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
`model.frame`

), used only if `x`

is a formula. By
default the variables are taken from `environment(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,

- nstart
The number of starting values to be tried if

`start = NULL`

. Default 1.- trace
logical. Output tracing information? Default

`FALSE`

.- lower
The lower bound for uniquenesses during optimization. Should be > 0. Default 0.005.

- opt
A list of control values to be passed to

`optim`

's`control`

argument.- rotate
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.
10.1111/j.2044-8295.1937.tb00863.x.

Bartlett, M. S. (1938).
Methods of estimating mental factors.
*Nature*, **141**, 609--610.
10.1038/141246a0.

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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