| 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 |
A data frame, used only if x is a formula. |
covmat |
A covariance matrix, or a covariance list as returned by
cov.wt. Of course, correlation matrices are covariance
matrices. |
n.obs |
The number of observations, used if covmat is a
covariance matrix. |
subset |
A specification of the cases to be used, if x is
used as a matrix or formula. |
na.action |
The na.action to be used if x is
used as a formula. |
start |
NULL or a matrix of starting values, each column
giving an initial set of uniquenesses. |
scores |
Type of scores to produce, if any. The default is none,
"regression" gives Thompson's scores, "Bartlett" given
Bartlett's weighted least-squares scores. Partial matching allows
these names to be abbreviated. |
rotation |
character. "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 |
A list of control values,
|
... |
Components of control can also be supplied as
named arguments to factanal. |
The factor analysis model is
x = Lambda f + e
for a p–element row-vector x, a p x 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'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' 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 factors. |
dof |
The number of degrees of freedom of the factor analysis model. |
method |
The method: always "mle". |
scores |
If requested, a matrix of scores. napredict is
applied to handle the treatment of values omitted by the na.action. |
n.obs |
The number of observations if available, or NA. |
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 Barthlomew (1987, pp. 61-65)
example(ability.cov)