Estimate factor scores and score-quality diagnostics for an EFA model

Computes factor-score weights (and, from raw data, the factor scores themselves) natively for an efa_fit() solution or a directly supplied loading matrix, together with the score-quality diagnostics that describe how well the estimated scores represent the factors: the score intercorrelations, the determinacy (validity) and univocality of each score, and Guttman's indeterminacy index. Factor scores are returned only when raw data are supplied; a correlation matrix yields the weights and diagnostics alone.

Usage

efa_scores(
  x,
  f,
  Phi = NULL,
  rho = NULL,
  method = c("regression", "Bartlett", "Anderson", "tenBerge", "Harman", "components")
)

Source

Thurstone, L. L. (1935). The vectors of mind. University of Chicago Press.

Bartlett, M. S. (1937). The statistical conception of mental factors. British Journal of Psychology, 28, 97-104.

Anderson, T. W., & Rubin, H. (1956). Statistical inference in factor analysis. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability (Vol. 5, pp. 111-150). University of California Press.

Guttman, L. (1955). The determinacy of factor score matrices with implications for five other basic problems of common-factor theory. British Journal of Statistical Psychology, 8, 65-81.

ten Berge, J. M. F., Krijnen, W. P., Wansbeek, T., & Shapiro, A. (1999). Some new results on correlation-preserving factor scores prediction methods. Linear Algebra and its Applications, 289, 311-318.

Grice, J. W. (2001). Computing and evaluating factor scores. Psychological Methods, 6, 430-450.

Arguments

x: data.frame or matrix. Raw data (needed to obtain factor scores) or a correlation matrix (yields weights and diagnostics only). When raw data carry column names, they are matched to the model variables by name (any extra columns are ignored, and a model variable missing from x is an error); unnamed data are matched by position.
f: object of class efa_fit(), an efa_loadings object, or a matrix of factor loadings.
Phi: matrix. Factor intercorrelations. Only used when a loading matrix is supplied directly in f; taken from the efa object otherwise. Default is NULL, in which case the factors are assumed uncorrelated.
rho: matrix. Correlation matrix used to derive the scoring weights. Defaults to NULL, in which case f$orig_R is used for an efa object and cor(x, use = "pairwise") otherwise. Pass a matrix here to score against a correlation other than the one implied by f/x.
method: character. The factor-score method: one of "regression" (default), "Bartlett", "Anderson", "tenBerge", "Harman", or "components".

Value

An object of class efa_scores, a list containing:

weights: The p by m factor-score weight matrix.
scores: The factor scores (n by m), or NULL when a correlation matrix was supplied.
r.scores: The m by m correlations of the factor-score estimates.
score_cor: The m by m score-factor correlation matrix; its diagonal is the determinacy (validity) of each score and its off-diagonals the univocality.
determinacy: A data frame with, per factor, the determinacy rho, the squared determinacy rho2, and Guttman's indeterminacy index guttman.
settings: A list of the settings used.

Details

The p by m weight matrix W (standardized scores are scale(X) %*% W) is computed from the structure matrix S = Lambda %*% Phi, the model uniquenesses Psi = diag(1 - h2), and the scoring correlation matrix R according to method:

"regression": Thurstone's (1935) regression scores, W = R^-1 S.
"Bartlett": Bartlett's (1937) conditionally unbiased scores.
"Anderson": Anderson & Rubin's (1956) uncorrelated, unit-variance scores; defined for orthogonal factors only.
"tenBerge": ten Berge, Krijnen, Wansbeek & Shapiro's (1999) scores, which preserve the factor intercorrelations.
"Harman": Harman's (1976) idealized-variable scores.
"components": component scores, W = Lambda.

The determinacy (validity) of a score is its correlation with the factor it estimates, computed from the returned weights; for regression scores it is the multiple correlation between the factor and the observed variables (Guttman, 1955; Grice, 2001). The off-diagonal score-factor correlations give the univocality (the correlation of a score with the other factors), and 2 rho^2 - 1 is Guttman's (1955) indeterminacy index, the minimum correlation between two equally valid sets of scores. For a method other than "regression" both quantities are specific to those scores: the determinacy is the method's own score-factor correlation (never larger than the regression value), and the reported guttman follows from it.

Examples

# Weights and score diagnostics from an EFA on a correlation matrix
efa <- efa_fit(test_models$baseline$cormat, n_factors = 3, N = 500,
               estimator = "PAF", rotation = "oblimin")
fs <- efa_scores(test_models$baseline$cormat, f = efa)
#> ℹ `x` is a correlation matrix; factor scores cannot be computed. Only factor
#>   weights and score diagnostics are returned. Enter raw data to get factor
#>   scores.
fs
#> 
#> ── Factor scores (regression) ──────────────────────────────────────────────────
#> 
#> Weights and diagnostics only (correlation-matrix input; no scores).
#> 
#> ── Score determinacy ───────────────────────────────────────────────────────────
#> 
#>      rho  rho2 guttman
#> F1 0.894 0.798   0.597
#> F2 0.888 0.788   0.576
#> F3 0.883 0.780   0.561
summary(fs)
#> 
#> ── Factor scores (regression) ──────────────────────────────────────────────────
#> 
#> Weights and diagnostics only (correlation-matrix input; no scores).
#> 
#> ── Score determinacy ───────────────────────────────────────────────────────────
#> 
#>      rho  rho2 guttman
#> F1 0.894 0.798   0.597
#> F2 0.888 0.788   0.576
#> F3 0.883 0.780   0.561
#> 
#> ── Factor weights ──────────────────────────────────────────────────────────────
#> 
#>        F1    F2     F3
#> V1  0.016 0.037  0.206
#> V2  0.023 0.036  0.146
#> V3  0.038 0.033  0.140
#> V4  0.060 0.025  0.194
#> V5  0.062 0.014  0.138
#> V6  0.009 0.013  0.247
#> V7  0.024 0.177  0.053
#> V8  0.017 0.187  0.031
#> V9  0.031 0.173  0.020
#> V10 0.016 0.222 -0.002
#> V11 0.026 0.115  0.084
#> V12 0.035 0.240  0.030
#> V13 0.202 0.051  0.010
#> V14 0.163 0.002  0.050
#> V15 0.177 0.059  0.007
#> V16 0.170 0.006  0.050
#> V17 0.214 0.013  0.016
#> V18 0.173 0.025  0.041
#> 
#> ── Score validity and univocality ──────────────────────────────────────────────
#> 
#> Diagonal: validity (score-factor correlation). Off-diagonal: univocality.
#> 
#>       F1    F2    F3
#> F1 0.894 0.638 0.668
#> F2 0.643 0.888 0.650
#> F3 0.676 0.653 0.883
#> 
#> ── Score intercorrelations ─────────────────────────────────────────────────────
#> 
#>       F1    F2    F3
#> F1 1.000 0.719 0.756
#> F2 0.719 1.000 0.735
#> F3 0.756 0.735 1.000

# Factor scores from raw data (Bartlett method)
# \donttest{
efa_raw <- efa_fit(GRiPS_raw, n_factors = 1, estimator = "PAF")
#> ℹ `x` is not a correlation matrix; computing correlations from the raw data.
efa_scores(GRiPS_raw, f = efa_raw, method = "Bartlett")
#> 
#> ── Factor scores (Bartlett) ────────────────────────────────────────────────────
#> 
#> Scored 810 observations on 1 factor.
#> 
#> ── Score determinacy ───────────────────────────────────────────────────────────
#> 
#>      rho  rho2 guttman
#> F1 0.972 0.946   0.891
# }

# Loadings supplied directly, with the factor intercorrelations
efa_scores(test_models$baseline$cormat, f = efa$rot_loadings, Phi = efa$Phi)
#> ℹ `x` is a correlation matrix; factor scores cannot be computed. Only factor
#>   weights and score diagnostics are returned. Enter raw data to get factor
#>   scores.
#> 
#> ── Factor scores (regression) ──────────────────────────────────────────────────
#> 
#> Weights and diagnostics only (correlation-matrix input; no scores).
#> 
#> ── Score determinacy ───────────────────────────────────────────────────────────
#> 
#>      rho  rho2 guttman
#> F1 0.894 0.798   0.597
#> F2 0.888 0.788   0.576
#> F3 0.883 0.780   0.561