| EFA_SCORES {EFA.dimensions} | R Documentation |
Exploratory factor analysis scores
Description
Factor scores, and factor score indeterminacy coefficients, for exploratory factor analysis
Usage
EFA_SCORES(loadings=NULL, loadings_type='structure', data=NULL, cormat=NULL,
corkind='pearson', phi=NULL, method = 'Thurstone', verbose = TRUE)Arguments
loadings |
The factor loadings. Required for all methods except PCA. |
loadings_type |
(optional) The kind of factor loadings. The options are 'structure' (the default) or 'pattern'. Use 'structure' for orthogonal loadings. |
data |
(optional) An all-numeric dataframe where the rows are cases & the columns are the variables. Required if factor scores for cases are desired. |
cormat |
(optional) The item/variable correlation matrix. Not required when "data" is provided. |
corkind |
(optional) The kind of correlation matrix to be used. The options are 'pearson', 'kendall', 'spearman', 'gamma', and 'polychoric'. The kind of correlation should be the same as the kind that was used to produce the "loadings". |
phi |
(optional) The factor correlations. |
method |
(optional) The method to be used for computing the factor scores (e.g., method = 'Thurstone'). The options are:
|
verbose |
(optional) Should detailed results be displayed in console? TRUE (default) or FALSE |
Details
Before using factor scores, it is important to establish that there is an acceptable degree of "determinacy" for the computed factor scores (Grice, 2001; Waller, 2023).
The following descriptions of factor score indeterminacy are either taken directly from, or adapted from, Grice (2001):
"As early as the 1920s researchers recognized that, even if the correlations among a set of ability tests could be reduced to a subset of factors, the scores on these factors would be indeterminate (Wilson, 1928). In other words, an infinite number of ways for scoring the individuals on the factors could be derived that would be consistent with the same factor loadings. Under certain conditions, for instance, an individual with a high ranking on g (general intelligence), according to one set of factor scores, could receive a low ranking on the same common factor according to another set of factor scores, and the researcher would have no way of deciding which ranking is "true" based on the results of the factor analysis. As startling as this possibility seems, it is a fact of the mathematics of the common factor model.
The indeterminacy problem is not that the factor scores cannot be directly and appropriately computed; it is that an infinite number of sets of such scores can be created for the same analysis that will all be equally consistent with the factor loadings.
The degree of indeterminacy will not be equivalent across studies and is related to the ratio between the number of items and factors in a particular design (Meyer, 1973; Schonemann, 1971). It may also be related to the magnitude of the communalities (Gorsuch, 1983). Small amounts of indeterminacy are obviously desirable, and the con- sequences associated with a high degree of indeterminacy are extremely unsettling. Least palatable is the fact that if the maximum possible proportion of indeterminacy in the scores for a particular factor meets or exceeds 50 struct two orthogonal or negatively correlated sets of factor scores that will be equally consistent with the same factor loadings (Guttman, 1955).
MULTR & RSQR MULTR is the multiple correlation between each factor and the original variables (Green, 1976; Mulaik, 1976). MULTR ranges from 0 to 1, with high values being desirable, and indicates the maximum possible degree of determinacy for factor scores. Some authors have suggested that MULTR values should be substantially higher than .707 which, when squared, would equal .50. RSQR is the square or MULTR and represents the maximum proportion of determinacy.
MINCOR
the minimum correlation that could be obtained between two sets of equally valid factor scores for each factor (Guttman, 1955; Mulaik, 1976; Schonemann, 1971). This index ranges from -1 to +1. High positive values are desirable. When MINCOR is zero, then two sets of competing factor scores can be constructed for the same common factor that are orthogonal or even negatively correlated. MINCOR values approaching zero are distressing, and negative values are disastrous. MINCOR values of zero or less occur when MULTR <= .707 (at least 50 indeterminacy). MULTR values that do not appreciably exceed .71 are therefore particularly problematic. High values that approach 1.0 indicate that the factors may be slightly indeterminate, but the infinite sets of factor scores that could be computed will yield highly similar rankings of the individuals. In other words, the practical impact of the indeterminacy is minimal. MINCOR is the "Guttman's Indeterminacy Index" that is provided by the fsIndeterminacy function in the fungible package.
VALIDITY
While the MULTR values represent the maximum correlation between the factor score estimates and the factors, the VALIDITY coefficients represent the actual correlations between the factor score estimates and their respective factors, which may be lower than MULTR. The VALIDITY coefficients may range from -1 to +1. They should be interpreted in the same manner as MULTR. Gorsuch (1983, p. 260) recommended values of at least .80, but much larger values (>.90) may be necessary if the factor score estimates are to serve as adequate substitutes for the factors themselves.
Correlational Accuracy
If the factor score estimates are adequate representations of the factors, then the correlations between the factor scores should be similar to the correlations between the factors."
Value
A list with the following elements:
FactorScores |
The factor scores |
FSCoef |
The factor score coefficients (W) |
MULTR |
The multiple correlation between each factor and the original variables |
RSQR |
The square or MULTR, representing the maximum proportion of determinacy |
MINCOR |
Guttmans indeterminacy index, the minimum correlation that could be obtained between two sets of equally valid factor scores for each factor. |
VALIDITY |
The correlations between the factor score estimates and their respective factors |
UNIVOCALITY |
The extent to which the estimated factor scores are excessively or insufficiently correlated with other factors in the same analysis |
FactorScore_Correls |
The correlations between the factor scores |
phi |
The correlations between the factors |
pattern |
The pattern matrix |
pattern |
The structure matrix |
Author(s)
Brian P. O'Connor
References
Anderson, R. D., & Rubin, H. (1956). Statistical inference in factor analysis.
Proceedings of the Third Berkeley Symposium of Mathematical Statistics and Probability, 5, 111-150.
Bartlett, M. S. (1937). The statistical conception of mental factors.
British Journal of Psychology, 28, 97-104.
Grice, J. (2001). Computing and evaluating factor scores.
Psychological Methods, 6(4), 430-450.
Harman, H. H. (1976). Modern factor analysis. University of Chicago press.
ten Berge, J. M. F., Krijnen, W. P., Wansbeek, T., and Shapiro, A. (1999).
Some new results on correlation-preserving factor scores prediction methods.
Linear Algebra and its Applications, 289(1-3), 311-318.
Thurstone, L. L. (1935). The vectors of mind. Chicago: University of Chicago Press.
Waller, N. G. (2023). Breaking our silence on factor score indeterminacy.
Journal of Educational and Behavioral Statistics, 48(2), 244-261.
Examples
efa_out <- EFA(data=data_RSE, extraction = 'ml', Nfactors=2, rotation='promax')
EFA_SCORES(loadings=efa_out$structure, loadings_type='structure', data=data_RSE,
phi=efa_out$phi, method = 'tenBerge')
# PCA scores
EFA_SCORES(data=data_NEOPIR, method = 'PCA')