EFAtools
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")Examples
Here are a few examples of EFAtools functionalities
# load the package
library(EFAtools)Factor Retention
Use N_FACTORS() 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
N_FACTORS(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: 1Skips some criteria on correlation input:
N_FACTORS(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): 4
#>
#> 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 suppliedEFA
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.
# ULS / MINRES estimation with oblimin rotation and sandwich SEs
mod <- EFA(DOSPERT_raw, n_factors = 5, method = "ULS", rotation = "oblimin",
se = "sandwich")
#> ℹ `x` is not a correlation matrix; computing correlations from the raw data.
mod
#>
#> EFA performed with type = 'EFAtools', method = '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
#>
#> Legend:
#> bold = |loading| >= .300
#> grey = below cutoff
#> red h2/u2 = Heywood-relevant value
#>
#> ── 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 ───────────────────────────────────────────────────────────────────
#>
#> scaled χ²(295) = 2912.75, p < .001
#> CFI: .90
#> TLI: .86
#> RMSEA [90% CI]: .05 [.05; .06]
#> AIC: NA
#> BIC: NA
#> CAF: .43
#> RMSR: .03
#> SRMR: .03
# detailed output with summary()
summary(mod)
#>
#> EFA performed with type = 'EFAtools', method = 'ULS', and rotation = 'oblimin'.
#>
#> ── Model Diagnostics ───────────────────────────────────────────────────────────
#>
#> Factors: 5
#> Variables: 30
#> N: 3123
#> Rotation local optima: 1 distinct from 100 random 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
#>
#> Legend:
#> bold = |loading| >= .300
#> grey = below cutoff
#> red h2/u2 = Heywood-relevant value
#>
#> ── 95% Wald CIs for salient rotated loadings ───────────────────────────────────
#>
#> Variable Factor est lower upper
#> ethR_1 F1 .513 .469 .557
#> ethR_2 F1 .518 .473 .562
#> ethR_3 F1 .639 .600 .679
#> ethR_4 F1 .586 .541 .630
#> ethR_5 F1 .477 .431 .522
#> ethR_6 F1 .621 .581 .661
#> heaR_1 F1 .426 .382 .471
#> heaR_2 F1 .453 .409 .497
#> heaR_3 F1 .415 .366 .463
#> heaR_4 F1 .362 .313 .411
#> heaR_5 F1 .382 .335 .428
#> heaR_6 F1 .430 .385 .475
#> recR_1 F2 .407 .369 .445
#> recR_2 F2 .531 .491 .572
#> recR_3 F2 .619 .584 .655
#> recR_4 F2 .861 .838 .885
#> recR_5 F2 .805 .778 .832
#> recR_6 F2 .637 .602 .672
#> finR_1 F3 .840 .805 .876
#> finR_3 F3 .856 .823 .888
#> finR_5 F3 .873 .843 .903
#> socR_1 F4 .646 .612 .679
#> socR_2 F4 .679 .651 .707
#> socR_3 F4 .640 .605 .675
#> socR_4 F4 .614 .582 .646
#> socR_5 F4 .379 .341 .416
#> socR_6 F4 .549 .512 .585
#> finR_2 F5 .688 .652 .723
#> finR_4 F5 .710 .674 .747
#> finR_6 F5 .683 .647 .720
#>
#> ── 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% Wald CIs for factor intercorrelations ───────────────────────────────────
#>
#> Factors est lower upper
#> F1 ~~ F2 .372 .341 .402
#> F1 ~~ F3 .448 .412 .485
#> F1 ~~ F4 .006 -.028 .039
#> F1 ~~ F5 .154 .104 .204
#> F2 ~~ F3 .319 .277 .362
#> F2 ~~ F4 .200 .169 .230
#> F2 ~~ F5 .290 .234 .346
#> F3 ~~ F4 -.042 -.091 .007
#> F3 ~~ F5 .344 .298 .390
#> F4 ~~ F5 .145 .099 .191
#>
#> ── 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 ───────────────────────────────────────────────────────────────────
#>
#> scaled χ²(295) = 2912.75, p < .001
#> CFI: .90
#> TLI: .86
#> RMSEA [90% CI]: .05 [.05; .06]
#> AIC: NA
#> BIC: NA
#> CAF: .43
#> RMSR: .03
#> SRMR: .03
#>
#> ── 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(GRiPS_raw, n_factors = 1, method = "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 type = 'EFAtools', method = '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
#>
#> Legend:
#> bold = |loading| >= .300
#> grey = below cutoff
#> red h2/u2 = Heywood-relevant value
#>
#> ── 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
#> RMSR: .02
#> SRMR: .02
summary(mod)
#>
#> EFA performed with type = 'EFAtools', method = '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
#>
#> Legend:
#> bold = |loading| >= .300
#> grey = below cutoff
#> red h2/u2 = Heywood-relevant value
#>
#> ── 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
#> RMSR: .02
#> 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(test_models$baseline$cormat, N = 500, n_factors = 3, method = "ML",
rotation = "oblimin", se = "information")
mod
#>
#> EFA performed with type = 'EFAtools', method = '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
#>
#> Legend:
#> bold = |loading| >= .300
#> grey = below cutoff
#> red h2/u2 = Heywood-relevant value
#>
#> ── 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
#> RMSR: .03
#> SRMR: .03
summary(mod)
#>
#> EFA performed with type = 'EFAtools', method = 'ML', and rotation = 'oblimin'.
#>
#> ── Model Diagnostics ───────────────────────────────────────────────────────────
#>
#> Factors: 3
#> Variables: 18
#> N: 500
#> Rotation local optima: 1 distinct from 100 random 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
#>
#> Legend:
#> bold = |loading| >= .300
#> grey = below cutoff
#> red h2/u2 = Heywood-relevant value
#>
#> ── 95% Wald CIs for salient rotated loadings ───────────────────────────────────
#>
#> Variable Factor est lower upper
#> V13 F1 .612 .476 .748
#> V14 F1 .540 .401 .679
#> V15 F1 .552 .413 .691
#> V16 F1 .550 .411 .689
#> V17 F1 .652 .521 .784
#> V18 F1 .549 .410 .689
#> V7 F2 .524 .385 .662
#> V8 F2 .562 .427 .697
#> V9 F2 .535 .398 .671
#> V10 F2 .661 .533 .788
#> V11 F2 .352 .208 .497
#> V12 F2 .649 .515 .783
#> V1 F3 .607 .463 .750
#> V2 F3 .458 .307 .610
#> V3 F3 .430 .276 .584
#> V4 F3 .536 .387 .686
#> V5 F3 .418 .264 .572
#> V6 F3 .687 .554 .821
#>
#> ── 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
#> RMSR: .03
#> 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.