factanal(x, factors, data = NULL, covmat = NULL, n.obs = NA,
subset, na.action, start = NULL,
scores = c("none", "regression", "Bartlett"),
rotation = "varimax", control = NULL, ...)model.frame), used only if x is a formula. By
default the variables are taken from environment(formula).cov.wt. Of course, correlation matrices are covariance
matrices.covmat is a
covariance matrix.x is
used as a matrix or formula.na.action to be used if x is
used as a formula.NULL or a matrix of starting values, each column
giving an initial set of uniquenesses."regression" gives Thompson's scores, "Bartlett" given
Bartlett's weighted least-squares scores. Partial matching allows
these names to be abbreviated."none" or the name of a function
to be used to rotate the factors: it will be called with first
argument the loadings matrix, and should return a list with component
loadings giving the rotated loadings, or just the rotated loadings.control can also be supplied as
named arguments to factanal."factanal" with components"loadings": see
loadings for its print method.factors."mle".napredict is
applied to handle the treatment of values omitted by the na.action.NA.Thus factor analysis is in essence a model for the correlation matrix of $x$, $$\Sigma = \Lambda\Lambda^\prime + \Psi$$ There is still some indeterminacy in the model for it is unchanged if $\Lambda$ is replaced by $G \Lambda$ for any orthogonal matrix $G$. Such matrices $G$ are known as rotations (although the term is applied also to non-orthogonal invertible matrices).
If covmat is supplied it is used. Otherwise x is used
if it is a matrix, or a formula x is used with data to
construct a model matrix, and that is used to construct a covariance
matrix. (It makes no sense for the formula to have a response, and
all the variables must be numeric.) Once a covariance matrix is found
or calculated from x, it is converted to a correlation matrix
for analysis. The correlation matrix is returned as component
correlation of the result.
The fit is done by optimizing the log likelihood assuming multivariate
normality over the uniquenesses. (The maximizing loadings for given
uniquenesses can be found analytically: Lawley & Maxwell (1971,
p.start are tried
in turn and the best fit obtained is used. If start = NULL
then the first fit is started at the value suggested by
control$nstart - 1 other values are
tried, randomly selected as equal values of the uniquenesses.
The uniquenesses are technically constrained to lie in $[0, 1]$,
but near-zero values are problematical, and the optimization is
done with a lower bound of control$lower, default 0.005
(Lawley & Maxwell, 1971, p.
Scores can only be produced if a data matrix is supplied and used. The first method is the regression method of Thomson (1951), the second the weighted least squares method of Bartlett (1937, 8). Both are estimates of the unobserved scores $f$. Thomson's method regresses (in the population) the unknown $f$ on $x$ to yield $$\hat f = \Lambda^\prime \Sigma^{-1} x$$ and then substitutes the sample estimates of the quantities on the right-hand side. Bartlett's method minimizes the sum of squares of standardized errors over the choice of $f$, given (the fitted) $\Lambda$.
If x is a formula then the standard NA-handling is
applied to the scores (if requested): see napredict.
The print method (documented under loadings)
follows the factor analysis convention of drawing attention to the
patterns of the results, so the default precision is three decimal
places, and small loadings are suppressed.
Bartlett, M. S. (1938) Methods of estimating mental factors. Nature, 141, 609--610.
Lawley, D. N. and Maxwell, A. E. (1971) Factor Analysis as a Statistical Method. Second edition. Butterworths.
Thomson, G. H. (1951) The Factorial Analysis of Human Ability. London University Press.
loadings (which explains some details of the
print method), varimax, princomp,
ability.cov, Harman23.cor,
Harman74.cor. Other rotation methods are available in various contributed packages,
including
# A little demonstration, v2 is just v1 with noise,
# and same for v4 vs. v3 and v6 vs. v5
# Last four cases are there to add noise
# and introduce a positive manifold (g factor)
v1 <- c(1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,5,6)
v2 <- c(1,2,1,1,1,1,2,1,2,1,3,4,3,3,3,4,6,5)
v3 <- c(3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,5,4,6)
v4 <- c(3,3,4,3,3,1,1,2,1,1,1,1,2,1,1,5,6,4)
v5 <- c(1,1,1,1,1,3,3,3,3,3,1,1,1,1,1,6,4,5)
v6 <- c(1,1,1,2,1,3,3,3,4,3,1,1,1,2,1,6,5,4)
m1 <- cbind(v1,v2,v3,v4,v5,v6)
cor(m1)
factanal(m1, factors = 3) # varimax is the default
factanal(m1, factors = 3, rotation = "promax")
# The following shows the g factor as PC1
prcomp(m1) # signs may depend on platform
## formula interface
factanal(~v1+v2+v3+v4+v5+v6, factors = 3,
scores = "Bartlett")$scores
## a realistic example from Bartholomew (1987, pp. 61-65)
utils::example(ability.cov)Run the code above in your browser using DataLab