Some factor analytic solutions produce correlated factors which may in turn be factored. If the solution has one higher order, the omega function is most appropriate. But, in the case of multi higher order factors, then the faMulti function will do a lower level factoring and then factor the resulting correlation matrix. Multi level factor diagrams are also shown.

```
fa.multi(r, nfactors = 3, nfact2 = 1, n.obs = NA, n.iter = 1, rotate = "oblimin",
scores = "regression", residuals = FALSE, SMC = TRUE, covar = FALSE, missing =
FALSE,impute = "median", min.err = 0.001, max.iter = 50, symmetric = TRUE, warnings
=TRUE, fm = "minres", alpha = 0.1, p = 0.05, oblique.scores = FALSE, np.obs = NULL,
use ="pairwise", cor = "cor", ...)
```fa.multi.diagram(multi.results,sort=TRUE,labels=NULL,flabels=NULL,cut=.2,gcut=.2,
simple=TRUE,errors=FALSE,
digits=1,e.size=.1,rsize=.15,side=3,main=NULL,cex=NULL,color.lines=TRUE
,marg=c(.5,.5,1.5,.5),adj=2, ...)

r

A correlation matrix or raw data matrix

nfactors

The desired number of factors for the lower level

nfact2

The desired number of factors for the higher level

n.obs

Number of observations used to find the correlation matrix if using a correlation matrix. Used for finding the goodness of fit statistics. Must be specified if using a correlaton matrix and finding confidence intervals.

np.obs

The pairwise number of observations. Used if using a correlation matrix and asking for a minchi solution.

rotate

"none", "varimax", "quartimax", "bentlerT", "equamax", "varimin", "geominT" and "bifactor" are orthogonal rotations. "promax", "oblimin", "simplimax", "bentlerQ, "geominQ" and "biquartimin" and "cluster" are possible oblique transformations of the solution. The default is to do a oblimin transformation, although versions prior to 2009 defaulted to varimax.

n.iter

Number of bootstrap interations to do in fa or fa.poly

residuals

Should the residual matrix be shown

scores

the default="regression" finds factor scores using regression. Alternatives for estimating factor scores include simple regression ("Thurstone"), correlaton preserving ("tenBerge") as well as "Anderson" and "Bartlett" using the appropriate algorithms (see factor.scores). Although scores="tenBerge" is probably preferred for most solutions, it will lead to problems with some improper correlation matrices.

SMC

Use squared multiple correlations (SMC=TRUE) or use 1 as initial communality estimate. Try using 1 if imaginary eigen values are reported. If SMC is a vector of length the number of variables, then these values are used as starting values in the case of fm='pa'.

covar

if covar is TRUE, factor the covariance matrix, otherwise factor the correlation matrix

missing

if scores are TRUE, and missing=TRUE, then impute missing values using either the median or the mean

impute

"median" or "mean" values are used to replace missing values

min.err

Iterate until the change in communalities is less than min.err

max.iter

Maximum number of iterations for convergence

symmetric

symmetric=TRUE forces symmetry by just looking at the lower off diagonal values

warnings

warnings=TRUE => warn if number of factors is too many

fm

factoring method fm="minres" will do a minimum residual (OLS), fm="wls" will do a weighted least squares (WLS) solution, fm="gls" does a generalized weighted least squares (GLS), fm="pa" will do the principal factor solution, fm="ml" will do a maximum likelihood factor analysis. fm="minchi" will minimize the sample size weighted chi square when treating pairwise correlations with different number of subjects per pair.

alpha

alpha level for the confidence intervals for RMSEA

p

if doing iterations to find confidence intervals, what probability values should be found for the confidence intervals

oblique.scores

When factor scores are found, should they be based on the structure matrix (default) or the pattern matrix (oblique.scores=TRUE).

use

How to treat missing data, use="pairwise" is the default". See cor for other options.

cor

How to find the correlations: "cor" is Pearson", "cov" is covariance, "tet" is tetrachoric, "poly" is polychoric, "mixed" uses mixed cor for a mixture of tetrachorics, polychorics, Pearsons, biserials, and polyserials, Yuleb is Yulebonett, Yuleq and YuleY are the obvious Yule coefficients as appropriate

multi.results

The results from fa.multi

labels

variable labels

flabels

Labels for the factors (not counting g)

size

size of graphics window

digits

Precision of labels

cex

control font size

color.lines

Use black for positive, red for negative

marg

The margins for the figure are set to be wider than normal by default

adj

Adjust the location of the factor loadings to vary as factor mod 4 + 1

main

main figure caption

…

additional parameters, specifically, keys may be passed if using the target rotation, or delta if using geominQ, or whether to normalize if using Varimax. In addition, for fa.multi.diagram, other options to pass into the graphics packages

e.size

the size to draw the ellipses for the factors. This is scaled by the number of variables.

cut

Minimum path coefficient to draw

gcut

Minimum general factor path to draw

simple

draw just one path per item

sort

sort the solution before making the diagram

side

on which side should errors be drawn?

errors

show the error estimates

rsize

size of the rectangles

The standard output from a factor analysis from `fa`

for the raw variables

The standard output from a factor analysis from `fa`

for the correlation matrix of the level 1 solution.

See `fa`

and `omega`

for a discussion of factor analysis and of the case of one higher order factor.

Revelle, William. (in prep) An introduction to psychometric theory with applications in R. Springer. Working draft available at https://personality-project.org/r/book/

```
# NOT RUN {
f31 <- fa.multi(Thurstone,3,1) #compare with \code{\link{omega}}
f31
fa.multi.diagram(f31)
# }
```

Run the code above in your browser using DataCamp Workspace