Maximum Likelihood Exploratory Factor Analysis and Confirmatory Factor Analysis

[hee sun]

Dear Statalist users,

I am using Stata 15 and comparing the results between Maximum likelihood exploratory factor analysis (ML EFA) and Confirmatory factor analysis (CFA) based on the one-factor model.

Here is an example of the data

clear
input float(item13 item14 item15 item16)
5 5 5 5
5 4 4 4
5 4 5 5
4 4 4 4
5 5 4 5
5 5 4 4
5 5 5 5
5 5 4 4
5 5 5 5
5 4 4 4
5 5 5 5
5 5 5 4
5 5 5 4
4 5 5 5
5 4 4 4
5 5 5 5
5 5 5 5
5 5 5 5
5 5 5 5
5 4 5 5
5 5 5 5
5 5 5 5
5 4 5 5
5 5 4 5
5 5 5 5
5 5 5 4
5 5 5 5
4 4 4 4
5 4 5 5
5 5 4 4
4 4 4 5
4 5 4 5
5 5 5 4
5 5 5 5
5 5 5 5
5 5 5 5
4 4 5 4
5 5 5 5
4 4 4 4
5 5 5 5
5 5 4 5
4 3 4 5
4 4 3 2
5 5 5 5
5 5 5 5
5 5 5 5
4 4 4 2
4 4 4 5
4 4 4 4
5 5 5 5
4 4 4 4
4 5 3 4
4 4 3 3
5 5 4 4
4 4 4 3
4 4 4 4
5 5 4 5
4 5 3 3
4 4 4 4
4 3 1 2
5 5 5 5
5 4 4 4
4 4 4 4
4 4 4 4
4 4 4 4
5 5 5 4
4 4 4 4
4 4 4 4
4 5 4 3
1 1 1 1
4 4 4 4
5 5 5 4
5 5 5 5
5 5 5 4
5 5 5 5
5 5 5 5
5 5 5 5
5 5 5 5
5 5 5 5
5 5 5 5
5 5 5 5
5 5 5 4
5 5 5 5
5 5 5 5
5 5 5 4
5 5 5 5
5 5 5 5
5 5 5 5
5 5 5 4
4 4 4 4
5 5 5 5
4 5 5 4
5 5 5 5
5 5 4 5
5 5 5 5
4 5 4 4
5 5 5 5
5 5 5 5
5 5 5 5
5 5 5 5
end

*Conducting ML EFA and CFA

factor item13-item16, factors(1) ml
sem(X->item13-item16), stand

It seems that (1) the factor loadings from ML EFA and the standardized coefficients from CFA and (2)
the uniqueness from ML EFA and the residual variances from CFA are same but their log likelihoods are different.
Given that CFA is more flexible than EFA in general, is there anyway to make the results of ML EFA same as the results of CFA in this situation?
I also want to know why their log-likelihoods are different.
Thank you

[Phil Bromiley]

I am not expert in this area, but as I understand it, the exploratory factor analysis model has what the documentation calls "an obvious freedom" - it is not fully identified which is why you can rotate it. In contrast, the sem is fully identified.

[hee sun]

Thank you for your answer. However, I think rotation should not change the results on the log likelihood and factor loadings (this is true because this model only has one factor).