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.
David Matheson
SPSS Statistical Support
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