fsi: Factor Score Indeterminacy estimates )

In psych: Procedures for Psychological, Psychometric, and Personality Research

Factor Score Indeterminacy estimates )

Description

Factor analysis either using Exploratory Factor Analysis fa or Confirmatory Factor analysis CFA estimates factor loadings to best fit a latent variable model that is underdetermined. An infinite number of factor scores from such a model all will fit equally well (Grice, 2001, Nicewander, 2020, Waller, 2023). fsi calculates an estimated R2 of observed factor scores with latent factors using either the Nicewander approach (default) or Grice approach (to match previous versions.)

Usage

fsi(f, phi = NULL, r = NULL, Grice = FALSE,short=TRUE)

Arguments

f	A factor loading matrix
phi	Factor correlations
r	The correlations of the raw data – needed for the Grice option
Grice	If TRUE, find the Grice (2001) solution.
short	By default, return a vector of R2 values for each factor. Alternatively report the R2 values and H matrix (see Nicewander, 2020).

Details

A dirty little secret of factor analysis is that the factor scores are not perfectly correlated with their corresponding factors. In fact, there are an infinite number of factor score solutions that all fit equally well. The factor scores are, in fact, indeterminant. Although known for almost 100 years, the lack of determinancy is rarely reported. “for any set of p, imperfectly measured variables that fit Spearman's model, a research can construct an infinite set of factor scores, each of which fits the model perfectly." Waller,2023, p 245.

Reviews of the factor score inderminancy (FSI) problem and possible solutions include Grice (2001), Nicewander (2020) and Waller (2023).

The Grice (2001) approach is highly cited, but recent work by Nicewander (2020) and Waller (2023) suggests an alternative approach, which is more highly grounded in statistical theory. fsi will report both of these solutions. Nicewander approach is used in the CFA function and has replaced (as of version 2.6.1) the Grice approach used in fa.

The fsi will report the R2 between putative factor scores and the latent factors using either the Nicewander (2020) or the Grice (2001) approach. Prior versions of fa in psych reported just the Grice solution. As of psych_2.6.1 the default for fa is to report the Nicewander estimate, but with an option to use the Grice estimate.

Value

R2	Muliple R2 of the factor model with the factors
H	The first nfactor elments of the H matrix of factor correlations in the nvar space. See Nicewander, 2020.

Author(s)

William Revelle

References

Grice, James W. (2001) Computing and evaluating factors scores. Psychological Methods, 6, p 430-450. 10.1037/1082-989X.6.4.430

Nicewander, W. Alan (2020) A pespective on the mathematical and psychometric aspects of factor indeterminacy. Multivariate Behavioral Research. 55 DOI 10.1080/00273171.2019.1684872)

Waller, N. G. (2023) Breaking our silence on factor score indeterminacy. Journal of Educational and Behavioral Statistics. 48, p 244-261. doi; 10.3102/1076998622112.

See Also

fa and CFA.

Examples

#matrix from Waller example for fsIndeterminacy from fungible
Lambda <- matrix(c(.8,  0,
                   .7,  0,
                   .6,  0,
                    0, .5,
                    0, .4,
                    0, .3), 6, 2, byrow=TRUE)
                       
 fsi(Lambda)  #should be .7675 and .3837
 
 A <- matrix( c(.8, 0,
                .8, 0,
                 0,  .6,
                 0,  .6), 4,2, byrow=TRUE)
                 
 fsi(A)   #should be .781 .529  (from Nicewander Appendix 1)                      

psych documentation built on Feb. 3, 2026, 9:08 a.m.