EXTENSION_FA {EFA.dimensions}

Extension factor analysis

Description

Extension factor analysis, which provides correlations between nonfactored items and the factors that exist in a set of core items. The extension item correlations are then used to decide which factor, if any, a prospective item belongs to.

Usage

EXTENSION_FA(data, Ncore, Next, higherorder, roottest,  
             corkind, 
             extraction, rotation, 
             Nfactors, NfactorsHO, 
             Ndatasets, percentile, 
             salvalue, numsals, 
             iterpaf, ppower, 
             verbose, factormodel, rotate)

Arguments

data

An all-numeric dataframe where the rows are cases & the columns are the variables.

Ncore

An integer indicating the number of core variables. The function will run the factor analysis on the data that appear in column #1 to column #Ncore of the data matrix.

Next

An integer indicting the number of extension variables, if any. The function will run extension factor analyses on the remaining columns in data, i.e., using column #Ncore+1 to the last column in data. Enter zero if there are no extension variables.

higherorder

Should a higher-order factor analysis be conducted? The options are TRUE or FALSE.

roottest

The method for determining the number of factors. The options are: 'Nsalient' for number of salient loadings (see salvalue & numsals below); 'parallel' for parallel analysis (see Ndatasets & percentile below); 'MAP' for Velicer's minimum average partial test; 'SEscree' for the standard error scree test; 'nevals>1' for the number of eigenvalues > 1; and 'user' for a user-specified number of factors (see Nfactors & NfactorsHO below).

corkind

The kind of correlation matrix to be used. The options are 'pearson', 'kendall', 'spearman', 'gamma', and 'polychoric'.

extraction

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

rotation

The factor rotation method. The options are: 'promax', 'varimax', and 'none'.

Nfactors

An integer indicating the user-determined number of factors (required only if roottest = 'user').

NfactorsHO

An integer indicating the user-determined number of higher order factors (required only if roottest = 'user' and higherorder = TRUE).

Ndatasets

An integer indicating the # of random data sets for parallel analyses (required only if roottest = 'parallel').

percentile

An integer indicating the percentile from the distribution of parallel analysis random eigenvalues to be used in determining the # of factors (required only if roottest = 'parallel'). Suggested value: 95

salvalue

The minimum value for a loading to be considered salient (required only if roottest = 'Nsalient'). Suggested value: .40

numsals

The number of salient loadings required for the existence of a factor i.e., the number of loadings > or = to salvalue (see above) for the function to identify a factor. Required only if roottest = 'Nsalient'. Gorsuch (1995a, p. 545) suggests: 3

iterpaf

The maximum # of iterations for a principal axis / common factor analysis (required only if extraction = 'PAF'). Suggested value: 100

ppower

The power value to be used in a promax rotation (required only if rotation = 'promax'). Suggested value: 3

verbose

Should detailed results be displayed in console? TRUE (default) or FALSE

factormodel

(Deprecated.) Use 'extraction' instead.

rotate

(Deprecated.) Use 'rotation' instead.

Details

Traditional scale development statistics can produce results that are baffling or misunderstood by many users, which can lead to inappropriate substantive interpretations and item selection decisions. High internal consistencies do not indicate unidimensionality; item-total correlations are inflated because each item is correlated with its own error as well as the common variance among items; and the default number-of-eigenvalues-greater-than-one rule, followed by principal components analysis and varimax rotation, produces inflated loadings and the possible appearance of numerous uncorrelated factors for items that measure the same construct (Gorsuch, 1997a, 1997b). Concerned investigators may then neglect the higher order general factor in their data as they use misleading statistical output to trim items and fashion unidimensional scales.

These problems can be circumvented in exploratory factor analysis by using more appropriate factor analytic procedures and by using extension analysis as the basis for adding items to scales. Extension analysis provides correlations between nonfactored items and the factors that exist in a set of core items. The extension item correlations are then used to decide which factor, if any, a prospective item belongs to. The decisions are unbiased because factors are defined without being influenced by the extension items. One can also examine correlations between extension items and any higher order factor(s) in the core items. The end result is a comprehensive, undisturbed, and informative picture of the correlational structure that exists in a set of core items and of the potential contribution and location of additional items to the structure.

Extension analysis is rarely used, at least partly because of limited software availability. Furthermore, when it is used, both traditional extension analysis and its variants (e.g., correlations between estimated factor scores and extension items) are prone to the same problems as the procedures mentioned above (Gorsuch, 1997a, 1997b). However, Gorusch (1997b) described how diagonal component analysis can be used to bypass the problems and uncover the noninflated and unbiased extension variable correlations – all without computing factor scores.

Value

A list with the following elements:

fits1

eigenvalues & fit coefficients for the first set of core variables

rff

factor intercorrelations

corelding

core variable loadings on the factors

extcorrel

extension variable correlations with the factors

fits2

eigenvalues & fit coefficients for the higher order factor analysis

rfflding

factor intercorrelations from the first factor analysis and the loadings on the higher order factor(s)

ldingsef

variable loadings on the lower order factors and their correlations with the higher order factor(s)

extsef

extension variable correlations with the lower order factor(s) and their correlations with the higher order factor(s)

Author(s)

Brian P. O'Connor

References

Dwyer, P. S. (1937) The determination of the factor loadings of a given test from the known factor loadings of other tests. Psychometrika, 3, 173-178.

Gorsuch, R. L. (1997a). Exploratory factor analysis: Its role in item analysis. Journal of Personality Assessment, 68, 532-560.

Gorsuch, R. L. (1997b). New procedure for extension analysis in exploratory factor analysis. Educational and Psychological Measurement, 57, 725-740.

Horn, J. L. (1973) On extension analysis and its relation to correlations between variables and factor scores. Multivariate Behavioral Research, 8(4), 477-489.

O'Connor, B. P. (2001). EXTENSION: SAS, SPSS, and MATLAB programs for extension analysis. Applied Psychological Measurement, 25, p. 88.

Examples

EXTENSION_FA(data_RSE, Ncore=7, Next=3, higherorder=TRUE, 
             roottest='MAP',  
             corkind='pearson', 
             extraction='PCA', rotation='promax', 
             Nfactors=2, NfactorsHO=1, 
             Ndatasets=100, percentile=95, 
             salvalue=.40, numsals=3, 
             iterpaf=200, 
             ppower=4, 
             verbose=TRUE)

EXTENSION_FA(data_NEOPIR, Ncore=12, Next=6, higherorder=TRUE, 
             roottest='MAP',  
             corkind='pearson', 
             extraction='PCA', rotation='promax', 
             Nfactors=4, NfactorsHO=1, 
             Ndatasets=100, percentile=95, 
             salvalue=.40, numsals=3, 
             iterpaf=200, 
             ppower=4, 
             verbose=TRUE)