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Why do I sometimes get negative eigenvalues when using the pf and ipf options of factor?

Why does the cumulative proportion of variance sometimes exceed 1 when using the pf and ipf options of factor?

Factor analysis estimated using the principal factor method (factor, pf) or iterated principal factor method (factor, ipf) can produce negative eigenvalues. This, in turn, can cause the cumulative proportion of variance to exceed 1. Why is this?

In factor analysis, we model the covariance matrix as

	S = Lambda * Lambda' + Psi

In the principal component method of estimating a factor analysis (factor, pcf), eigenvalues and eigenvector of S, the sample covariance, are computed, and then the elements of Psi are calculated. This method will not produce negative eigenvalues (or cumulative proportions above 1) since the sample covariance matrix will be positive semidefinite.

However, with the principal factor method of estimating a factor analysis (factor, pf), eigenvalues and eigenvectors of S − Psi are computed after first estimating initial values for Psi. S − Psi is not guaranteed to be positive semidefinite. When it is not, you will get some negative eigenvalues and will see cumulative proportions above 1. Since the iterated principal factor method of estimating factor analysis (factor, ipf) is simply an iteration of this process, it too can produce negative eigenvalues and cumulative proportions above 1.

See Rencher (2012) for details.

Reference

Rencher, A. C. and W.F. Christensen. 2012.

Methods of Multivariate Analysis. 3rd ed. New Jersey: Wiley.