Sample Size of Factor Analysis?

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Hello, I was asked to do a factor analysis of 40 variables but I only have 70 cases. Needless to say, I had to increase iterations to 100 to get the program to converge and I still believe that it makes no sense to do a factor analysis with less than 2 cases per variable. I was then asked to provide a citation for that. Could someone point me to a source discussing the minimum case per variable requirement for factor analysis that I can cite? Thanks a lot.

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关键词：Analysis Analysi Sample Analys alysis increase believe minimum provide someone

Really new to me, I am waiting for the right answer.

In response to your second question, the KMO and AIC (anti-image correlation) are not printed when the correlation matrix is nonpositive definite, which seems likely to apply from your description of the numbers of cases and variables in your study. I've pasted a related resolution from the support web site
( http://support.spss.com ) below.

David Matheson
SPSS Statistical Support

Resolution number: 20414  Created on: Aug 21 2001  Last Reviewed on: Feb 28 2009

Problem Subject:  FACTOR does not print KMO or Bartlett test for Nonpositive Definite Matrices

Problem Description:  I have run the SPSS FACTOR procedure with principal components analysis (PCA) as the extraction method. I requested the Kaiser-Mayer-Olkin (KMO) measure of sample adequacy and the Bartlett test of sphericity but neither of these measures was printed. The "Communalities", "Total Variance Explained" and "Component Matrix" tables were printed. Why was my request for KMO and Bartlett's sphericity test ignored?

Resolution Subject: KMO, Bartlett's sphericity, and anti-image correlation not printed for nonpositive definite matrices

Resolution Description:
It is likely the case that your correlation matrix is nonpositive definite (NPD), i.e., that some of the eigenvalues of your correlation matrix are not positive numbers. If this is the case, there will be a footnote to the correlation matrix that states "This matrix is not positive definite." Even if you did not request the correlation matrix as part of the FACTOR output, requesting the KMO or Bartlett test will cause the title "Correlation Matrix" to be printed. The footnote will be printed under this title if the correlation matrix was not requested. An NPD matrix will also result in suppression of other output from the 'Descriptives' dialog of the Factor dialog, namely the inverse of the correlation matrix, the anti-image correlation matrix, and the significance values for the correlations. If you had requested a factor extraction method other than PCA or unweighted least squares (ULS), an NPD matrix would have caused the procedure to stop without further analysis.

Matrices can be NPD as a result of various other properties. A correlation matrix will be NPD if there are linear dependencies among the variables, as reflected by one or more eigenvalues of 0. For example, if variable X12 can be reproduced by a weighted sum of variables X5, X7, and X10, then there is a linear dependency among those variables and the correlation matrix that includes them will be NPD. If there are more variables in the analysis than there are cases, then the correlation matrix will have linear dependencies and be NPD. Remember that FACTOR uses listwise deletion of cases with missing data by default. If you had more cases in the file than variables in the analysis but also had many missing values, listwise deletion could leave you with more variables than retained cases. Pairwise deletion of missing data can also lead to NPD matrices. Negative eigenvalues may be present in these situations. See the following chapter for a helpful discussion and illustration of!
  how this
can happen.

Wothke, W. (1993) Nonpositive definite matrices in structural modeling. In K.A. Bollen & J.S. Long (Eds.), Testing Structural Equation Models. Newbury Park NJ: Sage. (Chap. 11, pp. 256-293).

Elements of the KMO and Bartlett test statistic can not be calculated if the correlation matrix is NPD. See the formulae for these statistics in the current Statistical Algorithms documentation by clicking Help->Algorithms in SPSS, then scrolling down to the link for Factor Algorithms. Then click the link for Optional Statistics. . The formulae are also on page 20 of the Factor chapter at
http://support.spss.com/ProductsExt/SPSS/Documentation/Statistics/algorithms/14.0/factor.pdf

The Bartlett formula includes the log of the determinant of the correlation matrix. If there are linear dependencies, then the determinant of the matrix will be 0 and its log will be undefined. The KMO measure formula includes elements of the anti-image covariance matrix, whose calculation involves the inverse of the correlation matrix. If the correlation matrix has linear dependencies, then its inverse can not be computed.

Apart from the inability to print the KMO or Bartlett's test, the presence of an NPD correlation matrix may lead you to rethink the choice of variables or attempt to acquire data on a larger sample to achieve more reliable results.

I'm not sure the effort is worth it, but....

You can try to use Dwyer's extension analysis. You start by creating a set
of homogenous item packages or parcels - combine sets of 2-4 items into new
scales by reviewing the item correlations (combine those items with the
highest inter-item correlations). Then, factor analyze the item parcels (you
will have reduced the number of variables in the factor analysis to about
10-15 (instead of 40). Convergence and iterations should behave better.
Rotate and then use the Dwyer extension procedure described in Gorsuch
(1983) Factor Analysis (2nd Ed.) on pages 236-238. Essentially, the factor
solution of the parcels is projected onto the original set of items. You'll
get your factor structure and pattern matrix (if you rotate obliquely) of
the 40 items.

If you need some background on item parceling, you can find out more about
it by searching "item parcels." I know their use is controversial. You can
also check up on Andrew Comrey's work in developing his personality
inventory and Ray Cattell's work.

In addition to  the recommended ratios of 10 to 20 people per variable, the following has also been suggested:

Some Monte Carlo simulation research (Guadagnoli & Velincer, 1998) suggest ... replicable factors tend to be estimated if:
1. factors are each defined by four or more measured variables with structure coefficients each great than .6 [in absolute value], regardless or sample size; or
2. factors are each defined with 10 or more structure coefficients each around .4[in absolute value], if sample size is greater than 150; or
3. sample size is at least 300." (Thompson, 2004, p. 24)

Linda

Thompson, B. (2004). Exploratory and confirmatory factor analysis: Understanding concepts and applications. Washington, DC: American Psychological Association.

Perhaps not the most authoritative citation, but the APA publication Edited by Grimm and Yarnold, Reading and Understanding Multivariate Statistics, 8th Ed. 2003.  Washington DC.  Page 100.    Referred to as the subjects to variables ratio (STV), "the minimum number of observations in ones sample should be at least five times the number of variables."

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