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
xis a formula. By default the variables are taken from
- 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
covmatis a covariance matrix.
- A specification of the cases to be used, if
xis used as a matrix or formula.
na.actionto be used if
xis used as a formula.
NULLor 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.
"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
loadingsgiving 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
- The lower bound for uniquenesses during optimization. Should be > 0. Default 0.005.
- A list of control values to be passed to
- a list of additional arguments for the rotation function.
- Components of
controlcan also be supplied as named arguments to
The factor analysis model is x = Λ f + e for a p--element row-vector x, a p x k matrix Λ of loadings, a k--element vector f of scores and a p--element vector eof 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, Σ = Λ'Λ + Ψ There is still some indeterminacy in the model for it is unchanged if Λ is replaced by G Λ for any orthogonal matrix G. Such matrices G are known as rotations (although the term is applied also to non-orthogonal invertible matrices).
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 JoreskogTEXT (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 = Λ' Σ^-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) Λ.
x is a formula then the standard
NA-handling is applied to the scores (if requested): see
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.
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
- The uniquenesses computed.
- The correlation matrix used.
- The results of the optimization: the value of the negative log-likelihood and information on the iterations used.
- The argument
- The number of degrees of freedom of the factor analysis model.
- The method: always
- The rotation matrix if relevant.
- If requested, a matrix of scores.
napredictis applied to handle the treatment of values omitted by the
- The number of observations if available, or
- The matched call.
- If relevant.
- STATISTIC, PVAL
- The significance-test statistic and P value, if it can be computed.
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.
JoreskogTEXT, 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.
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 more common than most texts and some other programs would lead one to believe.
Other rotation methods are available in various contributed packages, including GPArotation and psych.
# 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)
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