principal(r, nfactors = 1, residuals = FALSE,rotate="varimax",n.obs=NULL, scores=FALSE,missing=FALSE,impute="median",digits=2)factanal:
$\chi^2 = (n.obs - 1 - (2 * p + 5)/6 - (2 * factors)/3)) * f$There are a number of data reduction techniques including principal components and factor analysis. Both PC and FA attempt to approximate a given correlation or covariance matrix of rank n with matrix of lower rank (p). $_nR_n \approx _{n}F_{kk}F_n'+ U^2$ where k is much less than n. For principal components, the item uniqueness is assumed to be zero and all elements of the correlation matrix are fitted. That is, $_nR_n \approx _{n}F_{kk}F_n'$ 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.
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 can be estimated by regression. This is done just to be parallel with the principal axis factor analysis function factor.pa
VSS,factor2cluster,factor.pa, factor.congruence#Four principal components of the Harmon 24 variable problem
#compare to a four factor principal axes solution using factor.congruence
pc <- principal(Harman74.cor$cov,4,rotate=TRUE)
pa <- factor.pa(Harman74.cor$cov,4,rotate=TRUE)
round(factor.congruence(pc,pa),2)Run the code above in your browser using DataLab