principal: Principal components analysis

Description

Does an eigen value decomposition and returns eigen values, loadings, and degree of fit for a specified number of components. Basically it is just doing a principal components for n principal components. Can show the residual correlations as well. The quality of reduction in the squared correlations is reported by comparing residual correlations to original correlations. Unlike princomp, this returns a subset of just the best nfactors. The eigen vectors are rescaled by the sqrt of the eigen values to produce the component loadings more typical in factor analysis.

Usage

principal(r, nfactors = 1, residuals = FALSE,rotate="varimax",n.obs=NULL, digits=2)

Arguments

a correlation matrix. If a raw data matrix is used, the correlations will be found using pairwise deletions for missing values.

nfactors

Number of components to extract

residuals

FALSE, do not show residuals, TRUE, report residuals

rotate

n.obs

Number of observations used to find the correlation matrix if using a correlation matrix. Used for finding the goodness of fit statistics.

digits

digits =2. Accuracy of answers as well as display

Value

valuesEigen Values of all components -- useful for a scree plot
rotationwhich rotation was requested?
n.obsnumber of observations specified or found
communalityCommunality estimates for each item. These are merely the sum of squared factor loadings for that item.
loadingsA standard loading matrix of class ``loadings"
fitFit of the model to the correlation matrix
fit.offhow well are the off diagonal elements reproduced?
residualResidual matrix -- if requested
communalityThe history of the communality estimates. Probably only useful for teaching what happens in the process of iterative fitting.
dofDegrees of Freedom for this model. This is the number of observed correlations minus the number of independent parameters (number of items * number of factors - nf*(nf-1)/2. That is, dof = niI * (ni-1)/2 - ni * nf + nf*(nf-1)/2.
objectivevalue of the function that is minimized by maximum likelihood procedures. This is reported for comparison purposes and as a way to estimate chi square goodness of fit. The objective function is $f = log(trace ((FF'+U2)^{-1} R) - log(|(FF'+U2)^{-1} R|) - n.items$.
STATISTICIf the number of observations is specified or found, this is a chi square based upon the objective function, f. Using the formula from factanal: $\chi^2 = (n.obs - 1 - (2 * p + 5)/6 - (2 * factors)/3)) * f$
PVALIf n.obs > 0, then what is the probability of observing a chisquare this large or larger?
phiIf oblique rotations (using oblimin from the GPArotation package) are requested, what is the interfactor correlation.

Details

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 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.

Examples

Run this code

#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)

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