EFA.dimensions (version 0.1.6)

RAWPAR: Parallel analysis of eigenvalues with real data as input

Description

Parallel analysis of eigenvalues, with real data as input, for deciding on the number of components or factors.

Usage

RAWPAR(data, randtype, extract, Ndatasets, percentile, 
       corkind, corkindRAND, Ncases, verbose)

Arguments

data

An all-numeric dataframe where the rows are cases & the columns are the variables, or a correlation matrix with ones on the diagonal. The function internally determines whether the data are a correlation matrix.

randtype

The kind of random data to be used in the parallel analysis: 'generated' for random normal data generation; 'permuted' for permutations of the raw data matrix.

extract

The factor extraction method. The options are: 'PAF' for principal axis / common factor analysis; 'PCA' for principal components analysis. 'image' for image analysis.

Ndatasets

An integer indicating the # of random data sets for parallel analyses.

percentile

An integer indicating the percentile from the distribution of parallel analysis random eigenvalues to be used in determining the # of factors. Suggested value: 95

corkind

The kind of correlation matrix to be used if data is not a correlation matrix. The options are 'pearson', 'kendall', 'spearman', and 'polychoric'. Required only if the entered data is not a correlation matrix.

corkindRAND

The kind of correlation matrix to be used for the random data analyses. The options are 'pearson', 'kendall', 'spearman', and 'polychoric'. The default is 'pearson'.

Ncases

The number of cases upon which a correlation matrix is based. Required only if data is a correlation matrix.

verbose

Should detailed results be displayed in console? The options are: TRUE (default) or FALSE.

Value

A list with:

eigenvalues

the eigenvalues for the real and random data

nfPA

the number of factors based on the parallel analysis

Details

The parallel analysis procedure for deciding on the number of components or factors involves extracting eigenvalues from random data sets that parallel the actual data set with regard to the number of cases and variables. For example, if the original data set consists of 305 observations for each of 8 variables, then a series of random data matrices of this size (305 by 8) would be generated, and eigenvalues would be computed for the correlation matrices for the original, real data and for each of the random data sets. The eigenvalues derived from the actual data are then compared to the eigenvalues derived from the random data. In Horn's original description of this procedure, the mean eigenvalues from the random data served as the comparison baseline, whereas the more common current practice is to use the eigenvalues that correspond to the desired percentile (typically the 95th) of the distribution of random data eigenvalues. Factors or components are retained as long as the ith eigenvalue from the actual data is greater than the ith eigenvalue from the random data.

The RAWPAR function permits users to specify PCA or PAF or image as the factor extraction method. Principal components eigenvalues are often used to determine the number of common factors. This is the default in most statistical software packages, and it is the primary practice in the literature. It is also the method used by many factor analysis experts, including Cattell, who often examined principal components eigenvalues in his scree plots to determine the number of common factors. Principal components eigenvalues are based on all of the variance in correlation matrices, including both the variance that is shared among variables and the variances that are unique to the variables. In contrast, principal axis eigenvalues are based solely on the shared variance among the variables. The procedures are qualitatively different. Some therefore claim that the eigenvalues from one extraction method should not be used to determine the number of factors for another extraction method. The PAF option in the extract argument for the PARALLEL function was included solely for research purposes. It is best to use PCA as the extraction method for regular data analyses. The MAP test (also in this package) is generally more suitable for determining the number of common factors.

Polychoric correlations are time-consuming to compute. While polychoric correlations should probably be specified for the real data eigenvalues when data consists of item-level responses, polychoric correlations should probably not be specified for the random data computations, even for item-level data. The procedure would take much time and it is unnecessary. Polychoric correlations are estimates of what the Pearson correlations would be had the real data been continuous. For item-level data, specify polychoric correlations for the real data eigenvalues (corkind='polychoric') and use the default for the random data eigenvalues (corkindRAND='pearson'). The option for using polychoric correlations for the random data computations (corkindRAND='polychoric') was provided solely for research purposes.

References

Horn, J. L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30, 179-185. O'Connor, B. P. (2000). SPSS and SAS programs for determining the number of components using parallel analysis and Velicer's MAP test. Behavior Research Methods, Instrumentation, and Computers, 32, 396-402. Zwick, W. R., & Velicer, W. F. (1986). Comparison of five rules for determining the number of components to retain. Psychological Bulletin, 99, 432-442.

Examples

# NOT RUN {
# parallel analysis of the WISC data
RAWPAR(data_TabFid, randtype='generated', extract='PCA', Ndatasets=100,
       percentile=95, corkind='pearson', verbose=TRUE)

# parallel analysis of the Harman correlation matrix
RAWPAR(data_Harman, randtype='generated', extract='PCA', Ndatasets=100, 
       percentile=95, corkind='pearson', Ncases=305, verbose=TRUE)

# parallel analysis of the Rosenberg Self-Esteem Scale (RSE)
RAWPAR(data_RSE, randtype='permuted', extract='PCA', Ndatasets=100,
       percentile=95, corkind='pearson', corkindRAND='pearson', verbose=TRUE)

# parallel analysis of the Rosenberg Self-Esteem Scale (RSE) using polychoric correlations
RAWPAR(data_RSE, randtype='generated', extract='PCA', Ndatasets=100,
       percentile=95, corkind='polychoric', verbose=TRUE)

# parallel analysis of the NEO-PI-R data
RAWPAR(data_NEOPIR, randtype='generated', extract='PCA', Ndatasets=100, 
       percentile=95, corkind='pearson', Ncases=305, verbose=TRUE)
# }