2. Presentation Outline
Theoretical Introduction to
Confirmatory Factor Analysis (CFA)
Methods of CFA
How to run CFA in Mplus
Interpreting Output of CFA in Mplus
3. Whether you are conducting
exploratory factor analysis
(EFA) or confirmatory factor
analysis (CFA) your basic aim
is the same.
To describe a large number of
observed variables in terms of
a smaller number of latent
variables (factors).
What then differentiates CFA
from EFA?
Confirmatory Factor Analysis
4. CFA is a theoretically driven approach - EFA is
statistically driven.
CFA has the ability to directly test specific
hypotheses – EFA does not.
One of the primary reasons psychologists use
CFA is to validate psychometric tests.
CFA vs. EFA
5. Construct validity involves determining whether a particular
psychometric tool measures what it claims to.
For example, we developed the Measure of Criminal Social
Identity to measure three distinct factors of Criminal Social
Identity.
If we found that the observed covariance matrix was best
explained by a 1 factor solution then this would undermine
the measure’s construct validity.
We found a three factor solution was the best fit! (Boduszek,
Adamson, Shevlin, & Hyland, 2012).
Construct Validity
6. Very often measures of psychological
constructs are used without any explicit
investigation that they are in fact measuring
what they claim to measure.
CFA procedures afford psychologists the
ability to explicitly investigate whether a
given measure is performing as they intend
it to.
Construct validity is only one element of the
overall validation process but it is a critical
element!
Construct Validity
7. Let’s look at our example for this section – The
Posttraumatic Stress Diagnostic Scale (PDS:
Foa, Cashman, Jaycox, & Perry, 1997)
The PDS contains questions that measure the
severity of 17 symptoms listed in the DSM-IV
which characterise PTSD.
The DSM-IV states that these 17 symptoms fall
into three symptom categories (Factors):
Intrusions, Avoidance & Emotional Numbing, and
Hyperarousal.
PTSD Factor Structure
8. However, a large body of evidence suggest that these
17 symptoms are better represented by two distinct
intercorrelated four-factor models.
Intrusions, Avoidance, Emotional Numbing, and
Hyperarousal (King, Leskin, King, and Weathers
1998).
Intrusions, Avoidance, Dysphoria, and Hyperarousal
(Simms, Watson, & Doebbeling, 2002).
PTSD Factor Structure
11. We now have a number of competing models
that attempt to explain the underlying structure
of these 17 symptoms.
Meaningless unless we have some empirical
method of testing these competing prediction.
CFA makes it possible to test these competing
theoretical predictions.
Competing Models
12. This testing procedure is related to the concept of model fit.
Let’s say we wanted to test the DSM’s 3-factor
conceptualization.
We could go out a collect data from a sufficiently large
sample of the population who have been exposed to a
trauma.
If the DSM’s model reflects reality then there should be a
covariance matrix within the obtained sample data
consistent with this idea of three underlying factors.
In other words the obtained data should match (‘fit’) our
theoretical prediction.
Model Fit
13. If our proposed model does not fit the data then our
model is a not an accurate representation of reality.
This gets to the crux of why CFA is so
valuable/powerful – Falsification.
A central feature of any scientific theory is that it must
be falsifiable.
“A theory that explains everything explains nothing.”
~ Karl Popper
Falsification
14. Certain theoretical models may contain multiple factors that
are proposed to be correlated.
If the correlations between factors are high, it is suggestive
that the correlations among these factors may be accounted
for by a superordinate factor, or factors.
An additional latent variable(s) may be responsible for the
observed factor correlations.
In our case the relationships between Intrusions, Avoidance,
Dysphoria, and Hyperarousal may be explained by a higher-
order latent variable - PTSD.
Higher Order Models
15. Assessing the accuracy of a theoretical model is
judged in relation to its ‘goodness of fit’.
Same procedure as in the EFA!
Range of goodness of fit statistics available in
Mplus.
Chi-Square, CFI, TLI, RMSEA, SRMR, AIC.
Model Fit
16. The 2 statistic is the most frequently cited index
of absolute fit.
The probability of the 2 should be greater than
the chosen alpha level (0.05).
Compares the observed covariance matrix with
our theoretically proposed covariance matrix.
A non-significant result indicates no statistically
significant difference between the actual
covariance matrix and our proposed model to
explain this covariance matrix.
Chi-Square (2)
17. The 2 statistic should be interpreted cautiously!
Most criticisms of the 2 test are concerned with the effects of
sample size.
The power of a test is positively related to sample size
Poor models produce non-significant results with low sample
sizes while good models can produce statistically significant
results when sample sizes are high.
Klein (1994) recommends evaluating the 2 result in relation to
the degrees of freedom (df).
2:df values of less than 3:1 suggest good model fit.
Despite the limitations of the 2 test researchers are advised to
always cite the value in their reports (Hoyle & Panter, 1995).
Chi-Square (2)
18. Comparative Fit Index (CFI; Bentler, 1990) and
Tucker Lewis Index (TLI; Tucker and Lewis, 1973) are
incremental fit indices.
CFI and TLI indicate how much better a model fits the
data compared to a baseline model where all variables
are uncorrelated.
Values can range from 0-1
For these indices values above .90 indicate reasonable
fit
Values above .95 indicated good model fit (Bentler,
1990; Hu & Bentler, 1999).
CFI & TLI
19. The Root Mean Square Error of Approximation (RMSEA) is a
measure of “discrepancy per degree of freedom” in a model
(Browne & Cudeck, 1993).
This fit index recognises that models can only ever be
approximately correct.
A flexible index based on chi-square yet takes parsimony into
account
The addition of a parameter which reduces the chi-square by a
substantial degree will cause a decrease in the RMSEA.
Produces calculation of confidence intervals and significance
tests.
Values < 0.05 suggest good model fit.
Values < 0.08 suggest reasonable model fit.
RMSEA
20. The standardized root mean-square residual (SRMR:
Joreskog & Sorborn, 1981) is an absolute measure of fit
Is defined as the standardized difference between the
observed correlation and the predicted correlation.
This measure tends to be smaller as sample size increases
and as the number of parameters in the model increases –
no penalties for model complexity.
Values < 0.05 indicate good model fit
Values < 0.08 indicate reasonable model fit
SRMR
21. The Akaike Information Criterion (AIC; Akaike, 1974)
is a comparative measure of model fit.
Only meaningful when multiple models are
estimated.
Lower values indicate a better fit and so the model
with the lowest AIC is the best fitting model.
The AIC also contains explicit penalties for model
complexity.
AIC
22. A non-significant 2, or a 2:df ratio of less than 3:1 (Kline,
2005)
CFI and TLI values above .95 (Hu &Bentler, 1999; Vandenberg
& Lance, 2000).
However, for CFI and TLI, values above .90 indicate adequate
fit (Bentler, 1990; Hu & Bentler, 1999).
RMSEA and SRMR values less than .05 suggest good fit and
values up to .08 indicate reasonable errors of approximation in
the population (Browne and Cudeck, 1989).
AIC is used to compare alternative models, with the smallest
value indicating the best fitting model.
The CFI, RMSEA and the AIC all have explicit penalties for
model complexity.
Goodness of Fit
23. The adequacy of any model can also be judge by
investigating the factor loadings.
In CFA, there are generally no cross-factor loadings as
in EFA, unless specifically indicated a priori.
These factor loadings represent the strength of the
association between the latent variable and the
observed variable.
Indicates how much of the variance in each indicator
is explained by the latent variable.
The remainder is due to measurement error.
Factor Loadings
24. CFA is a theoretically driven analytic procedure.
Allows researchers to determine the construct validity
of a measure.
Has the ability to falsify proposed theoretical models.
Cannot prove that a model is “correct” – only fail to
reject it. Always the possibility that a better model
exists which just hasn’t been tested.
Adequacy of a theoretical model is judged in terms of
how well it “fits” the observed data – a range of fit
statistics: 2, CFI, TLI, RMSEA, & SRMR.
Conclusion
26. We will be testing 4 alternative models theorised to
explain the underlying structure of the PDS.
A 1-factor model in which all 17 items load onto a single
PTSD factor.
An intercorrelated four-factor model consistent with King,
Leskin, King, and Weathers (1998)
An intercorrelated four-factor model consistent with
Simms, Watson, and Doebbeling (2002)
A high-order conceptualization of the Simms et al. (2002)
model.
Models
27. We will be using the data set entitled ‘PDS’
Unlike SPSS, Mplus does not allow you to use drop-down
commands to estimate the model - you must write the
syntax yourself (don’t panic!).
It is a good idea to create a shorter data set yourself for
your specific analysis in Mplus.
Saving Data for Use in Mplus
28. Mplus cannot directly read an SPSS file.
Mplus can easily read Tab delimited data, so we can save our
dataset as a .dat file. This can be done by choosing File, Save
as.
Be sure to untick the box “Write Variable Names to
Spreadsheet”
We will save the variable names quickly from SPSS by copying
them from the Variable View window and pasting them into a
new text editor or directly into an Mplus input file.
Ready to open a new Mplus window and start writing syntax.
Saving Data for Use in Mplus
33. First we have to provide a TITLE for our analysis (PDS Simms
CFA)
To read our DATA we indicate the location of the .dat file we
saved.
Under the VARIABLE heading after ‘names are’ you paste in
your variable names from your SPSS data set.
In the next line, we indicate which values should be considered
missing in each variable. In our example missing are all (99).
Mplus Syntax for CFA
34. In USEVAR enter those variables which are to be used for the
current analysis (PDS22-PDS38).
The CATEGORICAL option is used to specify which variables
are treated as binary or ordered categorical (ordinal) variables
in the model and its estimation.
Not applicable in this case so we place an ! in front – this
eliminates this option
Under the ANALYSIS heading we must indicate what
ESTIMATOR we will be using.
Mplus Syntax for CFA
35. Because our observed variables are measured on 5-point Likert
scale we will use Robust Maximum Likelihood (MLR)
estimation.
If your observed variables are categorical use Estimator =
WLSMV
Mplus Syntax for CFA
36. The MODEL statement specifies the particular model to be
estimated.
This is the place where you have to create your latent variables
(four factors in this example).
In CFA we use the command “by” to create latent variables.
The latent variable “Intrusions” is measured by items PDS22-
PDS26
The latent variable “Avoidance” is measured by items PDS 27-
PDS28
The latent variable “Dysphoria” is measured by items PDS29-
PDS36
The latent variable “Hyperarousal” is measured by items
PDS37-PDS38
Mplus Syntax for CFA
37. The metric of the factors is set automatically by the
program by fixing the first factor loading in each “by”
statement to 1.
The factor loadings (and error variances) for each
indicator are estimated – along with a significance test.
The error variances are not correlated as the default.
The factors are correlated as the default .
In the OUTPUT: we want MODINDICES and
STANDARDIZED.
Mplus Syntax for CFA
38. There are a few things to keep in mind when creating
Mplus syntax.
First, all commands end with a semicolon; omitting the
semicolon will lead to error messages.
Second, commands can take up more than one line, as the
semicolon marks the command end.
Finally, Mplus is not case sensitive; capital and lowercase
letters can be used interchangeably.
Mplus Syntax for CFA
39. Once you have created syntax for confirmatory factor
analysis press to run the model.
Save this as an input file under some name e.g., dysphoria
model.inp in the same folder as the PDS.dat file.
This produces a text output (.out) file stored in the
working directory with the results.
For this model the output file looks like the following:
Mplus Output for CFA
41. The first part of the output provides a summary of the
analysis including:
The number of groups (1)
The number of observations (participants included in the
analysis, N=310)
The number of items included in the confirmatory model
(number of dependent variables = 17)
The number of latent variables (4).
Furthermore, Mplus gives more info which you do not
need to report except what Estimator was used (in this
example it was MLR= robust maximum likelihood).
Mplus Output for CFA
42. The next step is to investigate how well the model fit our
data.
This model of the symptom structure was specified and
estimated in Mplus as an intercorrelated 4-factor solution.
Before we look at the factor structure we have to assess the
fit between the data and pre-established theoretical model.
Goodness-of-fit indices are used to assess model fit.
Assessing Model Fit
43.
44. If this 4-factor model fits the data, and better than other
models tested, then we are interested in more details about
this model.
Mplus output provides lots of information however you are
interested only in few of them.
Unstandardized Factor Loadings and Standard Errors
Standardized Factor Loadings and Significance Levels
Factor Correlations.
Assessing Model Fit
