factor.scores: Various ways to estimate factor scores for the factor analysis model

Description

A fundamental problem with factor analysis is that although the model is defined at the structural level, it is indeterminate at the data level. This problem of factor indeterminancy leads to alternative ways of estimating factor scores, none of which is ideal. Following Grice (2001) four different methods are available here.

Usage

factor.scores(x, f, Phi = NULL, method = c("Thurstone", "tenBerge", "Anderson", 
       "Bartlett", "Harman","components"))

Arguments

Either a matrix of data if scores are to be found, or a correlation matrix if just the factor weights are to be found.

The output from the fa function, or a factor loading matrix.

Phi

If a pattern matrix is provided, then what were the factor intercorrelations. Does not need to be specified if f is the output from the fa function.

method

Which of four factor score estimation procedures should be used. Defaults to "Thurstone" or regression based weights. See details below for the other four methods.

Value

- scores (the factor scores if the raw data is given)
- weights (the factor weights)

Details

Although the factor analysis model is defined at the structural level, it is undefined at the data level. This is a well known but little discussed problem with factor analysis.

Factor scores represent estimates of common part of the variables and should not be thought of as identical to the factors themselves. If a factor scores is thought of as a chop stick stuck into the center of an ice cream cone and factor scores are represented by straws anywhere along the edge of the cone the problem of factor indeterminacy becomes clear, for depending on the shape of the cone, two straws can be negatively correlated with each other. (The imagery is taken from Niels Waller, adapted from Stanley Mulaik). In a very clear discussion of the problem of factor score indeterminacy, Grice (2001) reviews several alternative ways of estimating factor scores and considers weighting schemes that will produce uncorrelated factor score estimates as well as the effect of using course coded (unit weighted) factor weights.

factor.scores uses four different ways of estimate factor scores. In all cases, the factor score estimates are based upon the data matrix, X, times a weighting matrix, W, which weights the observed variables.

method="Thurstone" finds the regression based weights:$W = R^{-1} F$where R is the correlation matrix and F is the factor loading matrix.
method="tenBerge" finds weights such that the correlation between factors for an oblique solution is preserved. Note that formula 8 in Grice has a typo in the formula for C and should be:$L = F \Phi^(1/2)$$C = R^(-1/2) L (L' R^(-1) L)^(-1/2)$$W = R ^(-1/2) C \Phi^(1/2)$
method="Anderson" finds weights such that the factor scores will be uncorrelated:$W = U^{-2}F (F' U^{-2} R U^{-2} F)^{-1/2}$where U is the diagonal matrix of uniquenesses. The Anderson method works for orthogonal factors only, while the tenBerge method works for orthogonal or oblique solutions.
method = "Bartlett" finds weights given$W = U^{-2}F (F' U^{-2}F)^{-1}$
method="Harman" finds weights based upon socalled "idealized" variables:$W = F (t(F) F)^{-1}$.
method="components" uses weights that are just component loadings.

References

Grice, James W.,2001, Computing and evaluating factor scores, Psychological Methods, 6,4, 430-450. (note the typo in equation 8)

ten Berge, Jos M.F., Wim P. Krijnen, Tom Wansbeek and Alexander Shapiro (1999) Some new results on correlation-preserving factor scores prediction methods. Linear Algebra and its Applications, 289, 311-318.

Revelle, William. (in prep) An introduction to psychometric theory with applications in R. Springer. Working draft available at http://personality-project.org/r/book/

Examples

Run this code

f3 <- fa(Thurstone)
f3$weights  #just the scoring weights
f5 <- fa(bfi,5)
round(cor(f5$scores,use="pairwise"),2)
#compare to the f5 solution

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