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, …)A formula or a numeric matrix or an object that can be coerced to a numeric matrix.
The number of factors to be fitted.
An optional data frame (or similar: see
model.frame), used only if x is a formula. By
default the variables are taken from environment(formula).
A covariance matrix, or a covariance list as returned by
cov.wt. Of course, correlation matrices are covariance
matrices.
The number of observations, used if covmat is a
covariance matrix.
A specification of the cases to be used, if x is
used as a matrix or formula.
The 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.
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.
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.
A list of control values,
The number of starting values to be tried if
start = NULL. Default 1.
logical. Output tracing information? Default FALSE.
The lower bound for uniquenesses during optimization. Should be > 0. Default 0.005.
A list of control values to be passed to
optim's control argument.
a list of additional arguments for the rotation function.
Components of control can also be supplied as
named arguments to factanal.
An object of class "factanal" with components
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. This is of class "loadings": see
loadings for its print method.
The uniquenesses computed.
The correlation matrix used.
The results of the optimization: the value of the criterion (a linear function of the negative log-likelihood) and information on the iterations used.
The argument factors.
The number of degrees of freedom of the factor analysis model.
The method: always "mle".
The rotation matrix if relevant.
If requested, a matrix of scores. napredict is
applied to handle the treatment of values omitted by the na.action.
The number of observations if available, or NA.
The matched call.
If relevant.
The significance-test statistic and P value, if it can be computed.
The factor analysis model is $$x = \Lambda f + e$$ for a \(p\)--element vector \(x\), a \(p \times k\) matrix \(\Lambda\) of loadings, a \(k\)--element vector \(f\) of scores and a \(p\)--element vector \(e\) 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 \(\Psi\), the uniquenesses. It is also common to scale the observed variables to unit variance, and done in this function.
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.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
J<U+00F6>reskog (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.
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. (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.
J<U+00F6>reskog, 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.
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 GPArotation and psych.
# NOT RUN {
# 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
# }
# NOT RUN {
factanal(m1, factors = 3, rotation = "promax")
# }
# NOT RUN {
# The following shows the g factor as PC1
# }
# NOT RUN {
prcomp(m1) # signs may depend on platform
# }
# NOT RUN {
## formula interface
factanal(~v1+v2+v3+v4+v5+v6, factors = 3,
scores = "Bartlett")$scores
# }
# NOT RUN {
## a realistic example from Bartholomew (1987, pp. 61-65)
utils::example(ability.cov)
# }
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