paran {paran}R Documentation

Horn's Parallel Analysis of Principal Components/Factors

Description

paran performs Horn's 'parallel analysis' to a principal components or factor analysis, so as to adjust for sample bias in the retention of components.

Usage

paran(x, iterations=0, centile=0, quietly=FALSE, status=TRUE, all=FALSE, cfa=FALSE, graph=FALSE, color=TRUE, col=c("black","red","blue"), lty=c(1,2,3), lwd=1, file="", width=640, height=640)

Arguments

x a numeric matrix or data frame for principal components analysis, or maximum likelihood factor analysis
iterations a whole number representing the number of random data sets to be produced in the analysis. The default, indicated by zero, is 30*P, where P is the number of variables or columns in x.
centile a whole number between 1 and 99 indicating the centile used in estimating bias. The default is to use the mean. By selecting a conservative number, such as 95 or 99, and a large number of iterations, paran can be used to perform the modified version of parallel analysis suggested by Glorfeld (1995).
quietly suppresses tabled output of the analysis, and only returns the vector of estimated biases.
status indicates progress in the computation. Parallel analysis can take some time to complete given a large data set and/or a large number of iterations. The cfa option may noticably increase the computational requirements of paran.
all outputs the results of the parallel analysis to the table for all components, not only those with unadjusted eigenvalues greater than 1.
cfa perform a common factor analysis instead of a principal components analysis. This provides only the unrotated eigenvalues from the common factor model.
graph requests that a plot of the unadjusted, adjusted, and random eigenvalues in a format similar to that presented by Horn in his 1965 paper. Retained components or factors are indicated by the solid circular markers on the adjusted eigenvaue plot, and non-retained components or factors are indicated with hollow circular markers.
color renders the graph in color with unadjusted eigenvalues drawn in red, adjusted eigenvalues drawn in black, and random eigenvalues drawn in blue if set to TRUE, and all lines drawn solid. If set to FALSE, the graph is rendered in black and white, and the line connecting the unadjusted eigenvalues is dashed, the line connecting the random eigenvalues is dotted, and the line connecting the adjusted eigenvalues is solid.
col a character vector with the color names of adjusted eigenvalues, unadjusted eigenvalues, and estimated random eigenvalues on the plot. These settings are used, only if color=TRUE
lty an integer vector with the line type codes for adjusted eigenvalues, unadjusted eigenvalues, and estimated random eigenvalues on the plot. These settings are used only if color=FALSE.
lwd the line width. The default is 1.
file the png file in which to save the graph output if the analysis is graphed and file is given a character string representing a valid path. The default is not to save the graph.
width the width in pixels of the png file. The default is 640.
height the height in pixels of the png file. The default is 640.

Details

paran is an implementation of Horn's (1965) technique for evaluating the components or factors retained in a principle components analysis (PCA) or factor analysis (FA). According to Horn, a common interpretation of non-correlated data is that they are perfectly non-colinear, and one would expect therefore to see eigenvalues equal to 1 in a PCA or FA of such data. However, Horn notes that multi-colinearity occurs due to sampling error and least-squares "bias," even in uncorrelated data, and therefore actual PCAs or FSs of such data will reveal eigenvalues of components or factors greater than and less than 1. His strategy is to contrast eigenvalues produced through a PCA or FA on a number of random data sets (of uncorrelated variables) with the same number of variables and observations as the experimental or observational dataset to produce eigenvalues for components or factors that are adjusted for the sample error-induced inflation. Values greater than zero are retained in the adjustment given by:

Observed data Eigenvalue_n - (Simulated Data Eigenvalue_n - 1)

paran performs a principal components analysis or common factor analysis with no rotation and performs Horn's adjustment. The user may also specify how many times to make the contrast with a random dataset (default is 30 per variable). Values less than 1 will be ignored, and the default value assumed. Random datasets are generated using the rnorm() function. The program returns a vector of length P of the estimated bias for each eigenvector, where P = the number of variables in the analysis. if centile is specified, paran may be thus be used to conduct parallel analysis following Glorfeld's suggestions to reduce the likelihood of over-retention. (Glorfeld, 1995)

Value

a list of objects relating to the parallel analysis:

Retained components/factors a scalar integer representing the number of components/factors retained
Adjusted eigenvalues a vector of the estimated eigenvalues adjusted for a finite sample size
Unadjusted eigenvalues a vector of the eigenvalues of the observed data from either princomp or factanal
Random eigenvalues
a vector of the estimated (mean or centile) eigenvalues from iterations number of N by P random data sets
Bias
a vector of the estimated bias of the unadjusted eigenvalues (i.e. the difference between the adjusted and unadjusted eigenvalues)
Simulated eigenvalues
an iterations by P matrix with each row containing the eigenvalues from princomp or factanal on an N by {P
data set of uncorrelated random data

Remarks

Hayton, et al. urge a parameterization of the random data to approximate the distribution of the observed data with respect to the middle ("mid-point") and the observed min and max. However, the PCA as I understand it is insensitive to standardizing transformations of each variable, and any linear transformation of all variables, and produces the same eigenvalues used in component or factor retention decisions. This is born by the notable lack of difference between analyses conducted using the a variety of simulated distributional assumptions (Dinno, 2007). The central limit theorem would seem to make the selection of a distributional form for the random data moot with any sizeable number of iterations. Former functionality implementing the recommendation by Hayton et al. has been removed, since parallel analysis is insensitive to it, and it only adds to the computation time required to conduct parallel analysis.

Author(s)

Alexis Dinno (adinno at post dot harvard dot edu)

References

Horn J. L. 1965. "A rationale and a test for the number of factors in factor analysis." Psychometrika. 30: 179–185

Zwick W. R., Velicer WF. 1986. "Comparison of Five Rules for Determining the Number of Components to Retain." Psychological Bulletin. 99: 432–442

Glorfeld, L. W. 1995. "An Improvement on Horn's Parallel Analysis Methodology for Selecting the Correct Number of Factors to Retain. Educational and Psychological Measurement. 55(3): 377–393

Hayton J. C., Allen D. G., and Scarpello V. 2004. "Factor Retention Decisions in Exploratory Factor Analysis: A Tutorial on Parallel Analysis" Organizational Research Methods. 7(2): 191–205

Dinno A. 2007 "Exploring the Sensitivity of Horn's Parallel Analysis to the Distributional Form of Simulated Data" Unpublished manuscript available upon request.

See Also

princomp.

Examples

## perform a standard parallel analysis on the US Arrest data
paran(USArrests, iterations=5000)

## a conservative analysis with different result!
paran(USArrests, iterations=5000, centile=95)

[Package paran version 1.3.6 Index]