Factor Analysis - CFA
Last updated on 2025-06-24 | Edit this page
Overview
Questions
- What is a confirmatory factor analysis?
- How can I run CFA in R?
- How can I specify CFA models using lavaan?
- How can I interpret the results of a CFA and EFA in R?
Objectives
- Understand the difference between exploratory and confirmatory factor analysis
- Learn how to conduct confirmatory factor analysis in R
- Learn how to interpret the results of confirmatory factor analysis in R
Confirmatory Factor Analysis (CFA)
In the previous lesson, we approached the DASS as if we had no knowledge about the underlying factor structure. However, this is not really the case, since there is a specific theory proposing three interrelated factors. We can use confirmatory factor analysis to test whether this theory can adequately explain the item correlations found here.
Importantly, we not only need a theory of the number of factors, but also which items are supposed to load on which factor. This will be necessary when establishing our CFA model. Furthermore, we will test whether a one-factor model also is supported by the data and compare these two models, evaluating which one is better.
The main R package we will be using is lavaan. This
package is widely used for Structural Equation Modeling (SEM) in R. CFA
is a specific type of model in the general SEM framework. The lavaan
homepage provides a great tutorial on CFA, as well (https://lavaan.ugent.be/tutorial/cfa.html). For a more
detailed tutorial in SEM and mathematics behind CFA, please refer to the
lavaan homepage or this
introduction.
For now, let’s walk through the steps of a CFA in our specific case. First of all, in order to do confirmatory factor analysis, we need to have a hypothesis to confirm. How many factors, which variables are supposed to span which factor? Secondly, we need to investigate the correlation matrix for any issues (low correlations, correlations = 1). Then, we need to define our model properly using lavaan syntax. And finally, we can run our model and interpret our results.
Model Specification
Let’s start by defining the theory and the model syntax.
The DASS has 42 items and measures three scales: depression, anxiety, and stress. These scale are said to be intercorrelated, as they share common causes (genetic, environmental etc.). Furthermore, the DASS provides a detailed manual as to which items belong to which scales. The sequence of items and their respective scales is:
R
library(dplyr)
OUTPUT
Attaching package: 'dplyr'OUTPUT
The following objects are masked from 'package:stats':
filter, lagOUTPUT
The following objects are masked from 'package:base':
intersect, setdiff, setequal, unionR
dass_data <- read.csv("data/kaggle_dass/data.csv")
fa_data <- dass_data %>%
filter(testelapse < 600) %>%
select(starts_with("Q")& ends_with("A"))
item_scales_dass <- c(
"S", "A", "D", "A", "D", "S", "A",
"S", "A", "D", "S", "S", "D", "S",
"A", "D", "D", "S", "A", "A", "D",
"S", "A", "D", "A", "D", "S", "A",
"S", "A", "D", "S", "S", "D", "S",
"A", "D", "D", "S", "A", "A", "D"
)
item_scales_dass_data <- data.frame(
item_nr = 1:42,
scale = item_scales_dass
)
item_scales_dass_data
OUTPUT
item_nr scale
1 1 S
2 2 A
3 3 D
4 4 A
5 5 D
6 6 S
7 7 A
8 8 S
9 9 A
10 10 D
11 11 S
12 12 S
13 13 D
14 14 S
15 15 A
16 16 D
17 17 D
18 18 S
19 19 A
20 20 A
21 21 D
22 22 S
23 23 A
24 24 D
25 25 A
26 26 D
27 27 S
28 28 A
29 29 S
30 30 A
31 31 D
32 32 S
33 33 S
34 34 D
35 35 S
36 36 A
37 37 D
38 38 D
39 39 S
40 40 A
41 41 A
42 42 DScales vs. Factors
There is an important difference between scales and factors. Again, factors represent mathematical constructs discovered or constructed during factor analysis. Scales represent a collection of items bound together by the authors of the test and said to measure a certain construct. These two words cannot be used synonymously. Scales refer to item collections whereas factors refer to solutions of factor analysis. A scale can build a factor, in fact its items should span a common factor if the scale is constructed well. But a scale is not the same thing as a factor.
Now, in order to define our model we need to keep the distinction between manifest and latent variables in mind. Our manifest, measured variables are the items. The factors are the latent variable. We can also define which manifest variables contribute to which latent variables. In lavaan, this can be specified in the model code using a special syntax.
R
library(lavaan)
OUTPUT
This is lavaan 0.6-19
lavaan is FREE software! Please report any bugs.R
model_cfa_3facs <- c(
"
# Model Syntax
# =~ is like in linear regression
# left hand is latent factor, right hand manifest variables contributing
# Define which items load on which factor
depression =~ Q3A + Q5A + Q10A + Q13A + Q16A + Q17A + Q21A + Q24A + Q26A + Q31A + Q34A + Q37A + Q38A + Q42A
anxiety =~ Q2A + Q4A + Q7A + Q9A + Q15A + Q19A + Q20A + Q23A + Q25A + Q28A + Q30A + Q36A + Q40A + Q41A
stress =~ Q1A + Q6A + Q8A + Q11A + Q12A + Q14A + Q18A + Q22A + Q27A + Q29A + Q32A + Q33A + Q35A + Q39A
# Define correlations between factors using ~~
depression ~~ anxiety
depression ~~ stress
anxiety ~~ stress
"
)
Now, we can fit this model using cfa().
R
fit_cfa_3facs <- cfa(
model = model_cfa_3facs,
data = fa_data,
)
To inspect the model results, we use summary().
R
summary(fit_cfa_3facs, fit.measures = TRUE, standardize = TRUE)
OUTPUT
lavaan 0.6-19 ended normally after 46 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 87
Number of observations 37242
Model Test User Model:
Test statistic 107309.084
Degrees of freedom 816
P-value (Chi-square) 0.000
Model Test Baseline Model:
Test statistic 1037764.006
Degrees of freedom 861
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.897
Tucker-Lewis Index (TLI) 0.892
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -1857160.022
Loglikelihood unrestricted model (H1) -1803505.480
Akaike (AIC) 3714494.044
Bayesian (BIC) 3715235.735
Sample-size adjusted Bayesian (SABIC) 3714959.249
Root Mean Square Error of Approximation:
RMSEA 0.059
90 Percent confidence interval - lower 0.059
90 Percent confidence interval - upper 0.059
P-value H_0: RMSEA <= 0.050 0.000
P-value H_0: RMSEA >= 0.080 0.000
Standardized Root Mean Square Residual:
SRMR 0.042
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
depression =~
Q3A 1.000 0.808 0.776
Q5A 0.983 0.006 155.298 0.000 0.794 0.742
Q10A 1.156 0.007 175.483 0.000 0.934 0.818
Q13A 1.077 0.006 173.188 0.000 0.870 0.810
Q16A 1.077 0.006 165.717 0.000 0.870 0.782
Q17A 1.158 0.007 172.607 0.000 0.936 0.807
Q21A 1.221 0.007 182.633 0.000 0.986 0.844
Q24A 0.987 0.006 159.408 0.000 0.797 0.758
Q26A 1.020 0.006 162.876 0.000 0.824 0.771
Q31A 0.966 0.006 156.243 0.000 0.780 0.746
Q34A 1.169 0.007 175.678 0.000 0.944 0.819
Q37A 1.118 0.007 168.220 0.000 0.903 0.791
Q38A 1.229 0.007 180.035 0.000 0.993 0.834
Q42A 0.826 0.006 131.716 0.000 0.668 0.646
anxiety =~
Q2A 1.000 0.563 0.506
Q4A 1.282 0.014 93.089 0.000 0.722 0.691
Q7A 1.303 0.014 94.229 0.000 0.734 0.707
Q9A 1.323 0.014 93.485 0.000 0.745 0.696
Q15A 1.116 0.013 89.050 0.000 0.629 0.636
Q19A 1.077 0.013 83.014 0.000 0.606 0.565
Q20A 1.473 0.015 96.526 0.000 0.830 0.742
Q23A 0.898 0.011 84.756 0.000 0.506 0.584
Q25A 1.244 0.014 89.922 0.000 0.701 0.647
Q28A 1.468 0.015 98.046 0.000 0.827 0.767
Q30A 1.260 0.014 90.684 0.000 0.710 0.657
Q36A 1.469 0.015 96.658 0.000 0.827 0.744
Q40A 1.374 0.015 93.607 0.000 0.774 0.698
Q41A 1.256 0.014 91.905 0.000 0.707 0.674
stress =~
Q1A 1.000 0.776 0.750
Q6A 0.943 0.007 137.700 0.000 0.732 0.695
Q8A 0.974 0.007 142.400 0.000 0.756 0.717
Q11A 1.030 0.007 152.388 0.000 0.799 0.761
Q12A 0.969 0.007 139.655 0.000 0.752 0.704
Q14A 0.797 0.007 111.328 0.000 0.619 0.572
Q18A 0.824 0.007 116.739 0.000 0.639 0.598
Q22A 0.947 0.007 141.490 0.000 0.735 0.713
Q27A 0.995 0.007 146.258 0.000 0.772 0.734
Q29A 1.020 0.007 149.022 0.000 0.791 0.746
Q32A 0.872 0.007 130.508 0.000 0.677 0.663
Q33A 0.970 0.007 142.051 0.000 0.753 0.715
Q35A 0.849 0.007 129.451 0.000 0.658 0.658
Q39A 0.956 0.007 144.729 0.000 0.742 0.727
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
depression ~~
anxiety 0.326 0.004 73.602 0.000 0.716 0.716
stress 0.491 0.005 93.990 0.000 0.784 0.784
anxiety ~~
stress 0.381 0.005 76.884 0.000 0.873 0.873
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.Q3A 0.431 0.003 128.109 0.000 0.431 0.398
.Q5A 0.515 0.004 129.707 0.000 0.515 0.450
.Q10A 0.431 0.003 125.297 0.000 0.431 0.331
.Q13A 0.398 0.003 125.958 0.000 0.398 0.345
.Q16A 0.481 0.004 127.782 0.000 0.481 0.389
.Q17A 0.467 0.004 126.117 0.000 0.467 0.348
.Q21A 0.394 0.003 122.826 0.000 0.394 0.288
.Q24A 0.471 0.004 129.017 0.000 0.471 0.426
.Q26A 0.463 0.004 128.367 0.000 0.463 0.405
.Q31A 0.486 0.004 129.556 0.000 0.486 0.444
.Q34A 0.439 0.004 125.238 0.000 0.439 0.330
.Q37A 0.487 0.004 127.220 0.000 0.487 0.374
.Q38A 0.430 0.003 123.805 0.000 0.430 0.304
.Q42A 0.622 0.005 132.512 0.000 0.622 0.583
.Q2A 0.922 0.007 133.250 0.000 0.922 0.744
.Q4A 0.572 0.004 127.940 0.000 0.572 0.523
.Q7A 0.539 0.004 127.113 0.000 0.539 0.500
.Q9A 0.590 0.005 127.667 0.000 0.590 0.515
.Q15A 0.581 0.004 130.109 0.000 0.581 0.595
.Q19A 0.785 0.006 132.085 0.000 0.785 0.681
.Q20A 0.561 0.004 124.984 0.000 0.561 0.449
.Q23A 0.493 0.004 131.617 0.000 0.493 0.658
.Q25A 0.681 0.005 129.719 0.000 0.681 0.581
.Q28A 0.478 0.004 123.083 0.000 0.478 0.412
.Q30A 0.662 0.005 129.348 0.000 0.662 0.568
.Q36A 0.551 0.004 124.837 0.000 0.551 0.446
.Q40A 0.631 0.005 127.580 0.000 0.631 0.513
.Q41A 0.601 0.005 128.683 0.000 0.601 0.546
.Q1A 0.467 0.004 126.294 0.000 0.467 0.437
.Q6A 0.571 0.004 129.082 0.000 0.571 0.516
.Q8A 0.541 0.004 128.138 0.000 0.541 0.486
.Q11A 0.464 0.004 125.600 0.000 0.464 0.421
.Q12A 0.574 0.004 128.705 0.000 0.574 0.504
.Q14A 0.785 0.006 132.627 0.000 0.785 0.672
.Q18A 0.733 0.006 132.077 0.000 0.733 0.642
.Q22A 0.523 0.004 128.331 0.000 0.523 0.492
.Q27A 0.510 0.004 127.255 0.000 0.510 0.461
.Q29A 0.498 0.004 126.551 0.000 0.498 0.443
.Q32A 0.585 0.004 130.300 0.000 0.585 0.561
.Q33A 0.541 0.004 128.213 0.000 0.541 0.489
.Q35A 0.568 0.004 130.460 0.000 0.568 0.567
.Q39A 0.491 0.004 127.618 0.000 0.491 0.471
depression 0.653 0.007 88.452 0.000 1.000 1.000
anxiety 0.317 0.006 50.766 0.000 1.000 1.000
stress 0.602 0.007 83.798 0.000 1.000 1.000Now, this output contains several key pieces of information:
- fit statistics for the model
- information on item loadings on the latent variables (factors)
- information on the correlations between the latent variables
- information on the variance of latent and manifest variables
Let’s walk through the output one-by-one.
Fit measures
The model fit is evaluated using three key criteria.
The first is the Chi^2 statistic. This evaluates whether the model adequately reproduces the covariance matrix in the real data or whether there is some deviation. Significant results indicate that the model does not reproduce the covariance matrix. However, this is largely dependent on sample size and not informative in practice. The CFA model is always a reduction of the underlying covariance matrix. That is the whole point of a model. Therefore, do not interpret the chi-square test and its associated p-value. Nonetheless, it is customary to report this information when presenting the model fit.
The following two fit statistics are independent of sample size and widely accepted in practice.
The CFI (Comparative Fit Index) is an indicator of fit. It ranges between 0 and 1 with 0 meaning horrible fit and 1 reflecting perfect fit. Values above 0.8 are acceptable and CFI > 0.9 is considered good. This index is independent of sample size and robust against violations of some assumptions CFA makes. It is however dependent on the number of free parameters as it does not include a penalty for additional parameters. Therefore, models with more parameter will generally have better CFI than parsimonious models with fewer parameters.
The RMSEA (Root Mean Squared Error Approximation) is an indicator of misfit that includes a penalty for the number of parameters. The higher the RMSEA, the worse. Generally, RMSEA < 0.05 is considered good and RMSEA < 0.08 acceptable. Models with RMSEA > 0.10 should not be interpreted. This fit statistic indicates whether the model fits the data well while controlling for the number of parameters specified.
It is customary to report all three of these measures when reporting the results of a CFA. Similarly, you should also interpret both the CFI and RMSEA, as bad fits in both statistics can indicate issues in the model specification. The above model would be reported as follows.
The model with three correlated factors showed acceptable fit to the data \(\chi^2(816) = 107309.08, p < 0.001, CFI = 0.90, RMSEA = 0.06\), 95% CI = \([0.06; 0.06]\).
In a future section, we will explore how you can compare models and select models based on these fit statistics. Here, it is important to focus on fit statistics that incorporate the number of free parameters like the RMSEA. Otherwise, more complex models will always outperform less complex models.
In addition to the statistics presented above, model comparison is often based on information criteria, like the AIC (Akaike Information Criterion) or the BIC (Bayesian Information Criterion). Here, lower values indicate better models. These information criteria also penalize complexity and are thus well suited for model comparison. More on this in the later section.
Loadings on latent variables
The next section in the model summary output display information on the latent variables. Specifically how highly the manifest variables “load” onto the latent variables they were assigned to. Here, loadings > 0.6 are considered high and at least one of the manifest variables should load highly on the latent variables. If all loadings are low, this might indicate that the factor you are trying to define does not really exist, as there is little relation between the manifest variables specified to load on this factor.
Importantly, the output shows two columns with loadings. The first column “Estimate” shows unstandardized loadings (like regression weights in a linear model). The last two columns “std.lv” and “std.all” show two different standardization techniques of the loadings. For now, focus on the “std.all” column. This can be interpreted in similar fashion as the factor loadings in the EFA.
In our example, these loadings are looking very good, especially for the depression scale. But for all scales, almost all items show standardized loadings of > 0.6, indicating that they reflect the underlying factor well.
Key Difference between EFA and CFA
These loadings highlight a key difference between CFA and a factor analysis with a fixed number of factors as conducted following an EFA.
In the EFA case, item cross-loadings are allowed. An item can load onto all factors. This is not the case in theory (unless explicitly stated). Here, the items only load onto the factor as specified in the model syntax. This is also why we obtain different loadings and correlations between factors in this CFA example compared to the three-factor FA from the previous episode.
Correlations between latent variables
The next section in the output informs us about the covariances between the latent variables. The “Estimate” column again shows the unstandardized solution while the “std.all” column reflects the standardized solution. The “std.all” column can be interpreted like a correlation (only on a latent level).
Here, we can see high correlations > 0.7 between our three factors. Anxiety and stress even correlate to 0.87! Maybe there is one underlying “g” factor of depression in our data after all? We will investigate this further in the section regarding model comparison.
Variances
The last section of the summary output displays the variances of the errors of manifest variables and the variance of the latent variables. Here, you should make sure that all values in “Estimate” are positive and that the variance of the latent variables differs significantly from 0, the results for this test are printed in “P(>|z|)”.
For CFA this section is not that relevant and plays a larger role in other SEM models.
Model Comparison
The last thing we will learn in this episode is how to leverage the fit statistics and other model information to compare different models. For example, remember that the EFA showed that the first factor had an extremely large eigenvalue, indicating that one factor already explained a substantial amount of variance.
A skeptic of the DASS might argue that it does not measure three distinct scales after all, but rather one unifying “I’m doing bad” construct. Therefore, this skeptic might generate a competing model of the DASS, with only one factor on which all items are loading.
Let’s first test the fit statistics of this model and then compare the two approaches.
R
model_cfa_1fac <- c(
"
# Model Syntax G Factor Model DASS
# Define only one general factor
g_dass =~ Q3A + Q5A + Q10A + Q13A + Q16A + Q17A + Q21A + Q24A + Q26A + Q31A + Q34A + Q37A + Q38A + Q42A +Q2A + Q4A + Q7A + Q9A + Q15A + Q19A + Q20A + Q23A + Q25A + Q28A + Q30A + Q36A + Q40A + Q41A + Q1A + Q6A + Q8A + Q11A + Q12A + Q14A + Q18A + Q22A + Q27A + Q29A + Q32A + Q33A + Q35A + Q39A
"
)
fit_cfa_1fac <- cfa(model_cfa_1fac, data = fa_data)
summary(fit_cfa_1fac, fit.measures = TRUE, standardize = TRUE)
OUTPUT
lavaan 0.6-19 ended normally after 25 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 84
Number of observations 37242
Model Test User Model:
Test statistic 235304.681
Degrees of freedom 819
P-value (Chi-square) 0.000
Model Test Baseline Model:
Test statistic 1037764.006
Degrees of freedom 861
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.774
Tucker-Lewis Index (TLI) 0.762
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -1921157.820
Loglikelihood unrestricted model (H1) -1803505.480
Akaike (AIC) 3842483.641
Bayesian (BIC) 3843199.757
Sample-size adjusted Bayesian (SABIC) 3842932.805
Root Mean Square Error of Approximation:
RMSEA 0.088
90 Percent confidence interval - lower 0.087
90 Percent confidence interval - upper 0.088
P-value H_0: RMSEA <= 0.050 0.000
P-value H_0: RMSEA >= 0.080 1.000
Standardized Root Mean Square Residual:
SRMR 0.067
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Latent Variables:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
g_dass =~
Q3A 1.000 0.761 0.731
Q5A 1.016 0.007 142.019 0.000 0.773 0.722
Q10A 1.100 0.008 144.449 0.000 0.837 0.733
Q13A 1.113 0.007 156.040 0.000 0.846 0.788
Q16A 1.060 0.007 142.723 0.000 0.806 0.725
Q17A 1.147 0.008 148.618 0.000 0.872 0.753
Q21A 1.168 0.008 150.256 0.000 0.889 0.760
Q24A 0.991 0.007 140.991 0.000 0.754 0.717
Q26A 1.057 0.007 148.520 0.000 0.804 0.752
Q31A 0.962 0.007 137.412 0.000 0.732 0.700
Q34A 1.156 0.008 150.634 0.000 0.879 0.762
Q37A 1.063 0.008 139.320 0.000 0.809 0.709
Q38A 1.166 0.008 147.074 0.000 0.887 0.745
Q42A 0.865 0.007 124.363 0.000 0.658 0.637
Q2A 0.663 0.008 87.349 0.000 0.504 0.453
Q4A 0.806 0.007 114.105 0.000 0.613 0.587
Q7A 0.806 0.007 114.965 0.000 0.613 0.591
Q9A 0.901 0.007 125.202 0.000 0.686 0.641
Q15A 0.748 0.007 111.994 0.000 0.569 0.576
Q19A 0.675 0.007 92.306 0.000 0.513 0.478
Q20A 1.008 0.007 134.599 0.000 0.767 0.686
Q23A 0.585 0.006 99.515 0.000 0.445 0.514
Q25A 0.787 0.007 107.302 0.000 0.599 0.553
Q28A 0.968 0.007 134.071 0.000 0.737 0.684
Q30A 0.907 0.007 124.927 0.000 0.690 0.639
Q36A 1.034 0.007 139.134 0.000 0.787 0.708
Q40A 0.935 0.007 125.352 0.000 0.711 0.641
Q41A 0.773 0.007 108.798 0.000 0.588 0.560
Q1A 0.956 0.007 138.241 0.000 0.727 0.704
Q6A 0.869 0.007 122.694 0.000 0.661 0.629
Q8A 0.961 0.007 136.064 0.000 0.731 0.693
Q11A 0.987 0.007 140.627 0.000 0.751 0.715
Q12A 0.943 0.007 131.722 0.000 0.718 0.672
Q14A 0.724 0.007 98.622 0.000 0.551 0.510
Q18A 0.784 0.007 108.318 0.000 0.596 0.558
Q22A 0.929 0.007 134.440 0.000 0.707 0.685
Q27A 0.950 0.007 134.779 0.000 0.723 0.687
Q29A 0.972 0.007 137.073 0.000 0.740 0.698
Q32A 0.826 0.007 119.966 0.000 0.628 0.615
Q33A 0.954 0.007 135.255 0.000 0.725 0.689
Q35A 0.812 0.007 120.437 0.000 0.618 0.618
Q39A 0.916 0.007 133.922 0.000 0.697 0.683
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.Q3A 0.505 0.004 132.069 0.000 0.505 0.466
.Q5A 0.550 0.004 132.300 0.000 0.550 0.479
.Q10A 0.602 0.005 132.010 0.000 0.602 0.463
.Q13A 0.439 0.003 130.205 0.000 0.439 0.380
.Q16A 0.587 0.004 132.219 0.000 0.587 0.475
.Q17A 0.582 0.004 131.451 0.000 0.582 0.433
.Q21A 0.576 0.004 131.207 0.000 0.576 0.422
.Q24A 0.539 0.004 132.416 0.000 0.539 0.486
.Q26A 0.496 0.004 131.465 0.000 0.496 0.434
.Q31A 0.559 0.004 132.788 0.000 0.559 0.511
.Q34A 0.557 0.004 131.148 0.000 0.557 0.419
.Q37A 0.648 0.005 132.595 0.000 0.648 0.498
.Q38A 0.629 0.005 131.668 0.000 0.629 0.444
.Q42A 0.635 0.005 133.849 0.000 0.635 0.595
.Q2A 0.985 0.007 135.470 0.000 0.985 0.795
.Q4A 0.717 0.005 134.451 0.000 0.717 0.656
.Q7A 0.702 0.005 134.407 0.000 0.702 0.651
.Q9A 0.675 0.005 133.792 0.000 0.675 0.589
.Q15A 0.652 0.005 134.557 0.000 0.652 0.668
.Q19A 0.889 0.007 135.325 0.000 0.889 0.771
.Q20A 0.662 0.005 133.053 0.000 0.662 0.529
.Q23A 0.551 0.004 135.083 0.000 0.551 0.736
.Q25A 0.814 0.006 134.773 0.000 0.814 0.694
.Q28A 0.619 0.005 133.100 0.000 0.619 0.533
.Q30A 0.689 0.005 133.811 0.000 0.689 0.591
.Q36A 0.616 0.005 132.614 0.000 0.616 0.499
.Q40A 0.724 0.005 133.782 0.000 0.724 0.588
.Q41A 0.755 0.006 134.707 0.000 0.755 0.686
.Q1A 0.539 0.004 132.706 0.000 0.539 0.505
.Q6A 0.669 0.005 133.958 0.000 0.669 0.605
.Q8A 0.577 0.004 132.918 0.000 0.577 0.520
.Q11A 0.539 0.004 132.455 0.000 0.539 0.489
.Q12A 0.624 0.005 133.301 0.000 0.624 0.548
.Q14A 0.864 0.006 135.115 0.000 0.864 0.740
.Q18A 0.786 0.006 134.728 0.000 0.786 0.689
.Q22A 0.564 0.004 133.067 0.000 0.564 0.530
.Q27A 0.584 0.004 133.037 0.000 0.584 0.528
.Q29A 0.576 0.004 132.822 0.000 0.576 0.513
.Q32A 0.648 0.005 134.127 0.000 0.648 0.621
.Q33A 0.581 0.004 132.993 0.000 0.581 0.525
.Q35A 0.620 0.005 134.099 0.000 0.620 0.619
.Q39A 0.556 0.004 133.113 0.000 0.556 0.534
g_dass 0.579 0.007 81.612 0.000 1.000 1.000The one factor model showed a poor fit to the data \(\chi^2(819) = 235304.68, p < 0.001, CFI = 0.77, RMSEA = 0.09\), 95% CI = \([0.09; 0.09]\).
This already shows that the idea of only one unifying factor in the
DASS does not reflect the data well. We can formally test the difference
in the \(\chi^2\) statistic using
anova(), not to be confused with the ANOVA.
R
anova(fit_cfa_1fac, fit_cfa_3facs)
OUTPUT
Chi-Squared Difference Test
Df AIC BIC Chisq Chisq diff RMSEA Df diff Pr(>Chisq)
fit_cfa_3facs 816 3714494 3715236 107309
fit_cfa_1fac 819 3842484 3843200 235305 127996 1.0703 3 < 2.2e-16
fit_cfa_3facs
fit_cfa_1fac ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1This output displays that the three factor model shows better information criteria (AIC and BIC) and that it also has lower (better) \(\chi^2\) statistics, significantly so. Coupled with an improved RMSEA for the three factor model, this evidence strongly suggests that the three factor model is better suited to explain the DASS data than the one factor model.
Challenge 1
Read in data and answer the question: do we have a hypothesis as to how the structure should be?
Challenge 2
Run a CFA with three factors
Challenge 3
Interpret some results
Challenge 4
Run a cfa with one factor, interpret some results
Challenge 5
Compare the models from challenge 3 and challenge 4, which one fits better?
Key Points
- Something