# 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:

1. Screening the data for factorability, multicollinearity, non-normality, and outliers (`efa_screen()`, with `efa_bartlett()` and `efa_kmo()` for the individual measures).
2. Retention: deciding how many factors to keep, with a comprehensive suite of factor retention criteria (`efa_retain()` and the individual criteria it wraps).
3. Extraction and rotation of the factor solution (`efa_fit()`), with a choice of estimators and orthogonal or oblique rotations.
4. 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:

    library(EFAtools)

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:

    plot(pa)

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:

    plot(ret)
    #> $EKC

    #>
    #> $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.

## Extracting Factors with `efa_fit()`

`efa_fit()` performs the factor extraction and rotation. For extraction you can choose principal axis factoring (PAF), maximum likelihood (ML), or unweighted least squares (ULS, also known as MINRES); diagonally weighted least squares (DWLS) is additionally available for ordinal data analysed with polychoric or tetrachoric correlations (see the EFA with ordinal and missing data vignette). For rotation the package offers varimax and promax as well as a range of further rotations (orthogonal rotations keep the factors uncorrelated, oblique ones let them correlate): quartimax and equamax are orthogonal; oblimin, quartimin, and simplimax are oblique; and geomin, bentler, and bifactor come in both an orthogonal form (`geominT`, `bentlerT`, `bifactorT`) and an oblique one (`geominQ`, `bentlerQ`, `bifactorQ`). The suffix is part of the name: `geomin` on its own matches both forms and is rejected. All are computed by rotation engines built into the package.

An EFA with PAF and no rotation is as simple as:

    efa_fit(DOSPERT_sub, n_factors = 6)
    #>
    #> EFA performed with estimator = 'PAF' and rotation = 'none'.
    #>
    #> ── Unrotated Loadings ──────────────────────────────────────────────────────────
    #>
    #>          F1     F2     F3     F4     F5     F6    h2    u2
    #> ethR_1  .558  -.193   .141   .232  -.129   .049  .442  .558
    #> ethR_2  .472  -.157   .073   .242   .037   .059  .316  .684
    #> ethR_3  .507  -.402   .164   .112  -.087   .062  .470  .530
    #> ethR_4  .308  -.278   .189   .294  -.103   .224  .356  .644
    #> ethR_5  .459  -.173   .202   .042  -.082   .045  .292  .708
    #> ethR_6  .412  -.308   .230   .201  -.066   .207  .405  .595
    #> finR_1  .602  -.304  -.461  -.045   .244   .028  .730  .270
    #> finR_2  .294   .281  -.255  -.066  -.341   .086  .359  .641
    #> finR_3  .626  -.285  -.471  -.031   .264   .114  .778  .222
    #> finR_4  .557  -.012  -.357  -.023  -.437  -.063  .633  .367
    #> finR_5  .657  -.342  -.491  -.064   .218   .028  .842  .158
    #> finR_6  .551   .157  -.341  -.007  -.429  -.082  .635  .365
    #> heaR_1  .464  -.079   .047   .133   .010   .046  .244  .756
    #> heaR_2  .409  -.020   .163   .229   .040   .068  .252  .748
    #> heaR_3  .497  -.101   .172   .127   .007  -.325  .409  .591
    #> heaR_4  .547  -.110   .183   .038   .070  -.322  .455  .545
    #> heaR_5  .382   .002   .175   .250   .028   .010  .240  .760
    #> heaR_6  .504   .063   .102   .015   .006  -.196  .307  .693
    #> recR_1  .443   .337   .057  -.064   .007  -.063  .322  .678
    #> recR_2  .617  -.012   .147  -.141   .126  -.232  .493  .507
    #> recR_3  .612   .178   .103  -.247   .080  -.153  .507  .493
    #> recR_4  .634   .228   .254  -.447   .033   .235  .774  .226
    #> recR_5  .640   .135   .267  -.463   .051   .261  .784  .216
    #> recR_6  .629   .238   .135  -.244  -.060  -.014  .534  .466
    #> socR_1  .075   .528  -.024   .233   .213   .124  .400  .600
    #> socR_2  .358   .428  -.083   .247   .147   .013  .401  .599
    #> socR_3  .118   .528  -.160   .122   .044   .116  .348  .652
    #> socR_4  .235   .481  -.057   .214   .166  -.093  .372  .628
    #> socR_5  .317   .330   .014   .154  -.030   .002  .234  .766
    #> socR_6  .260   .462  -.058   .267  -.023   .082  .363  .637
    #>
    #> ── Variances Accounted for ─────────────────────────────────────────────────────
    #>
    #>                      F1     F2     F3     F4    F5     F6
    #> SS loadings        7.013  2.417  1.529  1.246  .850   .641
    #> Prop Tot Var        .234   .081   .051   .042  .028   .021
    #> Cum Prop Tot Var    .234   .314   .365   .407  .435   .457
    #> Prop Comm Var       .512   .177   .112   .091  .062   .047
    #> Cum Prop Comm Var   .512   .689   .800   .891  .953  1.000
    #>
    #> ── Model Fit ───────────────────────────────────────────────────────────────────
    #>
    #> CAF: .48
    #> SRMR: .03
    #> df: 270

To rotate the loadings, set the `rotation` argument. Here we use a promax rotation. We assign the result, as later sections reuse this solution:

    efa_dospert <- efa_fit(DOSPERT_sub, n_factors = 6, rotation = "promax")
    efa_dospert
    #>
    #> EFA performed with estimator = 'PAF' and rotation = 'promax'.
    #>
    #> ── Rotated Loadings ────────────────────────────────────────────────────────────
    #>
    #>           F1     F2     F3     F4     F5     F6    h2    u2
    #> ethR_1   .547  -.025  -.020   .130   .009   .122  .442  .558
    #> ethR_2   .447  -.055   .132   .098   .115  -.043  .316  .684
    #> ethR_3   .532   .045   .068   .117  -.212   .030  .470  .530
    #> ethR_4   .695  -.035  -.039  -.162   .000  -.002  .356  .644
    #> ethR_5   .382   .159  -.051   .130  -.100   .036  .292  .708
    #> ethR_6   .664   .086  -.003  -.089  -.063  -.044  .405  .595
    #> finR_1  -.015  -.019   .853   .022   .003   .008  .730  .270
    #> finR_2  -.050   .121  -.044  -.211   .120   .599  .359  .641
    #> finR_3   .046   .035   .895  -.094   .061  -.011  .778  .222
    #> finR_4   .039  -.071   .098   .035  -.102   .768  .633  .367
    #> finR_5  -.006  -.010   .887   .023  -.038   .057  .842  .158
    #> finR_6  -.022  -.051   .023   .061   .031   .774  .635  .365
    #> heaR_1   .318   .042   .111   .091   .092   .019  .244  .756
    #> heaR_2   .416   .020  -.003   .093   .191  -.081  .252  .748
    #> heaR_3   .133  -.140  -.071   .674  -.036  -.009  .409  .591
    #> heaR_4   .074  -.031  -.006   .697  -.067  -.074  .455  .545
    #> heaR_5   .385  -.037  -.053   .170   .197  -.072  .240  .760
    #> heaR_6   .044   .062  -.026   .455   .059   .048  .307  .693
    #> recR_1  -.067   .253  -.060   .219   .247   .094  .322  .678
    #> recR_2  -.046   .231   .089   .574  -.046  -.090  .493  .507
    #> recR_3  -.155   .402   .053   .423   .035   .011  .507  .493
    #> recR_4   .042   .926  -.010  -.089   .002  -.017  .774  .226
    #> recR_5   .088   .945   .034  -.114  -.065  -.058  .784  .216
    #> recR_6  -.020   .510  -.067   .212   .053   .154  .534  .466
    #> socR_1   .016   .029   .025  -.137   .679  -.162  .400  .600
    #> socR_2   .057  -.035   .103   .072   .602  -.020  .401  .599
    #> socR_3  -.084   .061   .029  -.199   .559   .110  .348  .652
    #> socR_4  -.090  -.082   .023   .197   .583  -.057  .372  .628
    #> socR_5   .097   .032  -.086   .077   .370   .117  .234  .766
    #> socR_6   .134  -.035  -.072  -.079   .568   .142  .363  .637
    #>
    #> ── Factor Intercorrelations ────────────────────────────────────────────────────
    #>
    #>       F1     F2     F3     F4     F5     F6
    #> F1  1.000
    #> F2   .357  1.000
    #> F3   .455   .349  1.000
    #> F4   .567   .594   .502  1.000
    #> F5   .031   .316   .046   .287  1.000
    #> F6   .329   .441   .463   .486   .324  1.000
    #>
    #> ── Variances Accounted for ─────────────────────────────────────────────────────
    #>
    #>                      F1     F2     F3     F4     F5     F6
    #> SS loadings        2.565  2.518  2.457  2.225  2.213  1.717
    #> Prop Tot Var        .086   .084   .082   .074   .074   .057
    #> Cum Prop Tot Var    .086   .169   .251   .326   .399   .457
    #> Prop Comm Var       .187   .184   .179   .162   .162   .125
    #> Cum Prop Comm Var   .187   .371   .551   .713   .875  1.000
    #>
    #> ── Model Fit ───────────────────────────────────────────────────────────────────
    #>
    #> CAF: .48
    #> SRMR: .03
    #> df: 270

We can also use a different estimator and rotation, here for example ULS and an oblimin rotation:

    efa_uls <- efa_fit(DOSPERT_sub, n_factors = 6, rotation = "oblimin", estimator = "uls")
    efa_uls
    #>
    #> EFA performed with estimator = 'ULS' and rotation = 'oblimin'.
    #>
    #> ── Rotated Loadings ────────────────────────────────────────────────────────────
    #>
    #>           F1     F2     F3     F4     F5     F6    h2    u2
    #> ethR_1   .039   .009   .522  -.007   .137   .146  .441  .559
    #> ethR_2   .166  -.025   .419   .100   .097  -.008  .316  .684
    #> ethR_3   .120   .064   .511  -.220   .115   .052  .470  .530
    #> ethR_4  -.013  -.038   .644  -.031  -.104   .008  .356  .644
    #> ethR_5  -.003   .177   .371  -.103   .126   .059  .292  .708
    #> ethR_6   .029   .084   .620  -.086  -.048  -.025  .405  .595
    #> finR_1   .850  -.013  -.012   .003   .004   .028  .729  .271
    #> finR_2  -.022   .102  -.039   .106  -.137   .551  .358  .642
    #> finR_3   .883   .028   .039   .054  -.088   .007  .778  .222
    #> finR_4   .160  -.054   .063  -.109   .068   .725  .634  .366
    #> finR_5   .890  -.005  -.002  -.037   .006   .073  .842  .158
    #> finR_6   .085  -.028   .005   .023   .092   .737  .636  .364
    #> heaR_1   .144   .064   .302   .083   .091   .046  .244  .756
    #> heaR_2   .029   .047   .389   .176   .095  -.043  .252  .748
    #> heaR_3   .011  -.044   .154  -.016   .558   .051  .409  .591
    #> heaR_4   .073   .060   .101  -.041   .571  -.008  .455  .545
    #> heaR_5  -.016   .002   .362   .185   .158  -.031  .240  .760
    #> heaR_6   .035   .123   .064   .074   .382   .092  .307  .693
    #> recR_1  -.024   .282  -.050   .254   .193   .122  .322  .678
    #> recR_2   .153   .295  -.014  -.016   .467  -.028  .493  .507
    #> recR_3   .105   .441  -.118   .062   .348   .056  .507  .493
    #> recR_4   .010   .881   .053   .014  -.060   .001  .774  .226
    #> recR_5   .052   .897   .095  -.053  -.084  -.039  .786  .214
    #> recR_6  -.016   .524   .004   .068   .191   .178  .534  .466
    #> socR_1  -.004   .033  -.010   .656  -.104  -.134  .400  .600
    #> socR_2   .109   .000   .043   .585   .074   .018  .401  .599
    #> socR_3   .009   .054  -.095   .539  -.147   .110  .348  .652
    #> socR_4   .031  -.034  -.091   .576   .169  -.015  .372  .628
    #> socR_5  -.061   .059   .091   .359   .086   .137  .234  .766
    #> socR_6  -.064  -.017   .114   .542  -.036   .155  .363  .637
    #>
    #> ── Factor Intercorrelations ────────────────────────────────────────────────────
    #>
    #>       F1     F2     F3     F4     F5     F6
    #> F1  1.000
    #> F2   .303  1.000
    #> F3   .379   .272  1.000
    #> F4   .017   .245   .030  1.000
    #> F5   .367   .412   .436   .173  1.000
    #> F6   .353   .353   .186   .282   .238  1.000
    #>
    #> ── Variances Accounted for ─────────────────────────────────────────────────────
    #>
    #>                      F1     F2     F3     F4     F5     F6
    #> SS loadings        2.749  2.655  2.465  2.162  1.869  1.797
    #> Prop Tot Var        .092   .088   .082   .072   .062   .060
    #> Cum Prop Tot Var    .092   .180   .262   .334   .397   .457
    #> Prop Comm Var       .201   .194   .180   .158   .136   .131
    #> Cum Prop Comm Var   .201   .395   .574   .732   .869  1.000
    #>
    #> ── Model Fit ───────────────────────────────────────────────────────────────────
    #>
    #> χ²(270) = 618.06, p < .001
    #> CFI: .93
    #> TLI: .89
    #> RMSEA [90% CI]: .05 [.05; .06]
    #> AIC: 78.06
    #> BIC: -1059.88
    #> ECVI: 2.02
    #> CAF: .48
    #> SRMR: .03

Note that `estimator = "uls"` was written in lowercase although the canonical spelling is `"ULS"`. This holds throughout the `efa_*` interface: wherever an argument takes a fixed set of values, capitalization does not matter, while the result stores and prints the canonical spelling. For an argument that takes a single value an unambiguous abbreviation is enough as well, and a value matching nothing raises an error naming the argument and listing the choices. Arguments that take several values at once, such as the analytic choices `efa_average()` averages over, need the full spelling of each value.

## 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

## Schmid-Leiman Transformation and McDonald’s Omegas

For the Schmid-Leiman transformation and the omega coefficients we return to the PAF/promax solution (`efa_dospert`) from above. Its indicator names encode the risk domain each item belongs to (ethical, financial, health, recreational, and social), and its pattern coefficients recovered these theoretical domains fairly well: items from the same domain tend to load on the same factor. The factor intercorrelations also included some sizeable values, which raises the question of whether a general factor underlies the domains. A Schmid-Leiman transformation lets us examine that.

### Schmid-Leiman Transformation

The Schmid-Leiman (SL) transformation orthogonalises an oblique solution into a hierarchical one with a general factor and orthogonalised group factors. `efa_schmid_leiman()` performs it:

    sl_dospert <- efa_schmid_leiman(efa_dospert)
    sl_dospert
    #>
    #> EFA for second-order loadings performed with estimator = 'PAF'
    #>
    #> ── Schmid-Leiman Solution ──────────────────────────────────────────────────────
    #>
    #>           g     F1     F2     F3     F4     F5     F6    h2    u2
    #> ethR_1  .487   .440  -.018  -.016   .067   .009   .093  .445  .555
    #> ethR_2  .400   .360  -.041   .104   .050   .110  -.033  .318  .682
    #> ethR_3  .441   .428   .033   .054   .060  -.202   .023  .427  .573
    #> ethR_4  .225   .559  -.026  -.031  -.084   .000  -.001  .372  .628
    #> ethR_5  .406   .308   .119  -.041   .067  -.095   .028  .290  .710
    #> ethR_6  .325   .534   .064  -.003  -.046  -.060  -.034  .402  .598
    #> finR_1  .523  -.012  -.014   .676   .011   .003   .006  .731  .269
    #> finR_2  .266  -.041   .090  -.035  -.109   .114   .458  .316  .684
    #> finR_3  .528   .037   .026   .709  -.049   .058  -.008  .789  .211
    #> finR_4  .529   .032  -.053   .078   .018  -.097   .587  .643  .357
    #> finR_5  .574  -.005  -.008   .703   .012  -.036   .043  .827  .173
    #> finR_6  .528  -.018  -.038   .018   .031   .029   .592  .633  .367
    #> heaR_1  .403   .256   .031   .088   .047   .088   .014  .246  .754
    #> heaR_2  .344   .335   .015  -.002   .048   .182  -.062  .270  .730
    #> heaR_3  .503   .107  -.104  -.056   .347  -.034  -.007  .400  .600
    #> heaR_4  .548   .060  -.023  -.005   .359  -.064  -.057  .441  .559
    #> heaR_5  .331   .310  -.028  -.042   .088   .187  -.055  .254  .746
    #> heaR_6  .491   .036   .046  -.021   .235   .056   .037  .304  .696
    #> recR_1  .417  -.054   .189  -.047   .113   .235   .072  .288  .712
    #> recR_2  .601  -.037   .172   .071   .296  -.044  -.069  .492  .508
    #> recR_3  .589  -.125   .299   .042   .218   .033   .008  .503  .497
    #> recR_4  .550   .034   .689  -.008  -.046   .002  -.013  .780  .220
    #> recR_5  .549   .071   .704   .027  -.059  -.062  -.044  .812  .188
    #> recR_6  .585  -.016   .380  -.053   .109   .050   .118  .518  .482
    #> socR_1  .030   .013   .021   .020  -.071   .646  -.124  .439  .561
    #> socR_2  .306   .046  -.026   .081   .037   .573  -.015  .433  .567
    #> socR_3  .081  -.068   .045   .023  -.103   .532   .084  .315  .685
    #> socR_4  .217  -.073  -.061   .018   .102   .555  -.044  .376  .624
    #> socR_5  .282   .078   .023  -.068   .040   .352   .090  .225  .775
    #> socR_6  .211   .108  -.026  -.057  -.040   .540   .109  .366  .634
    #>
    #> ── Variances Accounted for ─────────────────────────────────────────────────────
    #>
    #>                      g      F1     F2     F3    F4     F5     F6
    #> SS loadings        5.710  1.550  1.327  1.518  .553  1.994  1.000
    #> Prop Tot Var        .190   .052   .044   .051  .018   .066   .033
    #> Cum Prop Tot Var    .190   .242   .286   .337  .355   .422   .455
    #> Prop Comm Var       .418   .113   .097   .111  .040   .146   .073
    #> Cum Prop Comm Var   .418   .532   .629   .740  .781   .927  1.000

The general-factor column shows that the items load substantially on the general factor across all domains except the social one, whose items load least strongly on it on average (mean general-factor loading around .19, against roughly .38 to .55 in the other domains) — that is, the non-social domains covary considerably, while the social domain stands somewhat apart. Individual social items vary considerably around that average, so this is a statement about the domain and not about every item in it.

### McDonald’s Omegas

Finally, `efa_reliability()` computes omega estimates and related indices from the SL solution. You can either specify the indicator-to-factor correspondences via the `factor_map` argument or let the function determine them automatically, in which case each indicator is assigned to its highest-loading factor. With a reasonably simple structure here, it is easiest to leave `factor_map` unspecified — a few items do land on a neighbouring domain’s factor, which is exactly what motivates the explicit map shown afterwards:

    efa_reliability(sl_dospert)
    #>
    #> Total variance from the correlation matrix.
    #>
    #> ── Reliability coefficients ────────────────────────────────────────────────────
    #>
    #>      tot  hier   sub  alpha    H
    #> g   .931  .718  .177   .886  .886
    #> F1  .797  .377  .415   .788  .652
    #> F2  .872  .474  .393   .845  .683
    #> F3  .912  .344  .568   .914  .739
    #> F4  .715  .531  .177   .716  .305
    #> F5  .742  .124  .615   .741  .726
    #> F6  .747  .295  .451   .742  .571
    #>
    #> ── Common-variance indices ─────────────────────────────────────────────────────
    #>
    #>     ECV   PUC
    #> g  .418  .828

To set the correspondences explicitly instead — for example to match a theoretical assignment — pass a `factor_map` matrix:

    efa_reliability(sl_dospert,
      factor_map = matrix(c(rep(1, 6), rep(0, 24),                  # F1: ethical
                            rep(0, 18), rep(1, 6), rep(0, 6),       # F2: recreational
                            rep(0, 6), 1, 0, 1, 0, 1, rep(0, 19),   # F3: financial (1, 3, 5)
                            rep(0, 12), rep(1, 6), rep(0, 12),      # F4: health
                            rep(0, 24), rep(1, 6),                  # F5: social
                            rep(0, 7), 1, 0, 1, 0, 1, rep(0, 18)),  # F6: financial (2, 4, 6)
                          ncol = 6, byrow = FALSE))
    #>
    #> Total variance from the correlation matrix.
    #>
    #> ── Reliability coefficients ────────────────────────────────────────────────────
    #>
    #>      tot  hier   sub  alpha    H
    #> g   .931  .718  .150   .886  .886
    #> F1  .743  .317  .420   .763  .611
    #> F2  .851  .535  .292   .844  .689
    #> F3  .912  .344  .568   .914  .739
    #> F4  .665  .479  .088   .697  .262
    #> F5  .744  .082  .660   .735  .722
    #> F6  .747  .295  .451   .742  .571
    #>
    #> ── Common-variance indices ─────────────────────────────────────────────────────
    #>
    #>     ECV   PUC
    #> g  .418  .848

The columns of `factor_map` are matched to the group factors by position, so the assignment has to be read off the solution rather than assumed: a column naming the wrong set of items yields a well-formed but meaningless subscale omega instead of an error. Comparing the two tables, the health row (F4) drops more in omega subscale than in omega total, because three of its six items actually load higher on the ethical factor and still contribute their true-score variance to the total through that factor.

Which coefficient to report depends on the question being asked. Omega hierarchical asks whether a total score measures a single construct. Omega subscale asks whether a subscale carries anything beyond the general factor. The H index asks whether a factor is well defined by its indicators. ECV and PUC together ask whether a unidimensional model would be defensible at all. Alpha is reported alongside them because it is still widely expected, but it assumes essentially tau-equivalent items — equal true-score contributions — whereas factor analysis yields congeneric solutions, for which alpha is only a lower bound. For a multidimensional scale such as the DOSPERT it is therefore rarely the coefficient to report, and the omegas answer the question alpha is usually asked to answer. See `?efa_reliability` for the definition of each coefficient and the references behind it.

## Where to Next

This vignette covered the core workflow. `EFAtools` goes further in several directions, each of which has its own vignette or article:

- EFA with ordinal and missing data: analysing ordinal items with polychoric correlations and handling missing data.
- Migrating to the efa_* interface: a guide for users coming from the older uppercase function names.
- Multigroup EFA (on the package website): comparing factor structures across groups.
- Simulation and power (on the package website): simulating data and running power analyses for EFA.

Run `browseVignettes("EFAtools")` to see the vignettes installed with the package, or visit the package website for the full set of articles.