| factanal {stats} | R Documentation |
Perform maximum-likelihood factor analysis on a covariance matrix or data matrix.
factanal(x, factors, data = NULL, covmat = NULL, n.obs = NA,
subset, na.action, start = NULL,
scores = c("none", "regression", "Bartlett"),
rotation = "varimax", control = NULL, ...)
x |
A formula or a numeric matrix or an object that can be coerced to a numeric matrix. |
factors |
The number of factors to be fitted. |
data |
An optional data frame (or similar: see
|
covmat |
A covariance matrix, or a covariance list as returned by
|
n.obs |
The number of observations, used if |
subset |
A specification of the cases to be used, if |
na.action |
The |
start |
|
scores |
Type of scores to produce, if any. The default is none,
|
rotation |
character. |
control |
A list of control values,
|
... |
Components of |
The factor analysis model is
x = \Lambda f + e
for a p–element row-vector x, a p \times k
matrix of loadings, a k–element vector of scores
and a p–element vector of errors. None of the components
other than x is observed, but the major restriction is that the
scores be uncorrelated and of unit variance, and that the errors be
independent with variances \Phi, the
uniquenesses. Thus factor analysis is in essence a model for
the covariance matrix of x,
\Sigma = \Lambda^\prime\Lambda + \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. 27).) All the starting values supplied in 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
Jöreskog (1963) and given by Lawley & Maxwell
(1971, p. 31), and then 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. 32).
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.
An object of class "factanal" with components
loadings |
A matrix of loadings, one column for each factor. The factors are ordered in decreasing order of sums of squares of loadings, and given the sign that will make the sum of the loadings positive. |
uniquenesses |
The uniquenesses computed. |
correlation |
The correlation matrix used. |
criteria |
The results of the optimization: the value of the negative log-likelihood and information on the iterations used. |
factors |
The argument |
dof |
The number of degrees of freedom of the factor analysis model. |
method |
The method: always |
scores |
If requested, a matrix of scores. |
n.obs |
The number of observations if available, or |
call |
The matched call. |
na.action |
If relevant. |
STATISTIC, PVAL |
The significance-test statistic and P value, if if can be computed. |
There are so many variations on factor analysis that it is hard to compare output from different programs. Further, the optimization in maximum likelihood factor analysis is hard, and many other examples we compared had less good fits than produced by this function. In particular, solutions which are Heywood cases (with one or more uniquenesses essentially zero) are much often common than most texts and some other programs would lead one to believe.
Bartlett, M. S. (1937) The statistical conception of mental factors. British Journal of Psychology, 28, 97–104.
Bartlett, M. S. (1938) Methods of estimating mental factors. Nature, 141, 609–610.
Jöreskog, K. G. (1963) Statistical Estimation in Factor Analysis. Almqvist and Wicksell.
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.
print.loadings,
varimax, princomp,
ability.cov, Harman23.cor,
Harman74.cor
# 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)
## 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)