Create a Zero df Model
In the first chapter, we created a model of text-speed from the HolzingerSwineford1939 dataset with six manifest variables
x4, x5, x6, x7, x8, x9. Create a model of only reading comprehension with x4, x5, and x6.
This model should have zero degrees of freedom because it is not identified, as we are estimating as many parameters as we
have options with only three manifest variables. Analyze the model to find zero degrees of freedom. lavaan and the
HolzingerSwineford1939 dataset have been loaded for you.
* Name your model text.model and your latent variable text.
* Use variables x4, x5, and x6 as the manifest variables.
* Use the cfa() function to analyze the model.
* Use the summary() function to view the degrees of freedom.
# Create your text model specification
text.model <- 'text =~ x4 + x5 + x6'
# Analyze the model and include data argument
text.fit <- cfa(model = text.model, data = HolzingerSwineford1939)
# Summarize the model
summary(text.fit, standardized = TRUE, fit.measures = TRUE)
lavaan 0.6-11 ended normally after 15 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 6
Number of observations 301
Model Test User Model:
Test statistic 0.000
Degrees of freedom 0
Model Test Baseline Model:
Test statistic 497.430
Degrees of freedom 3
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 1.000
Tucker-Lewis Index (TLI) 1.000
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -1181.065
Loglikelihood unrestricted model (H1) -1181.065
Akaike (AIC) 2374.130
Bayesian (BIC) 2396.372
Sample-size adjusted Bayesian (BIC) 2377.344
Root Mean Square Error of Approximation:
RMSEA 0.000
90 Percent confidence interval - lower 0.000
90 Percent confidence interval - upper 0.000
P-value RMSEA <= 0.05 NA
Standardized Root Mean Square Residual:
SRMR 0.000
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
text =~
x4 1.000 0.984 0.847
x5 1.133 0.067 16.906 0.000 1.115 0.866
x6 0.924 0.056 16.391 0.000 0.910 0.832
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.x4 0.382 0.049 7.805 0.000 0.382 0.283
.x5 0.416 0.059 7.038 0.000 0.416 0.251
.x6 0.369 0.044 8.367 0.000 0.369 0.308
text 0.969 0.112 8.640 0.000 1.000 1.000
This model has zero degrees of freedom, which means we need to fix the model identification.
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Fix the Zero df Model
Identification is a core component in SEM models and you should have at least one degree of freedom for any model.
Using the text.model from the last exercise, update the model to create a degree of freedom.
You should label two of the coefficient paths to a to set them equal to each other. lavaan and the
HolzingerSwineford1939 dataset have been loaded for you.
* Update the text.model to have x5 and x6 paths both set to a.
* Analyze the model with the cfa() function.
* View the updated model with the summary() function.
# Update the model specification by setting two paths to the label a
text.model <- 'text =~ x4 + a*x5 + a*x6'
# Analyze the model
text.fit <- cfa(model = text.model, data = HolzingerSwineford1939)
# Summarize the model
summary(text.fit, standardized = TRUE, fit.measures = TRUE)
lavaan 0.6-11 ended normally after 14 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 6
Number of equality constraints 1
Number of observations 301
Model Test User Model:
Test statistic 11.227
Degrees of freedom 1
P-value (Chi-square) 0.001
Model Test Baseline Model:
Test statistic 497.430
Degrees of freedom 3
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.979
Tucker-Lewis Index (TLI) 0.938
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -1186.678
Loglikelihood unrestricted model (H1) -1181.065
Akaike (AIC) 2383.357
Bayesian (BIC) 2401.892
Sample-size adjusted Bayesian (BIC) 2386.035
Root Mean Square Error of Approximation:
RMSEA 0.184
90 Percent confidence interval - lower 0.098
90 Percent confidence interval - upper 0.288
P-value RMSEA <= 0.05 0.007
Standardized Root Mean Square Residual:
SRMR 0.073
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
text =~
x4 1.000 0.983 0.846
x5 (a) 1.009 0.054 18.747 0.000 0.992 0.815
x6 (a) 1.009 0.054 18.747 0.000 0.992 0.866
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.x4 0.383 0.050 7.631 0.000 0.383 0.284
.x5 0.499 0.054 9.164 0.000 0.499 0.336
.x6 0.328 0.045 7.285 0.000 0.328 0.250
text 0.967 0.113 8.585 0.000 1.000 1.000
Our text model is now identified with one degree of freedom and two equal parameter estimates for x5 and x6.
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Build a Multi-Factor Model
Another way to improve our text-speed model would be to create a two-factor model with text and speed as latent
variables, rather than split the model into two one-factor models. In this exercise, write the model specification code
for a two-factor model.
If you want to view the dataset to see the columns, you can use head(HolzingerSwineford1939).
* Name your model twofactor.model.
* Use x4, x5, and x6 to create a text latent variable.
* Use x7, x8, and x9 to create a speed latent variable.
# Create a two-factor model of text and speed variables
twofactor.model <- 'text =~ x4 + x5 + x6
speed =~ x7 + x8 + x9'
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Summarize the Multi-Factor Model
You have now created a two-factor model of the reading comprehension and speeded addition factors. Is that better than
a one-factor model? Use the cfa() and summary() functions on your new two-factor model of the HolzingerSwineford1939
dataset to show the fit indices.
# Previous one-factor model output
summary(text.fit, standardized = TRUE, fit.measures = TRUE)
# Two-factor model specification
twofactor.model <- 'text =~ x4 + x5 + x6
speed =~ x7 + x8 + x9'
# Use cfa() to analyze the model and include data argument
twofactor.fit <- cfa(model = twofactor.model, data = HolzingerSwineford1939)
# Use summary() to view the fitted model
summary(twofactor.fit, standardized = TRUE, fit.measures = TRUE)
lavaan 0.6-11 ended normally after 24 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 13
Number of observations 301
Model Test User Model:
Test statistic 14.354
Degrees of freedom 8
P-value (Chi-square) 0.073
Model Test Baseline Model:
Test statistic 681.336
Degrees of freedom 15
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.990
Tucker-Lewis Index (TLI) 0.982
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -2408.414
Loglikelihood unrestricted model (H1) -2401.237
Akaike (AIC) 4842.828
Bayesian (BIC) 4891.021
Sample-size adjusted Bayesian (BIC) 4849.792
Root Mean Square Error of Approximation:
RMSEA 0.051
90 Percent confidence interval - lower 0.000
90 Percent confidence interval - upper 0.093
P-value RMSEA <= 0.05 0.425
Standardized Root Mean Square Residual:
SRMR 0.039
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
text =~
x4 1.000 0.984 0.847
x5 1.132 0.067 16.954 0.000 1.114 0.865
x6 0.925 0.056 16.438 0.000 0.911 0.833
speed =~
x7 1.000 0.674 0.619
x8 1.150 0.165 6.990 0.000 0.775 0.766
x9 0.878 0.123 7.166 0.000 0.592 0.587
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
text ~~
speed 0.173 0.052 3.331 0.001 0.261 0.261
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.x4 0.382 0.049 7.854 0.000 0.382 0.283
.x5 0.418 0.059 7.113 0.000 0.418 0.252
.x6 0.367 0.044 8.374 0.000 0.367 0.307
.x7 0.729 0.084 8.731 0.000 0.729 0.616
.x8 0.422 0.084 5.039 0.000 0.422 0.413
.x9 0.665 0.071 9.383 0.000 0.665 0.655
text 0.969 0.112 8.647 0.000 1.000 1.000
speed 0.454 0.096 4.728 0.000 1.000 1.000
In comparing the one- and two-factor models, you should see that the fit indices
are improved in the two-factor model.
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Three-Factor Model with Zero Correlation
In this exercise, you will use the Eysenck Personality Inventory dataset from the psychTools library to create a
three-factor model of personality. This inventory includes 57 questions that measure extraversion, neuroticism,
and lying.
Let's create a three factor model using the latent variables: extraversion, neuroticism, and lying with four manifest
variables on each item. Remember when you create multiple latent variables, these endogenous variables are automatically
correlated. Set the correlation between the extraversion latent variable and neuroticism latent variable to zero, by
using the ~~ in model specification code.
* Create a lying factor including items V6, V12, V18, and V24.
* Set the correlation between extraversion and neuroticism to zero (in that order).
* Analyze the three-factor model with the cfa() function.
* Summarize the three-factor model and look for the zero-correlation you set in the model specification.
# Load the library and data
library(psych)
data(epi)
# Specify a three-factor model with one correlation set to zero
epi.model <- 'extraversion =~ V1 + V3 + V5 + V8
neuroticism =~ V2 + V4 + V7 + V9
lying =~ V6 + V12 + V18 + V24
extraversion ~~ 0*neuroticism'
# Run the model
epi.fit <- cfa(model = epi.model, data = epi)
# Examine the output
summary(epi.fit, standardized = TRUE, fit.measures = TRUE)
lavaan 0.6-11 ended normally after 106 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 26
Used Total
Number of observations 3193 3570
Model Test User Model:
Test statistic 584.718
Degrees of freedom 52
P-value (Chi-square) 0.000
Model Test Baseline Model:
Test statistic 2196.019
Degrees of freedom 66
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.750
Tucker-Lewis Index (TLI) 0.683
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -23208.145
Loglikelihood unrestricted model (H1) -22915.787
Akaike (AIC) 46468.291
Bayesian (BIC) 46626.077
Sample-size adjusted Bayesian (BIC) 46543.464
Root Mean Square Error of Approximation:
RMSEA 0.057
90 Percent confidence interval - lower 0.053
90 Percent confidence interval - upper 0.061
P-value RMSEA <= 0.05 0.004
Standardized Root Mean Square Residual:
SRMR 0.058
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
extraversion =~
V1 1.000 0.052 0.115
V3 1.360 0.329 4.127 0.000 0.070 0.141
V5 -2.829 0.554 -5.109 0.000 -0.146 -0.391
V8 7.315 1.832 3.992 0.000 0.377 0.797
neuroticism =~
V2 1.000 0.228 0.457
V4 0.424 0.053 8.004 0.000 0.097 0.196
V7 1.395 0.093 15.023 0.000 0.318 0.648
V9 1.205 0.078 15.506 0.000 0.275 0.553
lying =~
V6 1.000 0.135 0.272
V12 -0.851 0.132 -6.435 0.000 -0.115 -0.291
V18 -0.785 0.122 -6.421 0.000 -0.106 -0.289
V24 1.086 0.161 6.734 0.000 0.147 0.339
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
extraversion ~~
neuroticism 0.000 0.000 0.000
lying -0.002 0.001 -3.313 0.001 -0.258 -0.258
neuroticism ~~
lying -0.014 0.002 -6.867 0.000 -0.469 -0.469
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.V1 0.198 0.005 39.567 0.000 0.198 0.987
.V3 0.243 0.006 39.278 0.000 0.243 0.980
.V5 0.118 0.005 23.900 0.000 0.118 0.847
.V8 0.082 0.026 3.084 0.002 0.082 0.364
.V2 0.197 0.006 32.516 0.000 0.197 0.791
.V4 0.235 0.006 38.906 0.000 0.235 0.962
.V7 0.140 0.007 19.412 0.000 0.140 0.580
.V9 0.172 0.006 26.591 0.000 0.172 0.694
.V6 0.228 0.007 34.520 0.000 0.228 0.926
.V12 0.143 0.004 33.670 0.000 0.143 0.916
.V18 0.124 0.004 33.753 0.000 0.124 0.917
.V24 0.166 0.005 31.021 0.000 0.166 0.885
extraversion 0.003 0.001 2.480 0.013 1.000 1.000
neuroticism 0.052 0.005 10.010 0.000 1.000 1.000
lying 0.018 0.004 4.500 0.000 1.000 1.000
In the summary output, you can see the correlation between extraversion and neuroticism is set to zero
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Create a Direct Path
In this exercise, edit the epi.model to include a direct regression path between lying and neuroticism. We might expect
that a person's level of neuroticism would predict their level of lying.
You can edit the model specification provided to include this path, and you can use the same model specification code
that you might use when defining a regression in lm.
* Add a new line to the model specification code where lying is predicted by neuroticism.
* Complete the cfa() function to analyze your new model.
* Summarize the new model to see that that output has changed to a regression.
# Load the library and data
library(psych)
data(epi)
# Specify a three-factor model where lying is predicted by neuroticism
epi.model <- 'extraversion =~ V1 + V3 + V5 + V8
neuroticism =~ V2 + V4 + V7 + V9
lying =~ V6 + V12 + V18 + V24
lying ~ neuroticism'
# Run the model
epi.fit <- cfa(model = epi.model, data = epi)
# Examine the output
summary(epi.fit, standardized = TRUE, fit.measures = TRUE)
lavaan 0.6-11 ended normally after 103 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 26
Used Total
Number of observations 3193 3570
Model Test User Model:
Test statistic 534.426
Degrees of freedom 52
P-value (Chi-square) 0.000
Model Test Baseline Model:
Test statistic 2196.019
Degrees of freedom 66
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.774
Tucker-Lewis Index (TLI) 0.713
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -23183.000
Loglikelihood unrestricted model (H1) -22915.787
Akaike (AIC) 46417.999
Bayesian (BIC) 46575.786
Sample-size adjusted Bayesian (BIC) 46493.173
Root Mean Square Error of Approximation:
RMSEA 0.054
90 Percent confidence interval - lower 0.050
90 Percent confidence interval - upper 0.058
P-value RMSEA <= 0.05 0.058
Standardized Root Mean Square Residual:
SRMR 0.053
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
extraversion =~
V1 1.000 0.052 0.115
V3 1.135 0.268 4.230 0.000 0.059 0.118
V5 -2.497 0.443 -5.638 0.000 -0.129 -0.346
V8 8.223 2.008 4.096 0.000 0.425 0.898
neuroticism =~
V2 1.000 0.223 0.447
V4 0.462 0.054 8.493 0.000 0.103 0.209
V7 1.435 0.093 15.368 0.000 0.320 0.652
V9 1.214 0.078 15.570 0.000 0.271 0.545
lying =~
V6 1.000 0.125 0.252
V12 -0.943 0.150 -6.274 0.000 -0.118 -0.298
V18 -0.905 0.143 -6.339 0.000 -0.113 -0.308
V24 1.187 0.182 6.509 0.000 0.148 0.342
Regressions:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
lying ~
neuroticism -0.298 0.043 -6.943 0.000 -0.532 -0.532
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
extraversion ~~
neuroticism 0.003 0.001 3.761 0.000 0.240 0.240
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.V1 0.198 0.005 39.671 0.000 0.198 0.987
.V3 0.244 0.006 39.651 0.000 0.244 0.986
.V5 0.123 0.004 28.256 0.000 0.123 0.881
.V8 0.043 0.033 1.302 0.193 0.043 0.193
.V2 0.200 0.006 33.262 0.000 0.200 0.800
.V4 0.233 0.006 38.804 0.000 0.233 0.956
.V7 0.139 0.007 20.087 0.000 0.139 0.575
.V9 0.174 0.006 27.907 0.000 0.174 0.703
.V6 0.231 0.007 35.398 0.000 0.231 0.936
.V12 0.143 0.004 33.349 0.000 0.143 0.911
.V18 0.122 0.004 32.825 0.000 0.122 0.905
.V24 0.166 0.005 30.854 0.000 0.166 0.883
extraversion 0.003 0.001 2.643 0.008 1.000 1.000
neuroticism 0.050 0.005 9.947 0.000 1.000 1.000
.lying 0.011 0.003 3.970 0.000 0.717 0.717
You can see that the correlation between lying and neuroticism is now a direct path listed under
regressions in the output
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Check Model Variance
In order to evaluate your three-factor model of the epi, you can examine the variance of the manifest variables to
check for potential problems with the model. Very large variances can indicate potential issues; however, this value
should be compared to the original scale of the data.
Use the var() function on V1 to compare the variance of the original manifest variable to the estimated variance in your
summary output. The libraries, model, and datasets have been loaded for you.
* Use var() on V1 to calculate the variance of the original manifest variable in the epi dataset.
* Compare this value to the V1 variance estimate of the model.
# Run the model
epi.fit <- cfa(model = epi.model, data = epi)
# Examine the output
summary(epi.fit, standardized = TRUE, fit.measures = TRUE)
# Calculate the variance of V1 in the epi data
var(epi$V1)
[1] 0.2017972
lavaan 0.6-11 ended normally after 108 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 27
Number of observations 2897
Model Test User Model:
Test statistic 462.095
Degrees of freedom 51
P-value (Chi-square) 0.000
Model Test Baseline Model:
Test statistic 1994.632
Degrees of freedom 66
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.787
Tucker-Lewis Index (TLI) 0.724
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -21035.420
Loglikelihood unrestricted model (H1) -20804.372
Akaike (AIC) 42124.840
Bayesian (BIC) 42286.068
Sample-size adjusted Bayesian (BIC) 42200.280
Root Mean Square Error of Approximation:
RMSEA 0.053
90 Percent confidence interval - lower 0.048
90 Percent confidence interval - upper 0.057
P-value RMSEA <= 0.05 0.147
Standardized Root Mean Square Residual:
SRMR 0.052
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
extraversion =~
V1 1.000 0.052 0.115
V3 1.299 0.308 4.217 0.000 0.067 0.135
V5 -2.614 0.494 -5.295 0.000 -0.135 -0.359
V8 8.108 1.975 4.105 0.000 0.418 0.884
neuroticism =~
V2 1.000 0.225 0.450
V4 0.443 0.056 7.871 0.000 0.100 0.202
V7 1.439 0.099 14.501 0.000 0.323 0.659
V9 1.197 0.081 14.848 0.000 0.269 0.541
lying =~
V6 1.000 0.140 0.281
V12 -0.817 0.129 -6.341 0.000 -0.114 -0.287
V18 -0.782 0.122 -6.424 0.000 -0.109 -0.297
V24 0.995 0.151 6.592 0.000 0.139 0.321
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
extraversion ~~
neuroticism 0.002 0.001 3.629 0.000 0.210 0.210
lying -0.002 0.001 -3.402 0.001 -0.283 -0.283
neuroticism ~~
lying -0.016 0.002 -6.998 0.000 -0.521 -0.521
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.V1 0.199 0.005 37.811 0.000 0.199 0.987
.V3 0.243 0.006 37.654 0.000 0.243 0.982
.V5 0.123 0.005 27.226 0.000 0.123 0.871
.V8 0.049 0.030 1.615 0.106 0.049 0.218
.V2 0.199 0.006 31.450 0.000 0.199 0.797
.V4 0.234 0.006 37.030 0.000 0.234 0.959
.V7 0.136 0.007 18.273 0.000 0.136 0.566
.V9 0.174 0.007 26.548 0.000 0.174 0.707
.V6 0.227 0.007 32.757 0.000 0.227 0.921
.V12 0.145 0.004 32.495 0.000 0.145 0.918
.V18 0.123 0.004 32.041 0.000 0.123 0.912
.V24 0.168 0.005 30.875 0.000 0.168 0.897
extraversion 0.003 0.001 2.541 0.011 1.000 1.000
neuroticism 0.051 0.005 9.508 0.000 1.000 1.000
lying 0.019 0.004 4.491 0.000 1.000 1.000
You can see that your variance from the model (0.199) is very similar to the real variance (0.201)
which indicates our model does not have variance issues.
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Examine Modification Indices
The fit indices for our epi.model are low (in the .70s) for CFI and TLI. You can use modification indices to find
potential parameters (paths) to add to the model specification to improve model fit.
Examine the modification indices for the epi.model and add the path with the largest mi to your model. The libraries and
model have been loaded for you. After you edit your model, rerun the model summary to examine the change in fit indices.
* Use modificationindices() on the epi.fit to find the mi values.
* Use sort = TRUE to see the highest values first.
* Update the model specification code to include the largest mi value.
* Reanalyze the model with the updated model specification to see how your fit indices improved.
* Use epi.model2 from the previous step as your model name.
# Original model summary
summary(epi.fit, standardized = TRUE, fit.measures = TRUE)
# Examine the modification indices
modificationindices(epi.fit, sort = TRUE)
lhs op rhs mi epc sepc.lv sepc.all sepc.nox
40 neuroticism =~ V3 168.998 -0.691 -0.155 -0.312 -0.312
39 neuroticism =~ V1 122.899 0.531 0.119 0.266 0.266
48 lying =~ V3 122.762 1.315 0.180 0.361 0.361
70 V3 ~~ V7 76.778 -0.034 -0.034 -0.185 -0.185
58 V1 ~~ V2 76.542 0.032 0.032 0.163 0.163
47 lying =~ V1 60.229 -0.830 -0.114 -0.253 -0.253
59 V1 ~~ V4 27.315 0.020 0.020 0.093 0.093
67 V3 ~~ V8 20.964 0.042 0.042 0.366 0.366
75 V3 ~~ V24 19.003 0.016 0.016 0.081 0.081
32 extraversion =~ V4 18.508 0.831 0.045 0.091 0.091
103 V4 ~~ V12 17.706 0.014 0.014 0.078 0.078
113 V9 ~~ V18 15.994 0.012 0.012 0.083 0.083
51 lying =~ V2 15.368 0.613 0.084 0.168 0.168
52 lying =~ V4 13.179 -0.552 -0.076 -0.153 -0.153
35 extraversion =~ V6 11.524 -0.792 -0.043 -0.086 -0.086
57 V1 ~~ V8 10.574 -0.027 -0.027 -0.261 -0.261
86 V8 ~~ V4 9.716 0.012 0.012 0.106 0.106
68 V3 ~~ V2 9.516 -0.013 -0.013 -0.058 -0.058
66 V3 ~~ V5 9.467 -0.010 -0.010 -0.061 -0.061
116 V6 ~~ V18 9.309 0.011 0.011 0.066 0.066
45 neuroticism =~ V18 8.918 0.199 0.045 0.122 0.122
64 V1 ~~ V18 8.768 0.008 0.008 0.054 0.054
74 V3 ~~ V18 8.347 -0.009 -0.009 -0.053 -0.053
84 V5 ~~ V24 8.334 0.008 0.008 0.056 0.056
46 neuroticism =~ V24 7.824 0.240 0.054 0.125 0.125
89 V8 ~~ V6 7.197 -0.011 -0.011 -0.103 -0.103
71 V3 ~~ V9 6.986 -0.011 -0.011 -0.051 -0.051
99 V2 ~~ V24 6.977 0.010 0.010 0.053 0.053
111 V9 ~~ V6 6.588 0.011 0.011 0.053 0.053
61 V1 ~~ V9 6.361 0.009 0.009 0.049 0.049
76 V5 ~~ V8 5.668 0.062 0.062 0.760 0.760
87 V8 ~~ V7 5.270 0.009 0.009 0.109 0.109
117 V6 ~~ V24 4.652 0.010 0.010 0.050 0.050
107 V7 ~~ V6 4.451 -0.009 -0.009 -0.049 -0.049
60 V1 ~~ V7 4.238 0.007 0.007 0.044 0.044
69 V3 ~~ V4 4.212 0.009 0.009 0.037 0.037
31 extraversion =~ V2 4.066 -0.393 -0.021 -0.042 -0.042
56 V1 ~~ V5 3.741 0.006 0.006 0.038 0.038
85 V8 ~~ V2 3.729 -0.007 -0.007 -0.071 -0.071
100 V4 ~~ V7 3.174 -0.008 -0.008 -0.047 -0.047
96 V2 ~~ V6 2.705 0.007 0.007 0.032 0.032
88 V8 ~~ V9 2.438 -0.006 -0.006 -0.062 -0.062
37 extraversion =~ V18 2.373 -0.270 -0.015 -0.040 -0.040
95 V2 ~~ V9 2.309 0.009 0.009 0.048 0.048
106 V7 ~~ V9 1.956 -0.013 -0.013 -0.084 -0.084
83 V5 ~~ V18 1.909 0.003 0.003 0.026 0.026
34 extraversion =~ V9 1.852 -0.272 -0.015 -0.029 -0.029
92 V8 ~~ V24 1.837 0.005 0.005 0.056 0.056
94 V2 ~~ V7 1.815 0.010 0.010 0.058 0.058
49 lying =~ V5 1.688 0.155 0.021 0.057 0.057
33 extraversion =~ V7 1.661 0.281 0.015 0.031 0.031
55 V1 ~~ V3 1.304 -0.004 -0.004 -0.020 -0.020
38 extraversion =~ V24 1.183 0.239 0.013 0.030 0.030
114 V9 ~~ V24 1.131 -0.004 -0.004 -0.023 -0.023
81 V5 ~~ V6 1.131 0.003 0.003 0.020 0.020
42 neuroticism =~ V8 1.060 0.155 0.035 0.074 0.074
110 V7 ~~ V24 1.032 0.004 0.004 0.025 0.025
120 V18 ~~ V24 1.003 -0.003 -0.003 -0.024 -0.024
63 V1 ~~ V12 0.997 0.003 0.003 0.018 0.018
65 V1 ~~ V24 0.969 0.003 0.003 0.018 0.018
91 V8 ~~ V18 0.954 -0.003 -0.003 -0.038 -0.038
101 V4 ~~ V9 0.932 -0.004 -0.004 -0.021 -0.021
98 V2 ~~ V18 0.909 -0.003 -0.003 -0.019 -0.019
115 V6 ~~ V12 0.906 0.004 0.004 0.021 0.021
53 lying =~ V7 0.903 -0.175 -0.024 -0.049 -0.049
54 lying =~ V9 0.892 -0.156 -0.021 -0.043 -0.043
77 V5 ~~ V2 0.700 0.002 0.002 0.016 0.016
43 neuroticism =~ V6 0.642 0.070 0.016 0.032 0.032
44 neuroticism =~ V12 0.584 0.055 0.012 0.031 0.031
72 V3 ~~ V6 0.561 -0.003 -0.003 -0.014 -0.014
62 V1 ~~ V6 0.557 0.003 0.003 0.014 0.014
119 V12 ~~ V24 0.520 -0.003 -0.003 -0.017 -0.017
78 V5 ~~ V4 0.379 0.002 0.002 0.011 0.011
90 V8 ~~ V12 0.325 -0.002 -0.002 -0.022 -0.022
36 extraversion =~ V12 0.267 -0.098 -0.005 -0.013 -0.013
105 V4 ~~ V24 0.228 0.002 0.002 0.009 0.009
93 V2 ~~ V4 0.175 0.002 0.002 0.008 0.008
79 V5 ~~ V7 0.141 0.001 0.001 0.008 0.008
80 V5 ~~ V9 0.119 -0.001 -0.001 -0.007 -0.007
118 V12 ~~ V18 0.118 -0.001 -0.001 -0.008 -0.008
41 neuroticism =~ V5 0.116 0.017 0.004 0.010 0.010
102 V4 ~~ V6 0.106 -0.001 -0.001 -0.006 -0.006
97 V2 ~~ V12 0.104 -0.001 -0.001 -0.006 -0.006
73 V3 ~~ V12 0.077 -0.001 -0.001 -0.005 -0.005
108 V7 ~~ V12 0.054 -0.001 -0.001 -0.005 -0.005
112 V9 ~~ V12 0.022 0.000 0.000 -0.003 -0.003
82 V5 ~~ V12 0.013 0.000 0.000 0.002 0.002
50 lying =~ V8 0.012 -0.041 -0.006 -0.012 -0.012
104 V4 ~~ V18 0.012 0.000 0.000 -0.002 -0.002
109 V7 ~~ V18 0.003 0.000 0.000 0.001 0.001
# Edit the model specification
epi.model2 <- 'extraversion =~ V1 + V3 + V5 + V8
neuroticism =~ V2 + V4 + V7 + V9
lying =~ V6 + V12 + V18 + V24
neuroticism =~ V3'
# Reanalyze the model
epi.fit <- cfa(model = epi.model2, data = epi)
# Summarize the updated model
summary(epi.fit, standardized = TRUE, fit.measures = TRUE)
lavaan 0.6-11 ended normally after 116 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 28
Used Total
Number of observations 3193 3570
Model Test User Model:
Test statistic 332.891
Degrees of freedom 50
P-value (Chi-square) 0.000
Model Test Baseline Model:
Test statistic 2196.019
Degrees of freedom 66
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.867
Tucker-Lewis Index (TLI) 0.825
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -23082.232
Loglikelihood unrestricted model (H1) -22915.787
Akaike (AIC) 46220.465
Bayesian (BIC) 46390.389
Sample-size adjusted Bayesian (BIC) 46301.421
Root Mean Square Error of Approximation:
RMSEA 0.042
90 Percent confidence interval - lower 0.038
90 Percent confidence interval - upper 0.046
P-value RMSEA <= 0.05 0.999
Standardized Root Mean Square Residual:
SRMR 0.040
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
extraversion =~
V1 1.000 0.068 0.152
V3 1.798 0.325 5.532 0.000 0.123 0.246
V5 -2.268 0.360 -6.291 0.000 -0.155 -0.414
V8 5.077 0.887 5.725 0.000 0.346 0.732
neuroticism =~
V2 1.000 0.222 0.445
V4 0.432 0.053 8.134 0.000 0.096 0.194
V7 1.493 0.093 16.025 0.000 0.331 0.675
V9 1.186 0.074 15.938 0.000 0.263 0.530
lying =~
V6 1.000 0.135 0.272
V12 -0.851 0.127 -6.699 0.000 -0.115 -0.290
V18 -0.799 0.119 -6.728 0.000 -0.108 -0.294
V24 1.115 0.157 7.087 0.000 0.151 0.347
neuroticism =~
V3 -0.732 0.066 -11.074 0.000 -0.163 -0.327
Covariances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
extraversion ~~
neuroticism 0.004 0.001 4.953 0.000 0.283 0.283
lying -0.003 0.001 -4.380 0.000 -0.346 -0.346
neuroticism ~~
lying -0.016 0.002 -7.337 0.000 -0.521 -0.521
Variances:
Estimate Std.Err z-value P(>|z|) Std.lv Std.all
.V1 0.196 0.005 39.250 0.000 0.196 0.977
.V3 0.217 0.006 34.642 0.000 0.217 0.878
.V5 0.116 0.004 29.066 0.000 0.116 0.828
.V8 0.104 0.014 7.603 0.000 0.104 0.465
.V2 0.200 0.006 33.875 0.000 0.200 0.802
.V4 0.235 0.006 39.046 0.000 0.235 0.962
.V7 0.131 0.007 19.577 0.000 0.131 0.544
.V9 0.178 0.006 29.830 0.000 0.178 0.720
.V6 0.228 0.007 34.969 0.000 0.228 0.926
.V12 0.144 0.004 34.186 0.000 0.144 0.916
.V18 0.123 0.004 34.035 0.000 0.123 0.914
.V24 0.166 0.005 31.188 0.000 0.166 0.879
extraversion 0.005 0.001 3.265 0.001 1.000 1.000
neuroticism 0.049 0.005 10.127 0.000 1.000 1.000
lying 0.018 0.004 4.651 0.000 1.000 1.000
Your fit indices should improve to the .80s by including this one extra parameter to the model.
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Compare Two Models
In the last exercises, you created two models of the epi, and these models have been loaded for you.
The original model epi.model and the updated model with the modified path epi.model1 can now be compared using the anova()
function to determine if the change in fit indices was a large change. We can use the anova() function because these models
are nested, which means they are the same manifest variables with different parameters.
* Analyze both models with the cfa() function and the epi dataset.
* Use the anova() function to compare these models.
# Analyze the original model
epi.fit <- cfa(model = epi.model, data = epi)
# Analyze the updated model
epi.fit1 <- cfa(model = epi.model1, data = epi)
# Compare those models
anova(epi.fit, epi.fit1)
Chi-Squared Difference Test
Df AIC BIC Chisq Chisq diff Df diff Pr(>Chisq)
epi.fit1 50 46220 46390 332.89
epi.fit 51 46396 46560 510.40 177.51 1 < 2.2e-16 ***
---
Signif. codes: 0 â€***’ 0.001 â€**’ 0.01 â€*’ 0.05 â€.’ 0.1 †’ 1
The updated model appears better, as the chi-square difference test is significant.
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Select Specific Fit Indices
You can also compare models by using the AIC or ECVI fit indices, rather than the anova() function. These fit indices are
very useful if your models include different manifest variables. When comparing sets of AIC or ECVI values, the best model
would have the smallest fit index. The saved models from the previous exercise have been loaded for you.
* Use fitmeasures() to calculate the fit indices for each model separately.
* Select only the aic and ecvi fit indices to shorten the output.
# Analyze the original model
epi.fit <- cfa(model = epi.model, data = epi)
# Find the fit indices for the original model
fitmeasures(epi.fit, c("aic", "ecvi"))
aic ecvi
46395.976 0.177
# Analyze the updated model
epi.fit1 <- cfa(model = epi.model1, data = epi)
# Find the fit indices for the updated model
fitmeasures(epi.fit1, c("aic", "ecvi"))
aic ecvi
46220.465 0.122
For both AIC and ECVI, the updated model included the smaller fit indices and would be considered the better model.