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I read that the reason an eigenvalue greater than 1.0 is used as a criterion in factor analysis extractions is that if the eigenvalue is less than 1.0, then the variable is explaining less variance than a single item.
My question is this: in the course of a higher-order factor analysis, does the same rationale for using the 1.0 criterion pertain, i.e. if the eigenvalue is less than one, is less varience explained by it than by a single lower-order factor?
The short answer to your question is "Yes". That is, the rationale for only retaining factors with eigenvalues larger than one holds for a higher order factor analysis just as for a lower order one.
One way of thinking about this rule of thumb is to realize that your p variables form a p-dimensional space. You want to rotate the axes of this space so that the new axes maximize the variance of the data points as they are projected onto the axes. The (normalized) eigenvectors of a matrix give the direction cosines determining the rotation, while the eigenvalues give the variance associated with each new axis.
When the matrix is a pxp correlation matrix, the variance of each variable is already standardized to 1, so things are particularly simple. An eigenvalue less than one represents a shrinking of an axis's importance in the new universe.
Similiarly, the Spectral Decomposition Theory says that any matrix of rank p can be broken down into the sum of p component matrices. These component matrices are just the outer product of each eigenvector (xx'), weighted by its eigenvalue. Again, a pxp correlation matrix has rank p, and p eigenvalues summing to p. So the factor associated with an eigenvalue of less than one is not pulling its own weight.
If you have further questions, send E-mail to stats@ssc.utexas.edu.