 EmpatheticAnger__5 0.215
## EmpatheticAnger__5 EmpatheticAnger__6 0.214
## EmpatheticAnger__1 EmpatheticAnger__2 0.212
## Moraloutrage__2 Moraloutrage__5 0.203
## EmpatheticAnger__3 EmpatheticAnger__6 0.201uva$keep_remove## $keep
## [1] "Moraloutrage__3" "Moraloutrage__5"
##
## $remove
## [1] "Moraloutrage__2" "Moraloutrage__5" "Moraloutrage__6" "Moraloutrage__6"Assumption Checking
Let’s check to see if the covariance matrix for ragland_r2 is positive definite. Note that if you are transforming your input matrix in some way (e.g., polychoric, tetrachoric, missing data), you should use that input matrix here.
factor_mat <- as.matrix(cov(Mydata_two))
det(factor_mat) #determinant is non-zero## [1] 1.09eigen(factor_mat) #all the eigenvalues are positive## eigen() decomposition
## $values
## [1] 20.446 4.326 1.146 0.790 0.738 0.613 0.533 0.456 0.365 0.338
##
## $vectors
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## [1,] -0.268 -0.281 -0.00352 -0.08708 0.4371 0.04152 -0.43743 -0.6428 -0.1299
## [2,] -0.260 -0.297 -0.05556 0.21257 0.3504 0.05342 -0.36944 0.7024 -0.1583
## [3,] -0.270 -0.284 0.12615 -0.03546 -0.0706 -0.19029 0.30163 -0.1812 -0.2229
## [4,] -0.273 -0.312 -0.15287 -0.01409 -0.8002 -0.00641 -0.38907 -0.0151 0.0614
## [5,] -0.273 -0.303 0.06935 0.00379 -0.0445 0.01828 0.55060 0.0766 -0.4724
## [6,] -0.277 -0.349 0.01673 -0.02648 0.1425 0.21180 0.30845 0.0231 0.7985
## [7,] -0.382 0.375 -0.72604 0.18691 0.0286 0.31879 0.14602 -0.0965 -0.0839
## [8,] -0.401 0.276 -0.09458 -0.14974 0.0892 -0.81572 -0.00925 0.0842 0.1717
## [9,] -0.346 0.331 0.55240 0.65463 -0.0930 0.09341 -0.05309 -0.1138 0.0482
## [10,] -0.369 0.339 0.33362 -0.67762 -0.0334 0.37178 -0.08533 0.1571 -0.0709
## [,10]
## [1,] -0.1653
## [2,] 0.1256
## [3,] 0.7839
## [4,] -0.0932
## [5,] -0.5428
## [6,] -0.0477
## [7,] 0.1071
## [8,] -0.1477
## [9,] -0.0367
## [10,] 0.0471multiv <- mvn(Mydata_two,mvnTest = "mardia",multivariatePlot = "qq",multivariateOutlierMethod="none")
The Q-Q plot plots the observed squared Mahalanobis distance (x-axis)
against the theoretical quantiles for a multivariate normal distribution
(y-axis). Departures from the line indicate departures from multivarite
normality.
The other Q-Q plot shows multivariate outliers based on the robust squared Mahalanobis distance. These indicate how far each case is from the center of the joint distribution.
multiv$multivariateNormality## Test Statistic
## 1 Mardia Skewness 2182.27740785489
## 2 Mardia Kurtosis 71.8296867199904
## 3 MVN <NA>
## p value
## 1 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000137888781097833
## 2 0
## 3 <NA>
## Result
## 1 NO
## 2 NO
## 3 NOCFA 2 Factor
CFA <- ' EmpAnger =~ EmpatheticAnger__1 + EmpatheticAnger__2 + EmpatheticAnger__3 + EmpatheticAnger__4 + EmpatheticAnger__5 +EmpatheticAnger__6
MoralOutrage =~ Moraloutrage__2 + Moraloutrage__3 + Moraloutrage__5 + Moraloutrage__6'
cfa_fit <- cfa(data = Mydata,CFA,std.lv=T,estimator="MLR")
summary(cfa_fit,std=T,fit.measures=T,rsquare=T)## lavaan 0.6-19 ended normally after 22 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 21
##
## Number of observations 482
##
## Model Test User Model:
## Standard Scaled
## Test Statistic 90.142 50.157
## Degrees of freedom 34 34
## P-value (Chi-square) 0.000 0.037
## Scaling correction factor 1.797
## Yuan-Bentler correction (Mplus variant)
##
## Model Test Baseline Model:
##
## Test statistic 5032.939 2344.732
## Degrees of freedom 45 45
## P-value 0.000 0.000
## Scaling correction factor 2.146
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.989 0.993
## Tucker-Lewis Index (TLI) 0.985 0.991
##
## Robust Comparative Fit Index (CFI) 0.994
## Robust Tucker-Lewis Index (TLI) 0.992
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -6899.620 -6899.620
## Scaling correction factor 2.123
## for the MLR correction
## Loglikelihood unrestricted model (H1) NA NA
## Scaling correction factor 1.922
## for the MLR correction
##
## Akaike (AIC) 13841.240 13841.240
## Bayesian (BIC) 13928.977 13928.977
## Sample-size adjusted Bayesian (SABIC) 13862.324 13862.324
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.059 0.031
## 90 Percent confidence interval - lower 0.044 0.016
## 90 Percent confidence interval - upper 0.073 0.045
## P-value H_0: RMSEA <= 0.050 0.159 0.992
## P-value H_0: RMSEA >= 0.080 0.008 0.000
##
## Robust RMSEA 0.042
## 90 Percent confidence interval - lower 0.011
## 90 Percent confidence interval - upper 0.066
## P-value H_0: Robust RMSEA <= 0.050 0.683
## P-value H_0: Robust RMSEA >= 0.080 0.003
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.019 0.019
##
## Parameter Estimates:
##
## Standard errors Sandwich
## Information bread Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## EmpAnger =~
## EmpthtcAngr__1 1.310 0.060 21.798 0.000 1.310 0.872
## EmpthtcAngr__2 1.288 0.063 20.552 0.000 1.288 0.869
## EmpthtcAngr__3 1.342 0.064 21.008 0.000 1.342 0.911
## EmpthtcAngr__4 1.333 0.064 20.961 0.000 1.333 0.839
## EmpthtcAngr__5 1.367 0.063 21.716 0.000 1.367 0.909
## EmpthtcAngr__6 1.421 0.060 23.714 0.000 1.421 0.914
## MoralOutrage =~
## Moraloutrag__2 1.797 0.061 29.338 0.000 1.797 0.867
## Moraloutrag__3 1.893 0.046 40.894 0.000 1.893 0.939
## Moraloutrag__5 1.627 0.058 28.065 0.000 1.627 0.854
## Moraloutrag__6 1.748 0.055 31.606 0.000 1.748 0.889
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## EmpAnger ~~
## MoralOutrage 0.692 0.025 27.238 0.000 0.692 0.692
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .EmpthtcAngr__1 0.542 0.073 7.418 0.000 0.542 0.240
## .EmpthtcAngr__2 0.539 0.097 5.545 0.000 0.539 0.245
## .EmpthtcAngr__3 0.371 0.055 6.765 0.000 0.371 0.171
## .EmpthtcAngr__4 0.746 0.111 6.748 0.000 0.746 0.296
## .EmpthtcAngr__5 0.395 0.054 7.349 0.000 0.395 0.174
## .EmpthtcAngr__6 0.398 0.080 4.980 0.000 0.398 0.165
## .Moraloutrag__2 1.067 0.159 6.707 0.000 1.067 0.248
## .Moraloutrag__3 0.481 0.062 7.757 0.000 0.481 0.118
## .Moraloutrag__5 0.980 0.103 9.501 0.000 0.980 0.270
## .Moraloutrag__6 0.814 0.117 6.980 0.000 0.814 0.210
## EmpAnger 1.000 1.000 1.000
## MoralOutrage 1.000 1.000 1.000
##
## R-Square:
## Estimate
## EmpthtcAngr__1 0.760
## EmpthtcAngr__2 0.755
## EmpthtcAngr__3 0.829
## EmpthtcAngr__4 0.704
## EmpthtcAngr__5 0.826
## EmpthtcAngr__6 0.835
## Moraloutrag__2 0.752
## Moraloutrag__3 0.882
## Moraloutrag__5 0.730
## Moraloutrag__6 0.790# lavInspect(cfa_fit)Inspecting residuals
stdlv <- ' EmpAnger =~ EmpatheticAnger__1 + EmpatheticAnger__2 + EmpatheticAnger__3 +
EmpatheticAnger__4 + EmpatheticAnger__5 + EmpatheticAnger__6
MoralOutrage =~ Moraloutrage__2 + Moraloutrage__3 + Moraloutrage__5 + Moraloutrage__6'
stdlv_fit <- cfa(data = Mydata, stdlv, std.lv=T, estimator="MLR")
summary(stdlv_fit,std=T,fit.measures=T)## lavaan 0.6-19 ended normally after 22 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 21
##
## Number of observations 482
##
## Model Test User Model:
## Standard Scaled
## Test Statistic 90.142 50.157
## Degrees of freedom 34 34
## P-value (Chi-square) 0.000 0.037
## Scaling correction factor 1.797
## Yuan-Bentler correction (Mplus variant)
##
## Model Test Baseline Model:
##
## Test statistic 5032.939 2344.732
## Degrees of freedom 45 45
## P-value 0.000 0.000
## Scaling correction factor 2.146
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.989 0.993
## Tucker-Lewis Index (TLI) 0.985 0.991
##
## Robust Comparative Fit Index (CFI) 0.994
## Robust Tucker-Lewis Index (TLI) 0.992
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -6899.620 -6899.620
## Scaling correction factor 2.123
## for the MLR correction
## Loglikelihood unrestricted model (H1) NA NA
## Scaling correction factor 1.922
## for the MLR correction
##
## Akaike (AIC) 13841.240 13841.240
## Bayesian (BIC) 13928.977 13928.977
## Sample-size adjusted Bayesian (SABIC) 13862.324 13862.324
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.059 0.031
## 90 Percent confidence interval - lower 0.044 0.016
## 90 Percent confidence interval - upper 0.073 0.045
## P-value H_0: RMSEA <= 0.050 0.159 0.992
## P-value H_0: RMSEA >= 0.080 0.008 0.000
##
## Robust RMSEA 0.042
## 90 Percent confidence interval - lower 0.011
## 90 Percent confidence interval - upper 0.066
## P-value H_0: Robust RMSEA <= 0.050 0.683
## P-value H_0: Robust RMSEA >= 0.080 0.003
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.019 0.019
##
## Parameter Estimates:
##
## Standard errors Sandwich
## Information bread Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## EmpAnger =~
## EmpthtcAngr__1 1.310 0.060 21.798 0.000 1.310 0.872
## EmpthtcAngr__2 1.288 0.063 20.552 0.000 1.288 0.869
## EmpthtcAngr__3 1.342 0.064 21.008 0.000 1.342 0.911
## EmpthtcAngr__4 1.333 0.064 20.961 0.000 1.333 0.839
## EmpthtcAngr__5 1.367 0.063 21.716 0.000 1.367 0.909
## EmpthtcAngr__6 1.421 0.060 23.714 0.000 1.421 0.914
## MoralOutrage =~
## Moraloutrag__2 1.797 0.061 29.338 0.000 1.797 0.867
## Moraloutrag__3 1.893 0.046 40.894 0.000 1.893 0.939
## Moraloutrag__5 1.627 0.058 28.065 0.000 1.627 0.854
## Moraloutrag__6 1.748 0.055 31.606 0.000 1.748 0.889
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## EmpAnger ~~
## MoralOutrage 0.692 0.025 27.238 0.000 0.692 0.692
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .EmpthtcAngr__1 0.542 0.073 7.418 0.000 0.542 0.240
## .EmpthtcAngr__2 0.539 0.097 5.545 0.000 0.539 0.245
## .EmpthtcAngr__3 0.371 0.055 6.765 0.000 0.371 0.171
## .EmpthtcAngr__4 0.746 0.111 6.748 0.000 0.746 0.296
## .EmpthtcAngr__5 0.395 0.054 7.349 0.000 0.395 0.174
## .EmpthtcAngr__6 0.398 0.080 4.980 0.000 0.398 0.165
## .Moraloutrag__2 1.067 0.159 6.707 0.000 1.067 0.248
## .Moraloutrag__3 0.481 0.062 7.757 0.000 0.481 0.118
## .Moraloutrag__5 0.980 0.103 9.501 0.000 0.980 0.270
## .Moraloutrag__6 0.814 0.117 6.980 0.000 0.814 0.210
## EmpAnger 1.000 1.000 1.000
## MoralOutrage 1.000 1.000 1.000lavInspect(stdlv_fit,what="implied")## $cov
## EmA__1 EmA__2 EmA__3 EmA__4 EmA__5 EmA__6 Mrl__2 Mrl__3
## EmpatheticAnger__1 2.26
## EmpatheticAnger__2 1.69 2.20
## EmpatheticAnger__3 1.76 1.73 2.17
## EmpatheticAnger__4 1.75 1.72 1.79 2.52
## EmpatheticAnger__5 1.79 1.76 1.84 1.82 2.27
## EmpatheticAnger__6 1.86 1.83 1.91 1.89 1.94 2.42
## Moraloutrage__2 1.63 1.60 1.67 1.66 1.70 1.77 4.29
## Moraloutrage__3 1.72 1.69 1.76 1.75 1.79 1.86 3.40 4.07
## Moraloutrage__5 1.48 1.45 1.51 1.50 1.54 1.60 2.92 3.08
## Moraloutrage__6 1.58 1.56 1.62 1.61 1.65 1.72 3.14 3.31
## Mrl__5 Mrl__6
## EmpatheticAnger__1
## EmpatheticAnger__2
## EmpatheticAnger__3
## EmpatheticAnger__4
## EmpatheticAnger__5
## EmpatheticAnger__6
## Moraloutrage__2
## Moraloutrage__3
## Moraloutrage__5 3.63
## Moraloutrage__6 2.84 3.87lavInspect(stdlv_fit,what="sampstat")## $cov
## EmA__1 EmA__2 EmA__3 EmA__4 EmA__5 EmA__6 Mrl__2 Mrl__3
## EmpatheticAnger__1 2.26
## EmpatheticAnger__2 1.76 2.20
## EmpatheticAnger__3 1.75 1.69 2.17
## EmpatheticAnger__4 1.71 1.71 1.82 2.52
## EmpatheticAnger__5 1.75 1.74 1.86 1.83 2.27
## EmpatheticAnger__6 1.88 1.85 1.90 1.88 1.96 2.42
## Moraloutrage__2 1.63 1.59 1.56 1.69 1.61 1.61 4.29
## Moraloutrage__3 1.85 1.76 1.89 1.84 1.84 1.80 3.45 4.07
## Moraloutrage__5 1.46 1.43 1.55 1.45 1.52 1.46 2.88 3.04
## Moraloutrage__6 1.62 1.47 1.64 1.58 1.63 1.61 3.11 3.28
## Mrl__5 Mrl__6
## EmpatheticAnger__1
## EmpatheticAnger__2
## EmpatheticAnger__3
## EmpatheticAnger__4
## EmpatheticAnger__5
## EmpatheticAnger__6
## Moraloutrage__2
## Moraloutrage__3
## Moraloutrage__5 3.63
## Moraloutrage__6 2.97 3.87#Residual = observed - implied
res <- residuals(stdlv_fit,type="raw")
res## $type
## [1] "raw"
##
## $cov
## EmA__1 EmA__2 EmA__3 EmA__4 EmA__5 EmA__6 Mrl__2 Mrl__3
## EmpatheticAnger__1 0.000
## EmpatheticAnger__2 0.072 0.000
## EmpatheticAnger__3 -0.012 -0.043 0.000
## EmpatheticAnger__4 -0.035 -0.007 0.028 0.000
## EmpatheticAnger__5 -0.046 -0.023 0.027 0.010 0.000
## EmpatheticAnger__6 0.015 0.020 -0.011 -0.014 0.015 0.000
## Moraloutrage__2 0.001 -0.015 -0.105 0.034 -0.087 -0.152 0.000
## Moraloutrage__3 0.133 0.071 0.132 0.090 0.054 -0.058 0.049 0.000
## Moraloutrage__5 -0.015 -0.016 0.036 -0.055 -0.017 -0.138 -0.039 -0.044
## Moraloutrage__6 0.040 -0.093 0.015 -0.032 -0.028 -0.111 -0.031 -0.032
## Mrl__5 Mrl__6
## EmpatheticAnger__1
## EmpatheticAnger__2
## EmpatheticAnger__3
## EmpatheticAnger__4
## EmpatheticAnger__5
## EmpatheticAnger__6
## Moraloutrage__2
## Moraloutrage__3
## Moraloutrage__5 0.000
## Moraloutrage__6 0.126 0.000res_cov <- residuals(stdlv_fit,type="cor.bentler")$cov
res_cov #note that the residuals are already standardized because we standardized the factor in estimation## EmA__1 EmA__2 EmA__3 EmA__4 EmA__5 EmA__6 Mrl__2 Mrl__3
## EmpatheticAnger__1 0.000
## EmpatheticAnger__2 0.032 0.000
## EmpatheticAnger__3 -0.006 -0.020 0.000
## EmpatheticAnger__4 -0.015 -0.003 0.012 0.000
## EmpatheticAnger__5 -0.020 -0.010 0.012 0.004 0.000
## EmpatheticAnger__6 0.006 0.009 -0.005 -0.006 0.006 0.000
## Moraloutrage__2 0.000 -0.005 -0.034 0.010 -0.028 -0.047 0.000
## Moraloutrage__3 0.044 0.024 0.044 0.028 0.018 -0.019 0.012 0.000
## Moraloutrage__5 -0.005 -0.006 0.013 -0.018 -0.006 -0.047 -0.010 -0.012
## Moraloutrage__6 0.014 -0.032 0.005 -0.010 -0.009 -0.036 -0.008 -0.008
## Mrl__5 Mrl__6
## EmpatheticAnger__1
## EmpatheticAnger__2
## EmpatheticAnger__3
## EmpatheticAnger__4
## EmpatheticAnger__5
## EmpatheticAnger__6
## Moraloutrage__2
## Moraloutrage__3
## Moraloutrage__5 0.000
## Moraloutrage__6 0.034 0.000print(modindices(stdlv_fit,standardized=T))## lhs op rhs mi epc sepc.lv sepc.all
## 24 EmpAnger =~ Moraloutrage__2 2.812 -0.131 -0.131 -0.063
## 25 EmpAnger =~ Moraloutrage__3 8.331 0.190 0.190 0.094
## 26 EmpAnger =~ Moraloutrage__5 0.737 -0.063 -0.063 -0.033
## 27 EmpAnger =~ Moraloutrage__6 1.069 -0.073 -0.073 -0.037
## 28 MoralOutrage =~ EmpatheticAnger__1 2.225 0.081 0.081 0.054
## 29 MoralOutrage =~ EmpatheticAnger__2 0.004 0.003 0.003 0.002
## 30 MoralOutrage =~ EmpatheticAnger__3 1.955 0.066 0.066 0.044
## 31 MoralOutrage =~ EmpatheticAnger__4 0.282 0.033 0.033 0.021
## 32 MoralOutrage =~ EmpatheticAnger__5 0.005 -0.003 -0.003 -0.002
## 33 MoralOutrage =~ EmpatheticAnger__6 9.006 -0.147 -0.147 -0.094
## 34 EmpatheticAnger__1 ~~ EmpatheticAnger__2 12.329 0.103 0.103 0.191
## 35 EmpatheticAnger__1 ~~ EmpatheticAnger__3 0.616 -0.021 -0.021 -0.046
## 36 EmpatheticAnger__1 ~~ EmpatheticAnger__4 1.980 -0.047 -0.047 -0.075
## 37 EmpatheticAnger__1 ~~ EmpatheticAnger__5 7.728 -0.075 -0.075 -0.162
## 38 EmpatheticAnger__1 ~~ EmpatheticAnger__6 0.799 0.025 0.025 0.053
## 39 EmpatheticAnger__1 ~~ Moraloutrage__2 0.057 0.010 0.010 0.013
## 40 EmpatheticAnger__1 ~~ Moraloutrage__3 0.970 0.032 0.032 0.062
## 41 EmpatheticAnger__1 ~~ Moraloutrage__5 1.413 -0.046 -0.046 -0.063
## 42 EmpatheticAnger__1 ~~ Moraloutrage__6 0.517 0.026 0.026 0.039
## 43 EmpatheticAnger__2 ~~ EmpatheticAnger__3 7.341 -0.071 -0.071 -0.158
## 44 EmpatheticAnger__2 ~~ EmpatheticAnger__4 0.078 -0.009 -0.009 -0.015
## 45 EmpatheticAnger__2 ~~ EmpatheticAnger__5 1.960 -0.038 -0.038 -0.081
## 46 EmpatheticAnger__2 ~~ EmpatheticAnger__6 1.514 0.034 0.034 0.073
## 47 EmpatheticAnger__2 ~~ Moraloutrage__2 2.227 0.060 0.060 0.079
## 48 EmpatheticAnger__2 ~~ Moraloutrage__3 0.473 0.022 0.022 0.043
## 49 EmpatheticAnger__2 ~~ Moraloutrage__5 0.172 0.016 0.016 0.022
## 50 EmpatheticAnger__2 ~~ Moraloutrage__6 6.404 -0.091 -0.091 -0.138
## 51 EmpatheticAnger__3 ~~ EmpatheticAnger__4 2.135 0.043 0.043 0.083
## 52 EmpatheticAnger__3 ~~ EmpatheticAnger__5 4.631 0.052 0.052 0.137
## 53 EmpatheticAnger__3 ~~ EmpatheticAnger__6 0.868 -0.023 -0.023 -0.060
## 54 EmpatheticAnger__3 ~~ Moraloutrage__2 12.370 -0.123 -0.123 -0.195
## 55 EmpatheticAnger__3 ~~ Moraloutrage__3 4.822 0.061 0.061 0.144
## 56 EmpatheticAnger__3 ~~ Moraloutrage__5 1.729 0.043 0.043 0.072
## 57 EmpatheticAnger__3 ~~ Moraloutrage__6 0.175 0.013 0.013 0.024
## 58 EmpatheticAnger__4 ~~ EmpatheticAnger__5 0.226 0.015 0.015 0.027
## 59 EmpatheticAnger__4 ~~ EmpatheticAnger__6 0.530 -0.023 -0.023 -0.041
## 60 EmpatheticAnger__4 ~~ Moraloutrage__2 4.300 0.097 0.097 0.108
## 61 EmpatheticAnger__4 ~~ Moraloutrage__3 0.094 0.011 0.011 0.019
## 62 EmpatheticAnger__4 ~~ Moraloutrage__5 1.714 -0.058 -0.058 -0.068
## 63 EmpatheticAnger__4 ~~ Moraloutrage__6 0.548 -0.031 -0.031 -0.040
## 64 EmpatheticAnger__5 ~~ EmpatheticAnger__6 1.354 0.030 0.030 0.075
## 65 EmpatheticAnger__5 ~~ Moraloutrage__2 0.645 -0.029 -0.029 -0.044
## 66 EmpatheticAnger__5 ~~ Moraloutrage__3 0.075 -0.008 -0.008 -0.018
## 67 EmpatheticAnger__5 ~~ Moraloutrage__5 0.547 0.025 0.025 0.040
## 68 EmpatheticAnger__5 ~~ Moraloutrage__6 0.140 0.012 0.012 0.021
## 69 EmpatheticAnger__6 ~~ Moraloutrage__2 0.068 0.010 0.010 0.015
## 70 EmpatheticAnger__6 ~~ Moraloutrage__3 3.203 -0.052 -0.052 -0.118
## 71 EmpatheticAnger__6 ~~ Moraloutrage__5 0.912 -0.033 -0.033 -0.053
## 72 EmpatheticAnger__6 ~~ Moraloutrage__6 0.355 0.019 0.019 0.034
## 73 Moraloutrage__2 ~~ Moraloutrage__3 8.980 0.194 0.194 0.270
## 74 Moraloutrage__2 ~~ Moraloutrage__5 1.193 -0.067 -0.067 -0.066
## 75 Moraloutrage__2 ~~ Moraloutrage__6 1.157 -0.067 -0.067 -0.071
## 76 Moraloutrage__3 ~~ Moraloutrage__5 7.109 -0.157 -0.157 -0.228
## 77 Moraloutrage__3 ~~ Moraloutrage__6 5.965 -0.154 -0.154 -0.246
## 78 Moraloutrage__5 ~~ Moraloutrage__6 19.024 0.250 0.250 0.279
## sepc.nox
## 24 -0.063
## 25 0.094
## 26 -0.033
## 27 -0.037
## 28 0.054
## 29 0.002
## 30 0.044
## 31 0.021
## 32 -0.002
## 33 -0.094
## 34 0.191
## 35 -0.046
## 36 -0.075
## 37 -0.162
## 38 0.053
## 39 0.013
## 40 0.062
## 41 -0.063
## 42 0.039
## 43 -0.158
## 44 -0.015
## 45 -0.081
## 46 0.073
## 47 0.079
## 48 0.043
## 49 0.022
## 50 -0.138
## 51 0.083
## 52 0.137
## 53 -0.060
## 54 -0.195
## 55 0.144
## 56 0.072
## 57 0.024
## 58 0.027
## 59 -0.041
## 60 0.108
## 61 0.019
## 62 -0.068
## 63 -0.040
## 64 0.075
## 65 -0.044
## 66 -0.018
## 67 0.040
## 68 0.021
## 69 0.015
## 70 -0.118
## 71 -0.053
## 72 0.034
## 73 0.270
## 74 -0.066
## 75 -0.071
## 76 -0.228
## 77 -0.246
## 78 0.279Bifactor Model
CFA_bifactor <- ' EmpAnger =~ EmpatheticAnger__1 + EmpatheticAnger__2 + EmpatheticAnger__3 +
EmpatheticAnger__4 + EmpatheticAnger__5 + EmpatheticAnger__6
MoralOutrage =~ Moraloutrage__2 + Moraloutrage__3 + Moraloutrage__5 + Moraloutrage__6
g =~ EmpatheticAnger__1 + EmpatheticAnger__2 + EmpatheticAnger__3 +
EmpatheticAnger__4 + EmpatheticAnger__5 + EmpatheticAnger__6 + Moraloutrage__2 + Moraloutrage__3 + Moraloutrage__5 + Moraloutrage__6'
CFA_bifactor_fit <- cfa(data = Mydata,CFA_bifactor,std.lv=T, orthogonal=T, estimator="MLR")## Warning: lavaan->lav_object_post_check():
## some estimated ov variances are negativesummary(CFA_bifactor_fit,std=T,fit.measures=T,rsquare=T)## lavaan 0.6-19 ended normally after 71 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 30
##
## Number of observations 482
##
## Model Test User Model:
## Standard Scaled
## Test Statistic 52.843 26.662
## Degrees of freedom 25 25
## P-value (Chi-square) 0.001 0.373
## Scaling correction factor 1.982
## Yuan-Bentler correction (Mplus variant)
##
## Model Test Baseline Model:
##
## Test statistic 5032.939 2344.732
## Degrees of freedom 45 45
## P-value 0.000 0.000
## Scaling correction factor 2.146
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.994 0.999
## Tucker-Lewis Index (TLI) 0.990 0.999
##
## Robust Comparative Fit Index (CFI) 0.999
## Robust Tucker-Lewis Index (TLI) 0.999
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -6880.970 -6880.970
## Scaling correction factor 1.871
## for the MLR correction
## Loglikelihood unrestricted model (H1) NA NA
## Scaling correction factor 1.922
## for the MLR correction
##
## Akaike (AIC) 13821.940 13821.940
## Bayesian (BIC) 13947.279 13947.279
## Sample-size adjusted Bayesian (SABIC) 13852.061 13852.061
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.048 0.012
## 90 Percent confidence interval - lower 0.030 0.000
## 90 Percent confidence interval - upper 0.066 0.032
## P-value H_0: RMSEA <= 0.050 0.543 1.000
## P-value H_0: RMSEA >= 0.080 0.001 0.000
##
## Robust RMSEA 0.017
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.055
## P-value H_0: Robust RMSEA <= 0.050 0.914
## P-value H_0: Robust RMSEA >= 0.080 0.001
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.011 0.011
##
## Parameter Estimates:
##
## Standard errors Sandwich
## Information bread Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## EmpAnger =~
## EmpthtcAngr__1 0.938 0.053 17.758 0.000 0.938 0.624
## EmpthtcAngr__2 0.955 0.056 17.107 0.000 0.955 0.644
## EmpthtcAngr__3 0.960 0.054 17.865 0.000 0.960 0.651
## EmpthtcAngr__4 0.978 0.063 15.561 0.000 0.978 0.616
## EmpthtcAngr__5 1.015 0.055 18.552 0.000 1.015 0.675
## EmpthtcAngr__6 1.116 0.050 22.437 0.000 1.116 0.718
## MoralOutrage =~
## Moraloutrag__2 0.110 0.131 0.841 0.401 0.110 0.053
## Moraloutrag__3 -0.819 1.213 -0.675 0.500 -0.819 -0.406
## Moraloutrag__5 0.323 0.298 1.086 0.277 0.323 0.170
## Moraloutrag__6 0.335 0.325 1.032 0.302 0.335 0.170
## g =~
## EmpthtcAngr__1 0.913 0.069 13.203 0.000 0.913 0.608
## EmpthtcAngr__2 0.865 0.072 12.063 0.000 0.865 0.583
## EmpthtcAngr__3 0.936 0.065 14.376 0.000 0.936 0.635
## EmpthtcAngr__4 0.903 0.075 11.983 0.000 0.903 0.569
## EmpthtcAngr__5 0.916 0.070 13.056 0.000 0.916 0.608
## EmpthtcAngr__6 0.894 0.075 11.908 0.000 0.894 0.575
## Moraloutrag__2 1.749 0.064 27.420 0.000 1.749 0.844
## Moraloutrag__3 2.024 0.064 31.722 0.000 2.024 1.004
## Moraloutrag__5 1.630 0.082 19.927 0.000 1.630 0.856
## Moraloutrag__6 1.755 0.074 23.655 0.000 1.755 0.892
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## EmpAnger ~~
## MoralOutrage 0.000 0.000 0.000
## g 0.000 0.000 0.000
## MoralOutrage ~~
## g 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .EmpthtcAngr__1 0.545 0.073 7.512 0.000 0.545 0.241
## .EmpthtcAngr__2 0.539 0.097 5.553 0.000 0.539 0.245
## .EmpthtcAngr__3 0.375 0.055 6.802 0.000 0.375 0.173
## .EmpthtcAngr__4 0.751 0.111 6.792 0.000 0.751 0.298
## .EmpthtcAngr__5 0.396 0.055 7.227 0.000 0.396 0.175
## .EmpthtcAngr__6 0.371 0.076 4.913 0.000 0.371 0.154
## .Moraloutrag__2 1.223 0.180 6.795 0.000 1.223 0.285
## .Moraloutrag__3 -0.703 2.138 -0.329 0.742 -0.703 -0.173
## .Moraloutrag__5 0.864 0.108 7.973 0.000 0.864 0.238
## .Moraloutrag__6 0.677 0.143 4.743 0.000 0.677 0.175
## EmpAnger 1.000 1.000 1.000
## MoralOutrage 1.000 1.000 1.000
## g 1.000 1.000 1.000
##
## R-Square:
## Estimate
## EmpthtcAngr__1 0.759
## EmpthtcAngr__2 0.755
## EmpthtcAngr__3 0.827
## EmpthtcAngr__4 0.702
## EmpthtcAngr__5 0.825
## EmpthtcAngr__6 0.846
## Moraloutrag__2 0.715
## Moraloutrag__3 NA
## Moraloutrag__5 0.762
## Moraloutrag__6 0.825Standardized loadings on the g factor indicate the semi-partial correlation between the item and a general factor underlying all the items.
Standardized loadings on the specific factors indicate the semi-partial correlation between the item and the specific factors underlying the subset of items.
Model Comparison
factorcomp_nested <- compareFit(cfa_fit, CFA_bifactor_fit, nested = TRUE)
summary(factorcomp_nested)## ################### Nested Model Comparison #########################
##
## Scaled Chi-Squared Difference Test (method = "satorra.bentler.2001")
##
## lavaan->unknown():
## 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)
## CFA_bifactor_fit 25 13822 13947 52.8
## cfa_fit 34 13841 13929 90.1 29 9 0.00064 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## ####################### Model Fit Indices ###########################
## chisq.scaled df.scaled pvalue.scaled rmsea.robust cfi.robust
## CFA_bifactor_fit 26.662† 25 .373 .017† 0.999†
## cfa_fit 50.157 34 .037 .042 .994
## tli.robust srmr aic bic
## CFA_bifactor_fit 0.999† .011† 13821.940† 13947.279
## cfa_fit .992 .019 13841.240 13928.977†
##
## ################## Differences in Fit Indices #######################
## df.scaled rmsea.robust cfi.robust tli.robust srmr
## cfa_fit - CFA_bifactor_fit 9 0.026 -0.005 -0.007 0.008
## aic bic
## cfa_fit - CFA_bifactor_fit 19.3 -18.3