EFA within a CFA framework, as the name implies, combines aspects of both EFA and CFA. It produces a factor solution that is close to an EFA solution while providing features found in CFA, such as standard errors, statistical tests and modification indices.
The trick to doing EFA within a CFA framework is to use the same number of constraints in CFA as are used in EFA. If m is the number of factors in an EFA model then the number of constraints is m2. By using m2 constraints in CFA we get a model with the same fit as an EFA model.
We will demonstrate EFA within a CFA framework using an artificial dataset with 500 observations on six variables. The data wee constructed to give a two factor solution.
We begin by loading the data and computing a correlation matrix.
use https://stats.idre.ucla.edu/stat/data/efa_cfa, clear
correlate
(obs=500)
| y1 y2 y3 y4 y5 y6
-------------+------------------------------------------------------
y1 | 1.0000
y2 | 0.5212 1.0000
y3 | 0.4719 0.5339 1.0000
y4 | -0.0008 -0.0345 -0.0101 1.0000
y5 | -0.0256 -0.0365 -0.0400 0.4297 1.0000
y6 | 0.0176 -0.0159 0.0390 0.3688 0.4268 1.0000
Now we are ready to run a maximum likelihood EFA model followed by an oblique rotation using the quartimin normalize option. We will also use estat common to get the correlation between the oblique facts.
factor y1 y2 y3 y4 y5 y6, ml
(obs=500)
number of factors adjusted to 3
Iteration 0: log likelihood = -31.62904
Iteration 1: log likelihood = -3.1615516
Iteration 2: log likelihood = -.34164269
Iteration 3: log likelihood = -.34066861 (backed up)
[output omitted]
Iteration 25: log likelihood = -.05290372
number of factors adjusted to 2
Iteration 0: log likelihood = -35.679836
Iteration 1: log likelihood = -1.103664
Iteration 2: log likelihood = -1.1024515
Factor analysis/correlation Number of obs = 500
Method: maximum likelihood Retained factors = 2
Rotation: (unrotated) Number of params = 11
Schwarz's BIC = 70.5656
Log likelihood = -1.102451 (Akaike's) AIC = 24.2049
--------------------------------------------------------------------------
Factor | Eigenvalue Difference Proportion Cumulative
-------------+------------------------------------------------------------
Factor1 | 1.54078 0.30931 0.5558 0.5558
Factor2 | 1.23147 . 0.4442 1.0000
--------------------------------------------------------------------------
LR test: independent vs. saturated: chi2(15) = 593.54 Prob>chi2 = 0.0000
LR test: 2 factors vs. saturated: chi2(4) = 2.19 Prob>chi2 = 0.7015
Factor loadings (pattern matrix) and unique variances
-------------------------------------------------
Variable | Factor1 Factor2 | Uniqueness
-------------+--------------------+--------------
y1 | 0.6763 0.0631 | 0.5387
y2 | 0.7657 0.0354 | 0.4124
y3 | 0.6944 0.0612 | 0.5141
y4 | -0.0675 0.6060 | 0.6282
y5 | -0.0975 0.6969 | 0.5048
y6 | -0.0290 0.6080 | 0.6295
-------------------------------------------------
rotate, oblique quartimin normalize
Factor analysis/correlation Number of obs = 500
Method: maximum likelihood Retained factors = 2
Rotation: oblique quartimin (Kaiser on) Number of params = 11
Schwarz's BIC = 70.5656
Log likelihood = -1.102451 (Akaike's) AIC = 24.2049
--------------------------------------------------------------------------
Factor | Variance Proportion Rotated factors are correlated
-------------+------------------------------------------------------------
Factor1 | 1.53686 0.5544
Factor2 | 1.23729 0.4463
--------------------------------------------------------------------------
LR test: independent vs. saturated: chi2(15) = 593.54 Prob>chi2 = 0.0000
LR test: 2 factors vs. saturated: chi2(4) = 2.19 Prob>chi2 = 0.7015
Rotated factor loadings (pattern matrix) and unique variances
-------------------------------------------------
Variable | Factor1 Factor2 | Uniqueness
-------------+--------------------+--------------
y1 | 0.6794 0.0117 | 0.5387
y2 | 0.7657 -0.0228 | 0.4124
y3 | 0.6972 0.0084 | 0.5141
y4 | -0.0073 0.6095 | 0.6282
y5 | -0.0281 0.7025 | 0.5048
y6 | 0.0313 0.6086 | 0.6295
-------------------------------------------------
Factor rotation matrix
--------------------------------
| Factor1 Factor2
-------------+------------------
Factor1 | 0.9971 -0.0990
Factor2 | 0.0758 0.9951
--------------------------------
estat common
Correlation matrix of the quartimin rotated common factors
----------------------------------
Factors | Factor1 Factor2
-------------+--------------------
Factor1 | 1
Factor2 | -.02325 1
----------------------------------
The rotated factor solution gives us a rather clean two factor model. We note that the model fit versus a saturated model has a chi-square of 2.19 with four degrees of freedom. This is a very good fit for an EFA and reflects the synthetic nature of the data. We also note the the two factors have a small correlation of -.02325.
Since the EFA solution has two factors we will need to make a total of four constraints in the CFA model. The easiest ones are to set the factor variances to one (2 factors, 2 constraints). The other constraints revolve around identifying anchor variables for each factor and constraining the cross loadings for the anchor variables to be zero (2 cross loadings, 2 more constraints for a total of 4).
An anchor item has a high loading on one factor and low loadings on the remaining factors. From the rotated solution above we see that y2 has the highest loading on Factor1 and has low loading on Factor2. We will make y2 the anchor item for Factor1 and will constrain the loading to equal zero in Factor2. We will use y5 as the anchor item for Factor2.
Here is the sem command for the EFA within a CFA framework.
sem (F1 -> y1 y2 y3 y4 y5@0 y6) ///
(F2 -> y1 y2@0 y3 y4 y5 y6) , ///
variance(F1@1 F2@1) standardized
Endogenous variables
Measurement: y1 y2 y3 y4 y5 y6
Exogenous variables
Latent: F1 F2
Fitting target model:
Iteration 0: log likelihood = -5064.3487 (not concave)
Iteration 1: log likelihood = -5005.6323 (not concave)
Iteration 2: log likelihood = -4997.4943 (not concave)
Iteration 3: log likelihood = -4982.0445 (not concave)
Iteration 4: log likelihood = -4978.5317 (not concave)
[output omitted]
Iteration 15: log likelihood = -4975.0886 (not concave)
Iteration 16: log likelihood = -4975.0731
Iteration 17: log likelihood = 2163798 (not concave)
[output omitted]
Iteration 40: log likelihood = 2163798 (not concave)
Iteration 41: log likelihood = 2163798 (not concave)
Iteration 42: log likelihood = 2163798 (not concave)
--Break--
r(1);
The sem command would have run forever if we had let it. However, since the log likelihood did not change from the 17th iteration on, we broke out of the program. There are two possible reasons for the endless iterations: 1) Either the model is not identified or 2) the starting values did not allow sem to converge on a solution. I’m betting that it is the second reason: bad starting values.
To investigate this we will run the sem command again but limit the iterations to 16 and see what the results look like.
sem (F1 -> y1 y2 y3 y4 y5@0 y6) ///
(F2 -> y1 y2@0 y3 y4 y5 y6) , ///
variance(F1@1 F2@1) standardized iterate(16)
Endogenous variables
Measurement: y1 y2 y3 y4 y5 y6
Exogenous variables
Latent: F1 F2
Fitting target model:
Iteration 0: log likelihood = -5064.3487 (not concave)
Iteration 1: log likelihood = -5005.6323 (not concave)
Iteration 2: log likelihood = -4997.4943 (not concave)
[output omitted]
Iteration 15: log likelihood = -4975.0886 (not concave)
Iteration 16: log likelihood = -4975.0731
convergence not achieved
Structural equation model Number of obs = 500
Estimation method = ml
Log likelihood = -4975.0731
( 1) [var(F1)]_cons = 1
( 2) [var(F2)]_cons = 1
------------------------------------------------------------------------------
| OIM
Standardized | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
Measurement |
y1 <- |
F1 | .6708773 .0557835 12.03 0.000 .5615436 .7802109
F2 | .0120947 .042074 0.29 0.774 -.0703689 .0945583
_cons | -.0157292 .0447242 -0.35 0.725 -.103387 .0719286
-----------+----------------------------------------------------------------
y2 chi2 = 0.0000
Warning: convergence not achieved
In the results above we see that all of the loadings are between 0 and 1 except for variable y6. Also the residual variance for y6 is much higher than any of the other residual variances. I think we can fix the problem by setting the initial starting value for the loading on Factor1 to 0 and on Factor2 to 0.5. We do this by including init in the sem command as seen below:
sem (F1 -> y1 y2 y3 y4 y5@0 (y6, init(0.0))) ///
(F2 -> y1 y2@0 y3 y4 y5 (y6, init(0.5))) , ///
variance(F1@1 F2@1) standardized
Endogenous variables
Measurement: y1 y2 y3 y4 y5 y6
Exogenous variables
Latent: F1 F2
Fitting target model:
Iteration 0: log likelihood = -5467.1006 (not concave)
Iteration 1: log likelihood = -5087.7064 (not concave)
[output omitted]
Iteration 8: log likelihood = -4905.7634
Iteration 9: log likelihood = -4905.7634
Structural equation model Number of obs = 500
Estimation method = ml
Log likelihood = -4905.7634
( 1) [var(F1)]_cons = 1
( 2) [var(F2)]_cons = 1
------------------------------------------------------------------------------
| OIM
Standardized | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
Measurement |
y1 chi2 = 0.6981
This time sem converged on a solution. All of the factor loadings and residual variances look reasonable. The model fit versus a saturated model has a chi-square of 2.20 with 4 df. This value is almost identical to the chi-square of 2.19 with 4 df from the maximum likelihood EFA model we ran at the beginning.
Before comparing the EFA within a CFA framework with the EFA solution, let’s look at some of the various indicators of goodness of fit.
estat gof, stat(all)
----------------------------------------------------------------------------
Fit statistic | Value Description
---------------------+------------------------------------------------------
Likelihood ratio |
chi2_ms(4) | 2.205 model vs. saturated
p > chi2 | 0.698
chi2_bs(15) | 596.921 baseline vs. saturated
p > chi2 | 0.000
---------------------+------------------------------------------------------
Population error |
RMSEA | 0.000 Root mean squared error of approximation
90% CI, lower bound | 0.000
upper bound | 0.051
pclose | 0.947 Probability RMSEA <= 0.05
---------------------+------------------------------------------------------
Information criteria |
AIC | 9857.527 Akaike's information criterion
BIC | 9954.463 Bayesian information criterion
---------------------+------------------------------------------------------
Baseline comparison |
CFI | 1.000 Comparative fit index
TLI | 1.012 Tucker-Lewis index
---------------------+------------------------------------------------------
Size of residuals |
SRMR | 0.007 Standardized root mean squared residual
CD | 0.925 Coefficient of determination
----------------------------------------------------------------------------
It is not surprising that the gof values all look good considering the artificial nature of the data.
Now, let’s compare the EFA within CFA solution to the EFA solution.
Summary of results
efa rotated maximum
efa within cfa loadings likelihood loading
F1 F2 Factor1 Factor2
y1 .6814311 .0319918 0.6794 0.0117
y2 .7665428 0 0.7657 -0.0228
y3 .6991976 .0292474 0.6972 0.0084
y4 .0171268 .6110693 -0.0073 0.6095
y5 0 .7036822 -0.0281 0.7025
y6 .0557532 .6112825 0.0313 0.6086
factor: LR test: 2 factors vs. saturated: chi2(4) = 2.19 Prob>chi2 = 0.7015
sem: LR test of model vs. saturated: chi2(4) = 2.20 Prob>chi2 = 0.6981
The EFA within a CFA framework results appear very close to the rotated maximum likelihood EFA results while having the added benefits of standard errors, statistical tests and access to modification indices, if needed.