One way to determine the number of factors or components in a data matrix or a correlation matrix is to examine the ``scree" plot of the successive eigenvalues. Sharp breaks in the plot suggest the appropriate number of components or factors to extract. ``Parallel" analysis is an alternative technique that compares the scree of factors of the observed data with that of a random data matrix of the same size as the original. This may be done for continuous , dichotomous, or polytomous data using Pearson, tetrachoric or polychoric correlations.

```
fa.parallel(x,n.obs=NULL,fm="minres",fa="both",nfactors=1,
main="Parallel Analysis Scree Plots",
n.iter=20,error.bars=FALSE,se.bars=FALSE,SMC=FALSE,ylabel=NULL,show.legend=TRUE,
sim=TRUE,quant=.95,cor="cor",use="pairwise",plot=TRUE,correct=.5,sqrt=FALSE)
paSelect(keys,x,cor="cor", fm="minres",plot=FALSE)
fa.parallel.poly(x ,n.iter=10,SMC=TRUE, fm = "minres",correct=TRUE,sim=FALSE,
fa="both",global=TRUE) #deprecated
# S3 method for poly.parallel
plot(x,show.legend=TRUE,fa="both",...)
```

A plot of the eigen values for the original data, ntrials of resampling of the original data, and of a equivalent size matrix of random normal deviates. If the data are a correlation matrix, specify the number of observations.

Also returned (invisibly) are:

- fa.values
The eigen values of the factor model for the real data.

- fa.sim
The descriptive statistics of the simulated factor models.

- pc.values
The eigen values of a principal components of the real data.

- pc.sim
The descriptive statistics of the simulated principal components analysis.

- nfact
Number of factors with eigen values > eigen values of random data

- ncomp
Number of components with eigen values > eigen values of random data

- values
The simulated values for all simulated trials

- x
A data.frame or data matrix of scores. If the matrix is square, it is assumed to be a correlation matrix. Otherwise, correlations (with pairwise deletion) will be found.

- n.obs
n.obs=0 implies a data matrix/data.frame. Otherwise, how many cases were used to find the correlations.

- fm
What factor method to use. (minres, ml, uls, wls, gls, pa) See

`fa`

for details.- fa
show the eigen values for a principal components (fa="pc") or a principal axis factor analysis (fa="fa") or both principal components and principal factors (fa="both")

- nfactors
The number of factors to extract when estimating the eigen values. Defaults to 1, which was the prior value used.

- main
a title for the analysis

- n.iter
Number of simulated analyses to perform

- 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. This matches the call to

`fa`

.- keys
A list of scoring keys to allow fa.parallel on multiple subsets of the data

- correct
For tetrachoric correlations, should a correction for continuity be applied. (See

`tetrachoric`

.) If set to 0, then no correction is applied, otherwise, the default is to add .5 observations to the cell.- sim
For continuous data, the default is to resample as well as to generate random normal data. If sim=FALSE, then just show the resampled results. These two results are very similar. This does not make sense in the case of correlation matrix, in which case resampling is impossible. In the case of polychoric or tetrachoric data, in addition to randomizing the real data, should we compare the solution to random simulated data. This will double the processing time, but will basically show the same result.

- error.bars
Should error.bars be plotted (default = FALSE)

- se.bars
Should the error bars be standard errors (se.bars=TRUE) or 1 standard deviation (se.bars=FALSE, the default). With many iterations, the standard errors are very small and some prefer to see the broader range. The default has been changed in version 1.7.8 to be se.bars=FALSE to more properly show the range.

- SMC
SMC=TRUE finds eigen values after estimating communalities by using SMCs. smc = FALSE finds eigen values after estimating communalities with the first factor.

- ylabel
Label for the y axis -- defaults to ``eigen values of factors and components", can be made empty to show many graphs

- show.legend
the default is to have a legend. For multiple panel graphs, it is better to not show the legend

- quant
if nothing is specified, the empirical eigen values are compared to the mean of the resampled or simulated eigen values. If a value (e.g., quant=.95) is specified, then the eigen values are compared against the matching quantile of the simulated data. Clearly the larger the value of quant, the few factors/components that will be identified. The default is to use quant=.95.

- global
If doing polychoric analyses (fa.parallel.poly) and the number of alternatives differ across items, it is necessary to turn off the global option. fa.parallel.poly is deprecated but this choice is still relevant.

- ...
additional plotting parameters, for plot.poly.parallel

- plot
By default, fa.parallel draws the eigen value plots. If FALSE, suppresses the graphic output

- sqrt
If TRUE, take the squareroot of the eigen values to more understandably show the scale. Note, although providing more useful graphics the results will be the same. See DelGuidice, 2022

William Revelle

Cattell's ``scree" test is one of most simple tests for the number of factors problem. Horn's (1965) ``parallel" analysis is an equally compelling procedure. Other procedures for determining the most optimal number of factors include finding the Very Simple Structure (VSS) criterion (`VSS`

) and Velicer's `MAP`

procedure (included in `VSS`

). Both the VSS and the MAP criteria are included in the `nfactors`

function which also reports the mean item complexity and the BIC for each of multiple solutions. fa.parallel plots the eigen values for a principal components and the factor solution (minres by default) and does the same for random matrices of the same size as the original data matrix. For raw data, the random matrices are 1) a matrix of univariate normal data and 2) random samples (randomized across rows) of the original data.

`fa.parallel`

with the cor=poly option will do what `fa.parallel.poly`

explicitly does: parallel analysis for polychoric and tetrachoric factors.
If the data are dichotomous, `fa.parallel.poly`

will find tetrachoric correlations for the real and simulated data, otherwise, if the number of categories is less than 10, it will find polychoric correlations.
Note that fa.parallel.poly is slower than fa.parallel because of the complexity of calculating the tetrachoric/polychoric correlations.
The functionality of `fa.parallel.poly`

is included in `fa.parallel`

with cor=poly option (etc.) option but the older `fa.parallel.poly`

is kept for those who call it directly.

That is, `fa.parallel`

now will do tetrachorics or polychorics directly if the cor option is set to "tet" or "poly". As with `fa.parallel.poly`

this will take longer.

The means of (ntrials) random solutions are shown. Error bars are usually very small and are suppressed by default but can be shown if requested. If the sim option is set to TRUE (default), then parallel analyses are done on resampled data as well as random normal data. In the interests of speed, the parallel analyses are done just on resampled data if sim=FALSE. Both procedures tend to agree.

As of version 1.5.4, I added the ability to specify the quantile of the simulated/resampled data, and to plot standard deviations or standard errors. By default, this is set to the 95th percentile.

Alternative ways to estimate the number of factors problem are discussed in the Very Simple Structure (Revelle and Rocklin, 1979) documentation (`VSS`

) and include Wayne Velicer's `MAP`

algorithm (Veicer, 1976).

Parallel analysis for factors is actually harder than it seems, for the question is what are the appropriate communalities to use. If communalities are estimated by the Squared Multiple Correlation (SMC) `smc`

, then the eigen values of the original data will reflect major as well as minor factors (see `sim.minor`

to simulate such data). Random data will not, of course, have any structure and thus the number of factors will tend to be biased upwards by the presence of the minor factors.

By default, fa.parallel estimates the communalities based upon a one factor minres solution. Although this will underestimate the communalities, it does seem to lead to better solutions on simulated or real (e.g., the `bfi`

or Harman74) data sets.

For comparability with other algorithms (e.g, the paran function in the paran package), setting smc=TRUE will use smcs as estimates of communalities. This will tend towards identifying more factors than the default option.

Yet another option (suggested by Florian Scharf) is to estimate the eigen values based upon a particular factor model (e.g., specify nfactors > 1).

Printing the results will show the eigen values of the original data that are greater than simulated values.

A sad observation about parallel analysis is that it is sensitive to sample size. That is, for large data sets, the eigen values of random data are very close to 1. This will lead to different estimates of the number of factors as a function of sample size. Consider factor structure of the bfi data set (the first 25 items are meant to represent a five factor model). For samples of 200 or less, parallel analysis suggests 5 factors, but for 1000 or more, six factors and components are indicated. This is not due to an instability of the eigen values of the real data, but rather the closer approximation to 1 of the random data as n increases.

Although with nfactors=1, 6 factors are suggested, when specifying nfactors =5, parallel analysis of the bfi suggests 12 factors should be extracted!

When simulating dichotomous data in fa.parallel.poly, the simulated data have the same difficulties as the original data. This functionally means that the simulated and the resampled results will be very similar. Note that fa.parallel.poly has functionally been replaced with fa.parallel with the cor="poly" option.

As with many psych functions, fa.parallel has been changed to allow for multicore processing. For running a large number of iterations, it is obviously faster to increase the number of cores to the maximum possible (using the options("mc.cores=n) command where n is determined from detectCores().

Del Giudice, M. (2022, October 21). The Square-Root Scree Plot: A Simple Improvement to a Classic Display. doi: 10.31234/osf.io/axubd

Floyd, Frank J. and Widaman, Keith. F (1995) Factor analysis in the development and refinement of clinical assessment instruments. Psychological Assessment, 7(3):286-299, 1995.

Horn, John (1965) A rationale and test for the number of factors in factor analysis. Psychometrika, 30, 179-185.

Humphreys, Lloyd G. and Montanelli, Richard G. (1975), An investigation of the parallel analysis criterion for determining the number of common factors. Multivariate Behavioral Research, 10, 193-205.

Revelle, William and Rocklin, Tom (1979) Very simple structure - alternative procedure for estimating the optimal number of interpretable factors. Multivariate Behavioral Research, 14(4):403-414.

Velicer, Wayne. (1976) Determining the number of components from the matrix of partial correlations. Psychometrika, 41(3):321-327, 1976.

`fa`

, `nfactors`

, `VSS`

, `VSS.plot`

, `VSS.parallel`

, `sim.minor`

```
#test.data <- Harman74.cor$cov #The 24 variable Holzinger - Harman problem
#fa.parallel(test.data,n.obs=145)
fa.parallel(Thurstone,n.obs=213) #the 9 variable Thurstone problem
#set.seed(123)
#minor <- sim.minor(24,4,400) #4 large and 12 minor factors
#ffa.parallel(minor$observed) #shows 5 factors and 4 components -- compare with
#fa.parallel(minor$observed,SMC=FALSE) #which shows 6 and 4 components factors
#a demonstration of parallel analysis of a dichotomous variable
#fp <- fa.parallel(psychTools::ability) #use the default Pearson correlation
#fpt <- fa.parallel(psychTools::ability,cor="tet") #do a tetrachoric correlation
#fpt <- fa.parallel(psychTools::ability,cor="tet",quant=.95) #do a tetrachoric correlation and
#use the 95th percentile of the simulated results
#apply(fp$values,2,function(x) quantile(x,.95)) #look at the 95th percentile of values
#apply(fpt$values,2,function(x) quantile(x,.95)) #look at the 95th percentile of values
#describe(fpt$values) #look at all the statistics of the simulated values
#paSelect(bfi.keys,bfi)#do 5 different runs, once for each subscale
#paSelect(ability.keys,ability,cor="poly",fm="minrank")
```

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