# How to use lavaan package to perform Confirmatory Factor Analysis

Manual
R
Lavaan
Author

Jihong

Published

October 19, 2017

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)

## 1 Background

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

## 2 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).

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

### 2.1 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")
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

### 2.2 Sample Mean and Variance

means <- dat2[,2:7] %>%
summarise_all(funs(mean)) %>% round(3) %>% t() %>% as.data.frame()
Warning: funs() was deprecated in dplyr 0.8.0.
ℹ 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))
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

### 2.3 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)
Warning: The dot-dot notation (..density..) was deprecated in ggplot2 3.4.0.
ℹ Please use after_stat(density) instead.

## 3 Estimation with CFA

### 3.1 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 0.6.13 ended normally after 14 iterations

Estimator                                         ML
Optimization method                           NLMINB
Number of model parameters                        18

Number of observations                           507
Number of missing patterns                         1

Model Test User Model:
Standard      Scaled
Test Statistic                               115.410      75.834
Degrees of freedom                                 9           9
P-value (Chi-square)                           0.000       0.000
Scaling correction factor                                  1.522
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 (SABIC)       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 H_0: RMSEA <= 0.050                    0.000       0.000
P-value H_0: RMSEA >= 0.080                    1.000       1.000

Robust RMSEA                                               0.149
90 Percent confidence interval - lower                     0.120
90 Percent confidence interval - upper                     0.181
P-value H_0: Robust RMSEA <= 0.050                         0.000
P-value H_0: Robust RMSEA >= 0.080                         1.000

Standardized Root Mean Square Residual:

SRMR                                           0.040       0.040

Parameter Estimates:

Standard errors                             Sandwich
Observed information based on                Hessian

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

### 3.2 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) 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) 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 24.228 -0.214 -0.214 -0.283 -0.283 Aff2 ~~ Aff6 22.494 -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 ### 3.3 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 0.6.13 ended normally after 19 iterations Estimator ML Optimization method NLMINB Number of model parameters 19 Number of observations 507 Number of missing patterns 1 Model Test User Model: Standard Scaled Test Statistic 32.332 21.512 Degrees of freedom 8 8 P-value (Chi-square) 0.000 0.006 Scaling correction factor 1.503 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 (SABIC) 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 H_0: RMSEA <= 0.050 0.046 0.268 P-value H_0: RMSEA >= 0.080 0.475 0.068 Robust RMSEA 0.071 90 Percent confidence interval - lower 0.036 90 Percent confidence interval - upper 0.108 P-value H_0: Robust RMSEA <= 0.050 0.148 P-value H_0: Robust RMSEA >= 0.080 0.375 Standardized Root Mean Square Residual: SRMR 0.022 0.022 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 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 ### 3.4 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)
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)
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 Diagram

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

## 5 Reliability in Factor Analysis

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 =~ L1*Aff1 + L2*Aff2 + L3*Aff3

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:

Standard errors                             Sandwich
Observed information based on                Hessian

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

### 7.1 Figure 1 : Factor Score Distribution

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

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

## 8 Tables

### 8.1 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 0.927 0.9 0.000 0.05 1.000
Two-Factor 6 19 21.512 1.503 8 0.006 0.979 0.9 0.046 0.05 0.475

### 8.2 Table 2: Model Comparisons

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

### 8.3 Table 3: Model Estimates

Unstandardized
Standardized
Estimate SE Estimate
Item 1 0.903 0.055 0.676
Item 2 1.153 0.042 0.865
Item 3 1.117 0.051 0.820
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