For data with more variables than observations (D>>N), the covariance matrix is singular
and a general inverse is used to determine the inverse correlation matrix and estimate scores.
Using the principal axes method of Factor Analysis, communalities are estimated by iteratively updating the diagonal of the correlation matrix
and solving the eigenvector decomposition. Communalities for each variable are estimated according to the number of factors and convergence is defined
by the stabalization of total communalities between iterations.
Covariance matrix or raw data matrix. A correlation matrix is computed using pairwise deletion.
nfactors
Number of factors to extract. Default is 1.
residuals
logical. If residual matrix is included in result
prerotate
logical. Rotate the loadings using a varimax orthogonal rotation before applying a different rotation.
rotate
"none", "varimax", "promax" rotation applied to the loadings
m
integer. power of the fitting function in a promax rotation. Default is 4.
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. If D>>N, ginv(r) is used during the calculation.
force
if TRUE, a square matrix r will be interpreted as a data matrix. The default is FALSE, and square matrices are assumed to represent covariance
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. Default is 0.001
digits
Number of digits to display in output
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 displays warning messages encountered during estimation
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.
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
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.
Note
This is a direct adaptation from the factor.pa function implemented in the psych package.
Author(s)
Lisa McFerrin
References
Gorsuch, Richard, (1983) Factor Analysis. Lawrence Erlebaum Associates.
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