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

Computes factor-score weights, and (from raw data) the factor scores themselves, for an efa_fit() solution or a directly supplied loading matrix. It also returns score-quality diagnostics: 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 f is a directly supplied loading matrix, a correlation-matrix x also supplies the correlations the weights are derived from; when f is an efa_fit() object, its own fitted correlations are used instead (supply rho to derive the weights from another matrix). x describes the model variables either way. 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). A named correlation matrix is likewise matched to the loading rows by name; its row and column names must use the same order, and it must carry one row and column for each model variable. Unnamed input is matched by position.

Raw data are scored as supplied: no imputation is performed, so a case with a missing value on any model variable receives NA scores (and is reported as not scored). A model variable that carries no usable spread in x – constant, infinite, or observed fewer than twice – is an error.

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, in which case a supplied Phi is ignored with a warning. Named rows and columns are matched to the loading columns and must use the same order. 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; for a directly supplied loading matrix, x itself when it is a correlation matrix, and cor(x, use = "pairwise") otherwise. Pass a matrix here to score against a correlation other than the one implied by f/x. Named rows and columns are matched to the loading rows; row and column names must use the same order.

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. A case with a missing value on any model variable is not scored and keeps NA in every column.
r.scores: The m by m correlations of the factor-score estimates (see Details for the "components"-method scale caveat).
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, including the number of supplied observations n_obs and the number of them that could be scored n_scored.

Details

Each method combines the loadings with some or all of the factor correlations and the scoring correlation matrix into weights in a different way:

"regression": Thurstone's (1935) regression scores.
"Bartlett": Bartlett's (1937) scores.
"Anderson": Anderson & Rubin's (1956) scores.
"tenBerge": ten Berge, Krijnen, Wansbeek & Shapiro's (1999) scores.
"Harman": Harman's (1976) scores, based on an idealized variable (a hypothetical variable that would correlate perfectly with the factor).
"components": Component scores. These are formed from the raw, uncentered data (X %*% W) rather than the standardized data, so unlike the other methods they are on the scale of the input variables. The diagnostics below describe the standardized combination scale(X) %*% W, and therefore differ from the realized correlations of the returned scores whenever the variables have unequal variances.

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). Guttman's (1955) indeterminacy index, 2 rho^2 - 1, is 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.

Determinacies close to 1 mean the scores stand in for the factor with little loss; Grice (2001) regards values of about .90 and above as the level required before scores are interpreted for individual cases, and treats lower values as usable only for group-level research. The Guttman index makes the same point more sharply, because a factor score is never the factor: at rho = .90 two equally valid sets of scores can still correlate as low as .62, and at rho = .80 as low as .28, so the rank order of cases is not unique.

Which method to prefer follows from what the scores are for. Regression scores correlate most highly with the factor, but they are biased towards it and correlate across factors even when the model is orthogonal. Bartlett scores are conditionally unbiased, which makes them the choice when the scores stand in for the factor in a later model. "tenBerge" reproduces the factor intercorrelations Phi, so it is the choice when the scores will be correlated with each other or with external variables. "Anderson" forces the scores to be uncorrelated with unit variance and is appropriate only when the factors themselves are orthogonal. "components" is a weighted sum of the observed variables rather than an estimate of a common factor.

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  .894  .798     .597
#> F2  .888  .788     .576
#> F3  .883  .780     .561
summary(fs)
#> 
#> ── Factor scores (regression) ──────────────────────────────────────────────────
#> 
#> Weights and diagnostics only (correlation-matrix input; no scores).
#> 
#> ── Score determinacy ───────────────────────────────────────────────────────────
#> 
#>      rho  rho2  guttman
#> F1  .894  .798     .597
#> F2  .888  .788     .576
#> F3  .883  .780     .561
#> 
#> ── Factor weights ──────────────────────────────────────────────────────────────
#> 
#>       F1    F2     F3
#> V1   .016  .037   .206
#> V2   .023  .036   .146
#> V3   .038  .033   .140
#> V4   .060  .025   .194
#> V5   .062  .014   .138
#> V6   .009  .013   .247
#> V7   .024  .177   .053
#> V8   .017  .187   .031
#> V9   .031  .173   .020
#> V10  .016  .222  -.002
#> V11  .026  .115   .084
#> V12  .035  .240   .030
#> V13  .202  .051   .010
#> V14  .163  .002   .050
#> V15  .177  .059   .007
#> V16  .170  .006   .050
#> V17  .214  .013   .016
#> V18  .173  .025   .041
#> 
#> ── Score validity and univocality ──────────────────────────────────────────────
#> 
#> Diagonal: validity (score-factor correlation). Off-diagonal: univocality.
#> 
#>      F1    F2    F3
#> F1  .894  .638  .668
#> F2  .643  .888  .650
#> F3  .676  .653  .883
#> 
#> ── Score intercorrelations ─────────────────────────────────────────────────────
#> 
#>       F1     F2     F3
#> F1  1.000   .719   .756
#> F2   .719  1.000   .735
#> F3   .756   .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 of 810 observations on 1 factor (see `$scores`).
#> 
#> ── Score determinacy ───────────────────────────────────────────────────────────
#> 
#>      rho  rho2  guttman
#> F1  .972  .946     .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  .894  .798     .597
#> F2  .888  .788     .576
#> F3  .883  .780     .561