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, ...)

- f1
The standard output from a factor analysis from

`fa`

for the raw variables- f2
The standard output from a factor analysis from

`fa`

for the correlation matrix of the level 1 solution.

The arguments match those of the fa function.

- 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

William Revelle

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/

`fa`

, `omega`

```
f31 <- fa.multi(Thurstone,3,1) #compare with \code{\link{omega}}
f31
fa.multi.diagram(f31)
```

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