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EFAtools

CRAN status R-CMD-check Codecov test coverage DOI

The EFAtools package provides functions to perform exploratory factor analysis (EFA) procedures and compare their solutions. The goal is to provide state-of-the-art factor retention methods and a high degree of flexibility in the EFA procedures. To speed up the analyses, some of the iterative procedures like principal axis factoring (PAF) are implemented in C++.

Installation

You can install the release version from CRAN with:

install.packages("EFAtools")

You can install the development version from GitHub with:

# install.packages("pak")
pak::pak("mdsteiner/EFAtools")

Package Overview

The efa_* functions cover the steps of an EFA workflow:

The uppercase names (EFA(), N_FACTORS(), …) are superseded by their efa_* equivalents, but remain exported and keep their arguments, so existing code keeps working.

The following vignettes and articles cover these in detail:

Examples

Here are a few examples of EFAtools functionalities

# load the package
library(EFAtools)

Factor Retention

Use efa_retain() to test the suitability of the data for factor analysis and to run multiple factor retention criteria with a single function call.

Works on raw-data:


# Run multiple factor retention methods
efa_retain(GRiPS_raw)
#> Warning: The suggested maximum number of factors was 2, but the Hull method needs at
#> least 3.
#> ℹ Setting it to 3.
#> ── Tests for the suitability of the data for factor analysis ───────────────────
#> 
#> ✔ The Bartlett's test of sphericity was significant at an alpha level of .05:
#>   χ²(28) = 5054.06, p < .001. These data are probably suitable for factor
#>   analysis.
#> ✔ The Kaiser-Meyer-Olkin criterion is marvellous (KMO = 0.955). These data are
#>   probably suitable for factor analysis.
#> 
#> ── Number of factors suggested by the factor retention criteria ────────────────
#> 
#> Comparison data
#> • Suggested number of factors: 1
#> 
#> Empirical Kaiser Criterion
#> • Original implementation (Braeken & van Assen, 2017): 1
#> 
#> Hull method
#> • CAF: 1
#> • CFI: 1
#> • RMSEA: 1
#> 
#> Minimum average partial
#> • Original implementation (TR2): 1
#> • Revised implementation (TR4): 1
#> 
#> Next Eigenvalue Sufficiency Test
#> • Suggested number of factors: 1
#> 
#> Parallel analysis
#> • SMC eigenvalues: 1

Skips some criteria on correlation input:


efa_retain(DOSPERT$cormat, N = DOSPERT$N)
#> Warning: `x` is a correlation matrix, but "CD" needs raw data.
#> ℹ Skipping "CD".
#> ── Tests for the suitability of the data for factor analysis ───────────────────
#> 
#> ✔ The Bartlett's test of sphericity was significant at an alpha level of .05:
#>   χ²(780) = 16071.13, p < .001. These data are probably suitable for factor
#>   analysis.
#> ✔ The Kaiser-Meyer-Olkin criterion is meritorious (KMO = 0.9). These data are
#>   probably suitable for factor analysis.
#> 
#> ── Number of factors suggested by the factor retention criteria ────────────────
#> 
#> Empirical Kaiser Criterion
#> • Original implementation (Braeken & van Assen, 2017): 10
#> 
#> Hull method
#> • CAF: 12
#> • CFI: 1
#> • RMSEA: 1
#> 
#> Minimum average partial
#> • Original implementation (TR2): 5
#> • Revised implementation (TR4): 6
#> 
#> Next Eigenvalue Sufficiency Test
#> • Suggested number of factors: 10
#> 
#> Parallel analysis
#> • SMC eigenvalues: 12
#> 
#> ── Criteria that could not be run ──────────────────────────────────────────────
#> 
#> ! CD: needs raw data, but a correlation matrix was supplied

EFA

Raw-Data Input

With raw-data input, you can use all functionalities, including sandwich and bootstrap standard errors and DWLS estimation with polychoric correlations or two-stage FIML estimation of correlations.

The bootstrap intervals below are percentile intervals over refitted resamples. For the loadings and factor correlations they are centred on the point estimate as you would expect; for the indices derived from the chi-square (RMSEA, AIC, BIC, ECVI) they sit above it, because each resample carries the sample’s own misfit plus fresh sampling noise. A point estimate lying below its own lower bound there is that shift, not a miscomputed interval. CFI and TLI are unaffected, being ratios in which the baseline chi-square shifts along with the model one.


# ULS / MINRES estimation with oblimin rotation and bootstrap SEs
mod <- efa_fit(DOSPERT_raw, n_factors = 5, estimator = "uls", rotation = "oblimin",
               se = "np-boot", seed = 1)
#> ℹ `x` is not a correlation matrix; computing correlations from the raw data.
mod
#> 
#> EFA performed with estimator = 'ULS' and rotation = 'oblimin'.
#> 
#> ── Rotated Loadings ────────────────────────────────────────────────────────────
#> 
#>           F1     F2     F3     F4     F5    h2    u2
#> ethR_1   .513  -.018   .030  -.016   .130  .309  .691
#> ethR_2   .518  -.044   .078   .019   .045  .304  .696
#> ethR_3   .639  -.001   .024  -.223   .081  .490  .510
#> ethR_4   .586  -.122  -.050  -.060   .046  .295  .705
#> ethR_5   .477   .065  -.010  -.127   .032  .267  .733
#> ethR_6   .621  -.098  -.007  -.016  -.021  .345  .655
#> finR_1  -.004  -.005   .840  -.021   .025  .717  .283
#> finR_2  -.066   .029  -.045   .068   .688  .476  .524
#> finR_3  -.005  -.013   .856   .010   .016  .730  .270
#> finR_4   .072   .041   .090  -.040   .710  .600  .400
#> finR_5  -.005  -.029   .873   .000   .040  .768  .232
#> finR_6   .054   .064   .093   .085   .683  .599  .401
#> heaR_1   .426   .087   .101   .087  -.036  .273  .727
#> heaR_2   .453   .053   .050   .136  -.050  .262  .738
#> heaR_3   .415   .130   .071   .019  -.052  .257  .743
#> heaR_4   .362   .163   .123  -.015  -.066  .254  .746
#> heaR_5   .382   .091  -.019   .123  -.057  .185  .815
#> heaR_6   .430   .206   .026   .138   .003  .338  .662
#> recR_1   .017   .407  -.035   .217   .026  .254  .746
#> recR_2   .117   .531   .111  -.101   .038  .410  .590
#> recR_3   .060   .619   .026   .003   .054  .452  .548
#> recR_4  -.072   .861  -.033  -.059   .027  .682  .318
#> recR_5  -.008   .805   .013  -.091  -.003  .628  .372
#> recR_6  -.020   .637   .031   .025   .102  .467  .533
#> socR_1  -.029  -.085  -.071   .646  -.004  .419  .581
#> socR_2   .093  -.027   .031   .679   .039  .474  .526
#> socR_3  -.133  -.058   .018   .640  -.005  .416  .584
#> socR_4  -.004   .018   .032   .614   .007  .383  .617
#> socR_5   .049   .103  -.045   .379   .051  .185  .815
#> socR_6   .004  -.008   .016   .549   .041  .308  .692
#> 
#> ── Factor Intercorrelations ────────────────────────────────────────────────────
#> 
#>       F1     F2     F3     F4     F5
#> F1  1.000
#> F2   .372  1.000
#> F3   .448   .319  1.000
#> F4   .006   .200  -.042  1.000
#> F5   .154   .290   .344   .145  1.000
#> 
#> ── Variances Accounted for ─────────────────────────────────────────────────────
#> 
#>                      F1     F2     F3     F4     F5
#> SS loadings        3.163  2.940  2.457  2.323  1.664
#> Prop Tot Var        .105   .098   .082   .077   .055
#> Cum Prop Tot Var    .105   .203   .285   .363   .418
#> Prop Comm Var       .252   .234   .196   .185   .133
#> Cum Prop Comm Var   .252   .486   .682   .867  1.000
#> 
#> ── Model Fit ───────────────────────────────────────────────────────────────────
#> 
#> χ²(295) = 3604.54, p < .001
#> CFI [95% bootstrap-CI]: .90 [.88, .90]
#> TLI [95% bootstrap-CI]: .85 [.82, .85]
#> RMSEA [90% CI] [95% bootstrap-CI]: .06 [.06; .06] [.06, .07]
#> AIC [95% bootstrap-CI]: 3014.54 [3034.98, 3700.48]
#> BIC [95% bootstrap-CI]: 1230.81 [1251.25, 1916.74]
#> ECVI [95% bootstrap-CI]: 1.26 [1.27, 1.48]
#> CAF [95% bootstrap-CI]: .43 [.43, .45]
#> SRMR [95% bootstrap-CI]: .03 [.03, .04]
#> 
#> Note: Bootstrap CIs based on 1000 bootstrap samples.
# detailed output with summary()
summary(mod)
#> 
#> EFA performed with estimator = 'ULS' and rotation = 'oblimin'.
#> 
#> ── Model Diagnostics ───────────────────────────────────────────────────────────
#> 
#> Factors: 5
#> Variables: 30
#> N: 3123
#> Bootstrap samples: 1000
#> Valid target-rotated samples: 1000 out of 1000
#> Rotation local optima: 1 distinct from 6 of 101 starts
#> Heywood cases: 0
#> Cross-loading items (|loading| >= .300): 0
#> Items without salient loading (|loading| >= .300): 0
#> Factors with fewer than 3 salient indicators: 0
#> Items with primary-loading gap < .200: 2
#> Largest |residual|: .246
#> Factor intercorrelations > .85: none
#> 
#> ── Rotated Loadings ────────────────────────────────────────────────────────────
#> 
#>           F1     F2     F3     F4     F5    h2    u2
#> ethR_1   .513  -.018   .030  -.016   .130  .309  .691
#> ethR_2   .518  -.044   .078   .019   .045  .304  .696
#> ethR_3   .639  -.001   .024  -.223   .081  .490  .510
#> ethR_4   .586  -.122  -.050  -.060   .046  .295  .705
#> ethR_5   .477   .065  -.010  -.127   .032  .267  .733
#> ethR_6   .621  -.098  -.007  -.016  -.021  .345  .655
#> finR_1  -.004  -.005   .840  -.021   .025  .717  .283
#> finR_2  -.066   .029  -.045   .068   .688  .476  .524
#> finR_3  -.005  -.013   .856   .010   .016  .730  .270
#> finR_4   .072   .041   .090  -.040   .710  .600  .400
#> finR_5  -.005  -.029   .873   .000   .040  .768  .232
#> finR_6   .054   .064   .093   .085   .683  .599  .401
#> heaR_1   .426   .087   .101   .087  -.036  .273  .727
#> heaR_2   .453   .053   .050   .136  -.050  .262  .738
#> heaR_3   .415   .130   .071   .019  -.052  .257  .743
#> heaR_4   .362   .163   .123  -.015  -.066  .254  .746
#> heaR_5   .382   .091  -.019   .123  -.057  .185  .815
#> heaR_6   .430   .206   .026   .138   .003  .338  .662
#> recR_1   .017   .407  -.035   .217   .026  .254  .746
#> recR_2   .117   .531   .111  -.101   .038  .410  .590
#> recR_3   .060   .619   .026   .003   .054  .452  .548
#> recR_4  -.072   .861  -.033  -.059   .027  .682  .318
#> recR_5  -.008   .805   .013  -.091  -.003  .628  .372
#> recR_6  -.020   .637   .031   .025   .102  .467  .533
#> socR_1  -.029  -.085  -.071   .646  -.004  .419  .581
#> socR_2   .093  -.027   .031   .679   .039  .474  .526
#> socR_3  -.133  -.058   .018   .640  -.005  .416  .584
#> socR_4  -.004   .018   .032   .614   .007  .383  .617
#> socR_5   .049   .103  -.045   .379   .051  .185  .815
#> socR_6   .004  -.008   .016   .549   .041  .308  .692
#> 
#> ── 95% bootstrap CIs for salient rotated loadings ──────────────────────────────
#> 
#> Variable  Factor  est    lower  upper
#> ethR_1    F1       .513   .465   .553
#> ethR_2    F1       .518   .472   .558
#> ethR_3    F1       .639   .589   .677
#> ethR_4    F1       .586   .532   .630
#> ethR_5    F1       .477   .431   .523
#> ethR_6    F1       .621   .572   .662
#> heaR_1    F1       .426   .381   .472
#> heaR_2    F1       .453   .407   .497
#> heaR_3    F1       .415   .361   .472
#> heaR_4    F1       .362   .306   .419
#> heaR_5    F1       .382   .326   .435
#> heaR_6    F1       .430   .382   .475
#> recR_1    F2       .407   .366   .447
#> recR_2    F2       .531   .487   .572
#> recR_3    F2       .619   .578   .654
#> recR_4    F2       .861   .825   .891
#> recR_5    F2       .805   .771   .835
#> recR_6    F2       .637   .600   .669
#> finR_1    F3       .840   .801   .868
#> finR_3    F3       .856   .819   .881
#> finR_5    F3       .873   .837   .899
#> socR_1    F4       .646   .612   .680
#> socR_2    F4       .679   .648   .708
#> socR_3    F4       .640   .602   .674
#> socR_4    F4       .614   .581   .648
#> socR_5    F4       .379   .339   .418
#> socR_6    F4       .549   .511   .587
#> finR_2    F5       .688   .646   .723
#> finR_4    F5       .710   .667   .740
#> finR_6    F5       .683   .642   .715
#> 
#> ── Factor Intercorrelations ────────────────────────────────────────────────────
#> 
#>       F1     F2     F3     F4     F5
#> F1  1.000
#> F2   .372  1.000
#> F3   .448   .319  1.000
#> F4   .006   .200  -.042  1.000
#> F5   .154   .290   .344   .145  1.000
#> 
#> ── 95% bootstrap CIs for factor intercorrelations ──────────────────────────────
#> 
#> Factors   est    lower  upper
#> F1 ~~ F2   .372   .327   .405
#> F1 ~~ F3   .448   .400   .481
#> F1 ~~ F4   .006  -.037   .049
#> F1 ~~ F5   .154   .108   .196
#> F2 ~~ F3   .319   .270   .359
#> F2 ~~ F4   .200   .160   .236
#> F2 ~~ F5   .290   .241   .326
#> F3 ~~ F4  -.042  -.082   .002
#> F3 ~~ F5   .344   .293   .373
#> F4 ~~ F5   .145   .098   .185
#> 
#> ── Structure Matrix ────────────────────────────────────────────────────────────
#> 
#>           F1    F2     F3     F4    F5
#> ethR_1   .540  .217   .299   .001  .212
#> ethR_2   .543  .190   .310   .017  .142
#> ethR_3   .661  .223   .348  -.209  .155
#> ethR_4   .525  .081   .192  -.072  .075
#> ethR_5   .500  .223   .240  -.106  .102
#> ethR_6   .578  .121   .233  -.035  .042
#> finR_1   .374  .265   .846  -.054  .309
#> finR_2   .031  .203   .168   .175  .681
#> finR_3   .376  .265   .854  -.027  .307
#> finR_4   .237  .294   .382   .068  .759
#> finR_5   .382  .260   .875  -.037  .331
#> finR_6   .225  .329   .369   .193  .754
#> heaR_1   .499  .284   .303   .097  .102
#> heaR_2   .488  .250   .247   .140  .072
#> heaR_3   .487  .296   .279   .037  .077
#> heaR_4   .467  .315   .315   .005  .077
#> heaR_5   .399  .235   .156   .136  .039
#> heaR_6   .519  .402   .279   .181  .158
#> recR_1   .158  .453   .103   .304  .166
#> recR_2   .370  .601   .350   .007  .234
#> recR_3   .310  .666   .269   .134  .252
#> recR_4   .237  .820   .221   .118  .246
#> recR_5   .297  .788   .270   .069  .221
#> recR_6   .247  .674   .260   .165  .298
#> socR_1  -.089  .010  -.140   .631  .036
#> socR_2   .107  .164   .049   .678  .155
#> socR_3  -.143  .025  -.089   .626  .057
#> socR_4   .022  .152   .013   .617  .111
#> socR_5   .078  .197   .012   .409  .128
#> socR_6   .018  .120   .006   .553  .124
#> 
#> ── Simple Structure Diagnostics ────────────────────────────────────────────────
#> 
#> Items with primary-loading gap < .200:
#> • heaR_4: F1 = .362, F2 = .163
#> • recR_1: F2 = .407, F4 = .217
#> 
#> 
#> ── Variances Accounted for ─────────────────────────────────────────────────────
#> 
#>                      F1     F2     F3     F4     F5
#> SS loadings        3.163  2.940  2.457  2.323  1.664
#> Prop Tot Var        .105   .098   .082   .077   .055
#> Cum Prop Tot Var    .105   .203   .285   .363   .418
#> Prop Comm Var       .252   .234   .196   .185   .133
#> Cum Prop Comm Var   .252   .486   .682   .867  1.000
#> 
#> ── Model Fit ───────────────────────────────────────────────────────────────────
#> 
#> χ²(295) = 3604.54, p < .001
#> CFI [95% bootstrap-CI]: .90 [.88, .90]
#> TLI [95% bootstrap-CI]: .85 [.82, .85]
#> RMSEA [90% CI] [95% bootstrap-CI]: .06 [.06; .06] [.06, .07]
#> AIC [95% bootstrap-CI]: 3014.54 [3034.98, 3700.48]
#> BIC [95% bootstrap-CI]: 1230.81 [1251.25, 1916.74]
#> ECVI [95% bootstrap-CI]: 1.26 [1.27, 1.48]
#> CAF [95% bootstrap-CI]: .43 [.43, .45]
#> SRMR [95% bootstrap-CI]: .03 [.03, .04]
#> 
#> Note: Bootstrap CIs based on 1000 bootstrap samples.
#> 
#> ── Residual Diagnostics ────────────────────────────────────────────────────────
#> 
#> Residual cutoff: |r| > .100
#> Number of large residuals: 7
#> Largest absolute residual: .246
#> 
#> Largest residuals:
#> • heaR_3 ~~ heaR_4: .246
#> • socR_5 ~~ socR_6: .190
#> • socR_2 ~~ socR_4: .145
#> • recR_4 ~~ recR_5: .138
#> • heaR_1 ~~ heaR_2: .135
#> • recR_2 ~~ recR_3: .126
#> • recR_1 ~~ recR_3: .112
#> 
#> Inspect the residual matrix for details (e.g., with residuals()).

#inspect residuals with residuals()
residuals(mod)
#>              ethR_1       ethR_2       ethR_3       ethR_4        ethR_5
#> ethR_1  0.000000000  0.001917900  0.040108892 -0.004369293  0.0398665165
#> ethR_2  0.001917900  0.000000000  0.051352421 -0.020431240 -0.0049432965
#> ethR_3  0.040108892  0.051352421  0.000000000  0.081671450  0.0288816866
#> ethR_4 -0.004369293 -0.020431240  0.081671450  0.000000000  0.0333134696
#> ethR_5  0.039866516 -0.004943296  0.028881687  0.033313470  0.0000000000
#> ethR_6  0.040582204  0.026283483  0.045451543  0.028148654  0.0187892739
#> finR_1  0.011127001 -0.006434037  0.014244992  0.009379816  0.0025756998
#> finR_2  0.007553570  0.003705828 -0.015346586  0.017745381  0.0176036738
#> finR_3  0.006233461  0.003308877  0.001315340  0.009312468  0.0034944277
#> finR_4 -0.016624811 -0.014663125 -0.017356120 -0.016698471 -0.0174307052
#> finR_5 -0.008253207  0.003043777 -0.004687718  0.004285814  0.0092380807
#> finR_6 -0.009286029 -0.003710525 -0.006965509 -0.013539656 -0.0301850113
#> heaR_1  0.004974977  0.008362143 -0.055877276  0.007132089  0.0010127739
#> heaR_2 -0.051382944  0.055166643 -0.070562208 -0.036727809 -0.0331374434
#> heaR_3 -0.009722443 -0.049225313 -0.048069933 -0.053699783 -0.0525275083
#> heaR_4 -0.033039266 -0.008960232 -0.015555165 -0.039907169 -0.0221866252
#> heaR_5 -0.032668820 -0.078643755 -0.046817302  0.042178950 -0.0501148566
#> heaR_6 -0.032441690 -0.012615309 -0.084461320 -0.064144574 -0.0056842306
#> recR_1  0.017147769 -0.020116010 -0.023082309 -0.018232276 -0.0215745628
#> recR_2 -0.007842344 -0.019216095  0.026180249 -0.018337407  0.0005967783
#> recR_3  0.021362240 -0.023522516  0.007777952  0.023306876 -0.0194801181
#> recR_4 -0.001484183  0.019319313  0.023063507  0.031611175  0.0222081232
#> recR_5 -0.005969560  0.014000848  0.020587445  0.031114695  0.0168276523
#> recR_6  0.020948069  0.027890355  0.015363854 -0.029059999  0.0169438153
#> socR_1  0.000806860 -0.012454706 -0.018900098  0.029253015  0.0035232684
#> socR_2  0.011740426  0.004707613  0.012154624 -0.002292166  0.0282698252
#> socR_3  0.022990996 -0.005414545  0.026565990  0.025314359 -0.0142255346
#> socR_4  0.022463180  0.006149984  0.042415994 -0.019458508  0.0157348216
#> socR_5 -0.015092819  0.014061796  0.002213433 -0.012377440  0.0357737850
#> socR_6 -0.014824816  0.030629268  0.024031868  0.039389116 -0.0112568151
#>              ethR_6        finR_1        finR_2        finR_3        finR_4
#> ethR_1  0.040582204  1.112700e-02  0.0075535698  6.233461e-03 -0.0166248106
#> ethR_2  0.026283483 -6.434037e-03  0.0037058276  3.308877e-03 -0.0146631253
#> ethR_3  0.045451543  1.424499e-02 -0.0153465862  1.315340e-03 -0.0173561198
#> ethR_4  0.028148654  9.379816e-03  0.0177453811  9.312468e-03 -0.0166984712
#> ethR_5  0.018789274  2.575700e-03  0.0176036738  3.494428e-03 -0.0174307052
#> ethR_6  0.000000000 -5.662428e-03 -0.0162175434 -1.021181e-03 -0.0043617571
#> finR_1 -0.005662428  0.000000e+00  0.0087004974  8.088778e-05 -0.0056161652
#> finR_2 -0.016217543  8.700497e-03  0.0000000000  2.498766e-03  0.0008717529
#> finR_3 -0.001021181  8.088778e-05  0.0024987657  0.000000e+00 -0.0069266835
#> finR_4 -0.004361757 -5.616165e-03  0.0008717529 -6.926684e-03  0.0000000000
#> finR_5  0.025056099  3.481751e-03  0.0004844848  3.513079e-03  0.0041242424
#> finR_6 -0.016470431 -1.355540e-02 -0.0082524593  2.963681e-03  0.0214543731
#> heaR_1 -0.018433909  1.792687e-02  0.0036649145  8.985145e-04  0.0137070190
#> heaR_2 -0.016470837 -2.387765e-02  0.0155659428  6.578785e-03  0.0077781862
#> heaR_3 -0.038071233 -1.429450e-02 -0.0123412149 -7.788520e-03  0.0336364625
#> heaR_4 -0.060952599 -3.754424e-03 -0.0399518832 -6.037476e-03  0.0222389062
#> heaR_5 -0.065972829 -6.846799e-03 -0.0034835004 -8.308590e-03  0.0246854909
#> heaR_6 -0.003199119 -7.804742e-03  0.0103726863 -5.704013e-03  0.0123496532
#> recR_1 -0.037810992  6.736902e-03  0.0258254704  5.928581e-03 -0.0112643878
#> recR_2 -0.032039516  1.226022e-02  0.0006634646 -1.310808e-02  0.0020501320
#> recR_3 -0.016986859 -3.360832e-03  0.0081716557 -1.803331e-03 -0.0009082108
#> recR_4  0.041721475 -9.301009e-03 -0.0062789188  9.980389e-03  0.0062924215
#> recR_5  0.035964591  4.251960e-04 -0.0071195012  9.512968e-03  0.0011034225
#> recR_6  0.010031264  6.759023e-03  0.0046234751 -3.828882e-03 -0.0324267549
#> socR_1  0.016903089  2.591499e-03  0.0336329043  7.609381e-03  0.0021701495
#> socR_2 -0.001469726  1.023220e-03  0.0135078560  6.453198e-03 -0.0085522565
#> socR_3  0.012457428 -3.306203e-03 -0.0127767504  9.893865e-03 -0.0147403651
#> socR_4 -0.001954184 -7.945136e-03 -0.0248892504 -1.207073e-02  0.0125020724
#> socR_5  0.043324097  2.112184e-02  0.0038245550 -1.807369e-02 -0.0084601186
#> socR_6  0.011257520  7.598876e-03 -0.0343169613 -1.676189e-03 -0.0055715620
#>               finR_5       finR_6        heaR_1       heaR_2       heaR_3
#> ethR_1 -0.0082532069 -0.009286029  0.0049749766 -0.051382944 -0.009722443
#> ethR_2  0.0030437766 -0.003710525  0.0083621430  0.055166643 -0.049225313
#> ethR_3 -0.0046877178 -0.006965509 -0.0558772764 -0.070562208 -0.048069933
#> ethR_4  0.0042858140 -0.013539656  0.0071320887 -0.036727809 -0.053699783
#> ethR_5  0.0092380807 -0.030185011  0.0010127739 -0.033137443 -0.052527508
#> ethR_6  0.0250560991 -0.016470431 -0.0184339094 -0.016470837 -0.038071233
#> finR_1  0.0034817509 -0.013555401  0.0179268693 -0.023877654 -0.014294500
#> finR_2  0.0004844848 -0.008252459  0.0036649145  0.015565943 -0.012341215
#> finR_3  0.0035130791  0.002963681  0.0008985145  0.006578785 -0.007788520
#> finR_4  0.0041242424  0.021454373  0.0137070190  0.007778186  0.033636462
#> finR_5  0.0000000000 -0.004350617 -0.0025399943  0.008688334 -0.019196171
#> finR_6 -0.0043506174  0.000000000 -0.0086512606  0.010828029  0.032742317
#> heaR_1 -0.0025399943 -0.008651261  0.0000000000  0.134744470 -0.037917070
#> heaR_2  0.0086883342  0.010828029  0.1347444696  0.000000000  0.011495127
#> heaR_3 -0.0191961710  0.032742317 -0.0379170701  0.011495127  0.000000000
#> heaR_4 -0.0181459685  0.042817995 -0.0879544260 -0.039913629  0.246048416
#> heaR_5  0.0024835494  0.017695385  0.0069554472  0.053177695  0.097486673
#> heaR_6 -0.0048411017  0.019469864  0.0400415933  0.048970234  0.042177257
#> recR_1 -0.0133250238 -0.010110995  0.0385149082  0.042181093 -0.028311370
#> recR_2 -0.0099501705 -0.006382334  0.0042557158 -0.019267980  0.014913165
#> recR_3 -0.0019269895 -0.019009983  0.0077593475 -0.004347081 -0.016209104
#> recR_4  0.0175468561 -0.011312022  0.0020271649 -0.007770894 -0.032917255
#> recR_5  0.0074957437 -0.001280454  0.0126984498 -0.010735757 -0.019892255
#> recR_6 -0.0004331486  0.016610161 -0.0481764605 -0.020457964 -0.015608539
#> socR_1  0.0077225903 -0.026656401  0.0290091620  0.018385833 -0.016464714
#> socR_2 -0.0080895488 -0.008773788 -0.0175637498 -0.010818655 -0.007281367
#> socR_3  0.0009902625  0.013650169 -0.0130447670 -0.033653184 -0.007074495
#> socR_4  0.0071285248 -0.004181445 -0.0559959050 -0.022965887  0.002606382
#> socR_5 -0.0009945018 -0.010899600  0.0065224503 -0.018230312 -0.021953959
#> socR_6  0.0002060683  0.019249072 -0.0149986602 -0.047981027 -0.004447033
#>              heaR_4       heaR_5        heaR_6       recR_1        recR_2
#> ethR_1 -0.033039266 -0.032668820 -3.244169e-02  0.017147769 -0.0078423438
#> ethR_2 -0.008960232 -0.078643755 -1.261531e-02 -0.020116010 -0.0192160948
#> ethR_3 -0.015555165 -0.046817302 -8.446132e-02 -0.023082309  0.0261802488
#> ethR_4 -0.039907169  0.042178950 -6.414457e-02 -0.018232276 -0.0183374073
#> ethR_5 -0.022186625 -0.050114857 -5.684231e-03 -0.021574563  0.0005967783
#> ethR_6 -0.060952599 -0.065972829 -3.199119e-03 -0.037810992 -0.0320395160
#> finR_1 -0.003754424 -0.006846799 -7.804742e-03  0.006736902  0.0122602201
#> finR_2 -0.039951883 -0.003483500  1.037269e-02  0.025825470  0.0006634646
#> finR_3 -0.006037476 -0.008308590 -5.704013e-03  0.005928581 -0.0131080830
#> finR_4  0.022238906  0.024685491  1.234965e-02 -0.011264388  0.0020501320
#> finR_5 -0.018145969  0.002483549 -4.841102e-03 -0.013325024 -0.0099501705
#> finR_6  0.042817995  0.017695385  1.946986e-02 -0.010110995 -0.0063823343
#> heaR_1 -0.087954426  0.006955447  4.004159e-02  0.038514908  0.0042557158
#> heaR_2 -0.039913629  0.053177695  4.897023e-02  0.042181093 -0.0192679799
#> heaR_3  0.246048416  0.097486673  4.217726e-02 -0.028311370  0.0149131647
#> heaR_4  0.000000000  0.055890567  5.212824e-02 -0.017062157  0.0235172951
#> heaR_5  0.055890567  0.000000000  8.871116e-02  0.064718883  0.0069543311
#> heaR_6  0.052128236  0.088711156  1.110223e-16  0.027034056  0.0055630715
#> recR_1 -0.017062157  0.064718883  2.703406e-02  0.000000000  0.0421381687
#> recR_2  0.023517295  0.006954331  5.563072e-03  0.042138169  0.0000000000
#> recR_3 -0.007880619 -0.013439695 -4.224870e-03  0.111561108  0.1261749511
#> recR_4 -0.049565941 -0.032687726 -3.943762e-02 -0.055618813 -0.0780319200
#> recR_5 -0.034115104 -0.030578343 -3.527430e-02 -0.068601722 -0.0419724053
#> recR_6  0.025719839 -0.016720314  1.185086e-02  0.001496346 -0.0186880385
#> socR_1 -0.033283387  0.022437892 -2.180810e-02  0.044539945 -0.0412132854
#> socR_2 -0.001017746 -0.031734973  8.396766e-03 -0.039516034 -0.0069896378
#> socR_3 -0.010388111 -0.003901592 -2.037909e-02  0.006296804  0.0025600487
#> socR_4  0.021094419 -0.028690233 -3.687410e-02 -0.047504630  0.0200855865
#> socR_5 -0.021543520 -0.046676390  1.558664e-02 -0.010229610  0.0044973282
#> socR_6  0.014141631 -0.015473321 -3.594436e-02 -0.016774432  0.0080663135
#>               recR_3       recR_4       recR_5        recR_6       socR_1
#> ethR_1  0.0213622398 -0.001484183 -0.005969560  0.0209480687  0.000806860
#> ethR_2 -0.0235225161  0.019319313  0.014000848  0.0278903547 -0.012454706
#> ethR_3  0.0077779521  0.023063507  0.020587445  0.0153638536 -0.018900098
#> ethR_4  0.0233068757  0.031611175  0.031114695 -0.0290599989  0.029253015
#> ethR_5 -0.0194801181  0.022208123  0.016827652  0.0169438153  0.003523268
#> ethR_6 -0.0169868588  0.041721475  0.035964591  0.0100312645  0.016903089
#> finR_1 -0.0033608324 -0.009301009  0.000425196  0.0067590234  0.002591499
#> finR_2  0.0081716557 -0.006278919 -0.007119501  0.0046234751  0.033632904
#> finR_3 -0.0018033305  0.009980389  0.009512968 -0.0038288824  0.007609381
#> finR_4 -0.0009082108  0.006292421  0.001103423 -0.0324267549  0.002170150
#> finR_5 -0.0019269895  0.017546856  0.007495744 -0.0004331486  0.007722590
#> finR_6 -0.0190099832 -0.011312022 -0.001280454  0.0166101608 -0.026656401
#> heaR_1  0.0077593475  0.002027165  0.012698450 -0.0481764605  0.029009162
#> heaR_2 -0.0043470811 -0.007770894 -0.010735757 -0.0204579636  0.018385833
#> heaR_3 -0.0162091043 -0.032917255 -0.019892255 -0.0156085389 -0.016464714
#> heaR_4 -0.0078806194 -0.049565941 -0.034115104  0.0257198393 -0.033283387
#> heaR_5 -0.0134396946 -0.032687726 -0.030578343 -0.0167203136  0.022437892
#> heaR_6 -0.0042248703 -0.039437617 -0.035274302  0.0118508646 -0.021808098
#> recR_1  0.1115611075 -0.055618813 -0.068601722  0.0014963460  0.044539945
#> recR_2  0.1261749511 -0.078031920 -0.041972405 -0.0186880385 -0.041213285
#> recR_3  0.0000000000 -0.053595869 -0.055836248 -0.0213043578 -0.010111162
#> recR_4 -0.0535958689  0.000000000  0.137872527  0.0330606581  0.029515654
#> recR_5 -0.0558362477  0.137872527  0.000000000 -0.0109558586  0.023221232
#> recR_6 -0.0213043578  0.033060658 -0.010955859  0.0000000000 -0.026975186
#> socR_1 -0.0101111620  0.029515654  0.023221232 -0.0269751862  0.000000000
#> socR_2 -0.0123831450  0.012569973  0.014029092  0.0146278708  0.020978578
#> socR_3  0.0060424204  0.007252597  0.008578265 -0.0056401178  0.016892154
#> socR_4  0.0079666116  0.009953444  0.013416691  0.0030085594 -0.007201901
#> socR_5 -0.0098534841  0.005302550  0.005317443  0.0096070343 -0.035147268
#> socR_6 -0.0087257570 -0.004799737 -0.005134521  0.0245081413 -0.041987581
#>              socR_2        socR_3       socR_4        socR_5        socR_6
#> ethR_1  0.011740426  0.0229909965  0.022463180 -0.0150928195 -0.0148248156
#> ethR_2  0.004707613 -0.0054145448  0.006149984  0.0140617960  0.0306292675
#> ethR_3  0.012154624  0.0265659898  0.042415994  0.0022134330  0.0240318676
#> ethR_4 -0.002292166  0.0253143591 -0.019458508 -0.0123774402  0.0393891165
#> ethR_5  0.028269825 -0.0142255346  0.015734822  0.0357737850 -0.0112568151
#> ethR_6 -0.001469726  0.0124574284 -0.001954184  0.0433240972  0.0112575195
#> finR_1  0.001023220 -0.0033062026 -0.007945136  0.0211218369  0.0075988760
#> finR_2  0.013507856 -0.0127767504 -0.024889250  0.0038245550 -0.0343169613
#> finR_3  0.006453198  0.0098938651 -0.012070727 -0.0180736894 -0.0016761891
#> finR_4 -0.008552257 -0.0147403651  0.012502072 -0.0084601186 -0.0055715620
#> finR_5 -0.008089549  0.0009902625  0.007128525 -0.0009945018  0.0002060683
#> finR_6 -0.008773788  0.0136501691 -0.004181445 -0.0108995999  0.0192490722
#> heaR_1 -0.017563750 -0.0130447670 -0.055995905  0.0065224503 -0.0149986602
#> heaR_2 -0.010818655 -0.0336531842 -0.022965887 -0.0182303123 -0.0479810272
#> heaR_3 -0.007281367 -0.0070744948  0.002606382 -0.0219539593 -0.0044470326
#> heaR_4 -0.001017746 -0.0103881110  0.021094419 -0.0215435205  0.0141416308
#> heaR_5 -0.031734973 -0.0039015921 -0.028690233 -0.0466763899 -0.0154733205
#> heaR_6  0.008396766 -0.0203790939 -0.036874099  0.0155866360 -0.0359443552
#> recR_1 -0.039516034  0.0062968042 -0.047504630 -0.0102296095 -0.0167744324
#> recR_2 -0.006989638  0.0025600487  0.020085587  0.0044973282  0.0080663135
#> recR_3 -0.012383145  0.0060424204  0.007966612 -0.0098534841 -0.0087257570
#> recR_4  0.012569973  0.0072525969  0.009953444  0.0053025504 -0.0047997375
#> recR_5  0.014029092  0.0085782650  0.013416691  0.0053174428 -0.0051345215
#> recR_6  0.014627871 -0.0056401178  0.003008559  0.0096070343  0.0245081413
#> socR_1  0.020978578  0.0168921541 -0.007201901 -0.0351472685 -0.0419875805
#> socR_2  0.000000000 -0.0474000959  0.145138599 -0.0500357905 -0.0589002183
#> socR_3 -0.047400096  0.0000000000 -0.007743223 -0.0054711753  0.0775383760
#> socR_4  0.145138599 -0.0077432234  0.000000000 -0.0483059742 -0.0487076115
#> socR_5 -0.050035791 -0.0054711753 -0.048305974  0.0000000000  0.1897359371
#> socR_6 -0.058900218  0.0775383760 -0.048707612  0.1897359371  0.0000000000

# DWLS estimation based on polychoric correlations, with robust sandwich SEs
mod <- efa_fit(GRiPS_raw, n_factors = 1, estimator = "dwls", cor_method = "poly",
               se = "sandwich")
#> ℹ `x` is not a correlation matrix; computing correlations from the raw data.
#> Warning: Some response-category combinations are empty despite a non-negligible expected
#> count.
#> ℹ The polychoric asymptotic covariance (and any DWLS weights or robust standard
#>   errors derived from it) can be unreliable for such structurally sparse cells;
#>   interpret them with caution.
mod
#> 
#> EFA performed with estimator = 'DWLS' and rotation = 'none'.
#> 
#> ── Unrotated Loadings ──────────────────────────────────────────────────────────
#> 
#>             F1    h2    u2
#> fun        .818  .669  .331
#> friends    .855  .731  .269
#> enjoy      .893  .797  .203
#> hurt       .775  .601  .399
#> part       .824  .679  .321
#> commonly   .843  .711  .289
#> chances    .817  .668  .332
#> attracted  .859  .738  .262
#> 
#> ── Variances Accounted for ─────────────────────────────────────────────────────
#> 
#>                 F1
#> SS loadings   5.594
#> Prop Tot Var   .699
#> 
#> ── Model Fit ───────────────────────────────────────────────────────────────────
#> 
#> scaled χ²(20) = 237.21, p < .001
#> CFI: .99
#> TLI: .99
#> RMSEA [90% CI]: .12 [.10; .13]
#> AIC: NA
#> BIC: NA
#> CAF: .49
#> SRMR: .02
summary(mod)
#> 
#> EFA performed with estimator = 'DWLS' and rotation = 'none'.
#> 
#> ── Model Diagnostics ───────────────────────────────────────────────────────────
#> 
#> Factors: 1
#> Variables: 8
#> N: 810
#> Heywood cases: 0
#> Cross-loading items (|loading| >= .300): 0
#> Items without salient loading (|loading| >= .300): 0
#> Factors with fewer than 3 salient indicators: 0
#> Items with primary-loading gap < .200: 0
#> Largest |residual|: .038
#> 
#> ── Unrotated Loadings ──────────────────────────────────────────────────────────
#> 
#>             F1    h2    u2
#> fun        .818  .669  .331
#> friends    .855  .731  .269
#> enjoy      .893  .797  .203
#> hurt       .775  .601  .399
#> part       .824  .679  .321
#> commonly   .843  .711  .289
#> chances    .817  .668  .332
#> attracted  .859  .738  .262
#> 
#> ── 95% Wald CIs for salient unrotated loadings ─────────────────────────────────
#> 
#> Variable   Factor  est    lower  upper
#> fun        F1       .818   .798   .838
#> friends    F1       .855   .835   .875
#> enjoy      F1       .893   .875   .910
#> hurt       F1       .775   .750   .800
#> part       F1       .824   .802   .846
#> commonly   F1       .843   .825   .861
#> chances    F1       .817   .795   .840
#> attracted  F1       .859   .842   .877
#> 
#> ── Variances Accounted for ─────────────────────────────────────────────────────
#> 
#>                 F1
#> SS loadings   5.594
#> Prop Tot Var   .699
#> 
#> ── Model Fit ───────────────────────────────────────────────────────────────────
#> 
#> scaled χ²(20) = 237.21, p < .001
#> CFI: .99
#> TLI: .99
#> RMSEA [90% CI]: .12 [.10; .13]
#> AIC: NA
#> BIC: NA
#> CAF: .49
#> SRMR: .02
#> 
#> Note: Wald CIs from the robust (Godambe) sandwich covariance.
#> 
#> ── Residual Diagnostics ────────────────────────────────────────────────────────
#> 
#> Residual cutoff: |r| > .100
#> Number of large residuals: 0
#> Largest absolute residual: .038
#> 
#> No absolute residuals > .100 occurred.
#> 
#> Inspect the residual matrix for details (e.g., with residuals()).

Correlation Input

When you don’t have raw data available, you can enter a correlation matrix and sample size. With ML-estimation, you can still get information SEs, but note that they assume multivariate normality.


# ML estimation with oblimin rotation and information SEs, based on correlation
# matrix and N
mod <- efa_fit(test_models$baseline$cormat, N = 500,  n_factors = 3, estimator = "ml",
           rotation = "oblimin", se = "information")
mod
#> 
#> EFA performed with estimator = 'ML' and rotation = 'oblimin'.
#> 
#> ── Rotated Loadings ────────────────────────────────────────────────────────────
#> 
#>        F1     F2     F3    h2    u2
#> V1   -.036   .043   .607  .373  .627
#> V2    .013   .087   .458  .274  .726
#> V3    .074   .074   .430  .280  .720
#> V4    .111   .007   .536  .379  .621
#> V5    .164   .005   .418  .290  .710
#> V6   -.055  -.036   .687  .402  .598
#> V7    .017   .524   .095  .355  .645
#> V8   -.003   .562   .044  .345  .655
#> V9    .044   .535   .017  .328  .672
#> V10  -.019   .661  -.051  .385  .615
#> V11   .030   .352   .230  .296  .704
#> V12   .034   .649  -.015  .437  .563
#> V13   .612   .095  -.068  .397  .603
#> V14   .540  -.053   .086  .320  .680
#> V15   .552   .137  -.065  .363  .637
#> V16   .550  -.039   .092  .345  .655
#> V17   .652  -.035  -.013  .390  .610
#> V18   .549   .012   .052  .349  .651
#> 
#> ── Factor Intercorrelations ────────────────────────────────────────────────────
#> 
#>       F1     F2     F3
#> F1  1.000
#> F2   .591  1.000
#> F3   .621   .596  1.000
#> 
#> ── Variances Accounted for ─────────────────────────────────────────────────────
#> 
#>                      F1     F2     F3
#> SS loadings        2.225  2.088  1.994
#> Prop Tot Var        .124   .116   .111
#> Cum Prop Tot Var    .124   .240   .350
#> Prop Comm Var       .353   .331   .316
#> Cum Prop Comm Var   .353   .684  1.000
#> 
#> ── Model Fit ───────────────────────────────────────────────────────────────────
#> 
#> χ²(102) = 123.75, p = .070
#> CFI: .99
#> TLI: .98
#> RMSEA [90% CI]: .02 [.00; .03]
#> AIC: -80.25
#> BIC: -510.14
#> ECVI: 0.52
#> CAF: .50
#> SRMR: .03
summary(mod)
#> 
#> EFA performed with estimator = 'ML' and rotation = 'oblimin'.
#> 
#> ── Model Diagnostics ───────────────────────────────────────────────────────────
#> 
#> Factors: 3
#> Variables: 18
#> N: 500
#> Rotation local optima: 1 distinct from 6 of 101 starts
#> Heywood cases: 0
#> Cross-loading items (|loading| >= .300): 0
#> Items without salient loading (|loading| >= .300): 0
#> Factors with fewer than 3 salient indicators: 0
#> Items with primary-loading gap < .200: 1
#> Largest |residual|: .069
#> Factor intercorrelations > .85: none
#> 
#> ── Rotated Loadings ────────────────────────────────────────────────────────────
#> 
#>        F1     F2     F3    h2    u2
#> V1   -.036   .043   .607  .373  .627
#> V2    .013   .087   .458  .274  .726
#> V3    .074   .074   .430  .280  .720
#> V4    .111   .007   .536  .379  .621
#> V5    .164   .005   .418  .290  .710
#> V6   -.055  -.036   .687  .402  .598
#> V7    .017   .524   .095  .355  .645
#> V8   -.003   .562   .044  .345  .655
#> V9    .044   .535   .017  .328  .672
#> V10  -.019   .661  -.051  .385  .615
#> V11   .030   .352   .230  .296  .704
#> V12   .034   .649  -.015  .437  .563
#> V13   .612   .095  -.068  .397  .603
#> V14   .540  -.053   .086  .320  .680
#> V15   .552   .137  -.065  .363  .637
#> V16   .550  -.039   .092  .345  .655
#> V17   .652  -.035  -.013  .390  .610
#> V18   .549   .012   .052  .349  .651
#> 
#> ── 95% Wald CIs for salient rotated loadings ───────────────────────────────────
#> 
#> Variable  Factor  est    lower  upper
#> V13       F1       .612   .488   .736
#> V14       F1       .540   .411   .669
#> V15       F1       .552   .423   .681
#> V16       F1       .550   .421   .679
#> V17       F1       .652   .535   .769
#> V18       F1       .549   .420   .679
#> V7        F2       .524   .394   .653
#> V8        F2       .562   .437   .687
#> V9        F2       .535   .407   .662
#> V10       F2       .661   .548   .773
#> V11       F2       .352   .211   .493
#> V12       F2       .649   .529   .770
#> V1        F3       .607   .474   .739
#> V2        F3       .458   .313   .604
#> V3        F3       .430   .282   .578
#> V4        F3       .536   .395   .677
#> V5        F3       .418   .269   .567
#> V6        F3       .687   .570   .805
#> 
#> ── Factor Intercorrelations ────────────────────────────────────────────────────
#> 
#>       F1     F2     F3
#> F1  1.000
#> F2   .591  1.000
#> F3   .621   .596  1.000
#> 
#> ── 95% Wald CIs for factor intercorrelations ───────────────────────────────────
#> 
#> Factors   est    lower  upper
#> F1 ~~ F2   .591   .499   .683
#> F1 ~~ F3   .621   .531   .712
#> F2 ~~ F3   .596   .503   .690
#> 
#> ── Structure Matrix ────────────────────────────────────────────────────────────
#> 
#>       F1    F2    F3
#> V1   .366  .384  .610
#> V2   .349  .367  .518
#> V3   .385  .374  .520
#> V4   .448  .392  .609
#> V5   .427  .351  .523
#> V6   .350  .341  .631
#> V7   .385  .590  .418
#> V8   .357  .587  .377
#> V9   .371  .571  .363
#> V10  .339  .619  .331
#> V11  .381  .507  .459
#> V12  .409  .661  .393
#> V13  .626  .416  .369
#> V14  .562  .317  .390
#> V15  .593  .425  .360
#> V16  .584  .341  .410
#> V17  .623  .343  .371
#> V18  .589  .368  .401
#> 
#> ── Simple Structure Diagnostics ────────────────────────────────────────────────
#> 
#> Items with primary-loading gap < .200:
#> • V11: F2 = .352, F3 = .230
#> 
#> 
#> ── Variances Accounted for ─────────────────────────────────────────────────────
#> 
#>                      F1     F2     F3
#> SS loadings        2.225  2.088  1.994
#> Prop Tot Var        .124   .116   .111
#> Cum Prop Tot Var    .124   .240   .350
#> Prop Comm Var       .353   .331   .316
#> Cum Prop Comm Var   .353   .684  1.000
#> 
#> ── Model Fit ───────────────────────────────────────────────────────────────────
#> 
#> χ²(102) = 123.75, p = .070
#> CFI: .99
#> TLI: .98
#> RMSEA [90% CI]: .02 [.00; .03]
#> AIC: -80.25
#> BIC: -510.14
#> ECVI: 0.52
#> CAF: .50
#> SRMR: .03
#> 
#> ── Residual Diagnostics ────────────────────────────────────────────────────────
#> 
#> Residual cutoff: |r| > .100
#> Number of large residuals: 0
#> Largest absolute residual: .069
#> 
#> No absolute residuals > .100 occurred.
#> 
#> Inspect the residual matrix for details (e.g., with residuals()).

Citation

If you use this package in your research, please acknowledge it by citing:

Steiner, M.D., & Grieder, S.G. (2020). EFAtools: An R package with fast and flexible implementations of exploratory factor analysis tools. Journal of Open Source Software, 5(53), 2521. https://doi.org/10.21105/joss.02521

Contribute or Report Bugs

If you want to contribute or report bugs, please open an issue on GitHub or email us at markus.d.steiner@gmail.com or silvia.steiner.grieder@gmail.com.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
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