Aaron Schlegel's Notebook of Interesting Things

The iterated principal factor method is an extension of the principal

As discussed in a previous post on the principal component method of factor analysis, the \(\hat{\Psi}\) term in the estimated covariance matrix \(S\), \(S = \hat{\Lambda} \hat{\Lambda}' + \hat{\Psi}\), was excluded and we proceeded directly to factoring \(S\) and \(R\). The principal factor method of factor analysis (also called the principal axis method) finds an initial estimate of \(\hat{\Psi}\) and factors \(S - \hat{\Psi}\), or \(R - \hat{\Psi}\) for the correlation matrix.

In the first post on factor analysis, we examined computing the estimated covariance matrix \(S\) of the rootstock data and proceeded to find two factors that fit most of the variance of the data. However, the variables in the data are not on the same scale of measurement, which can cause variables with comparatively large variances to dominate the diagonal of the covariance matrix and the resulting factors. The correlation matrix, therefore, makes more intuitive sense to employ in factor analysis.

The goal of factor analysis, similar to principal component analysis, is to reduce the original variables into a smaller number of factors that allows for easier interpretation. PCA and factor analysis still defer in several respects. One difference is principal components are defined as linear combinations of the variables while factors are defined as linear combinations of the underlying latent variables.