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How can I decide how many factors I should extract from a factor analysis solution?
There are a number of methods you can use, either individually or in concert to aid you in selecting the number of factors to retain from a factor analysis. Among them are:
1. The eigenvalue greater than or equal to 1.00 rule
Only factors with eigenvalues greater than or equal to 1.00 are retained, since one way to view this situation is that only factors with eigenvalues greater than or equal to 1.00 "pull their own weight" in explaining the common variance shared among your measures.
2. The scree plot
You can request that this be output from SPSS or SAS. You would retain the number of factors up to the "elbow" - 1. For example, consider the following scree plot:
Eigenvalue |* | | * | * * * | |____________________ 1 2 3 4 5 Factor Number
Here the "elbow" or bend is at factor 3, but you would retain 3 - 1 factors, or the first 2 factors.
3. Proportion of variance accounted for by factors
Decide a priori what a sufficient proportion of variance accounted for is, and retain only enough factors to cross that threshold.
4. The low error approach
Continue extracting factors until all residual values are .10 or lower.
5. Use a chi-square test
SAS and SPSS provide tests of overall goodness-of-fit of the factor analysis model to the data when you choose maximum-likelihood (ML) or generalized least-squares (GLS) factor extraction methods. If you choose to use one of these extraction methods (ML is generally more commonly used than GLS), you also must tell the software package how many factors you expect to be present. It then uses that number of factors as its null hypothesis. That is, the null hypothesis of the chi-square test is that the factor analysis model fits the data. So, a non-significant model test is desirable, whereas a statistically significant chi-square test means that more factors are needed to account for the structure of your data.
You should recognize two important caveats in using the chi-square method to help you decide how many factors to retain. The first caveat is that these test statistics are computed under the assumption of joint multivariate normality. If your data do not meet this assumption, it may not be appropriate to use these chi-square tests. The second caveat is that these tests are very sensitive to sample size. A factor analysis model which otherwise fits the data well may be statistically significant due to a large sample. If you use only the chi-square results to determine the number of factors to retain, you will probably retain too many factors.
You can always use more than one of these methods to help you decide which solution is optimal, but, as always, theory should be your foremost consideration. SAS and SPSS anticipate that theory can guide your decision to extract a given number of factors, so each package provides a method to limit the number of factors extracted to be a specific number (e.g., NFACTORS=2 to extract two factors).
Also, there is the problem that "a person with one watch always knows what time it is; a person with more than one watch never knows the exact time". In other words, using the information from all of these methods may lead to a situation where they conflict--say you retain only factors with eigenvalues greater than 1.00, but you have some residuals with values greater than .10.
In this type of situation, theory provides your first guideline, and the other rules of thumb can provide some additional guidance, but it is important not to follow any one of the rules of thumb by rote or too strictly but instead to evaluate the solution as a complete picture, including how it meshes with prior findings, your own theoretical models, etc.
If you have further questions, send E-mail to stats@ssc.utexas.edu.