Factor Analysis - Advanced Measurement Models
Last updated on 2025-06-24 | Edit this page
Estimated time: 25 minutes
Overview
Questions
- What is a bifactor model?
- What is a hierachical model?
- How can I specify different types of measurement models?
Objectives
- Understand the difference between hierarchical, non-hierarchical and bifactor models
- Learn how to specify a hierarchical model
- Learn how to specify a bifactor model
- Learn how to evaluate hierarchical and bifactor models
Measurement Models
So far, we have looked at only 1-factor and correlated multiple factor models in CFAs. This lesson, we will extend this approach to more general models. For example, CFA models can not only incorporate a number of factors correlating with each other, but also support a kind of “second CFA” on the levels of factors. If you have multiple factors that correlate highly with each other, they might themselves have a higher order factor that can account for their covariation.
This is called a “hierarchical” model. We can simply extend our three factor model as follows.
Hierarchical Models
R
library(lavaan)
OUTPUT
This is lavaan 0.6-19
lavaan is FREE software! Please report any bugs.R
model_cfa_hierarch <- c(
"
# Model Syntax
# 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
# Now, not correlation between factors, but hierarchy
high_level_fac =~ depression + anxiety + stress
"
)
Let’s load the data again and investigate this model:
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"))
fit_cfa_hierarch <- cfa(
model = model_cfa_hierarch,
data = fa_data,
)
summary(fit_cfa_hierarch, fit.measures = TRUE, standardized = TRUE)
OUTPUT
lavaan 0.6-19 ended normally after 38 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 0.006 162.877 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 93.090 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 89.051 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 0.007 141.491 0.000 0.735 0.713
Q27A 0.995 0.007 146.259 0.000 0.772 0.734
Q29A 1.020 0.007 149.023 0.000 0.791 0.746
Q32A 0.872 0.007 130.509 0.000 0.677 0.663
Q33A 0.970 0.007 142.052 0.000 0.753 0.715
Q35A 0.849 0.007 129.452 0.000 0.658 0.658
Q39A 0.956 0.007 144.730 0.000 0.742 0.727
high_level_fac =~
depression 1.000 0.802 0.802
anxiety 0.776 0.009 88.349 0.000 0.893 0.893
stress 1.171 0.009 124.409 0.000 0.978 0.978
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 0.233 0.003 77.473 0.000 0.357 0.357
.anxiety 0.064 0.002 41.710 0.000 0.202 0.202
.stress 0.027 0.002 14.464 0.000 0.044 0.044
high_level_fac 0.420 0.006 70.881 0.000 1.000 1.000Again, let’s look at the fit indices first. The hierarchical model with three factors showed acceptable fit to the data \(\chi^2(816) = 107309.08, p < 0.001, CFI = 0.90, RMSEA = 0.06\), 95% CI = \([0.06; 0.06]\). Similarly, the loadings on the first-order factors (depression, anxiety, stress) also look good and we do not observe any issues in the variance.
If you have a good memory, you might notice that the fit statistics are identical to those in the correlated 3-factors model. This is because the higher order factor in this case only restructures the bivariate correlations into factor loadings. However, this is not true for all number of factors. In models with more than 3 first-order factors, establishing a second-order factor loading on all of them introduces more constraints than allowing the factors to correlate freely.
R
# Semplot here?
The hierarchical model allows us to investigate two very important things. Firstly, we can investigate the presence of a higher order factor. This is only the case if 1) the model fit is acceptable, 2) the factor loadings on this second-order factor are high enough and 3) the variance of the second-order factor is significantly different from 0. In this case, all three things are true, supporting the conclusion that a model with a higher order factor that contributes variance to all lower factors fits the data well.
Secondly, they are extremely useful in Structural Equation Models (SEM), as this higher-order factor can then itself be correlated to other outcomes, and allows researchers to investigate how much the general factor is associated with other items.
Bi-Factor Models
In this hierarchical case, we assume that the higher-order factor captures the shared variance of all first-order factors. So the higher-order factor only contributes variance to the observed variables indirectly through the first-order factors (depression etc.).
Bi-factor models change this behavior significantly. They do not impose a hierarchical structure but rather partition the variance in the observed variables into a general factor and specific factors. In the case of the DASS, a bifactor model would assert that each item is mostly determined by a general “I’m doing bad factor” and depending on the item context, depression, anxiety, or stress specifically contribute to the response on the specific item.
In contrast to a hierarchical model, this g-factor is directly influencing responses in the items and is not related to the specific factors. These models are often employed when researchers have good reason to believe that there are some specific factors unrelated to general response, as for example when using multiple tests measuring the same construct. One might introduce a general factor for construct and several specific factors capturing measurement specific variance (e.g. g factor for Openness, specific factors for self-rating vs. other-rating).
The model syntax displays this direct loading and the uncorrelated latent variables:
R
model_cfa_bifac <- c(
"
# Model Syntax
# 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
# Direct loading of the items on the 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
# Define correlations between factors using ~~
depression ~~ 0*anxiety
depression ~~ 0*stress
anxiety ~~ 0*stress
g_dass ~~ 0*depression
g_dass ~~ 0*stress
g_dass ~~ 0*anxiety
"
)
R
fit_cfa_bifac <- cfa(model_cfa_bifac, data = fa_data)
summary(fit_cfa_bifac, fit.measures = TRUE, standardized = TRUE)
OUTPUT
lavaan 0.6-19 ended normally after 90 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 126
Number of observations 37242
Model Test User Model:
Test statistic 72120.789
Degrees of freedom 777
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) 0.931
Tucker-Lewis Index (TLI) 0.924
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -1839565.874
Loglikelihood unrestricted model (H1) -1803505.480
Akaike (AIC) 3679383.748
Bayesian (BIC) 3680457.922
Sample-size adjusted Bayesian (SABIC) 3680057.494
Root Mean Square Error of Approximation:
RMSEA 0.050
90 Percent confidence interval - lower 0.049
90 Percent confidence interval - upper 0.050
P-value H_0: RMSEA <= 0.050 0.968
P-value H_0: RMSEA >= 0.080 0.000
Standardized Root Mean Square Residual:
SRMR 0.031
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 0.476 0.457
Q5A 0.838 0.011 77.509 0.000 0.399 0.373
Q10A 1.408 0.014 104.191 0.000 0.670 0.587
Q13A 0.930 0.010 88.624 0.000 0.443 0.412
Q16A 1.143 0.012 92.973 0.000 0.544 0.489
Q17A 1.242 0.013 97.712 0.000 0.591 0.510
Q21A 1.494 0.014 108.013 0.000 0.711 0.609
Q24A 0.960 0.011 85.644 0.000 0.457 0.434
Q26A 0.862 0.011 81.459 0.000 0.410 0.384
Q31A 0.967 0.011 85.275 0.000 0.461 0.440
Q34A 1.252 0.013 99.210 0.000 0.596 0.517
Q37A 1.380 0.014 101.533 0.000 0.657 0.576
Q38A 1.546 0.014 107.851 0.000 0.736 0.618
Q42A 0.655 0.011 61.266 0.000 0.312 0.302
anxiety =~
Q2A 1.000 0.288 0.258
Q4A 1.573 0.037 42.513 0.000 0.452 0.433
Q7A 1.940 0.044 44.187 0.000 0.558 0.538
Q9A 0.232 0.018 12.886 0.000 0.067 0.062
Q15A 1.207 0.030 39.724 0.000 0.347 0.351
Q19A 1.288 0.033 38.640 0.000 0.370 0.345
Q20A 0.442 0.020 22.556 0.000 0.127 0.114
Q23A 1.100 0.028 39.596 0.000 0.316 0.365
Q25A 1.342 0.034 40.033 0.000 0.386 0.357
Q28A 0.667 0.021 31.767 0.000 0.192 0.178
Q30A 0.170 0.019 9.086 0.000 0.049 0.045
Q36A 0.402 0.019 21.113 0.000 0.116 0.104
Q40A 0.292 0.019 15.254 0.000 0.084 0.076
Q41A 1.866 0.043 43.679 0.000 0.537 0.511
stress =~
Q1A 1.000 0.381 0.368
Q6A 0.790 0.016 49.154 0.000 0.301 0.286
Q8A -0.056 0.014 -4.124 0.000 -0.021 -0.020
Q11A 1.079 0.017 62.272 0.000 0.411 0.391
Q12A -0.463 0.015 -29.980 0.000 -0.176 -0.165
Q14A 0.800 0.018 44.430 0.000 0.304 0.282
Q18A 0.754 0.017 43.691 0.000 0.287 0.269
Q22A 0.187 0.013 13.917 0.000 0.071 0.069
Q27A 0.993 0.017 58.180 0.000 0.378 0.359
Q29A 0.605 0.014 42.099 0.000 0.230 0.217
Q32A 0.737 0.016 46.342 0.000 0.281 0.275
Q33A -0.525 0.016 -33.597 0.000 -0.200 -0.190
Q35A 0.680 0.015 44.095 0.000 0.259 0.259
Q39A 0.538 0.014 38.401 0.000 0.205 0.201
g_dass =~
Q3A 1.000 0.647 0.621
Q5A 1.058 0.008 127.074 0.000 0.685 0.639
Q10A 1.034 0.008 131.792 0.000 0.669 0.586
Q13A 1.158 0.008 140.588 0.000 0.749 0.697
Q16A 1.036 0.008 127.946 0.000 0.670 0.603
Q17A 1.128 0.008 135.443 0.000 0.730 0.630
Q21A 1.100 0.008 139.112 0.000 0.711 0.609
Q24A 0.996 0.008 126.030 0.000 0.644 0.612
Q26A 1.103 0.008 133.013 0.000 0.713 0.668
Q31A 0.959 0.008 122.467 0.000 0.620 0.592
Q34A 1.138 0.008 137.814 0.000 0.736 0.638
Q37A 0.996 0.008 125.672 0.000 0.644 0.565
Q38A 1.086 0.008 135.445 0.000 0.703 0.590
Q42A 0.914 0.008 110.771 0.000 0.592 0.572
Q2A 0.764 0.010 78.619 0.000 0.494 0.444
Q4A 0.956 0.010 100.570 0.000 0.618 0.592
Q7A 0.947 0.009 100.271 0.000 0.613 0.590
Q9A 1.152 0.010 114.816 0.000 0.745 0.696
Q15A 0.856 0.009 96.201 0.000 0.554 0.560
Q19A 0.802 0.009 84.718 0.000 0.519 0.483
Q20A 1.245 0.011 117.876 0.000 0.805 0.720
Q23A 0.667 0.008 86.982 0.000 0.431 0.498
Q25A 0.953 0.010 97.541 0.000 0.617 0.570
Q28A 1.238 0.010 120.648 0.000 0.800 0.743
Q30A 1.095 0.010 109.651 0.000 0.709 0.656
Q36A 1.247 0.011 118.553 0.000 0.807 0.726
Q40A 1.173 0.010 113.292 0.000 0.759 0.684
Q41A 0.910 0.009 96.144 0.000 0.589 0.561
Q1A 1.111 0.010 113.901 0.000 0.719 0.695
Q6A 1.058 0.010 108.405 0.000 0.684 0.651
Q8A 1.187 0.010 118.873 0.000 0.768 0.729
Q11A 1.139 0.010 114.602 0.000 0.737 0.702
Q12A 1.260 0.010 122.913 0.000 0.815 0.764
Q14A 0.871 0.010 90.195 0.000 0.563 0.521
Q18A 0.916 0.010 95.101 0.000 0.593 0.555
Q22A 1.118 0.010 115.483 0.000 0.723 0.702
Q27A 1.091 0.010 110.804 0.000 0.706 0.671
Q29A 1.171 0.010 116.850 0.000 0.757 0.715
Q32A 0.968 0.009 103.368 0.000 0.626 0.613
Q33A 1.271 0.010 124.894 0.000 0.822 0.781
Q35A 0.945 0.009 103.042 0.000 0.611 0.611
Q39A 1.091 0.010 113.992 0.000 0.706 0.692
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
depression ~~
anxiety 0.000 0.000 0.000
stress 0.000 0.000 0.000
anxiety ~~
stress 0.000 0.000 0.000
depression ~~
g_dass 0.000 0.000 0.000
stress ~~
g_dass 0.000 0.000 0.000
anxiety ~~
g_dass 0.000 0.000 0.000
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.Q3A 0.439 0.003 129.116 0.000 0.439 0.405
.Q5A 0.519 0.004 131.501 0.000 0.519 0.452
.Q10A 0.406 0.003 120.568 0.000 0.406 0.312
.Q13A 0.398 0.003 128.596 0.000 0.398 0.345
.Q16A 0.492 0.004 128.021 0.000 0.492 0.398
.Q17A 0.460 0.004 125.804 0.000 0.460 0.343
.Q21A 0.354 0.003 115.937 0.000 0.354 0.259
.Q24A 0.484 0.004 130.257 0.000 0.484 0.437
.Q26A 0.465 0.004 130.540 0.000 0.465 0.407
.Q31A 0.499 0.004 130.430 0.000 0.499 0.455
.Q34A 0.433 0.003 124.935 0.000 0.433 0.325
.Q37A 0.456 0.004 122.826 0.000 0.456 0.350
.Q38A 0.381 0.003 115.828 0.000 0.381 0.269
.Q42A 0.621 0.005 133.654 0.000 0.621 0.581
.Q2A 0.912 0.007 132.163 0.000 0.912 0.736
.Q4A 0.506 0.004 117.060 0.000 0.506 0.463
.Q7A 0.391 0.004 96.997 0.000 0.391 0.363
.Q9A 0.586 0.005 130.242 0.000 0.586 0.512
.Q15A 0.549 0.004 126.066 0.000 0.549 0.563
.Q19A 0.746 0.006 127.751 0.000 0.746 0.647
.Q20A 0.585 0.005 129.861 0.000 0.585 0.468
.Q23A 0.463 0.004 126.185 0.000 0.463 0.618
.Q25A 0.642 0.005 125.465 0.000 0.642 0.548
.Q28A 0.484 0.004 128.458 0.000 0.484 0.417
.Q30A 0.661 0.005 131.247 0.000 0.661 0.567
.Q36A 0.571 0.004 129.586 0.000 0.571 0.463
.Q40A 0.647 0.005 130.851 0.000 0.647 0.526
.Q41A 0.466 0.004 105.809 0.000 0.466 0.424
.Q1A 0.407 0.004 113.121 0.000 0.407 0.381
.Q6A 0.548 0.004 125.196 0.000 0.548 0.495
.Q8A 0.521 0.004 126.156 0.000 0.521 0.469
.Q11A 0.391 0.004 108.324 0.000 0.391 0.355
.Q12A 0.444 0.004 109.048 0.000 0.444 0.390
.Q14A 0.757 0.006 128.517 0.000 0.757 0.649
.Q18A 0.708 0.005 128.759 0.000 0.708 0.620
.Q22A 0.535 0.004 130.122 0.000 0.535 0.503
.Q27A 0.466 0.004 116.394 0.000 0.466 0.421
.Q29A 0.497 0.004 127.087 0.000 0.497 0.442
.Q32A 0.572 0.004 127.129 0.000 0.572 0.548
.Q33A 0.392 0.004 101.742 0.000 0.392 0.354
.Q35A 0.561 0.004 128.205 0.000 0.561 0.560
.Q39A 0.501 0.004 128.729 0.000 0.501 0.481
depression 0.227 0.004 56.333 0.000 1.000 1.000
anxiety 0.083 0.004 23.169 0.000 1.000 1.000
stress 0.145 0.004 38.869 0.000 1.000 1.000
g_dass 0.418 0.006 65.244 0.000 1.000 1.000This model shows good fit to the data \(\chi^2(777) = 72120.79, p < 0.001, CFI = 0.93, RMSEA = 0.05\), 95% CI = \([0.05; 0.05]\). We can additionally observe no issues in the loadings on the general factor. For the specific factors, there are items that do not seem to measure anything factor-specific in addition to the general factor, as they show loadings near zero on their specific factor.
From this model, we can learn two things. First, a general factor seems to be well supported by the data. Indicating that one could take all the items of the DASS scale and compute one general “distress” score that can be meaningfully interpreted. Second, there remains some specific variance in the items attributable to each scale. Together with the findings from the 3 factor model, this supports the interpretation that the DASS measures general distress and a small specific contribution of stress, anxiety and depression. However, some of the items load considerably more on general distress than on the specific factors.
Model Comparison - Nested models
Lets formally compare the four models we tested here to declare a “winner”, the model that fits our present data best. To do this, we will investigate the fit index RMSEA, the information criteria and conduct a chi-square differences test. The model with the best RMSEA and AIC/BIC and acceptable CFI will be crowned our winner.
Let’s recompute the one-factor and three-factor model.
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)
model_cfa_3facs <- c(
"
# Model Syntax
# 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
"
)
fit_cfa_3facs <- cfa(
model = model_cfa_3facs,
data = fa_data,
)
Now, lets look at the fit indices.
R
fit_results <- data.frame(
c("1fac", "3fac", "hierarch", "bifac")
)
fit_results[, 2:6] <- 0
colnames(fit_results) <- c("model", "df", "chisq", "pvalue", "cfi", "rmsea")
fit_results[1, 2:6] <- fitmeasures(fit_cfa_1fac, c("df", "chisq", "pvalue", "cfi", "rmsea"))
fit_results[2, 2:6] <- fitmeasures(fit_cfa_3facs, c("df", "chisq", "pvalue", "cfi", "rmsea"))
fit_results[3, 2:6] <- fitmeasures(fit_cfa_hierarch, c("df", "chisq", "pvalue", "cfi", "rmsea"))
fit_results[4, 2:6] <- fitmeasures(fit_cfa_bifac, c("df", "chisq", "pvalue", "cfi", "rmsea"))
fit_results
OUTPUT
model df chisq pvalue cfi rmsea
1 1fac 819 235304.68 0 0.7738596 0.08767983
2 3fac 816 107309.08 0 0.8972970 0.05919692
3 hierarch 816 107309.08 0 0.8972970 0.05919692
4 bifac 777 72120.79 0 0.9311953 0.04965364In this comparison, the bifactor model clearly winds out. It shows
the lowest RMSEA despite having more free parameters and shows good CFI.
Using anova(), we can formally test this.
R
anova(fit_cfa_1fac, fit_cfa_3facs, fit_cfa_hierarch, fit_cfa_bifac)
WARNING
Warning: lavaan->lavTestLRT():
some models have the same degrees of freedomOUTPUT
Chi-Squared Difference Test
Df AIC BIC Chisq Chisq diff RMSEA Df diff
fit_cfa_bifac 777 3679384 3680458 72121
fit_cfa_3facs 816 3714494 3715236 107309 35188 0.15556 39
fit_cfa_hierarch 816 3714494 3715236 107309 0 0.00000 0
fit_cfa_1fac 819 3842484 3843200 235305 127996 1.07032 3
Pr(>Chisq)
fit_cfa_bifac
fit_cfa_3facs < 2.2e-16 ***
fit_cfa_hierarch
fit_cfa_1fac < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Important - CFA vs. EFA
Important note, do not confuse confirmatory with exploratory research! You cannot use the same data and first run an EFA, then define some models and run a CFA to confirm your “best” model. It should be clearly stated then that this is exploratory research only. True model difference tests to confirm hypothesis can only be conducted if these hypothesis allow specification of the model before the data is seen!
A look to SEM
What we have investigated so far is also called a measurement model, because it allows the specification of latent factors that are supposed to measure certain constructs. In the broader context of Structural Equation Modelling (SEM), we can extend this CFA measurement model by combining multiple measurement models and investigating the relationship between latent factors of different measurement models in a regression model.
For example, we can use the bifactor model as a measurement model for general distress and then relate this factor to the measures of the big 5.
R
big5_distress_model <- c(
"
# Model Syntax
# 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
# Direct loading of the items on the 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
# Define correlations between factors using ~~
depression ~~ 0*anxiety
depression ~~ 0*stress
anxiety ~~ 0*stress
g_dass ~~ 0*depression
g_dass ~~ 0*stress
g_dass ~~ 0*anxiety
"
)
Key Points
- Something