SALIENT {EFA.dimensions}

Salient loadings criterion for the number of factors

Description

Salient loadings criterion for determining the number of factors, as recommended by Gorsuch. Factors are retained when they consist of a specified minimum number (or more) variables that have a specified minimum (or higher) loading value.

Usage

SALIENT(data, salvalue=.4, numsals=3, max_cross=NULL, min_eigval=.7, corkind='pearson', 
        extraction = 'paf', rotation='promax', loading_mat = 'structure',  
         ppower = 3, iterpaf=100, Ncases=NULL, verbose=TRUE, factormodel, rotate)

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.

salvalue

(optional) The loading value that is considered salient. Default = .40. This can also be a vector of up to three values, e.g., salvalue = c(.4, .5, .6).

numsals

(optional) The required number of salient loadings for a factor. Default = 3. This can also be a vector of up to three values, e.g., numsals = c(3, 2, 1).

max_cross

(optional) The maximum value for cross-loadings.

min_eigval

(optional) The minimum eigenvalue for including a factor in the analyses. Default = .7

corkind

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

extraction

(optional) The factor extraction method for the analysis. The options are 'pca', 'paf' (the default), 'ml', 'image', 'minres', 'uls', 'ols', 'wls', 'gls', 'alpha', and 'fullinfo'.

rotation

(optional) The factor rotation method for the analysis. The orthogonal rotation options are: 'varimax' (the default), 'quartimax', 'bentlerT', 'equamax', 'geominT', 'bifactorT', 'entropy', and 'none'. The oblique rotation options are: 'promax' (the default), 'quartimin', 'oblimin', 'oblimax', 'simplimax', 'bentlerQ', 'geominQ', 'bifactorQ', and 'none'.

iterpaf

(optional) The maximum number of iterations for paf. Default value = 100

loading_mat

(optional) The kind of factor rotation matrix for an oblique rotation. The options are 'structure' (the default) or 'pattern'.

ppower

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

Ncases

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

verbose

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

factormodel

(Deprecated.) Use 'extraction' instead.

rotate

(Deprecated.) Use 'rotation' instead.

Details

In this procedure for determining the number of factors, factors are retained when each factor has at least a specified minimum number variables (e.g., 3) that have loadings that are greater than or equal to a specified minimum loading value (e.g., .40). Factor are considered trivial when they do not contain a sufficient number of salient loadings (Gorsuch, 1997, 2015; Boyd, 2011).

The procedure begins by extracting and rotating (if requested) an excessive number of factors. If the initial factor loadings do not meet the specified criteria, then the factor analysis is conducted again with one less factor and the loadings are again examined to determine whether the factor loadings meet the specified criteria. The procedure stops when a loading matrix meets the criteria, in which case the number of columns in the loading matrix is the number of factors according to the salient loadings criteria.

The initial, excessive number of factors for the procedure is determined using the min_eigval argument (for minimum eigenvalue). The default is .70, which can be adjusted (raised) when analyses produce an error caused by there being too few variables.

Although there is no consensus on what constitutes a 'salient' loading, an absolute value of .40 is common in the literature.

There are different versions of the salient loadings criterion method, which has not been extensively tested to date. The procedure is nevertheless considered promising by Gorsuch and others.

Some versions involve the use of multiple salient loading values, each with its own minimum number of variables. This can be done in the SALIENT function by providing a vector of values for the salvalue argument and a corresponding vector of values for the numsals argument. The maximum number of possible values is three, and there should be a logical order in the values, i.e., increasing values for salvalue and decreasing values for numsals.

It is also possible to place a restriction of the maximum value of the cross-loadings for the salient variables, e.g., requiring that a salient loading is not accompanied by cross-loadings on other variables that are greater than .15. Use the max_cross argument for this purpose, although it may be difficult to claim that cross-loadings should be small when the factors are correlated.

Value

The number of factors according to the salient loadings criterion.

Author(s)

Brian P. O'Connor

References

Boyd, K. C. (2011). Factor analysis. In M. Stausberg & S. Engler (Eds.), The Routledge Handbook of Research Methods in the Study of Religion (pp. 204-216). New York: Routledge.

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

Gorsuch, R. L. (2015). Factor analysis. Routledge/Taylor & Francis Group.

Examples

# the Harman (1967) correlation matrix
SALIENT(data_Harman, salvalue=.4, numsals=3, Ncases=305)

# Rosenberg Self-Esteem scale items, using Pearson correlations
SALIENT(data_RSE, salvalue=.4, numsals=3, corkind='pearson')

# NEO-PI-R scales
SALIENT(data_NEOPIR, salvalue = c(.4, .5, .6), numsals = c(3, 2, 1), extraction = 'paf', 
        rotation='promax', loading_mat = 'pattern')