psych (version 1.0-72)

fa: Factor analysis by Principal Axis, MinRes (minimum residual) or Weighted Least Squares

Description

Among the many ways to do factor analysis, one of the most conventional is principal axes. An eigen value decomposition of a correlation matrix is done and then the communalities for each variable are estimated by the first n factors. These communalities are entered onto the diagonal and the procedure is repeated until the sum(diag(r)) does not vary. Another technique is to use Ordinary Least Squares to find the minimum residual (minres) solution. A variation on minres is to do weighed least squares. For well behaved matrices, maximum likelihood factor analysis (factanal) is probably preferred.

Usage

fa(r, nfactors=1, residuals = FALSE, rotate = "varimax",n.obs = NA,
scores = FALSE,SMC=TRUE, missing=FALSE,impute="median",min.err = 0.001, digits = 2, max.iter = 50,symmetric=TRUE,warnings=TRUE,fm="minres")
factor.pa(r, nfactors=1, residuals = FALSE, rotate = "varimax",n.obs = NA,
scores = FALSE,SMC=TRUE, missing=FALSE,impute="median",min.err = 0.001, digits = 2, max.iter = 50,symmetric=TRUE,warnings=TRUE,fm="pa")
factor.minres(r, nfactors=1, residuals = FALSE, rotate = "varimax",n.obs = NA,
scores = FALSE,SMC=TRUE, missing=FALSE,impute="median",min.err = 0.001, digits = 2, max.iter = 50,symmetric=TRUE,warnings=TRUE,fm="minres")
factor.wls(r,nfactors=1,residuals=FALSE,rotate="varimax",n.obs = NA,
scores=FALSE,SMC=TRUE,missing=FALSE,impute="median", min.err = .001,digits=2,max.iter=50,symmetric=TRUE,warnings=TRUE,fm="wls")

Arguments

r

A correlation matrix or a raw data matrix. If raw data, the correlation matrix will be found using pairwise deletion.

nfactors

Number of factors to extract, default is 1

residuals

Should the residual matrix be shown

rotate

"none", "varimax", "quartimax", "bentlerT", and "geominT" are orthogonal rotations. "promax", "oblimin", "simplimax", "bentlerQ, and "geominQ" or "cluster" are possible rotations or transformations of the solution.

n.obs

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

scores

If TRUE, estimate factor scores

SMC

Use squared multiple correlations (SMC=TRUE) or use 1 as initial communality estimate. Try using 1 if imaginary eigen values are reported.

missing

if scores are TRUE, and missing=TRUE, then impute missing values using either the median or the mean

impute

"median" or "mean" values are used to replace missing values

min.err

Iterate until the change in communalities is less than min.err

digits

How many digits of output should be returned

max.iter

Maximum number of iterations for convergence

symmetric

symmetric=TRUE forces symmetry by just looking at the lower off diagonal values

warnings

warnings=TRUE => warn if number of factors is too many

fm

factoring method fm="minres" will do a minimum residual (OLS), fm="wls" will do a weighted least squares (WLS) solution, fm="pa" will do the principal factor solution

Value

• values
  Eigen values of the final solution
• communality
  Communality estimates for each item. These are merely the sum of squared factor loadings for that item.
• rotation
  which rotation was requested?
• n.obs
  number of observations specified or found
• loadings
  An item by factor loading matrix of class ``loadings" Suitable for use in other programs (e.g., GPA rotation or factor2cluster. To show these by sorted order, use print.psych with sort=TRUE
• fit
  How well does the factor model reproduce the correlation matrix. (See VSS, ICLUST, and principal for this fit statistic.
• fit.off
  how well are the off diagonal elements reproduced?
• dof
  Degrees of Freedom for this model. This is the number of observed correlations minus the number of independent parameters. Let n=Number of items, nf = number of factors then $dof = n * (n-1)/2 - n * nf + nf*(nf-1)/2$
• objective
  value 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 $f = log(trace ((FF'+U2)^{-1} R) - log(|(FF'+U2)^{-1} R|) - n.items$.
• STATISTIC
  If the number of observations is specified or found, this is a chi square based upon the objective function, f. Using the formula from factanal(which seems to be Bartlett's test) : $\chi^2 = (n.obs - 1 - (2 * p + 5)/6 - (2 * factors)/3)) * f$
• PVAL
  If n.obs > 0, then what is the probability of observing a chisquare this large or larger?
• Phi
  If oblique rotations (using oblimin from the GPArotation package or promax) are requested, what is the interfactor correlation.
• communality.iterations
  The history of the communality estimates. Probably only useful for teaching what happens in the process of iterative fitting.
• residual
  If residuals are requested, this is the matrix of residual correlations after the factor model is applied.
• R2
  The multiple R square between the factors and factor score estimates, if they were to be found. (From Grice, 2001)
• r.scores
  The correlations of the factor score estimates, if they were to be found.
• weights
  The beta weights to find the factor score estimates
• valid
  The validity coffiecient of course coded (unit weighted) factor score estimates (From Grice, 2001)
• score.cor
  The correlation matrix of course coded (unit weighted) factor score estimates, if they were to be found, based upon the loadings matrix rather than the weights matrix.

Details

Factor analysis is an attempt to approximate a correlation or covariance matrix with one of lesser rank. The basic model is that $_nR_n \approx _{n}F_{kk}F_n'+ U^2$ where k is much less than n. There are many ways to do factor analysis, and maximum likelihood procedures are probably the most preferred (see factanal ). The existence of uniquenesses is what distinguishes factor analysis from principal components analysis (e.g., principal).

Principal axes factor analysis has a long history in exploratory analysis and is a straightforward procedure. Successive eigen value decompositions are done on a correlation matrix with the diagonal replaced with diag (FF') until sum(diag(FF')) does not change (very much). The current limit of max.iter =50 seems to work for most problems, but the Holzinger-Harmon 24 variable problem needs about 203 iterations to converge for a 5 factor solution.

Principal axes may be used in cases when maximum likelihood solutions fail to converge.

A problem in factor analysis is to find the best estimate of the original communalities. Using the Squared Multiple Correlation (SMC) for each variable will underestimate the communalities, using 1s will over estimate. By default, the SMC estimate is used. If, however, a solution fails to be achieved, it is useful to try again using ones (SMC =FALSE).

The algorithm does not attempt to find the best (as defined by a maximum likelihood criterion) solution, but rather one that converges rapidly using successive eigen value decompositions. The maximum likelihood criterion of fit and the associated chi square value are reported, and will be worse than that found using maximum likelihood procedures.

The minimum residual (minres) solution is an unweighted least squares solution that takes a slightly different approach. It uses the optim function and adjusts the diagonal elements of the correlation matrix to mimimize the squared residual when the factor model is the eigen value decomposition of the reduced matrix. MINRES and PA will both work when ML will not, for they can be used when the matrix is singular. At least on a number of test cases, the MINRES solution is slightly more similar to the ML solution than is the PA solution. To a great extent, the minres and wls solutions follow ideas in the factanal function.

The weighted least squares (wls) solution weights the residual matrix by 1/ diagonal of the inverse of the correlation matrix. This has the effect of weighting items with low communalities more than those with high commumnalities.

Test cases comparing the output to SPSS suggest that the PA algorithm matches what SPSS calls uls, and that the wls solutions are equivalent in their fits.

Although for items, it is typical to find factor scores by scoring the salient items (using, e.g.,score.items factor scores can be estimated by regression.

Of the various rotation/transformation options, varimax, quatimax, bentlerT and geominT do orthogonal rotations. Promax transforms obliquely with a target matix equal to the varimax solution. oblimin, quartimin, simplimax, bentlerQ, and geominQ 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.

References

Gorsuch, Richard, (1983) Factor Analysis. Lawrence Erlebaum Associates.

Harman, Harry and Jones, Wayne (1966) Factor analysis by minimizing residuals (minres), Psychometrika, 31, 3, 351-368.

Revelle, William. (in prep) An introduction to psychometric theory with applications in R. Springer. Working draft available at http://personality-project.org/r/book.html

See Also

principal, VSS, ICLUST

Examples

#using the Harman 24 mental tests, compare a principal factor with a principal components solution
pc <- principal(Harman74.cor$cov,4,rotate="varimax")
pa <- fa(Harman74.cor$cov,4,fm="pa" ,rotate="varimax")
uls <- fa(Harman74.cor$cov,4,rotate="varimax")
wls <- fa(Harman74.cor$cov,4,fm="wls")

#to show the loadings sorted by absolute value
print(uls,sort=TRUE)

#then compare with a maximum likelihood solution using factanal
mle <- factanal(covmat=Harman74.cor$cov,factors=4)
factor.congruence(list(mle,pa,pc,uls,wls))
#note that the order of factors and the sign of some of factors differ 

#finally, compare the unrotated factor, ml, uls, and  wls solutions
wls <- factor.wls(Harman74.cor$cov,4,rotate="none")
pa <- factor.pa(Harman74.cor$cov,4,rotate="none")
mle <-  factanal(factors=4,covmat=Harman74.cor$cov,rotation="none")
uls <- factor.minres(Harman74.cor$cov,4,rotate="none")

factor.congruence(list(mle,pa,uls,wls))
#note that the order of factors and the sign of some of factors differ 



#an example of where the ML and PA and MR models differ is found in Thurstone.33.
#compare the first two factors with the 3 factor solution 
data(bifactor)
Thurstone.33 <- as.matrix(Thurstone.33)
mle2 <- factanal(covmat=Thurstone.33,factors=2,rotation="none")
mle3 <- factanal(covmat=Thurstone.33,factors=3 ,rotation="none")
pa2 <- factor.pa(Thurstone.33,2,rotate="none")
pa3 <- factor.pa(Thurstone.33,3,rotate="none")
mr2 <- fa(Thurstone.33,2,rotate="none")
mr3 <- fa(Thurstone.33,3,rotate="none")
factor.congruence(list(mle2,mle3,pa2,pa3,mr2,mr3))