Useful for those cases where the correlation matrix is improper (perhaps because of SAPA techniques).There are a number of data reduction techniques including principal components analysis (PCA) and factor analysis (EFA). Both PC and FA attempt to approximate a given correlation or covariance matrix of rank n with matrix of lower rank (p). $nRn = nFk kFn' + U2$ where k is much less than n. For principal components, the item uniqueness is assumed to be zero and all elements of the correlation or covariance matrix are fitted. That is, $nRn = nFk kFn' $ The primary empirical difference between a components versus a factor model is the treatment of the variances for each item. Philosophically, components are weighted composites of observed variables while in the factor model, variables are weighted composites of the factors.
For a n x n correlation matrix, the n principal components completely reproduce the correlation matrix. However, if just the first k principal components are extracted, this is the best k dimensional approximation of the matrix.
It is important to recognize that rotated principal components are not principal components (the axes associated with the eigen value decomposition) but are merely components. To point this out, unrotated principal components are labelled as PCi, while rotated PCs are now labeled as RCi (for rotated components) and obliquely transformed components as TCi (for transformed components). (Thanks to Ulrike Gromping for this suggestion.)
Rotations and transformations are either part of psych (Promax and cluster), of base R (varimax), or of GPArotation (simplimax, quartimax, oblimin, etc.).
Of the various rotation/transformation options, varimax, Varimax, quartimax, bentlerT, geominT, and bifactor do orthogonal rotations. Promax transforms obliquely with a target matix equal to the varimax solution. oblimin, quartimin, simplimax, bentlerQ, geominQ and biquartimin are oblique transformations. Most of these are just calls to the GPArotation package. The ``cluster'' option does a targeted rotation to a structure defined by the cluster representation of a varimax solution. With the optional "keys" parameter, the "target" option will rotate to a target supplied as a keys matrix. (See target.rot.)
The rotation matrix (rot.mat) is returned from all of these options. This is the inverse of the Th (theta?) object returned by the GPArotation package. The correlations of the factors may be found by $Phi = Th' Th$
Some of the statistics reported are more appropriate for (maximum likelihood) factor analysis rather than principal components analysis, and are reported to allow comparisons with these other models.
Although for items, it is typical to find component scores by scoring the salient items (using, e.g., score.items) component scores are found by regression where the regression weights are $R^(-1) lambda$ where $lambda$ is the matrix of component loadings. The regression approach is done to be parallel with the factor analysis function fa. The regression weights are found from the inverse of the correlation matrix times the component loadings. This has the result that the component scores are standard scores (mean=0, sd = 1) of the standardized input. A comparison to the scores from princomp shows this difference. princomp does not, by default, standardize the data matrix, nor are the components themselves standardized. The regression weights are found from the Structure matrix, not the Pattern matrix. If the scores are found with the covar option = TRUE, then the scores are not standardized but are just mean centered.
Jolliffe (2002) discusses why the interpretation of rotated components is complicated. The approach used here is consistent with the factor analytic tradition. The correlations of the items with the component scores closely matches (as it should) the component loadings (as reported in the structure matrix).
The output from the print.psych function displays the component loadings (from the pattern matrix), the h2 (communalities) the u2 (the uniquenesses), com (the complexity of the component loadings for that variable (see below). In the case of an orthogonal solution, h2 is merely the row sum of the squared component loadings. But for an oblique solution, it is the row sum of the (squared) orthogonal component loadings (remember, that rotations or transformations do not change the communality).