How to use Lavaan for Confirmatory Factor Analysis

This is one of my homework in Structural Equation Modeling in Fall 2017. Dr. Templin provided a excelent example showing how to perform Confirmatory Factor Analysis (CFA) using Lavaan Package. I elaborated each step as following.

  • First, load packages needed: If you don’t have the packages installed below, please use install.packages() to install them.
library(tidyverse)
library(lavaan)
#library(semPlot)
library(psych)
library(knitr)
library(kableExtra)

0. Background

CFA on Attitude towards Inclusive Education Survey (N = 507)

The affective dimension of attitudes subscale includes 6 items on a 6-point likert scale (1 = Strongly Agree, 6 = Strongly Disagree), measuring teachers’ feelings and emotions associated with inclusive education:

  1. I get frustrated when I have difficulty communicating with students with a disability.
  2. I get upset when students with a disability cannot keep up with the day-to-day curriculum in my classroom.
  3. I get irritated when I am unable to understand students with a disability.
  4. I am uncomfortable including students with a disability in a regular classroom with other students without a disability.
  5. I am disconcerted that students with a disability are included in the regular classroom, regardless of the severity of the disability.
  6. I get frustrated when I have to adapt the curriculum to meet the individual needs of all students.

The sample size (N) is 507, which includes 6 males and 501 females. I used one-factor model as first step. All items are loaded on one general factor - affective attitude towards inclusive education. Higher response score means more positive attitude towards inclusive education.

dat <- read.csv("../../static/files/AttitudeForInclusiveEducation.csv")
# head(dat)
dat2 <- dat %>% select(X,Aff.1:Aff.6)
colnames(dat2) <- c("PersonID", paste0("Aff",1:6))

1. Descriptive Statistics

The descriptive statistics for all items are provided below. It appears that item 4 is the least difficult item as it has the highest mean (\(\mu = 4.189, sd = 1.317\)); item 5 is the most difficult item as it has lowest mean score (\(\mu = 3.604, sd = 1.423\)). All responses for each item range from 1 to 6 (1 = Strongly agree, 6 = Strongly disagree). Thus, all categories are responded. In term of item discrimination, as item 3 has the largest standard deviation (\(sd = 1.364\)) and item 6 has the smallest, item 3 has highest discrimination whearas item 6 has lowest in CTT.

vars n mean sd median trimmed mad min max range skew kurtosis se
PersonID 1 507 254.000 146.503 254 254.000 188.290 1 507 506 0.000 -1.207 6.506
Aff1 2 507 3.765 1.337 4 3.779 1.483 1 6 5 -0.131 -0.927 0.059
Aff2 3 507 3.635 1.335 4 3.636 1.483 1 6 5 -0.026 -0.963 0.059
Aff3 4 507 3.493 1.364 3 3.472 1.483 1 6 5 0.124 -0.969 0.061
Aff4 5 507 4.189 1.317 4 4.287 1.483 1 6 5 -0.589 -0.327 0.058
Aff5 6 507 3.604 1.423 4 3.590 1.483 1 6 5 0.000 -0.939 0.063
Aff6 7 507 4.018 1.313 4 4.061 1.483 1 6 5 -0.356 -0.733 0.058
Item-total correlation table was provided below. All item-total correlation coefficients are higher than 0.7, which suggests good internal consistence. Item 1 has lowest item-total correlation (\(r = 0.733, sd = 1.337\)).
Table 1: Item-total Correlation Table
n raw.r std.r r.cor r.drop mean sd
Aff1 507 0.733 0.735 0.652 0.611 3.765 1.337
Aff2 507 0.835 0.836 0.806 0.753 3.635 1.335
Aff3 507 0.813 0.812 0.771 0.718 3.493 1.364
Aff4 507 0.789 0.790 0.742 0.689 4.189 1.317
Aff5 507 0.774 0.769 0.702 0.658 3.604 1.423
Aff6 507 0.836 0.838 0.805 0.755 4.018 1.313

Sample Correlation Matrix

According to Pearson Correlation Matrix below, we can see all items have fairly high pearson correlation coefficients ranging from 0.44 to 0.72. This provides the evidence of dimensionality. Item 2 and item 3 has highest correlation coefficient(\(r_{23} = 0.717\)). The lowest correlations lies between item 1 and item 4 as well as item 1 and item 5.

cor(dat2[2:7]) %>% round(3) %>% kable(caption = "Pearson Correlation Matrix")
Table 2: Pearson Correlation Matrix
Aff1 Aff2 Aff3 Aff4 Aff5 Aff6
Aff1 1.000 0.590 0.525 0.448 0.411 0.538
Aff2 0.590 1.000 0.717 0.534 0.553 0.602
Aff3 0.525 0.717 1.000 0.505 0.527 0.608
Aff4 0.448 0.534 0.505 1.000 0.609 0.682
Aff5 0.411 0.553 0.527 0.609 1.000 0.577
Aff6 0.538 0.602 0.608 0.682 0.577 1.000

Sample Mean and Variance

means <- dat2[,2:7] %>% 
  summarise_all(funs(mean)) %>% round(3) %>% t() %>% as.data.frame()
## Warning: funs() is soft deprecated as of dplyr 0.8.0
## Please use a list of either functions or lambdas: 
## 
##   # Simple named list: 
##   list(mean = mean, median = median)
## 
##   # Auto named with `tibble::lst()`: 
##   tibble::lst(mean, median)
## 
##   # Using lambdas
##   list(~ mean(., trim = .2), ~ median(., na.rm = TRUE))
## This warning is displayed once per session.
sds <- dat2[,2:7] %>% 
  summarise_all(funs(sd)) %>% round(3) %>% t() %>% as.data.frame()
table1 <- cbind(means,sds)

colnames(table1) <- c("Mean", "SD")
table1
##       Mean    SD
## Aff1 3.765 1.337
## Aff2 3.635 1.335
## Aff3 3.493 1.364
## Aff4 4.189 1.317
## Aff5 3.604 1.423
## Aff6 4.018 1.313

Sample Item Response Distributions

Those items did not exactly match normal distribution but acceptable.

# stack data
dat2_melted <- dat2 %>% gather(key, value,Aff1:Aff6) %>% arrange(PersonID)

# plot by variable
ggplot(dat2_melted, aes(value)) + 
  geom_histogram(aes(y=..density..), colour="black", fill="white", binwidth = 1) +
  geom_density(alpha=.2, fill="#FF6666") +
  scale_x_continuous(breaks = 1:6) +
  facet_wrap(~ key)

3. Estimation with CFA

One-factor Model

One-factor model was conducted as first step. The model has one latent facor - affective attitude and six indicators. In general, one-factor model does not provide great model fit except SRMR. The test statistics for chi-square is 75.835 (\(p < 0.05\)). CFI is 0.929, which larger than 0.95 suggests good model fit. RMSEA is 0.121, which lower than 0.05 suggest good model fit. SRMR is 0.04, which lower than 0.08. The standardized factor loadings range from 0.66 to 0.8. All factor loadings are significant at the level of alpha equals 0.05.

model1.syntax <- '
  AA =~ Aff1 + Aff2 + Aff3 + Aff4 + Aff5 + Aff6
'
model1 <- cfa(model1.syntax, data = dat2,std.lv = TRUE, mimic = "mplus", estimator = "MLR")
summary(model1, fit.measures = TRUE, standardized = TRUE)
## lavaan 0.6-5 ended normally after 13 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of free parameters                         18
##                                                       
##   Number of observations                           507
##   Number of missing patterns                         1
##                                                       
## Model Test User Model:
##                                               Standard      Robust
##   Test Statistic                               115.410      75.834
##   Degrees of freedom                                 9           9
##   P-value (Chi-square)                           0.000       0.000
##   Scaling correction factor                                  1.522
##     for the Yuan-Bentler correction (Mplus variant) 
## 
## Model Test Baseline Model:
## 
##   Test statistic                              1573.730     960.267
##   Degrees of freedom                                15          15
##   P-value                                        0.000       0.000
##   Scaling correction factor                                  1.639
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.932       0.929
##   Tucker-Lewis Index (TLI)                       0.886       0.882
##                                                                   
##   Robust Comparative Fit Index (CFI)                         0.934
##   Robust Tucker-Lewis Index (TLI)                            0.891
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -4492.146   -4492.146
##   Scaling correction factor                                  1.138
##       for the MLR correction                                      
##   Loglikelihood unrestricted model (H1)      -4434.440   -4434.440
##   Scaling correction factor                                  1.266
##       for the MLR correction                                      
##                                                                   
##   Akaike (AIC)                                9020.291    9020.291
##   Bayesian (BIC)                              9096.404    9096.404
##   Sample-size adjusted Bayesian (BIC)         9039.270    9039.270
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.153       0.121
##   90 Percent confidence interval - lower         0.129       0.101
##   90 Percent confidence interval - upper         0.178       0.142
##   P-value RMSEA <= 0.05                          0.000       0.000
##                                                                   
##   Robust RMSEA                                               0.149
##   90 Percent confidence interval - lower                     0.119
##   90 Percent confidence interval - upper                     0.181
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.040       0.040
## 
## Parameter Estimates:
## 
##   Information                                      Observed
##   Observed information based on                     Hessian
##   Standard errors                        Robust.huber.white
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   AA =~                                                                 
##     Aff1              0.886    0.055   16.183    0.000    0.886    0.663
##     Aff2              1.078    0.046   23.284    0.000    1.078    0.808
##     Aff3              1.066    0.055   19.499    0.000    1.066    0.783
##     Aff4              0.968    0.061   15.934    0.000    0.968    0.736
##     Aff5              1.004    0.055   18.291    0.000    1.004    0.706
##     Aff6              1.057    0.049   21.644    0.000    1.057    0.805
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Aff1              3.765    0.059   63.455    0.000    3.765    2.818
##    .Aff2              3.635    0.059   61.382    0.000    3.635    2.726
##    .Aff3              3.493    0.060   57.741    0.000    3.493    2.564
##    .Aff4              4.189    0.058   71.689    0.000    4.189    3.184
##    .Aff5              3.604    0.063   57.057    0.000    3.604    2.534
##    .Aff6              4.018    0.058   68.948    0.000    4.018    3.062
##     AA                0.000                               0.000    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Aff1              1.000    0.081   12.285    0.000    1.000    0.560
##    .Aff2              0.617    0.066    9.283    0.000    0.617    0.347
##    .Aff3              0.719    0.089    8.089    0.000    0.719    0.387
##    .Aff4              0.794    0.095    8.390    0.000    0.794    0.459
##    .Aff5              1.014    0.090   11.226    0.000    1.014    0.501
##    .Aff6              0.605    0.068    8.898    0.000    0.605    0.351
##     AA                1.000                               1.000    1.000

Local Misfit for One-factor Model

By looking into local misfit with residual variance-covariance matrix we can get the clues to improve the model. According to the model residuals, item 4 has relatively high positive residual covariance with item 5 and item 6. It suggests that the one-factor model underestimates the correlations among item 4, item 5 and item 6. In other words, another latent factor may be needed to explain the strong relations among item 4, 5, 6 which cannot be explained by a general Affective attitude factor.

Moreover, modification indices below also suggest that adding the error covariances among item 4, 5 and 6 will improve chi-square much better.

Thus, I decided to add one more factor - AAE. AAE was labeled as affective attitude towards educational environment which indicated by item 4, 5, 6. The other latent factor - AAC which was indicated by item 1, 2, 3 was labeled as Affective Attitude towards communication.

resid(model1)$cov %>% kable(caption = "Normalized Residual Variance-Covariance Matrix",digits = 3)
Table 3: Normalized Residual Variance-Covariance Matrix
Aff1 Aff2 Aff3 Aff4 Aff5 Aff6
Aff1 0.000 0.095 0.011 -0.070 -0.109 0.006
Aff2 0.095 0.000 0.153 -0.106 -0.034 -0.085
Aff3 0.011 0.153 0.000 -0.128 -0.051 -0.041
Aff4 -0.070 -0.106 -0.128 0.000 0.168 0.155
Aff5 -0.109 -0.034 -0.051 0.168 0.000 0.015
Aff6 0.006 -0.085 -0.041 0.155 0.015 0.000
modificationindices(model1, standardized = TRUE,sort. = TRUE) %>% slice(1:10) %>% kable(caption = "Modification Indices", digits = 3)
Table 3: Modification Indices
lhs op rhs mi epc sepc.lv sepc.all sepc.nox
Aff2 ~~ Aff3 55.906 0.319 0.319 0.479 0.479
Aff4 ~~ Aff6 45.674 0.279 0.279 0.403 0.403
Aff4 ~~ Aff5 25.673 0.243 0.243 0.270 0.270
Aff3 ~~ Aff4 24.229 -0.214 -0.214 -0.283 -0.283
Aff2 ~~ Aff6 22.493 -0.194 -0.194 -0.318 -0.318
Aff2 ~~ Aff4 21.303 -0.193 -0.193 -0.276 -0.276
Aff1 ~~ Aff2 11.974 0.153 0.153 0.195 0.195
Aff1 ~~ Aff5 7.918 -0.145 -0.145 -0.144 -0.144
Aff1 ~~ Aff4 4.309 -0.096 -0.096 -0.108 -0.108
Aff3 ~~ Aff6 4.018 -0.084 -0.084 -0.128 -0.128

Two-factor Model

The neccessity of adding another factor was tested by specifying a two-factor model.

In term of model fit indices, it appears that the global model fit indices are great with two-factor model (CFI = 0.986; RMSEA = 0.058; SRMR = 0.022). Ideally, two latent factors could be labeled as moderately correlated aspects of attitudes towards inclusive education. Thus, the first factor (AAC) could be labeled as how teachers feel about communicating with students with disability. The second (AAE) could be labeled as how teachers feel about evironment of inclusive education. All standardized factor loadings are statistically significant ranging from 0.676 to 0.865. The factor correlation between 2 factors is high (\(r = 0.838,p = 0.00\)).

model2.syntax <- '
  AAC =~ Aff1 + Aff2 + Aff3
  AAE =~ Aff4 + Aff5 + Aff6
'
model2 <- cfa(model2.syntax, data = dat2, std.lv = TRUE, mimic = "mplus", estimator = "MLR")
summary(model2, fit.measures = TRUE, standardized = TRUE)
## lavaan 0.6-5 ended normally after 19 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of free parameters                         19
##                                                       
##   Number of observations                           507
##   Number of missing patterns                         1
##                                                       
## Model Test User Model:
##                                               Standard      Robust
##   Test Statistic                                32.332      21.512
##   Degrees of freedom                                 8           8
##   P-value (Chi-square)                           0.000       0.006
##   Scaling correction factor                                  1.503
##     for the Yuan-Bentler correction (Mplus variant) 
## 
## Model Test Baseline Model:
## 
##   Test statistic                              1573.730     960.267
##   Degrees of freedom                                15          15
##   P-value                                        0.000       0.000
##   Scaling correction factor                                  1.639
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.984       0.986
##   Tucker-Lewis Index (TLI)                       0.971       0.973
##                                                                   
##   Robust Comparative Fit Index (CFI)                         0.987
##   Robust Tucker-Lewis Index (TLI)                            0.975
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)              -4450.606   -4450.606
##   Scaling correction factor                                  1.166
##       for the MLR correction                                      
##   Loglikelihood unrestricted model (H1)      -4434.440   -4434.440
##   Scaling correction factor                                  1.266
##       for the MLR correction                                      
##                                                                   
##   Akaike (AIC)                                8939.212    8939.212
##   Bayesian (BIC)                              9019.554    9019.554
##   Sample-size adjusted Bayesian (BIC)         8959.246    8959.246
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.077       0.058
##   90 Percent confidence interval - lower         0.051       0.034
##   90 Percent confidence interval - upper         0.106       0.082
##   P-value RMSEA <= 0.05                          0.046       0.268
##                                                                   
##   Robust RMSEA                                               0.071
##   90 Percent confidence interval - lower                     0.035
##   90 Percent confidence interval - upper                     0.108
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.022       0.022
## 
## Parameter Estimates:
## 
##   Information                                      Observed
##   Observed information based on                     Hessian
##   Standard errors                        Robust.huber.white
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   AAC =~                                                                
##     Aff1              0.903    0.055   16.357    0.000    0.903    0.676
##     Aff2              1.153    0.042   27.771    0.000    1.153    0.865
##     Aff3              1.117    0.051   22.079    0.000    1.117    0.820
##   AAE =~                                                                
##     Aff4              1.047    0.053   19.604    0.000    1.047    0.796
##     Aff5              1.036    0.053   19.366    0.000    1.036    0.728
##     Aff6              1.107    0.044   25.112    0.000    1.107    0.844
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   AAC ~~                                                                
##     AAE               0.838    0.028   29.829    0.000    0.838    0.838
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Aff1              3.765    0.059   63.455    0.000    3.765    2.818
##    .Aff2              3.635    0.059   61.382    0.000    3.635    2.726
##    .Aff3              3.493    0.060   57.741    0.000    3.493    2.564
##    .Aff4              4.189    0.058   71.689    0.000    4.189    3.184
##    .Aff5              3.604    0.063   57.057    0.000    3.604    2.534
##    .Aff6              4.018    0.058   68.948    0.000    4.018    3.062
##     AAC               0.000                               0.000    0.000
##     AAE               0.000                               0.000    0.000
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .Aff1              0.970    0.083   11.753    0.000    0.970    0.543
##    .Aff2              0.449    0.057    7.860    0.000    0.449    0.253
##    .Aff3              0.607    0.084    7.244    0.000    0.607    0.327
##    .Aff4              0.635    0.085    7.504    0.000    0.635    0.367
##    .Aff5              0.950    0.090   10.539    0.000    0.950    0.470
##    .Aff6              0.497    0.061    8.191    0.000    0.497    0.288
##     AAC               1.000                               1.000    1.000
##     AAE               1.000                               1.000    1.000

Local Misfit for two-factor model

The local misfit indices for two-factor model also suggest that the model fits data well. The largest normalized residuals is 1.215. Modification indices suggest that add covariance between item 5 and item 6. These local misfit is not theoretically defensible. Thus, the final model is two-factor model.

resid(model2)$cov %>% kable(caption = "Normalized Residual Variance-Covariance Matrix",digits = 3)
Table 4: Normalized Residual Variance-Covariance Matrix
Aff1 Aff2 Aff3 Aff4 Aff5 Aff6
Aff1 0.000 0.010 -0.053 -0.005 -0.003 0.105
Aff2 0.010 0.000 0.014 -0.075 0.048 -0.016
Aff3 -0.053 0.014 0.000 -0.076 0.050 0.049
Aff4 -0.005 -0.075 -0.076 0.000 0.056 0.019
Aff5 -0.003 0.048 0.050 0.056 0.000 -0.070
Aff6 0.105 -0.016 0.049 0.019 -0.070 0.000
modificationindices(model2, standardized = TRUE,sort. = TRUE) %>% slice(1:10) %>% kable(caption = "Modification Indices", digits = 3)
Table 4: Modification Indices
lhs op rhs mi epc sepc.lv sepc.all sepc.nox
Aff5 ~~ Aff6 16.894 -0.224 -0.224 -0.326 -0.326
AAC =~ Aff4 16.893 -0.578 -0.578 -0.439 -0.439
Aff1 ~~ Aff6 7.064 0.108 0.108 0.155 0.155
Aff4 ~~ Aff5 5.977 0.128 0.128 0.164 0.164
AAC =~ Aff6 5.976 0.368 0.368 0.280 0.280
Aff1 ~~ Aff3 5.093 -0.112 -0.112 -0.146 -0.146
AAE =~ Aff2 5.093 -0.362 -0.362 -0.272 -0.272
Aff3 ~~ Aff4 4.045 -0.077 -0.077 -0.124 -0.124
Aff2 ~~ Aff3 3.160 0.119 0.119 0.228 0.228
AAE =~ Aff1 3.160 0.236 0.236 0.176 0.176

4. Path Diagrams

#semPlot::semPaths(model2, what = "est")

5. Reliability

To get the estimates of reliabilities, Omega coefficients were calculated for each factor(\(\Omega_{AAC} = 0.832, p < 0.01; \Omega_{AAE} = 0.830, p < 0.01\)).

model03SyntaxOmega = "
  # AAC loadings (all estimated)
  AAC =~ L1*Aff1 + L2*Aff2 + L3*Aff3
  
  # AAE loadings (all estimated)
  AAE =~ L4*Aff4 + L5*Aff5 + L6*Aff6
  
  # Unique Variances:
  Aff1 ~~ E1*Aff1; Aff2 ~~ E2*Aff2; Aff3 ~~ E3*Aff3; Aff4 ~~ E4*Aff4; Aff5 ~~ E5*Aff5; Aff6 ~~ E6*Aff6; 
  
  
  # Calculate Omega Reliability for Sum Scores:
  OmegaAAC := ((L1 + L2 + L3)^2) / ( ((L1 + L2 + L3)^2) + E1 + E2 + E3)
  OmegaAAE := ((L4 + L5 + L6)^2) / ( ((L4 + L5 + L6)^2) + E4 + E5 + E6)
"

model03EstimatesOmega = sem(model = model03SyntaxOmega, data = dat2, estimator = "MLR", mimic = "mplus", std.lv = TRUE)
summary(model03EstimatesOmega, fit.measures = FALSE, rsquare = FALSE, standardized = FALSE, header = FALSE)
## 
## Parameter Estimates:
## 
##   Information                                      Observed
##   Observed information based on                     Hessian
##   Standard errors                        Robust.huber.white
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   AAC =~                                              
##     Aff1      (L1)    0.903    0.055   16.357    0.000
##     Aff2      (L2)    1.153    0.042   27.771    0.000
##     Aff3      (L3)    1.117    0.051   22.079    0.000
##   AAE =~                                              
##     Aff4      (L4)    1.047    0.053   19.604    0.000
##     Aff5      (L5)    1.036    0.053   19.366    0.000
##     Aff6      (L6)    1.107    0.044   25.112    0.000
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   AAC ~~                                              
##     AAE               0.838    0.028   29.829    0.000
## 
## Intercepts:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .Aff1              3.765    0.059   63.455    0.000
##    .Aff2              3.635    0.059   61.382    0.000
##    .Aff3              3.493    0.060   57.741    0.000
##    .Aff4              4.189    0.058   71.689    0.000
##    .Aff5              3.604    0.063   57.057    0.000
##    .Aff6              4.018    0.058   68.948    0.000
##     AAC               0.000                           
##     AAE               0.000                           
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .Aff1      (E1)    0.970    0.083   11.753    0.000
##    .Aff2      (E2)    0.449    0.057    7.860    0.000
##    .Aff3      (E3)    0.607    0.084    7.244    0.000
##    .Aff4      (E4)    0.635    0.085    7.504    0.000
##    .Aff5      (E5)    0.950    0.090   10.539    0.000
##    .Aff6      (E6)    0.497    0.061    8.191    0.000
##     AAC               1.000                           
##     AAE               1.000                           
## 
## Defined Parameters:
##                    Estimate  Std.Err  z-value  P(>|z|)
##     OmegaAAC          0.832    0.016   51.512    0.000
##     OmegaAAE          0.830    0.017   49.000    0.000

6. Factor Scores and Distributions

The AAC factor scores have an estimated mean of 0 with a variance of 0.88 due to the effect of the prior distribution. The SE for each person’s AAC factor score is 0.347; 95% confidence interval for AAC factor score is \(Score \pm 2*0.347\) = \(Score \pm 0.694\). The AAE factor scores have an estimated mean of 0 with a variance of 0.881 due to the effect of the prior distribution. The SE for each person’s AAC factor score is 0.357; 95% confidence interval for AAC factor score is \(Score \pm 2*0.357\) = \(Score \pm 0.714\).

Factor Realiability for AAC is 0.892 and factor realibility for AAE is 0.887. Both factor reliability are larger than omega.

The resulting distribution of the EAP estimates of factor score as shown in Figure 1. Figure 2 shows the predicted response for each item as a linear function of the latent factor based on the estimated model parameters. As shown, for AAE factor, the predicted item response goes above the highest response option just before a latent factor score of 2 (i.e., 2 SDs above the mean), resulting in a ceiling effect for AAE factor, as also shown in Figure 1.

The extent to which the items within each factor could be seen as exchangeable was then examined via an additional set of nested model comparisons, as reported in Table 1 (for fit) and Table 2 (for comparisons of fit). Two-factor has better model fit than one-facor model. Moreover, according to chi-square difference test, two-factor is significantly better than one-factor in model fit.

##       AAC       AAE 
## 0.8925361 0.8867031

7. Figures

Figure 1 : Factor Score Distribution

## Warning: Removed 2 rows containing missing values (geom_bar).

## Warning: Removed 2 rows containing missing values (geom_bar).

Figure 2 : Expected Item Response Plots

8. Tables

Table 1: Model Fit Statistics Using MLR

# Items # Parameters Scaled Chi-Square Chi-Square Scale Factor DF p-value CFI RMSEA RMSEA Lower RMSEA Upper RMSEA p-value
One-Factor 6 18 75.834 1.522 9 0.000 0.929 0.121 0.101 0.142 0.000
Two-Factor 6 19 21.512 1.503 8 0.006 0.986 0.058 0.034 0.082 0.268

Table 2: Model Comparisons

Df Chisq diff Df diff Pr(>Chisq)
One-Factor vs. Two-Factor 9 49.652 1 0

Table 3: Model Estimates

Unstandardized
Standardized
Estimate SE Estimate
Forgiveness Factor Loadings
Item 1 0.903 0.055 0.676
Item 2 1.153 0.042 0.865
Item 3 1.117 0.051 0.820
Not Unforgiveness Factor Loadings
Item 4 1.047 0.053 0.796
Item 5 1.036 0.053 0.728
Item 6 1.107 0.044 0.844
Factor Covariance
Factor Covariance 0.838 0.028 0.838
Item Intercepts
Item 1 3.765 0.059 2.818
Item 2 3.635 0.059 2.726
Item 3 3.493 0.060 2.564
Item 4 4.189 0.058 3.184
Item 5 3.604 0.063 2.534
Item 6 4.018 0.058 3.062
Item Unique Variances
Item 1 0.970 0.083 0.543
Item 2 0.449 0.057 0.253
Item 3 0.607 0.084 0.327
Item 4 0.635 0.085 0.367
Item 5 0.950 0.090 0.470
Item 6 0.497 0.061 0.288