EFAtools
EFAtools provides a complete workflow for exploratory
factor analysis (EFA). Starting from raw data or a correlation matrix,
it takes you through every stage of a typical analysis:
- Screening the data for factorability,
multicollinearity, non-normality, and outliers
(
efa_screen(), with efa_bartlett() and
efa_kmo() for the individual measures).
- Retention: deciding how many factors to keep, with
a comprehensive suite of factor retention criteria
(
efa_retain() and the individual criteria it wraps).
- Extraction and rotation of the factor solution
(
efa_fit()), with a choice of estimators and orthogonal or
oblique rotations.
- Post-processing: comparing solutions
(
efa_compare()), averaging across analytic choices
(efa_average()), estimating factor scores
(efa_scores()), a Schmid-Leiman transformation
(efa_schmid_leiman()), and reliability coefficients
including McDonald’s omegas (efa_reliability()).
The computationally intensive routines are implemented in C++ for
speed. This vignette walks through the whole workflow on a single data
set; it is an overview rather than a tutorial on the methods themselves,
so please consult the individual help pages for the statistical details
and literature references.
The package can be installed from CRAN using
install.packages("EFAtools"), or from GitHub using
pak::pak("mdsteiner/EFAtools"), and then loaded using:
We use the DOSPERT_raw data set throughout, which
contains responses of 3123 participants to the Domain Specific Risk
Taking Scale (DOSPERT); see ?DOSPERT_raw for details. The
data are rather large, so, purely to keep this vignette quick to build,
we work with the first 500 observations. In a real analysis you would of
course use the full data set.
# only use a subset to make analyses faster
DOSPERT_sub <- DOSPERT_raw[1:500, ]
Screening the Data
Before extracting factors it is worth checking whether the data are
suitable for factor analysis, and whether they satisfy the assumptions
of the estimator you plan to use. efa_screen() collects the
relevant diagnostics in one place. From raw data it reports the
Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy (overall and per
variable), Bartlett’s test of sphericity, the determinant and condition
number of the correlation matrix, the squared multiple correlation (SMC)
and per-variable diagnostics for each item, tests of multivariate
normality (Mardia and Henze-Zirkler), and multivariate outliers, and it
closes with a set of recommendations.
efa_screen(DOSPERT_sub, seed = 2)
#>
#> ── Sampling adequacy and sphericity ────────────────────────────────────────────
#>
#> ✔ The overall KMO value for your data is meritorious (Overall KMO = 0.87).
#> These data are probably suitable for factor analysis (verbal bands: Kaiser &
#> Rice, 1974).
#>
#> ✔ The Bartlett's test of sphericity was significant at an alpha level of .05.
#> These data are probably suitable for factor analysis.
#> χ²(435) = 5843.05, p < .001
#>
#> ── Multicollinearity ───────────────────────────────────────────────────────────
#>
#> ℹ Determinant: 0.00000634. It falls as variables are added, so the condition
#> index below carries the verdict.
#> ✔ Condition number: 46.561 (condition index 6.824). An index of 10 or less is
#> rarely of interest (Belsley, 1991).
#>
#> ── Per-variable diagnostics ────────────────────────────────────────────────────
#>
#> variance missing% SMC MSA flags
#> ethR_1 2.785 0 0.428 0.908
#> ethR_2 2.469 0 0.312 0.927
#> ethR_3 1.274 0 0.444 0.906 sparse
#> ethR_4 2.011 0 0.317 0.818 sparse
#> ethR_5 2.202 0 0.310 0.926
#> ethR_6 3.457 0 0.398 0.837
#> finR_1 1.962 0 0.683 0.880
#> finR_2 3.223 0 0.319 0.816
#> finR_3 2.356 0 0.716 0.853
#> finR_4 2.874 0 0.519 0.857
#> finR_5 2.458 0 0.760 0.853
#> finR_6 3.118 0 0.515 0.861
#> heaR_1 4.016 0 0.296 0.912
#> heaR_2 4.532 0 0.277 0.898
#> heaR_3 2.888 0 0.366 0.892
#> heaR_4 2.185 0 0.423 0.889
#> heaR_5 4.099 0 0.254 0.882
#> heaR_6 3.356 0 0.313 0.935
#> recR_1 4.570 0 0.374 0.888
#> recR_2 2.829 0 0.465 0.927
#> recR_3 3.664 0 0.511 0.905
#> recR_4 4.362 0 0.687 0.849
#> recR_5 3.741 0 0.685 0.840
#> recR_6 4.076 0 0.541 0.920
#> socR_1 1.237 0 0.351 0.714 sparse
#> socR_2 2.774 0 0.434 0.784
#> socR_3 2.148 0 0.343 0.735
#> socR_4 2.751 0 0.359 0.791
#> socR_5 3.562 0 0.284 0.840
#> socR_6 2.529 0 0.379 0.786
#>
#> ── Multivariate normality ──────────────────────────────────────────────────────
#>
#> ✖ Mardia's skewness: χ²(4960) = 11659.84, p < .001.
#> ✖ Mardia's kurtosis: z = 48.3, p < .001.
#> ✖ Henze-Zirkler: HZ = 1.03, p < .001.
#> These data depart from multivariate normality: 3 of the 3 tests reject it.
#>
#> ── Outliers ────────────────────────────────────────────────────────────────────
#>
#> ! A robust (MCD) covariance could not be computed; classical Mahalanobis
#> distances were used.
#> At least half the complete cases lie exactly on a lower-dimensional hyperplane
#> (an "exact fit"). This is common with coarse discrete items, where many
#> respondents give identical answers on an item pair; it does not mean the data
#> are collinear at the correlation level.
#> These distances come from a covariance the outliers themselves inflate, so the
#> diagnostic is no longer high-breakdown and tends to under-flag.
#> ℹ 60 of 500 observations were flagged as multivariate outliers (Mahalanobis
#> distance > 6.85).
#>
#> ── Recommendations ─────────────────────────────────────────────────────────────
#>
#> ! These data depart from multivariate normality; normal-theory standard errors
#> and fit statistics may be biased - prefer robust (sandwich) or bootstrapped
#> standard errors.
#> ! Bartlett's test is significant, but it assumes multivariate normality and
#> grows more sensitive as N increases; because these data are non-normal, treat
#> it as uninformative here and rely on the KMO.
#> ! 3 variables have a sparse response category (< 5 responses): ethR_3, ethR_4,
#> and socR_1; a low-frequency category can destabilise polychoric estimates -
#> consider collapsing it into an adjacent category.
#> ! 60 observations were flagged as potential multivariate outliers; inspect them
#> (see `$outliers$flagged`) before down-weighting or excluding.
KMO measures, on a 0-1 scale, how well each variable is predicted by
the others, and Bartlett’s test checks that the correlations are not all
zero. The overall KMO is meritorious and Bartlett’s test is clearly
significant, so the data are factorable. The determinant looks small,
but it is a product of 30 eigenvalues and falls as variables are added;
the condition index (6.8) is the measure that does not move with the
number of variables, and it is well below the value of 30 that flags a
near linear dependency. More interesting are the last few sections:
three items have a sparse response category, and both the Mardia and the
Henze-Zirkler tests reject multivariate normality. A consequence is that
normal-theory standard errors and fit indices may be biased, so robust
(sandwich) or bootstrapped standard errors are preferable, and
Bartlett’s test should be read with caution. For a fully ordinal
treatment you would move to polychoric correlations with a diagonally
weighted least-squares (DWLS) estimator.
The outlier section reports something other than the usual case, and
it says so. The diagnostic normally measures each case against a
high-breakdown location and scatter, estimated from a minimum covariance
determinant (MCD) subset found by a randomized search. For these data
the search keeps landing on a subset where one item is constant (several
DOSPERT items have a very common lowest response), so the covariance is
singular and no valid subset exists.
efa_screen() falls back to classical Mahalanobis
distances. It names the reason and labels the distances it reports. A
classical covariance uses every observation, outliers included, so the
outliers inflate the scatter that measures them. The diagnostic is no
longer high-breakdown and it tends to under-flag. Read the 60 flagged
cases as a lower bound, and as a sign of the non-normality the tests
above report, rather than as a list of cases to remove. The MCD does not
run, so its random subset search does not run either and
seed changes nothing here. seed matters only
when the robust estimate is available, as in the ordinal example in the
EFA with ordinal and missing
data vignette. Inspect the flagged cases (via
$outliers$flagged) rather than deleting them
automatically.
Bartlett’s test and the KMO criterion, introduced above, are also
available as standalone functions if you only want the two classic
factorability tests:
# Bartlett's test of sphericity
efa_bartlett(DOSPERT_sub)
#>
#> ✔ The Bartlett's test of sphericity was significant at an alpha level of .05.
#> These data are probably suitable for factor analysis.
#>
#> χ²(435) = 5843.05, p < .001
# KMO criterion
efa_kmo(DOSPERT_sub)
#>
#> ── Kaiser-Meyer-Olkin criterion (KMO) ──────────────────────────────────────────
#>
#> ✔ The overall KMO value for your data is meritorious.
#> These data are probably suitable for factor analysis.
#>
#> Overall: 0.87
#>
#> For each variable:
#> ethR_1 ethR_2 ethR_3 ethR_4 ethR_5 ethR_6 finR_1 finR_2 finR_3 finR_4 finR_5
#> 0.908 0.927 0.906 0.818 0.926 0.837 0.880 0.816 0.853 0.857 0.853
#> finR_6 heaR_1 heaR_2 heaR_3 heaR_4 heaR_5 heaR_6 recR_1 recR_2 recR_3 recR_4
#> 0.861 0.912 0.898 0.892 0.889 0.882 0.935 0.888 0.927 0.905 0.849
#> recR_5 recR_6 socR_1 socR_2 socR_3 socR_4 socR_5 socR_6
#> 0.840 0.920 0.714 0.784 0.735 0.791 0.840 0.786
Both tests can also be requested directly inside
efa_retain(), alongside the factor retention criteria
discussed next.
Deciding How Many Factors to Retain
One of the most consequential decisions in an EFA is how many factors
to extract. There is a large number of factor retention criteria, and no
single one dominates the others: which criterion works best depends on
the structure of the data — how many indicators there are, how strong
the factors are, and how strongly they intercorrelate. The current
recommendation is therefore to examine several criteria together and
check whether they converge (see, for example, Auerswald and Moshagen,
2019).
EFAtools implements a broad set of these criteria. They
can be called individually, or all (or a selection) of them together
through efa_retain().
Calling Individual Criteria
Each criterion has its own function. For a parallel analysis based on
squared multiple correlations (SMC; also called parallel analysis with
principal factors), for example, you would run:
# parallel analysis based on SMC eigenvalues
pa <- efa_parallel(DOSPERT_sub, eigen_type = "smc")
pa
#> ── Parallel analysis ───────────────────────────────────────────────────────────
#> Eigenvalues found using SMC; 1000 simulated datasets.
#>
#> • SMC: 10
#>
#> ℹ Number of factors retained using the "means" decision rule.
Printing the result reports the suggested number of factors. Each
criterion also has a plot() method that draws the figure
behind the decision — for a parallel analysis, the observed eigenvalues
against the reference eigenvalues from the simulated data:
Other criteria work the same way. The empirical Kaiser criterion, for
instance:
# empirical Kaiser criterion
efa_ekc(DOSPERT_sub)
#> ── Empirical Kaiser Criterion ──────────────────────────────────────────────────
#>
#> • Braeken & van Assen (2017): 8
The following criteria are currently implemented: comparison data
(efa_cd()), the empirical Kaiser criterion
(efa_ekc()), the hull method (efa_hull()), the
Kaiser-Guttman criterion (efa_kgc()), the minimum average
partial test (efa_map()), the next eigenvalue sufficiency
test (efa_nest()), parallel analysis
(efa_parallel()), the scree test
(efa_scree()), and sequential model tests
(efa_smt()). Many of them offer several variants of the
underlying method. For instance, parallel analysis, can be based on
eigenvalues from unity (principal components), SMCs, or a full EFA. See
the individual help pages for the options.
Running Several Criteria at Once with efa_retain()
To compare criteria, it is easier to call efa_retain(),
a wrapper around the implemented retention criteria that also runs
Bartlett’s test and computes the KMO. You can pick a subset with the
criteria argument:
ret <- efa_retain(DOSPERT_sub,
criteria = c("parallel", "ekc", "kgc", "smt", "map"))
#> Warning: The MAP criterion could only be evaluated up to 27 partialled components of 29.
#> ℹ A residual variance reached zero there, so the criterion is `NA` beyond that
#> point and the suggestion is the minimum over the range that could be
#> computed.
#> ℹ A stop well before the end of the grid usually indicates a near-singular
#> correlation matrix.
#> Warning: The sequential model tests selected an inadmissible solution: chi: 13 factors
#> (Heywood case) and AIC: 13 factors (Heywood case).
#> ℹ The selected solution has a Heywood case or did not converge, so the
#> suggested number of factors may be unreliable; interpret it with caution and
#> cross-check with other criteria.
ret
#> ── 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(435) = 5843.05, p < .001. These data are probably suitable for factor
#> analysis.
#> ✔ The Kaiser-Meyer-Olkin criterion is meritorious (KMO = 0.87). These data are
#> probably suitable for factor analysis.
#>
#> ── Suggested number of factors ─────────────────────────────────────────────────
#>
#> 8 suggestions from 5 criteria, ranging from 4 to 13 factors.
#>
#> Empirical Kaiser Criterion
#> • Braeken & van Assen (2017): 8
#>
#> Kaiser-Guttman criterion
#> Eigenvalues found using SMC.
#> • SMC: 4
#>
#> Minimum average partial
#> • Original implementation (TR2): 6
#> • Revised implementation (TR4): 6
#>
#> Parallel analysis
#> Eigenvalues found using SMC; 1000 simulated datasets.
#> • SMC: 10
#>
#> Sequential model tests
#> • Sequential chi-square model tests: 13
#> • Lower bound of RMSEA 90% CI: 6
#> • Akaike Information Criterion: 13
plot() on the result draws the figure for every
criterion that has one:
#>
#> $KGC
#>
#> $PARALLEL
Omitting criteria does not run every criterion listed
above. The default is a subset of six: comparison data, the empirical
Kaiser criterion, the hull method, the minimum average partial test, the
next eigenvalue sufficiency test, and parallel analysis. The call above
instead adds the Kaiser-Guttman criterion and the sequential model
tests, and drops comparison data (slow on larger data sets), the hull
method, and the next eigenvalue sufficiency test.
This is the situation one would rather avoid, but it does happen: the
criteria disagree, with suggestions ranging from four to well above ten
factors. When there is no clear convergence, the choice becomes partly a
matter of judgement. We proceed with six factors, consistent with the
minimum average partial test and the RMSEA-based sequential model test
above (this version of the DOSPERT scale is usually modeled with five
factors; the sixth factor here splits the financial-risk items into
two).
All criteria except comparison data (which needs raw data) can also
be applied to a correlation matrix, in which case the sample size must
be supplied. On a cleaner data set the criteria often agree:
efa_retain(test_models$baseline$cormat, N = 500, estimator = "uls",
criteria = c("parallel", "ekc", "smt"),
eigen_type_other = c("smc", "pca"))
#> ── Tests for the suitability of the data for factor analysis ───────────────────
#>
#> ✔ The Bartlett's test of sphericity was significant at an alpha level of .05:
#> χ²(153) = 2173.28, p < .001. These data are probably suitable for factor
#> analysis.
#> ✔ The Kaiser-Meyer-Olkin criterion is marvellous (KMO = 0.916). These data are
#> probably suitable for factor analysis.
#>
#> ── Suggested number of factors ─────────────────────────────────────────────────
#>
#> 6 suggestions from 3 criteria, ranging from 2 to 3 factors (most common: 3).
#>
#> Empirical Kaiser Criterion
#> • Braeken & van Assen (2017): 3
#>
#> Parallel analysis
#> Eigenvalues found using SMC and PCA; 1000 simulated datasets.
#> • PCA: 3
#> • SMC: 3
#>
#> Sequential model tests
#> • Sequential chi-square model tests: 3
#> • Lower bound of RMSEA 90% CI: 2
#> • Akaike Information Criterion: 3
Here the criteria agree on three factors, with only the conservative
RMSEA lower-bound variant of the sequential model test suggesting
two.
Comparing Solutions with efa_compare()
efa_compare() provides a quick way to see how similar
two loading (pattern) matrices are — whether they come from different
implementations, different estimators, or different rotations. For
instance, to compare the PAF/promax solution with the ULS/oblimin
one:
efa_compare(efa_dospert$rot_loadings, efa_uls$rot_loadings,
x_labels = c("PAF and promax", "ULS and oblimin"))
#>
#> ── Summary statistics ──────────────────────────────────────────────────────────
#>
#> Mean [min, max] absolute difference: .0278 [ .0001, .1262]
#> Median absolute difference: .0226
#> Root mean squared distance (RMSE): .0359
#> Max decimals where all numbers agree in absolute value: 0
#> Differing indicator-to-factor correspondences: 1 (highest loading),
#> 0 (all |loadings| >= 0.3)
#>
#> ── Elementwise differences ─────────────────────────────────────────────────────
#>
#> Differences: PAF and promax - ULS and oblimin.
#>
#> F1 F2 F3 F4 F5 F6
#> ethR_1 .0244 -.0335 -.0590 -.0073 .0157 -.0248
#> ethR_2 .0274 -.0298 -.0345 .0007 .0153 -.0348
#> ethR_3 .0208 -.0194 -.0522 .0025 .0073 -.0224
#> ethR_4 .0509 .0033 -.0253 -.0578 .0306 -.0099
#> ethR_5 .0110 -.0174 -.0482 .0045 .0028 -.0227
#> ethR_6 .0434 .0018 -.0325 -.0408 .0224 -.0190
#> finR_1 -.0027 -.0061 .0021 .0175 -.0001 -.0201
#> finR_2 -.0116 .0192 -.0214 -.0742 .0142 .0478
#> finR_3 .0072 .0077 .0114 -.0060 .0069 -.0179
#> finR_4 -.0240 -.0170 -.0625 -.0334 .0071 .0427
#> finR_5 -.0046 -.0054 -.0032 .0162 -.0006 -.0168
#> finR_6 -.0267 -.0230 -.0624 -.0316 .0075 .0379
#> heaR_1 .0156 -.0223 -.0331 .0003 .0095 -.0269
#> heaR_2 .0272 -.0276 -.0316 -.0019 .0148 -.0376
#> heaR_3 -.0213 -.0955 -.0820 .1154 -.0202 -.0595
#> heaR_4 -.0269 -.0912 -.0791 .1262 -.0266 -.0660
#> heaR_5 .0226 -.0393 -.0376 .0123 .0119 -.0404
#> heaR_6 -.0202 -.0618 -.0610 .0728 -.0150 -.0437
#> recR_1 -.0171 -.0288 -.0354 .0253 -.0074 -.0280
#> recR_2 -.0327 -.0634 -.0637 .1076 -.0301 -.0612
#> recR_3 -.0370 -.0393 -.0520 .0750 -.0277 -.0450
#> recR_4 -.0107 .0446 -.0201 -.0293 -.0118 -.0187
#> recR_5 -.0075 .0487 -.0177 -.0300 -.0117 -.0186
#> recR_6 -.0249 -.0134 -.0510 .0203 -.0149 -.0238
#> socR_1 .0259 -.0045 .0287 -.0333 .0225 -.0286
#> socR_2 .0137 -.0351 -.0068 -.0020 .0166 -.0377
#> socR_3 .0111 .0072 .0203 -.0524 .0206 .0005
#> socR_4 .0005 -.0482 -.0079 .0283 .0065 -.0417
#> socR_5 .0058 -.0272 -.0242 -.0084 .0114 -.0191
#> socR_6 .0199 -.0179 -.0076 -.0427 .0257 -.0124
Averaging Across Analytic Choices with
efa_average()
Comparing two solutions by hand answers the question for those two.
efa_average() asks it across a whole grid at once: it runs
an EFA for every combination of the settings you allow it to vary,
averages the loadings across the solutions that converged, and reports
the range each loading spans. That range is the useful part — it shows
how much of the result is the data and how much is the analytic choice.
Here we vary only the estimator, keeping one implementation each of PAF,
ULS, and ML with a promax rotation:
avg_dospert <- efa_average(DOSPERT_sub, n_factors = 6,
estimator = c("paf", "uls", "ml"),
type = "EFAtools", start_method = "psych")
avg_dospert
#>
#> Averaging performed with averaging method mean (trim = 0) across 3 EFAs,
#> varying the following settings: estimator.
#>
#> The error rate is at 0%. Of the solutions that did not result in an error, 100%
#> converged. Of the solutions that converged, 0% contained Heywood cases and 100%
#> were admissible.
#>
#> ══ Indicator-to-Factor Correspondences ═════════════════════════════════════════
#>
#> For each cell, the proportion of solutions including the respective
#> indicator-to-factor correspondence. A salience threshold of 0.3 was used to
#> determine indicator-to-factor correspondences.
#>
#> F1 F2 F3 F4 F5 F6
#> ethR_1 1.00 .00 .00 .00 .00 .00
#> ethR_2 1.00 .00 .00 .00 .00 .00
#> ethR_3 1.00 .00 .00 .00 .00 .00
#> ethR_4 1.00 .00 .00 .00 .00 .00
#> ethR_5 1.00 .00 .00 .00 .00 .00
#> ethR_6 1.00 .00 .00 .00 .00 .00
#> finR_1 .00 .00 1.00 .00 .00 .00
#> finR_2 .00 .00 .00 .00 .00 1.00
#> finR_3 .00 .00 1.00 .00 .00 .00
#> finR_4 .00 .00 .00 .00 .00 1.00
#> finR_5 .00 .00 1.00 .00 .00 .00
#> finR_6 .00 .00 .00 .00 .00 1.00
#> heaR_1 1.00 .00 .00 .00 .00 .00
#> heaR_2 1.00 .00 .00 .00 .00 .00
#> heaR_3 .00 .00 .00 1.00 .00 .00
#> heaR_4 .00 .00 .00 1.00 .00 .00
#> heaR_5 1.00 .00 .00 .00 .00 .00
#> heaR_6 .00 .00 .00 1.00 .00 .00
#> recR_1 .00 .00 .00 .33 .00 .00
#> recR_2 .00 .00 .00 1.00 .00 .00
#> recR_3 .00 .67 .00 1.00 .00 .00
#> recR_4 .00 1.00 .00 .00 .00 .00
#> recR_5 .00 1.00 .00 .00 .00 .00
#> recR_6 .00 1.00 .00 .00 .00 .00
#> socR_1 .00 .00 .00 .00 1.00 .00
#> socR_2 .00 .00 .00 .00 1.00 .00
#> socR_3 .00 .00 .00 .00 1.00 .00
#> socR_4 .00 .00 .00 .00 1.00 .00
#> socR_5 .00 .00 .00 .00 1.00 .00
#> socR_6 .00 .00 .00 .00 1.00 .00
#>
#> ══ Loadings ════════════════════════════════════════════════════════════════════
#>
#> ── Mean ────────────────────────────────────────────────────────────────────────
#>
#> F1 F2 F3 F4 F5 F6
#> ethR_1 .548 -.039 -.024 .142 .011 .117
#> ethR_2 .454 -.048 .133 .081 .117 -.042
#> ethR_3 .535 .038 .064 .122 -.213 .030
#> ethR_4 .691 -.018 -.038 -.173 .001 -.006
#> ethR_5 .395 .160 -.051 .119 -.094 .035
#> ethR_6 .672 .089 -.003 -.100 -.058 -.043
#> finR_1 -.017 -.022 .850 .035 .002 .004
#> finR_2 -.060 .116 -.045 -.193 .120 .597
#> finR_3 .044 .037 .893 -.087 .062 -.011
#> finR_4 .041 -.064 .095 .026 -.103 .771
#> finR_5 -.004 -.008 .886 .022 -.037 .056
#> finR_6 -.018 -.052 .020 .057 .032 .777
#> heaR_1 .312 .034 .106 .104 .088 .023
#> heaR_2 .417 .024 -.004 .084 .191 -.076
#> heaR_3 .148 -.144 -.065 .649 -.037 -.006
#> heaR_4 .095 -.032 -.001 .665 -.067 -.070
#> heaR_5 .383 -.041 -.052 .175 .194 -.074
#> heaR_6 .053 .040 -.024 .461 .058 .049
#> recR_1 -.079 .216 -.065 .273 .242 .089
#> recR_2 -.049 .196 .085 .617 -.054 -.096
#> recR_3 -.160 .357 .052 .478 .030 .004
#> recR_4 .041 .916 -.010 -.065 .004 -.013
#> recR_5 .086 .929 .034 -.082 -.062 -.055
#> recR_6 -.012 .495 -.063 .220 .056 .157
#> socR_1 .013 .025 .024 -.128 .679 -.160
#> socR_2 .068 -.031 .098 .062 .604 -.015
#> socR_3 -.087 .065 .035 -.196 .560 .109
#> socR_4 -.084 -.089 .018 .201 .579 -.053
#> socR_5 .094 .023 -.081 .094 .368 .106
#> socR_6 .132 -.029 -.068 -.077 .564 .138
#>
#> ── Range (max − min) ───────────────────────────────────────────────────────────
#>
#> F1 F2 F3 F4 F5 F6
#> ethR_1 .004 .044 .011 .037 .006 .015
#> ethR_2 .022 .022 .005 .051 .006 .003
#> ethR_3 .010 .021 .013 .015 .002 .001
#> ethR_4 .011 .051 .003 .032 .004 .013
#> ethR_5 .038 .002 .002 .033 .017 .005
#> ethR_6 .025 .011 .000 .034 .015 .002
#> finR_1 .008 .010 .009 .040 .005 .012
#> finR_2 .029 .015 .003 .053 .001 .004
#> finR_3 .008 .004 .007 .022 .005 .002
#> finR_4 .006 .022 .009 .026 .003 .010
#> finR_5 .007 .006 .002 .000 .002 .001
#> finR_6 .012 .005 .008 .010 .004 .007
#> heaR_1 .016 .024 .015 .039 .011 .012
#> heaR_2 .003 .015 .004 .027 .001 .015
#> heaR_3 .045 .013 .017 .074 .005 .010
#> heaR_4 .063 .002 .015 .094 .000 .013
#> heaR_5 .006 .011 .005 .013 .008 .008
#> heaR_6 .026 .064 .006 .016 .003 .003
#> recR_1 .037 .112 .016 .162 .014 .016
#> recR_2 .007 .106 .012 .128 .022 .018
#> recR_3 .015 .136 .003 .163 .013 .021
#> recR_4 .003 .029 .001 .072 .006 .014
#> recR_5 .006 .052 .002 .097 .009 .008
#> recR_6 .027 .046 .011 .024 .011 .007
#> socR_1 .010 .012 .002 .027 .003 .008
#> socR_2 .034 .012 .015 .028 .006 .016
#> socR_3 .009 .014 .018 .008 .001 .004
#> socR_4 .018 .018 .015 .012 .012 .012
#> socR_5 .010 .025 .013 .049 .006 .035
#> socR_6 .008 .017 .012 .005 .010 .014
#>
#> ══ Factor Intercorrelations from Oblique Solutions ═════════════════════════════
#>
#> ── Mean ────────────────────────────────────────────────────────────────────────
#>
#> F1 F2 F3 F4 F5 F6
#> F1 1.000
#> F2 .342 1.000
#> F3 .456 .334 1.000
#> F4 .565 .592 .496 1.000
#> F5 .026 .307 .040 .291 1.000
#> F6 .331 .429 .463 .493 .320 1.000
#>
#> ── Range (max − min) ───────────────────────────────────────────────────────────
#>
#> F1 F2 F3 F4 F5 F6
#> F1 .000
#> F2 .045 .000
#> F3 .003 .045 .000
#> F4 .005 .005 .017 .000
#> F5 .016 .027 .017 .013 .000
#> F6 .005 .036 .000 .019 .011 .000
#>
#> ══ Variances Accounted for ═════════════════════════════════════════════════════
#>
#> ── Mean ────────────────────────────────────────────────────────────────────────
#>
#> F1 F2 F3 F4 F5 F6
#> SS loadings 2.599 2.405 2.447 2.320 2.206 1.716
#> Prop Tot Var .087 .080 .082 .077 .074 .057
#> Prop Comm Var .190 .176 .179 .169 .161 .125
#>
#> ── Range (max − min) ───────────────────────────────────────────────────────────
#>
#> F1 F2 F3 F4 F5 F6
#> SS loadings .099 .336 .030 .284 .022 .005
#> Prop Tot Var .003 .011 .001 .009 .001 .000
#> Prop Comm Var .007 .024 .002 .021 .001 .000
#>
#> ══ Model Fit ═══════════════════════════════════════════════════════════════════
#>
#> The fit indices are the mean of the per-solution fit indices, not the fit of
#> the averaged loadings printed above, which are a cell-wise summary rather than
#> a fitted solution.
#>
#> Chi-square-based indices averaged over 2 of 3 solutions.
#> CAF, RMSR, and SRMR averaged over 3 of 3 solutions.
#>
#> M (SD) [Min; Max]
#> χ²: 613.61 ( 6.30) [609.15; 618.06]
#> df: 270
#> p: .000 (.000) [.000; .000]
#> CFI: .93 (.00) [.93; .94]
#> TLI: .89 (.00) [.89; .90]
#> RMSEA: .05 (.00) [.05; .05]
#> AIC: 73.61 ( 6.30) [69.15; 78.06]
#> BIC: -1064.34 ( 6.30) [-1068.80; -1059.88]
#> ECVI: 2.01 ( 0.01) [ 2.00; 2.02]
#> CAF: .48 (.00) [.48; .48]
#> RMSR: .03 (.00) [.03; .03]
#> SRMR: .03 (.00) [.03; .03]
Left unrestricted, the defaults sweep the different implementations
of principal axis factoring and promax rotation instead, which is
considerably more work. See ?efa_average for the settings
that can be varied, and for plot(), which shows the
minimum, maximum, and average loading of every indicator.
Factor Scores with efa_scores()
If you need factor scores, efa_scores() estimates them
directly from an efa_fit() solution and reports
score-quality diagnostics. Printing the object gives the determinacy of
each score and the Guttman indeterminacy index. summary()
adds the factor weights, the score intercorrelations, and a score
validity/univocality matrix. The off-diagonals of that matrix are the
univocality — how strongly each score correlates with the other
factors:
fac_scores <- efa_scores(DOSPERT_sub, f = efa_dospert)
fac_scores
#>
#> ── Factor scores (regression) ──────────────────────────────────────────────────
#>
#> Scored 500 of 500 observations on 6 factors (see `$scores`).
#>
#> ── Score determinacy ───────────────────────────────────────────────────────────
#>
#> rho rho2 guttman
#> F1 .903 .815 .629
#> F2 .951 .905 .810
#> F3 .961 .923 .846
#> F4 .910 .829 .658
#> F5 .887 .787 .574
#> F6 .907 .822 .644
summary(fac_scores)
#>
#> ── Factor scores (regression) ──────────────────────────────────────────────────
#>
#> Scored 500 of 500 observations on 6 factors.
#>
#> ── Score determinacy ───────────────────────────────────────────────────────────
#>
#> rho rho2 guttman
#> F1 .903 .815 .629
#> F2 .951 .905 .810
#> F3 .961 .923 .846
#> F4 .910 .829 .658
#> F5 .887 .787 .574
#> F6 .907 .822 .644
#>
#> ── Factor weights ──────────────────────────────────────────────────────────────
#>
#> F1 F2 F3 F4 F5 F6
#> ethR_1 .191 .023 .012 .050 -.001 .050
#> ethR_2 .123 -.005 .013 .037 .032 -.004
#> ethR_3 .192 .009 .019 .041 -.088 .004
#> ethR_4 .195 -.027 -.005 -.024 -.010 .005
#> ethR_5 .098 .016 -.001 .036 -.031 .017
#> ethR_6 .197 .000 -.005 -.005 -.030 -.016
#> finR_1 -.004 -.005 .238 .039 .000 .018
#> finR_2 -.018 .031 -.012 -.041 .052 .161
#> finR_3 .053 .014 .311 -.044 .027 -.006
#> finR_4 .033 -.016 .029 .031 -.032 .368
#> finR_5 .011 -.002 .436 .068 -.064 .064
#> finR_6 -.007 .020 .010 .051 .045 .381
#> heaR_1 .079 .004 .015 .034 .025 -.001
#> heaR_2 .109 -.002 .007 .035 .050 -.015
#> heaR_3 .062 -.011 -.004 .189 .001 .001
#> heaR_4 .041 -.005 .003 .223 -.010 -.019
#> heaR_5 .094 -.003 -.009 .047 .053 -.011
#> heaR_6 .022 .033 -.001 .115 .031 .020
#> recR_1 -.011 .057 .006 .061 .088 .045
#> recR_2 -.002 .054 .028 .200 .001 -.010
#> recR_3 -.060 .109 .006 .164 .043 .028
#> recR_4 .007 .373 -.006 -.001 .038 .017
#> recR_5 .047 .398 .016 -.033 -.041 -.012
#> recR_6 -.012 .102 -.015 .105 .045 .075
#> socR_1 .000 .025 .002 -.027 .231 -.030
#> socR_2 .003 -.014 .014 .044 .208 -.001
#> socR_3 -.042 .006 -.018 -.033 .183 .035
#> socR_4 -.028 .003 .008 .057 .202 -.005
#> socR_5 .022 .013 -.011 .026 .107 .051
#> socR_6 .031 .000 -.013 -.006 .195 .052
#>
#> ── Score validity and univocality ──────────────────────────────────────────────
#>
#> Diagonal: validity (score-factor correlation). Off-diagonal: univocality.
#>
#> F1 F2 F3 F4 F5 F6
#> F1 .903 .402 .501 .613 .043 .362
#> F2 .381 .951 .367 .608 .322 .453
#> F3 .470 .363 .961 .515 .051 .482
#> F4 .607 .635 .543 .910 .298 .523
#> F5 .044 .345 .056 .306 .887 .349
#> F6 .360 .475 .510 .525 .341 .907
#>
#> ── Score intercorrelations ─────────────────────────────────────────────────────
#>
#> F1 F2 F3 F4 F5 F6
#> F1 1.000 .423 .521 .673 .049 .399
#> F2 .423 1.000 .382 .667 .363 .499
#> F3 .521 .382 1.000 .566 .058 .531
#> F4 .673 .667 .566 1.000 .336 .577
#> F5 .049 .363 .058 .336 1.000 .385
#> F6 .399 .499 .531 .577 .385 1.000
Where to Next
This vignette covered the core workflow. EFAtools goes
further in several directions, each of which has its own vignette or
article:
Run browseVignettes("EFAtools") to see the vignettes
installed with the package, or visit the package website for the
full set of articles.