# How to conduct confirmatory factor analysis with small sample size?

I plan to conduct a confirmatory factor analysis with 12 observed variables and 3 latent variables. My sample size is 30. However, I've read that for factor analysis, a larger sample size is typically recommended.

Is there a method to perform confirmatory factor analysis with a small sample size? I've heard that regularized exploratory factor analysis is suitable for small sample sizes. Is there an equivalent approach for confirmatory factor analysis?

Additionally, which software can I use for these analyses? Currently, I have access to SPSS, AMOS, and LISREL.

With 12 observed variables and 3 latent variables, you are going to struggle with CFA if your sample size is only 30, as you will most likely have insufficient power.

There are many rules of thumb for determining adequate samples size, $N$, in CFA. For $p$ variables and $q$ model parameters, some of these are:

• $N \geq 200$
• $N/p \geq 10$
• $N/q \geq 5$

Another approach would be to determine statistical power by simulation.

Myers, N.D., Ahn, S. and Jin, Y., 2011. Sample size and power estimates for a confirmatory factor analytic model in exercise and sport: A Monte Carlo approach. Research Quarterly for Exercise and Sport, 82(3), pp.412-423.

https://doi.org/10.1080/02701367.2011.10599773

Wolf, E.J., Harrington, K.M., Clark, S.L. and Miller, M.W., 2013. Sample size requirements for structural equation models: An evaluation of power, bias, and solution propriety. Educational and psychological measurement, 73(6), pp.913-934.

https://doi.org/10.1177/0013164413495237

I agree with @Robert Long's sentiment that you should be skeptical about results (assuming your model even converges) with your data characteristics, especially your sample size of 30. The issue of what to do when one wants to fit a confirmatory factor analysis (CFA) model with a small sample has been addressed a number of times on this site (e.g., here), though yours is unique in that I don't believe I have seen an application of FA with a sample size smaller than 50 (though that does not mean it is not worth a shot!).

Is there a way to conduct confirmatory factor analysis with a small sample size? I heard about regularized exploratory factor analysis is used for small sample sizes. Is there an equivalent of it with confirmatory factor analysis?

You are correct to point out that regularization can help with small samples. For example, Jung & Lee (2011) found evidence that samples of less than 50 may be sufficient for fitting a regularized exploratory factor analysis (EFA) model. Though from your question, it seems like you are more interested in fitting a regularized CFA model, which can be fit using the regSEM (Jacobucci, Grimm, & McArdle, 2016)$$^1$$ R package.

Also, what softwares can I use to run these? I currently have SPSS, AMOS, and LISREL.

Regarding software, besides the regSEM package, the only other software I have seen used to fit a regularized FA model is MATLAB (e.g., Jung & Lee, 2011) and mplus (see Asparouhov & Muthén, 2023 for more information).

$$^1$$ Also see Li, Jacobucci, & Ammerman (2021) for a good tutorial on the method.

References

Asparouhov, T., & Muthén, B. (2023). Penalized structural equation models. Technical Report. 2023. Available online: https://bit. ly/3TlbxdC (accessed on 4 March 2023).

Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized structural equation modeling. Structural equation modeling: a multidisciplinary journal, 23(4), 555-566.

Jung, S., & Lee, S. (2011). Exploratory factor analysis for small samples. Behavior research methods, 43, 701-709.

Li, X., Jacobucci, R., & Ammerman, B. A. (2021). Tutorial on the use of the regsem package in R. Psych, 3(4), 579-592.