Basically, there are two ways to determine the appropriate sample size; by determining the minimum number of samples needed (N), or by determining the sample size as a function of the number of variables, also known as the subjects-to-variable ratio, (N:p) (Beavers, Lounsbury, Richards, Huck, Skolits, & Esquivel,, 2013; Guadagnoli & Velicer 1988; Hogarty, Hines, Kromrey, Ferron, & Mumford, 2005).

Hogarty et al. (2005) assessed the appropriate minimum sample size, N, versus the minimum N:p ratio. They found that a higher number of samples were necessary when the goal of the study goal was to understand which factors underlie which variables. When the study goal was to ensure that sample loadings correlated highly with population loadings, fewer samples were necessary.

Jung and Lee (2011) analyzed the outcomes of the factor extraction methods maximum likelihood factor analysis (MLFA), principle component analysis (PCA), and regularized exploratory factor analysis (REFA) using sample sizes less than 50. They found that REFA recovered good factor loadings, had smaller mean absolute differences and mean square errors, and provided stable factor-loading estimates with samples of 50 or less. Fabrigar et al. (1999) determined that samples as low as 100 could yield stable solutions. Beavers et al. (2013) recommended using samples of at least 150 for multivariate tools, such as EFA. Guadagnoli and Velicer (1988) concluded that the subjects-to-variable ratio, or sample size as a function of the number of variables (N:p), was not an accurate method to establish sample size. Guadagnoli and Velicer also determined that when researchers selected variables that were good indicators of a component, 150 observations yielded accurate solutions. At the high end of the minimum number of samples, Guadagnoli and Velicer (1988) found that they needed 300 when few variables defined factors with moderate to low loadings.

REFERENCES

Beavers, A. S., Lounsbury, J. W., Richards, J. K., Huck, S. W., Skolits, G J., & Esquivel, S. L. (2013). Practical considerations for using exploratory factor analysis in educational research. Practical Assessment, Research & Evaluation, 18(6), 1-13. Retrieved from http://www.pareonline.net/pdf/v18n6.pdf

Guadagnoli, E. & Velicer, W. F. (1988). Relation of sample size to the stability of component patterns. Psychological Bulletin, 103, 265-275. doi:10.1037/0033-2909.103.2.265

Hogarty, K Y., Hines, C. V., Kromrey, J. D., Ferron, J. M., & Mumford, K. R. (2005). The quality of factor solutions in exploratory factor analysis: The influence of sample size, communality, and overdetermination. Educational and Psychological Measurement, 65, 202-226. doi:10.1177/0013164404267287

Jung, S., & Lee, S. (2011). Exploratory factor analysis for small samples. Behavioral Resources, 43, 701-709. doi:10.3758/s13428-011-0077-9

There are many answers to this question, and really there are no perfect ones. It is up to the researcher to do what is best in that particular case. I would suggest having an item:participant ratio of at least 1:10.  This way you have a roughly 60% chance of obtaining the correct factor structure, which is fair enough.

A rule of thumb is, the larger the sample - the better.

The most helpful article I've found was this one:

• Anna B. Costello and Jason W. Osborne (2005) Best Practices in Exploratory Factor Analysis: Four Recommendations for Getting the Most  From Your Analysis

Remember, the danger you are trying to avoid is that a small sample size yields a solution that predicts a misleadingly large percent of the variance, and it will not replicate.  If you are forced to have a small sample, you need to at least confess those weaknesses.

I agree with the rule at least 10 cases for each item.

Nathan Zhao has compiled a huge collection of advice given on this issue here. There is no that much agreement. https://www.encorewiki.org/display/~nzhao/The+Minimum+Sample+Size+in+Factor+Analysis

There is this paper by MacCallum et al. (1999) that address this problem, you can find it in this psycnet link:

http://psycnet.apa.org/journals/met/4/1/84/

Also, I think the paper has a free access in this page

I agree with @John Antonakis, if you have a theoretical basis for the questionnaire is more recommended to use Confirmatory Factor Analysis

A very important parameter is to do with the actual concepts under consideration. To this end, we have to look carefully at the Thesaurus Tree, specifically, the root concepts, their level, equivalent and related terms, in relation to the age / academic level / geographic area.

Prof. E. J. Yannakoudakis

Dear all, 

I would  lilke to agree that most rules of thumb are useless. In a recent paper we found that 10.000 Persons for 25 Items are not enough to recover all loadings with adequate accuracy (https://www.researchgate.net/publication/265129980_Selecting_items_for_Big_Five_questionnaires_At_what_sample_size_do_factor_loadings_stabilize).

Article Selecting items for Big Five questionnaires: At what sample ...

Thank you for all your informative answers and literature suggestions!

Good Morning,

I also found several references on this topic in the literature. And some lack of agreement!

Variations from the ideal sample size ranging 100 to 1000 individuals.

http://ro.ecu.edu.au/cgi/viewcontent.cgi?article=1373&context=jephc

The sample of the our study of validation process of the EPEA was

composed of 128 nurses.

The use of PCA was corroborated by the KMO test, which was 0.726, showing that the studied sample had a good fit.

Bartlett’s sphericity test presented a significant value with c2(153)=614.477 (p

Nathan Zhao has a very nice compilation of recommendations here. https://www.encorewiki.org/display/~nzhao/The+Minimum+Sample+Size+in+Factor+Analysis

I'm afraid you will not find what you seek. There is a huge variety in recommended practice.

I agree with you Emil, there is a great variety of recommendations, but the disucussion and literautre suggestions has enriched my knowledge and my arguments for the intended sample size of my coming research protocol.

Kristina, You already have several good answers but let me add two points.

First, the psychometric approach shows that item analysis improves with test length (e.g., Fred Lord's old LOGIST recommendation of 50 items and 1000 people). So, translating that into EFA/CFA terms, you need many item indicators. Any rule like 10 people per indicator is actually backwards -- if you have fewer people then more items will help (a little). Of course, you asked about EFA and this assumes that you know what constructs you're measuring and that you are effective at adding indicators that measure that construct; that assumption is often (but not always) true.

Second, I think people lose sight or what's being analyzed, which is the intercorrelation or covariance matrix. Your question boils down to how well one estimates the correlation or covariance with a given sample size.  The standard error of a correlation is (1-rr)/sqrt(N-1). I'm attaching a LibreOffice spreadsheet showing the sample sizes required for various correlations given a desire to have 0.02 or 0.002 accuracy (i.e., fix the first or second decimal place of the correlation). To fix the second decimal place of typical correlations, you need samples in the tens of thousands but 2,500 will suffice to fix the first decimal.

If you think about it, one of the consequences of this is that you will need a much larger sample to accurately estimate model parameters in matrices with a lot of small, important intercorrelations (because the wide, asymmetric confidence interval of small-sample estimates of small correlations will contain many negative values). Goldberg and Velicer (2006) present a table based on Monte Carlo simulation showing that if you want the mean estimated loading to be within 0.05 of its population value, then you need N > 1000 if the mean loading is 0.40, about N=375 if the mean loading is 0.60, and about n=160 if the mean loading is 0.80.

I believe that these points imply that there is no "one-size-fits-all" minimum sample size and I think that's what you're finding out.

Goldberg, L. R., & Velicer, W. F. (2006). Principles of exploratory factor analysis. In S. Strack (Ed.), Differentiating normal and abnormal personality: Second edition. New York, NY: Springer. (pp. 209-237).

There are several rule of thumb to address the problem of sample size in EFA. You can find the review in Costello & Osborne (2005) (http://pareonline.net/getvn.asp?v=10&n=7). However the most accurate would be to perform a Monte Carlo simulation study to determine the statistical power. You can find guide for this analysis in Muthen & Muthen (2002) (http://www.statmodel.com/bmuthen/articles/Article_096.pdf). If You don't own Mplus You could perform Monte Carlo study in R using lavaan package (it is a separate chapter in great R book: http://blogs.baylor.edu/rlatentvariable/).

If sample size small, you can used Rasch Model.

I think that 150 - 200 sample size should give reliable estimates

Basically, there are two ways to determine the appropriate sample size; by determining the minimum number of samples needed (N), or by determining the sample size as a function of the number of variables, also known as the subjects-to-variable ratio, (N:p) (Beavers, Lounsbury, Richards, Huck, Skolits, & Esquivel,, 2013; Guadagnoli & Velicer 1988; Hogarty, Hines, Kromrey, Ferron, & Mumford, 2005).

Hogarty et al. (2005) assessed the appropriate minimum sample size, N, versus the minimum N:p ratio. They found that a higher number of samples were necessary when the goal of the study goal was to understand which factors underlie which variables. When the study goal was to ensure that sample loadings correlated highly with population loadings, fewer samples were necessary.

Jung and Lee (2011) analyzed the outcomes of the factor extraction methods maximum likelihood factor analysis (MLFA), principle component analysis (PCA), and regularized exploratory factor analysis (REFA) using sample sizes less than 50. They found that REFA recovered good factor loadings, had smaller mean absolute differences and mean square errors, and provided stable factor-loading estimates with samples of 50 or less. Fabrigar et al. (1999) determined that samples as low as 100 could yield stable solutions. Beavers et al. (2013) recommended using samples of at least 150 for multivariate tools, such as EFA. Guadagnoli and Velicer (1988) concluded that the subjects-to-variable ratio, or sample size as a function of the number of variables (N:p), was not an accurate method to establish sample size. Guadagnoli and Velicer also determined that when researchers selected variables that were good indicators of a component, 150 observations yielded accurate solutions. At the high end of the minimum number of samples, Guadagnoli and Velicer (1988) found that they needed 300 when few variables defined factors with moderate to low loadings.

REFERENCES

Beavers, A. S., Lounsbury, J. W., Richards, J. K., Huck, S. W., Skolits, G J., & Esquivel, S. L. (2013). Practical considerations for using exploratory factor analysis in educational research. Practical Assessment, Research & Evaluation, 18(6), 1-13. Retrieved from http://www.pareonline.net/pdf/v18n6.pdf

Guadagnoli, E. & Velicer, W. F. (1988). Relation of sample size to the stability of component patterns. Psychological Bulletin, 103, 265-275. doi:10.1037/0033-2909.103.2.265

Hogarty, K Y., Hines, C. V., Kromrey, J. D., Ferron, J. M., & Mumford, K. R. (2005). The quality of factor solutions in exploratory factor analysis: The influence of sample size, communality, and overdetermination. Educational and Psychological Measurement, 65, 202-226. doi:10.1177/0013164404267287

Jung, S., & Lee, S. (2011). Exploratory factor analysis for small samples. Behavioral Resources, 43, 701-709. doi:10.3758/s13428-011-0077-9

How about if you use CFA? Does the sample size affect differently than using EFA?

Using a formula to determine the appropriate sample size is the easiest to explain and defend for both CFA and EFA. I suggest using the subject-to-variable method (N:p). Comrey and Lee (1992) recommend a minimum of 5 observations per variable, or ideally 20 observations per variable. According to this empirical rule, assume that you have an initial survey with 20 Likert questions (20 variables), at a minimum 20*5 = 100 observations will be needed to perform an EFA or CFA analysis adequately (20 questions aka variables * 5 observations per variable). Ideally, 20*20 = 400 observations will be needed (20 questions aka variables * 20 observations per variable). So how many surveys do you need to send out? Assuming a 10% rate for invalid surveys and an average industry response rate of 20%, then you will need to administer a minimum of (100*1.10)/0.20 = 550 surveys (to get the minimum number of 5 observations per variable). Ideally you will need to administer (400*1.10)/0.20=2200 surveys (to get the ideal number of 20 observations per variable).

REFERENCE

Comrey, A. L., & Lee, H. B. (1992). A first course in factor analysis (2nd ed.). New York, NY: Psychology Press.