This vignette provides an overview for the functionalities of theEFAtools package. The general aim of the package is to provide flexibleimplementations of different algorithms for an exploratory factoranalyses (EFA) procedure, including factor retention methods, factorextraction and rotation methods, as well as the computation of aSchmid-Leiman solution and McDonald’s omega coefficients.
The package was first designed to enable a comparison of EFA(specifically, principal axis factoring with subsequent promax rotation)performed in R using thepsychpackage and EFA performed in SPSS. That is why some functions allow thespecification of a type, including"psych" and"SPSS", such that the respective procedure will be executedto match the output of these implementations (which do not always leadto the same results; see separate vignetteReplicate_SPSS_psychfor a demonstration of the replication of original results). Thisvignette will go through a complete example, that is, we will first showhow to determine the number of factors to retain, then perform differentfactor extraction methods, run a Schmid-Leiman transformation andcompute omegas.
Factor Retention Methods
As the goal of EFA is to determine the underlying factors from a setof multiple variables, one of the most important decisions is how manyfactors can or should be extracted. There exists a plethora of factorretention methods to use for this decision. The problem is that there isno method that consistently outperforms all other methods. Rather, whichfactor retention method to use depends on the structure of the data: arethere few or many indicators, are factors strong or weak, are the factorintercorrelations weak or strong. For rules on which methods to use,see, for example,Auerswaldand Moshagen, (2019).
There are multiple factor retention methods implemented in theEFAtools package. They can either be called with separatefunctions, or all (or a selection) of them using theN_FACTORS() function.
Calling Separate Functions
Let’s first look at how to determine the number of factors to retainby calling separate functions. For example, if you would like to performa parallel analysis based on squared multiple correlations (SMC;sometimes also called a parallel analysis with principal factors), youcan do the following:
# determine the number of factors to retain using parallel analysisPARALLEL(DOSPERT_sub,eigen_type ="SMC")#> ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.#> Parallel Analysis performed using 1000 simulated random data sets#> Eigenvalues were found using SMC#>#> Decision rule used: means#>#> ── Number of factors to retain according to ────────────────────────────────────#>#> • SMC-determined eigenvalues: 10
Generating the plot can also be suppressed if the output is printedexplicitly:
# determine the number of factors to retain using parallel analysisprint(PARALLEL(DOSPERT_sub,eigen_type ="SMC"),plot =FALSE)#> ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.#> Parallel Analysis performed using 1000 simulated random data sets#> Eigenvalues were found using SMC#>#> Decision rule used: means#>#> ── Number of factors to retain according to ────────────────────────────────────#>#> • SMC-determined eigenvalues: 10
Other factor retention methods can be used accordingly. For example,to use the empirical Kaiser criterion, use theEKCfunction:
# determine the number of factors to retain using parallel analysisprint(EKC(DOSPERT_sub),plot =FALSE)#> ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.#> ℹ The default implementation of EKC has changed compared to EFAtools version <= 0.5.0 to reflect the original version by Braeken and van Assen (2017). The previous version (which often yields different results from the original) is available with type = 'AM2019' for comparison purposes. See details in the help page.#> ── Number of factors suggested by EKC ──────────────────────────────────────────#>#> • Original implementation (Braeken & van Assen, 2017): 8#> ℹ Different implementations of EKC exist. Make sure to report which one you relied on (see EKC help page for details)
The following factor retention methods are currently implemented:comparison data (CD()), empirical Kaiser criterion(EKC()), the hull method (HULL()), theKaiser-Guttman criterion (KGC()), parallel analysis(PARALLEL()), scree test (SCREE()), andsequential model tests (SMT()). Many of these functionshave multiple versions of the respective factor retention methodimplemented, for example, the parallel analysis can be done based oneigenvalues found using unity (principal components) or SMCs, or on anEFA procedure. Another example is the hull method, which can be usedwith different fitting methods (principal axis factoring [PAF], maximumlikelihood [ML], or unweighted least squares [ULS]), and differentgoodness of fit indices. Please see the respective functiondocumentations for details.
Run Multiple Factor Retention Methods WithN_FACTORS()
If you want to use multiple factor retention methods, for example, tocompare whether different methods suggest the same number of factors, itis easier to use theN_FACTORS() function. This is awrapper around all the implemented factor retention methods. Moreover,it also enables to run the Bartlett’s test of sphericity and compute theKMO criterion.
For example, to test the suitability of the data for factor analysisand to determine the number of factors to retain based on parallelanalysis (but only using eigen values based on SMCs and PCA), the EKC,and the sequential model test, we can run the following code:
N_FACTORS(DOSPERT_sub,criteria =c("PARALLEL","EKC","SMT"),eigen_type_other =c("SMC","PCA"))#> ℹ The default implementation of EKC has changed compared to EFAtools version <= 0.5.0 to reflect the original version by Braeken and van Assen (2017). The previous version (which often yields different results from the original) is available with type = 'AM2019' for comparison purposes. See details in the help page.#>#> ── Tests for the suitability of the data for factor analysis ───────────────────#>#> Bartlett's test of sphericity#>#> ✔ 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#>#> Kaiser-Meyer-Olkin criterion (KMO)#>#> ✔ The overall KMO value for your data is meritorious with 0.87.#> These data are probably suitable for factor analysis.#>#> ── Number of factors suggested by the different factor retention criteria ──────#>#> • EKC (original implementation, type 'BvA2017'): 8#> • Parallel analysis with PCA: 5#> • Parallel analysis with SMC: 10#> • Sequential 𝜒² model tests: 14#> • Lower bound of RMSEA 90% confidence interval: 6#> • Akaike Information Criterion: 13
If all possible factor retention methods should be used, it issufficient to provide the data object (note that this takes a while, asthe comparison data is computationally expensive and thereforerelatively slow method, especially if larger datasets are used). Weadditionally specify the method argument to use unweighted least squares(ULS) estimation. This is a bit faster than using principle axisfactoring (PAF) and it enables the computation of more goodness of fitindices:
N_FACTORS(DOSPERT_sub,method ="ULS")#> ℹ The default implementation of EKC has changed compared to EFAtools version <= 0.5.0 to reflect the original version by Braeken and van Assen (2017). The previous version (which often yields different results from the original) is available with type = 'AM2019' for comparison purposes. See details in the help page.#>#> ── Tests for the suitability of the data for factor analysis ───────────────────#>#> Bartlett's test of sphericity#>#> ✔ 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#>#> Kaiser-Meyer-Olkin criterion (KMO)#>#> ✔ The overall KMO value for your data is meritorious with 0.87.#> These data are probably suitable for factor analysis.#>#> ── Number of factors suggested by the different factor retention criteria ──────#>#> • Comparison data: 5#> • EKC (original implementation, type 'BvA2017'): 8#> • Hull method with CAF: 6#> • Hull method with CFI: 6#> • Hull method with RMSEA: 6#> • Parallel analysis with SMC: 10#> • NEST: 10
Now, this is not the scenario one is happy about, but it still doeshappen: There is no obvious convergence between the methods and thus thechoice of the number of factors to retain becomes rather difficult (andto some extend arbitrary). We will proceed with 6 factors, as it is whatis typically used with DOSPERT data, but this does not mean that othernumber of factors are not just as plausible.
Note that all factor retention methods, except comparison data (CD),can also be used with correlation matrices. We usemethod = "ULS" andeigen_type_other = c("SMC", "PCA") to skip the slowercriteria. In this case, the sample size has to be specified:
N_FACTORS(test_models$baseline$cormat,N =500,method ="ULS",eigen_type_other =c("SMC","PCA"))#> Warning in N_FACTORS(test_models$baseline$cormat, N = 500, method = "ULS", : ! 'x' was a correlation matrix but CD needs raw data. Skipping CD.#> ℹ The default implementation of EKC has changed compared to EFAtools version <= 0.5.0 to reflect the original version by Braeken and van Assen (2017). The previous version (which often yields different results from the original) is available with type = 'AM2019' for comparison purposes. See details in the help page.#>#> ── Tests for the suitability of the data for factor analysis ───────────────────#>#> Bartlett's test of sphericity#>#> ✔ The Bartlett's test of sphericity was significant at an alpha level of .05.#> These data are probably suitable for factor analysis.#>#> 𝜒²(153) = 2173.28, p < .001#>#> Kaiser-Meyer-Olkin criterion (KMO)#>#> ✔ The overall KMO value for your data is marvellous with 0.916.#> These data are probably suitable for factor analysis.#>#> ── Number of factors suggested by the different factor retention criteria ──────#>#> • Comparison data: NA#> • EKC (original implementation, type 'BvA2017'): 3#> • Hull method with CAF: 3#> • Hull method with CFI: 3#> • Hull method with RMSEA: 3#> • Parallel analysis with PCA: 3#> • Parallel analysis with SMC: 3#> • NEST: 3
Exploratory Factor Analysis: Factor Extraction
Multiple algorithms to perform an EFA and to rotate the foundsolutions are implemented in theEFAtools package. All ofthem can be used using theEFA() function. To perform theEFA, you can use one of principal axis factoring (PAF), maximumlikelihood estimation (ML), and unweighted least squares (ULS; alsosometimes referred to as MINRES). To rotate the solutions, theEFAtools package offers varimax and promax rotations, aswell as the orthogonal and oblique rotations provided by theGPArotation package (i.e., theGPArotationfunctions are called in theEFA() function in thiscase).
You can run an EFA with PAF and no rotation like this:
EFA(DOSPERT_sub,n_factors =6)#> ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.#>#> EFA performed with type = 'EFAtools', method = '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 0.850 0.641#> Prop Tot Var 0.234 0.081 0.051 0.042 0.028 0.021#> Cum Prop Tot Var 0.234 0.314 0.365 0.407 0.435 0.457#> Prop Comm Var 0.512 0.177 0.112 0.091 0.062 0.047#> Cum Prop Comm Var 0.512 0.689 0.800 0.891 0.953 1.000#>#> ── Model Fit ───────────────────────────────────────────────────────────────────#>#> CAF: .48#> df: 270
To rotate the loadings (e.g., using a promax rotation) adapt therotation argument:
EFA(DOSPERT_sub,n_factors =6,rotation ="promax")#> ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.#>#> EFA performed with type = 'EFAtools', method = 'PAF', and rotation = 'promax'.#>#> ── Rotated Loadings ────────────────────────────────────────────────────────────#>#> F3 F4 F1 F2 F6 F5 h2 u2#> ethR_1 -.020 -.025 .547 .009 .130 .122 .442 .558#> ethR_2 .132 -.055 .447 .115 .098 -.043 .316 .684#> ethR_3 .068 .045 .532 -.212 .117 .030 .470 .530#> ethR_4 -.039 -.035 .695 .000 -.162 -.002 .356 .644#> ethR_5 -.051 .159 .382 -.100 .130 .036 .292 .708#> ethR_6 -.003 .086 .664 -.063 -.089 -.044 .405 .595#> finR_1 .853 -.019 -.015 .003 .022 .008 .730 .270#> finR_2 -.044 .121 -.050 .120 -.211 .599 .359 .641#> finR_3 .895 .035 .046 .061 -.094 -.011 .778 .222#> finR_4 .098 -.071 .039 -.102 .035 .768 .633 .367#> finR_5 .887 -.010 -.006 -.038 .023 .057 .842 .158#> finR_6 .023 -.051 -.022 .031 .061 .774 .635 .365#> heaR_1 .111 .042 .318 .092 .091 .019 .244 .756#> heaR_2 -.003 .020 .416 .191 .093 -.081 .252 .748#> heaR_3 -.071 -.140 .133 -.036 .674 -.009 .409 .591#> heaR_4 -.006 -.031 .074 -.067 .697 -.074 .455 .545#> heaR_5 -.053 -.037 .385 .197 .170 -.072 .240 .760#> heaR_6 -.026 .062 .044 .059 .455 .048 .307 .693#> recR_1 -.060 .253 -.067 .247 .219 .094 .322 .678#> recR_2 .089 .231 -.046 -.046 .574 -.090 .493 .507#> recR_3 .053 .402 -.155 .035 .423 .011 .507 .493#> recR_4 -.010 .926 .042 .002 -.089 -.017 .774 .226#> recR_5 .034 .945 .088 -.065 -.114 -.058 .784 .216#> recR_6 -.067 .510 -.020 .053 .212 .154 .534 .466#> socR_1 .025 .029 .016 .679 -.137 -.162 .400 .600#> socR_2 .103 -.035 .057 .602 .072 -.020 .401 .599#> socR_3 .029 .061 -.084 .559 -.199 .110 .348 .652#> socR_4 .023 -.082 -.090 .583 .197 -.057 .372 .628#> socR_5 -.086 .032 .097 .370 .077 .117 .234 .766#> socR_6 -.072 -.035 .134 .568 -.079 .142 .363 .637#>#> ── Factor Intercorrelations ────────────────────────────────────────────────────#>#> F1 F2 F3 F4 F5 F6#> F1 1.000 0.349 0.455 0.046 0.502 0.463#> F2 0.349 1.000 0.357 0.316 0.594 0.441#> F3 0.455 0.357 1.000 0.031 0.567 0.329#> F4 0.046 0.316 0.031 1.000 0.287 0.324#> F5 0.502 0.594 0.567 0.287 1.000 0.486#> F6 0.463 0.441 0.329 0.324 0.486 1.000#>#> ── Variances Accounted for ─────────────────────────────────────────────────────#>#> F3 F4 F1 F2 F6 F5#> SS loadings 2.457 2.518 2.565 2.213 2.225 1.717#> Prop Tot Var 0.082 0.084 0.086 0.074 0.074 0.057#> Cum Prop Tot Var 0.082 0.166 0.251 0.325 0.399 0.457#> Prop Comm Var 0.179 0.184 0.187 0.162 0.162 0.125#> Cum Prop Comm Var 0.179 0.363 0.551 0.712 0.875 1.000#>#> ── Model Fit ───────────────────────────────────────────────────────────────────#>#> CAF: .48#> df: 270
This now performed PAF with promax rotation with the specification,on average, we found to produce the most accurate results in asimulation analysis (see function documentation). If you want toreplicate the implementation of thepsych R package, you canset thetype argument to"psych":
EFA(DOSPERT_sub,n_factors =6,rotation ="promax",type ="psych")#> ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.#>#> EFA performed with type = 'psych', method = 'PAF', and rotation = 'promax'.#>#> ── Rotated Loadings ────────────────────────────────────────────────────────────#>#> F1 F2 F6 F4 F3 F5 h2 u2#> ethR_1 .554 -.002 -.029 .116 -.024 .136 .442 .558#> ethR_2 .441 .119 -.054 .088 .124 -.035 .316 .684#> ethR_3 .556 -.227 .052 .113 .063 .048 .470 .530#> ethR_4 .704 -.005 .003 -.195 -.042 -.006 .356 .644#> ethR_5 .395 -.111 .156 .131 -.052 .044 .292 .708#> ethR_6 .677 -.070 .117 -.109 -.007 -.047 .405 .595#> finR_1 -.002 .017 -.018 .040 .827 .029 .730 .270#> finR_2 -.034 .103 .116 -.229 -.041 .593 .359 .641#> finR_3 .057 .079 .050 -.082 .868 -.000 .778 .222#> finR_4 .078 -.134 -.100 .026 .096 .801 .633 .367#> finR_5 .013 -.027 -.010 .042 .860 .081 .842 .158#> finR_6 .005 .003 -.088 .051 .023 .801 .635 .365#> heaR_1 .315 .093 .038 .087 .105 .024 .244 .756#> heaR_2 .401 .197 .018 .082 -.007 -.083 .252 .748#> heaR_3 .125 -.044 -.210 .700 -.072 .036 .409 .591#> heaR_4 .068 -.074 -.103 .732 -.009 -.032 .455 .545#> heaR_5 .367 .201 -.048 .162 -.056 -.068 .240 .760#> heaR_6 .036 .054 .008 .477 -.028 .071 .307 .693#> recR_1 -.086 .250 .215 .233 -.059 .089 .322 .678#> recR_2 -.051 -.049 .170 .618 .085 -.065 .493 .507#> recR_3 -.160 .032 .348 .464 .051 .019 .507 .493#> recR_4 .051 -.002 .933 -.063 -.010 -.059 .774 .226#> recR_5 .102 -.069 .961 -.087 .033 -.100 .784 .216#> recR_6 -.018 .044 .476 .236 -.065 .147 .534 .466#> socR_1 -.043 .713 .032 -.155 .022 -.200 .400 .600#> socR_2 .007 .627 -.057 .062 .097 -.034 .401 .599#> socR_3 -.123 .580 .061 -.217 .028 .079 .348 .652#> socR_4 -.144 .608 -.121 .197 .020 -.065 .372 .628#> socR_5 .070 .377 .010 .068 -.085 .109 .234 .766#> socR_6 .094 .583 -.044 -.104 -.073 .122 .363 .637#>#> ── Factor Intercorrelations ────────────────────────────────────────────────────#>#> F1 F2 F3 F4 F5 F6#> F1 1.000 0.149 0.333 0.585 0.435 0.306#> F2 0.149 1.000 0.398 0.387 0.083 0.436#> F3 0.333 0.398 1.000 0.637 0.316 0.501#> F4 0.585 0.387 0.637 1.000 0.463 0.483#> F5 0.435 0.083 0.316 0.463 1.000 0.422#> F6 0.306 0.436 0.501 0.483 0.422 1.000#>#> ── Variances Accounted for ─────────────────────────────────────────────────────#>#> F1 F2 F6 F4 F3 F5#> SS loadings 2.612 2.256 2.332 2.353 2.364 1.778#> Prop Tot Var 0.087 0.075 0.078 0.078 0.079 0.059#> Cum Prop Tot Var 0.087 0.162 0.240 0.318 0.397 0.457#> Prop Comm Var 0.191 0.165 0.170 0.172 0.173 0.130#> Cum Prop Comm Var 0.191 0.355 0.526 0.698 0.870 1.000#>#> ── Model Fit ───────────────────────────────────────────────────────────────────#>#> CAF: .48#> df: 270
If you want to use theSPSS implementation, you can set thetype argument to"SPSS":
EFA(DOSPERT_sub,n_factors =6,rotation ="promax",type ="SPSS")#> ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.#>#> EFA performed with type = 'SPSS', method = 'PAF', and rotation = 'promax'.#>#> ── Rotated Loadings ────────────────────────────────────────────────────────────#>#> F1 F4 F3 F2 F6 F5 h2 u2#> ethR_1 .547 -.025 -.020 .009 .130 .122 .442 .558#> ethR_2 .447 -.055 .132 .115 .098 -.043 .316 .684#> ethR_3 .532 .045 .068 -.212 .117 .030 .470 .530#> ethR_4 .695 -.035 -.039 .000 -.162 -.002 .356 .644#> ethR_5 .382 .159 -.051 -.100 .130 .036 .292 .708#> ethR_6 .664 .086 -.003 -.063 -.089 -.044 .405 .595#> finR_1 -.015 -.019 .853 .003 .022 .008 .730 .270#> finR_2 -.050 .121 -.044 .120 -.211 .599 .359 .641#> finR_3 .046 .035 .895 .061 -.094 -.011 .778 .222#> finR_4 .039 -.071 .098 -.102 .035 .768 .633 .367#> finR_5 -.006 -.010 .887 -.038 .023 .057 .842 .158#> finR_6 -.022 -.051 .023 .031 .061 .774 .635 .365#> heaR_1 .318 .042 .111 .092 .091 .019 .244 .756#> heaR_2 .416 .020 -.003 .191 .093 -.081 .252 .748#> heaR_3 .133 -.140 -.071 -.036 .674 -.009 .409 .591#> heaR_4 .074 -.031 -.006 -.067 .697 -.074 .455 .545#> heaR_5 .385 -.037 -.053 .197 .170 -.072 .240 .760#> heaR_6 .044 .062 -.026 .059 .455 .048 .307 .693#> recR_1 -.067 .253 -.060 .247 .219 .094 .322 .678#> recR_2 -.046 .231 .089 -.046 .574 -.090 .493 .507#> recR_3 -.155 .402 .053 .035 .423 .011 .507 .493#> recR_4 .042 .926 -.010 .002 -.089 -.017 .774 .226#> recR_5 .088 .945 .034 -.065 -.114 -.058 .784 .216#> recR_6 -.020 .510 -.067 .053 .212 .154 .534 .466#> socR_1 .016 .029 .025 .679 -.137 -.162 .400 .600#> socR_2 .057 -.035 .103 .602 .072 -.020 .401 .599#> socR_3 -.084 .061 .029 .559 -.199 .110 .348 .652#> socR_4 -.090 -.082 .023 .583 .197 -.057 .372 .628#> socR_5 .097 .032 -.086 .370 .077 .117 .234 .766#> socR_6 .134 -.035 -.072 .568 -.079 .142 .363 .637#>#> ── Factor Intercorrelations ────────────────────────────────────────────────────#>#> F1 F2 F3 F4 F5 F6#> F1 1.000 0.357 0.455 0.031 0.567 0.329#> F2 0.357 1.000 0.349 0.316 0.594 0.441#> F3 0.455 0.349 1.000 0.046 0.502 0.463#> F4 0.031 0.316 0.046 1.000 0.287 0.324#> F5 0.567 0.594 0.502 0.287 1.000 0.486#> F6 0.329 0.441 0.463 0.324 0.486 1.000#>#> ── Variances Accounted for ─────────────────────────────────────────────────────#>#> F1 F4 F3 F2 F6 F5#> SS loadings 2.565 2.518 2.457 2.213 2.225 1.717#> Prop Tot Var 0.086 0.084 0.082 0.074 0.074 0.057#> Cum Prop Tot Var 0.086 0.169 0.251 0.325 0.399 0.457#> Prop Comm Var 0.187 0.184 0.179 0.162 0.162 0.125#> Cum Prop Comm Var 0.187 0.371 0.551 0.712 0.875 1.000#>#> ── Model Fit ───────────────────────────────────────────────────────────────────#>#> CAF: .48#> df: 270
This enables comparisons of different implementations. TheCOMPARE() function provides an easy way to compare howsimilar two loading (pattern) matrices are:
COMPARE(EFA(DOSPERT_sub,n_factors =6,rotation ="promax",type ="psych")$rot_loadings,EFA(DOSPERT_sub,n_factors =6,rotation ="promax",type ="SPSS")$rot_loadings)#> ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.#>#> ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.#> Mean [min, max] absolute difference: 0.0159 [ 0.0001, 0.0712]#> Median absolute difference: 0.0102#> Max decimals where all numbers are equal: 0#> Minimum number of decimals provided: 18#>#> F1 F2 F6 F4 F3 F5#> ethR_1 0.0078 -0.0106 -0.0042 -0.0138 -0.0035 0.0148#> ethR_2 -0.0059 0.0038 0.0008 -0.0100 -0.0078 0.0082#> ethR_3 0.0245 -0.0146 0.0066 -0.0037 -0.0052 0.0186#> ethR_4 0.0090 -0.0054 0.0375 -0.0325 -0.0028 -0.0045#> ethR_5 0.0128 -0.0116 -0.0035 0.0007 -0.0011 0.0073#> ethR_6 0.0129 -0.0067 0.0315 -0.0208 -0.0038 -0.0033#> finR_1 0.0131 0.0143 0.0012 0.0183 -0.0257 0.0209#> finR_2 0.0162 -0.0171 -0.0048 -0.0177 0.0025 -0.0055#> finR_3 0.0110 0.0182 0.0150 0.0121 -0.0272 0.0107#> finR_4 0.0392 -0.0319 -0.0286 -0.0093 -0.0023 0.0331#> finR_5 0.0191 0.0113 0.0007 0.0192 -0.0266 0.0243#> finR_6 0.0265 -0.0279 -0.0376 -0.0097 -0.0002 0.0271#> heaR_1 -0.0024 0.0009 -0.0042 -0.0038 -0.0062 0.0053#> heaR_2 -0.0153 0.0059 -0.0011 -0.0105 -0.0039 -0.0020#> heaR_3 -0.0075 -0.0086 -0.0702 0.0258 -0.0012 0.0450#> heaR_4 -0.0064 -0.0065 -0.0712 0.0356 -0.0028 0.0424#> heaR_5 -0.0182 0.0048 -0.0112 -0.0087 -0.0024 0.0034#> heaR_6 -0.0085 -0.0045 -0.0537 0.0222 -0.0015 0.0230#> recR_1 -0.0191 0.0028 -0.0382 0.0146 0.0004 -0.0053#> recR_2 -0.0045 -0.0025 -0.0615 0.0438 -0.0046 0.0251#> recR_3 -0.0054 -0.0023 -0.0546 0.0413 -0.0024 0.0080#> recR_4 0.0091 -0.0036 0.0078 0.0261 0.0001 -0.0421#> recR_5 0.0148 -0.0042 0.0154 0.0270 -0.0012 -0.0417#> recR_6 0.0021 -0.0090 -0.0342 0.0247 0.0011 -0.0068#> socR_1 -0.0595 0.0345 0.0032 -0.0180 -0.0031 -0.0379#> socR_2 -0.0494 0.0255 -0.0217 -0.0095 -0.0061 -0.0144#> socR_3 -0.0390 0.0208 0.0001 -0.0185 -0.0017 -0.0311#> socR_4 -0.0543 0.0254 -0.0381 -0.0005 -0.0032 -0.0083#> socR_5 -0.0270 0.0072 -0.0213 -0.0092 0.0003 -0.0082#> socR_6 -0.0401 0.0156 -0.0089 -0.0251 -0.0005 -0.0205
Why would you want to do this? One of us has had theexperience that a reviewer asked whether the results can be reproducedin another statistical program than R. We therefore implemented thispossibility in the package for an easy application of large scale,systematic comparisons.
Note that thetype argument of theEFA()function only affects the implementations of principal axis factoring(PAF), varimax and promax rotations. The other procedures are notaffected (except the order of the rotated factors for the other rotationmethods).
As indicated previously, it is also possible to use differentestimation and rotation methods. For example, to perform an EFA with ULSand an oblimin rotation, you can use the following code:
EFA(DOSPERT_sub,n_factors =6,rotation ="oblimin",method ="ULS")#> ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.#>#> EFA performed with type = 'EFAtools', method = 'ULS', and rotation = 'oblimin'.#>#> ── Rotated Loadings ────────────────────────────────────────────────────────────#>#> F1 F2 F3 F4 F5 F6 h2 u2#> ethR_1 .039 .009 .522 -.007 .146 .137 .441 .559#> ethR_2 .166 -.025 .419 .100 -.008 .097 .316 .684#> ethR_3 .120 .064 .511 -.220 .052 .115 .470 .530#> ethR_4 -.013 -.038 .644 -.031 .008 -.104 .356 .644#> ethR_5 -.003 .177 .371 -.103 .059 .126 .292 .708#> ethR_6 .029 .084 .620 -.086 -.025 -.048 .405 .595#> finR_1 .850 -.013 -.012 .003 .028 .004 .729 .271#> finR_2 -.022 .102 -.039 .106 .551 -.137 .358 .642#> finR_3 .883 .028 .039 .054 .007 -.088 .778 .222#> finR_4 .160 -.054 .063 -.109 .725 .068 .634 .366#> finR_5 .890 -.005 -.002 -.037 .073 .006 .842 .158#> finR_6 .085 -.028 .005 .023 .737 .092 .636 .364#> heaR_1 .144 .064 .302 .083 .046 .091 .244 .756#> heaR_2 .029 .047 .389 .176 -.043 .095 .252 .748#> heaR_3 .011 -.044 .154 -.016 .051 .558 .409 .591#> heaR_4 .073 .060 .101 -.041 -.008 .571 .455 .545#> heaR_5 -.016 .002 .362 .185 -.031 .158 .240 .760#> heaR_6 .035 .123 .064 .074 .092 .382 .307 .693#> recR_1 -.024 .282 -.050 .254 .122 .193 .322 .678#> recR_2 .153 .295 -.014 -.016 -.028 .467 .493 .507#> recR_3 .105 .441 -.118 .062 .056 .348 .507 .493#> recR_4 .010 .881 .053 .014 .001 -.060 .774 .226#> recR_5 .052 .897 .095 -.053 -.039 -.084 .786 .214#> recR_6 -.016 .524 .004 .068 .178 .191 .534 .466#> socR_1 -.004 .033 -.010 .656 -.134 -.104 .400 .600#> socR_2 .109 -.000 .043 .585 .018 .074 .401 .599#> socR_3 .009 .054 -.095 .539 .110 -.147 .348 .652#> socR_4 .031 -.034 -.091 .576 -.015 .169 .372 .628#> socR_5 -.061 .059 .091 .359 .137 .086 .234 .766#> socR_6 -.064 -.017 .114 .542 .155 -.036 .363 .637#>#> ── Factor Intercorrelations ────────────────────────────────────────────────────#>#> F1 F2 F3 F4 F5 F6#> F1 1.000 0.303 -0.379 -0.017 0.353 0.367#> F2 0.303 1.000 -0.272 -0.245 0.353 0.412#> F3 -0.379 -0.272 1.000 0.030 -0.186 -0.436#> F4 -0.017 -0.245 0.030 1.000 -0.282 -0.173#> F5 0.353 0.353 -0.186 -0.282 1.000 0.238#> F6 0.367 0.412 -0.436 -0.173 0.238 1.000#>#> ── Variances Accounted for ─────────────────────────────────────────────────────#>#> F1 F2 F3 F4 F5 F6#> SS loadings 2.570 2.467 1.800 1.992 1.670 1.528#> Prop Tot Var 0.086 0.082 0.060 0.066 0.056 0.051#> Cum Prop Tot Var 0.086 0.168 0.228 0.294 0.350 0.401#> Prop Comm Var 0.214 0.205 0.150 0.166 0.139 0.127#> Cum Prop Comm Var 0.214 0.419 0.568 0.734 0.873 1.000#>#> ── Model Fit ───────────────────────────────────────────────────────────────────#>#> 𝜒²(270) = 213.54, p = .995#> CFI = 1.00#> RMSEA [90% CI] = .00 [.00; .00]#> AIC = -326.46#> BIC = -1464.40#> CAF = .48
Of course,COMPARE() can also be used to compare resultsfrom different estimation or rotation methods (in fact, to compare anytwo matrices), not just from different implementations:
COMPARE(EFA(DOSPERT_sub,n_factors =6,rotation ="promax")$rot_loadings,EFA(DOSPERT_sub,n_factors =6,rotation ="oblimin",method ="ULS")$rot_loadings,x_labels =c("PAF and promax","ULS and oblimin"))#> ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.#>#> ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.#> Mean [min, max] absolute difference: 0.0278 [ 0.0001, 0.1262]#> Median absolute difference: 0.0226#> Max decimals where all numbers are equal: 0#> Minimum number of decimals provided: 18#>#> F3 F4 F1 F2 F6 F5#> ethR_1 -0.0590 -0.0335 0.0244 0.0157 -0.0073 -0.0248#> ethR_2 -0.0345 -0.0298 0.0274 0.0153 0.0006 -0.0348#> ethR_3 -0.0522 -0.0194 0.0208 0.0073 0.0025 -0.0224#> ethR_4 -0.0253 0.0033 0.0509 0.0306 -0.0578 -0.0099#> ethR_5 -0.0482 -0.0174 0.0110 0.0028 0.0045 -0.0227#> ethR_6 -0.0325 0.0018 0.0434 0.0224 -0.0408 -0.0190#> finR_1 0.0021 -0.0061 -0.0027 -0.0001 0.0175 -0.0201#> finR_2 -0.0214 0.0192 -0.0116 0.0142 -0.0742 0.0478#> finR_3 0.0114 0.0077 0.0072 0.0069 -0.0059 -0.0179#> finR_4 -0.0625 -0.0170 -0.0240 0.0071 -0.0334 0.0427#> finR_5 -0.0032 -0.0054 -0.0046 -0.0006 0.0162 -0.0168#> finR_6 -0.0624 -0.0230 -0.0267 0.0075 -0.0316 0.0379#> heaR_1 -0.0331 -0.0223 0.0156 0.0095 0.0003 -0.0269#> heaR_2 -0.0316 -0.0276 0.0272 0.0147 -0.0019 -0.0376#> heaR_3 -0.0820 -0.0955 -0.0213 -0.0202 0.1154 -0.0595#> heaR_4 -0.0791 -0.0912 -0.0269 -0.0266 0.1262 -0.0660#> heaR_5 -0.0376 -0.0393 0.0227 0.0119 0.0123 -0.0404#> heaR_6 -0.0610 -0.0618 -0.0202 -0.0150 0.0728 -0.0437#> recR_1 -0.0354 -0.0289 -0.0171 -0.0074 0.0253 -0.0280#> recR_2 -0.0637 -0.0634 -0.0327 -0.0301 0.1076 -0.0612#> recR_3 -0.0520 -0.0393 -0.0370 -0.0277 0.0750 -0.0450#> recR_4 -0.0201 0.0446 -0.0108 -0.0118 -0.0292 -0.0187#> recR_5 -0.0177 0.0487 -0.0075 -0.0117 -0.0300 -0.0186#> recR_6 -0.0510 -0.0134 -0.0249 -0.0149 0.0203 -0.0238#> socR_1 0.0287 -0.0045 0.0259 0.0225 -0.0333 -0.0286#> socR_2 -0.0068 -0.0351 0.0137 0.0166 -0.0020 -0.0377#> socR_3 0.0203 0.0072 0.0111 0.0206 -0.0524 0.0005#> socR_4 -0.0079 -0.0483 0.0005 0.0065 0.0283 -0.0417#> socR_5 -0.0242 -0.0272 0.0058 0.0114 -0.0084 -0.0191#> socR_6 -0.0076 -0.0179 0.0199 0.0257 -0.0427 -0.0124
Finally, if you are interested in factor scores from the EFAsolution, these can be obtained withFACTOR_SCORES(), awrapper forpsych::factor.scores() to be used directly withan output fromEFA():
EFA_mod<-EFA(DOSPERT_sub,n_factors =6,rotation ="promax")#> ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.fac_scores<-FACTOR_SCORES(DOSPERT_sub,f = EFA_mod)
Performance
To improve performance of the iterative procedures (currently theparallel analysis, and the PAF, ML, and ULS methods) we implemented someof them in C++. For example, the following code compares the EFAtoolsparallel analysis with the corresponding one implemented in the psychpackage (the default ofPARALLEL() is to use 1000 datasets,but 25 is enough to show the difference):
microbenchmark::microbenchmark(PARALLEL(DOSPERT_sub,eigen_type ="SMC",n_datasets =25), psych::fa.parallel(DOSPERT_sub,SMC =TRUE,plot =FALSE,n.iter =25))#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 5#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Parallel analysis suggests that the number of factors = 10 and the number of components = 4#> Unit: milliseconds#> expr#> PARALLEL(DOSPERT_sub, eigen_type = "SMC", n_datasets = 25)#> psych::fa.parallel(DOSPERT_sub, SMC = TRUE, plot = FALSE, n.iter = 25)#> min lq mean median uq max neval cld#> 55.0763 91.33885 92.61304 93.47655 97.64655 116.3787 100 a#> 199.2443 318.68955 330.36341 327.24535 334.44940 450.5079 100 b
Moreover, the following code compares the PAF implementation (of type“psych”) of the EFAtools package with the one from the psychpackage:
microbenchmark::microbenchmark(EFA(DOSPERT_raw,6), psych::fa(DOSPERT_raw,6,rotate ="none",fm ="pa"))#> Unit: milliseconds#> expr min lq#> EFA(DOSPERT_raw, 6) 12.6800 20.07165#> psych::fa(DOSPERT_raw, 6, rotate = "none", fm = "pa") 40.4386 51.23070#> mean median uq max neval cld#> 20.73336 20.47315 21.041 27.5929 100 a#> 56.58892 54.47820 58.141 236.1078 100 b
While these differences are not large, they grow larger the moreiterations the procedures need, which is usually the case if solutionsare more tricky to find. Especially for simulations this might come inhandy. For example, in one simulation analysis we ran over 10,000,000EFAs, thus a difference of about 25 milliseconds per EFA leads to adifference in runtime of almost three days.
Model Averaging
Instead of relying on one of the many possible implementations of,for example, PAF, and of using just one rotation (e.g., promax), it maybe desirable to average different solutions to potentially arrive at amore robust, average solution. TheEFA_AVERAGE() functionprovides this possibility. In addition to the average solution itprovides the variation across solutions, a matrix indicating therobustness of indicator-to-factor correspondences, and a visualisationof the average solution and the variability across solutions. Forexample, to average across all available factor extraction methods andacross all available oblique rotations, the following code can berun:
# Average solution across many different EFAs with oblique rotationsEFA_AV<-EFA_AVERAGE(test_models$baseline$cormat,n_factors =3,N =500,method =c("PAF","ML","ULS"),rotation ="oblique",show_progress =FALSE)# look at solutionEFA_AV#>#> Averaging performed with averaging method mean (trim = 0) across 162 EFAs, varying the following settings: method, init_comm, criterion_type, start_method, rotation, k_promax, P_type, and varimax_type.#>#> The error rate is at 0%. Of the solutions that did not result in an error, 100% 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#> V1 .11 .00 1.00#> V2 .11 .00 1.00#> V3 .11 .00 .94#> V4 .11 .00 1.00#> V5 .11 .00 .94#> V6 .11 .00 1.00#> V7 .11 .94 .00#> V8 .11 1.00 .00#> V9 .11 .94 .00#> V10 .11 1.00 .00#> V11 .11 .89 .00#> V12 .11 1.00 .00#> V13 1.00 .00 .00#> V14 1.00 .00 .00#> V15 1.00 .00 .00#> V16 1.00 .00 .00#> V17 1.00 .00 .00#> V18 1.00 .00 .00#>#>#> ══ Loadings ════════════════════════════════════════════════════════════════════#>#> ── Mean ────────────────────────────────────────────────────────────────────────#>#> F1 F2 F3#> V1 .025 .048 .576#> V2 .060 .077 .451#> V3 .115 .066 .425#> V4 .157 .007 .518#> V5 .198 -.002 .412#> V6 .002 -.028 .658#> V7 .074 .497 .102#> V8 .056 .538 .046#> V9 .100 .510 .018#> V10 .048 .625 -.046#> V11 .082 .336 .228#> V12 .094 .606 .007#> V13 .597 .083 -.047#> V14 .531 -.056 .093#> V15 .548 .122 -.049#> V16 .540 -.041 .097#> V17 .633 -.033 -.009#> V18 .542 .009 .060#>#> ── Range ───────────────────────────────────────────────────────────────────────#>#> F1 F2 F3#> V1 0.513 0.086 0.239#> V2 0.431 0.093 0.186#> V3 0.394 0.108 0.179#> V4 0.415 0.110 0.214#> V5 0.315 0.122 0.177#> V6 0.514 0.104 0.267#> V7 0.527 0.255 0.089#> V8 0.520 0.275 0.078#> V9 0.470 0.276 0.080#> V10 0.533 0.313 0.097#> V11 0.482 0.176 0.102#> V12 0.548 0.324 0.103#> V13 0.081 0.289 0.114#> V14 0.063 0.220 0.117#> V15 0.091 0.280 0.107#> V16 0.072 0.230 0.122#> V17 0.108 0.270 0.124#> V18 0.081 0.246 0.118#>#>#> ══ Factor Intercorrelations from Oblique Solutions ═════════════════════════════#>#> ── Mean ────────────────────────────────────────────────────────────────────────#>#> F1 F2 F3#> F1 1.000 0.431 0.518#> F2 0.431 1.000 0.454#> F3 0.518 0.454 1.000#>#> ── Range ───────────────────────────────────────────────────────────────────────#>#> F1 F2 F3#> F1 0.000 1.276 0.679#> F2 1.276 0.000 1.316#> F3 0.679 1.316 0.000#>#>#> ══ Variances Accounted for ═════════════════════════════════════════════════════#>#> ── Mean ────────────────────────────────────────────────────────────────────────#>#> F1 F2 F3#> SS loadings 2.443 1.929 1.904#> Prop Tot Var 0.136 0.107 0.106#> Prop Comm Var 0.389 0.307 0.303#>#> ── Range ───────────────────────────────────────────────────────────────────────#>#> F1 F2 F3#> SS loadings 2.831 1.356 1.291#> Prop Tot Var 0.157 0.075 0.072#> Prop Comm Var 0.419 0.215 0.215#>#>#> ══ Model Fit ═══════════════════════════════════════════════════════════════════#>#> M (SD) [Min; Max]#> 𝜒²: 101.73 (34.62) [53.23; 125.98]#> df: 102#> p: .369 (.450) [.054; 1.000]#> CFI: 1.00 (.00) [1.00; 1.00]#> RMSEA: .01 (.01) [.00; .02]#> AIC: -102.27 (34.62) [-150.77; -78.02]#> BIC: -532.16 (34.62) [-580.66; -507.91]#> CAF: .50 (.00) [.50; .50]
The first matrix of the output tells us that the indicators aremostly allocated to the same factors. However, that some rowsums arelarger than one also tells as that there likely are some cross loadingspresent in some solutions. Moreover, the relatively high percentages ofsalient pattern coefficients all loading on the first factor mayindicate that some rotation methods failed to achieve simple structureand it might be desirable to exclude these from the model averagingprocedure. The rest of the output is similar to the normalEFA() outputs shown above, only that in addition to theaverage coefficients their range is also shown. Finally, the plot showsthe average pattern coefficients and their ranges.
Important disclaimer: While it is possible that thisapproach provides more robust results, we are unaware of simulationstudies that have investigated and shown this. Therefore, it might makesense to for now use this approach mainly to test the robustness of theresults obtained with one single EFA implementation.
Exploratory Factor Analysis: Schmid-Leiman transformation andMcDonald’s Omegas
For the Schmid-Leiman transformation and computation of omegas, wewill use PAF and promax rotation:
efa_dospert<-EFA(DOSPERT_sub,n_factors =6,rotation ="promax")#> ℹ 'x' was not a correlation matrix. Correlations are found from entered raw data.efa_dospert#>#> EFA performed with type = 'EFAtools', method = 'PAF', and rotation = 'promax'.#>#> ── Rotated Loadings ────────────────────────────────────────────────────────────#>#> F3 F4 F1 F2 F6 F5 h2 u2#> ethR_1 -.020 -.025 .547 .009 .130 .122 .442 .558#> ethR_2 .132 -.055 .447 .115 .098 -.043 .316 .684#> ethR_3 .068 .045 .532 -.212 .117 .030 .470 .530#> ethR_4 -.039 -.035 .695 .000 -.162 -.002 .356 .644#> ethR_5 -.051 .159 .382 -.100 .130 .036 .292 .708#> ethR_6 -.003 .086 .664 -.063 -.089 -.044 .405 .595#> finR_1 .853 -.019 -.015 .003 .022 .008 .730 .270#> finR_2 -.044 .121 -.050 .120 -.211 .599 .359 .641#> finR_3 .895 .035 .046 .061 -.094 -.011 .778 .222#> finR_4 .098 -.071 .039 -.102 .035 .768 .633 .367#> finR_5 .887 -.010 -.006 -.038 .023 .057 .842 .158#> finR_6 .023 -.051 -.022 .031 .061 .774 .635 .365#> heaR_1 .111 .042 .318 .092 .091 .019 .244 .756#> heaR_2 -.003 .020 .416 .191 .093 -.081 .252 .748#> heaR_3 -.071 -.140 .133 -.036 .674 -.009 .409 .591#> heaR_4 -.006 -.031 .074 -.067 .697 -.074 .455 .545#> heaR_5 -.053 -.037 .385 .197 .170 -.072 .240 .760#> heaR_6 -.026 .062 .044 .059 .455 .048 .307 .693#> recR_1 -.060 .253 -.067 .247 .219 .094 .322 .678#> recR_2 .089 .231 -.046 -.046 .574 -.090 .493 .507#> recR_3 .053 .402 -.155 .035 .423 .011 .507 .493#> recR_4 -.010 .926 .042 .002 -.089 -.017 .774 .226#> recR_5 .034 .945 .088 -.065 -.114 -.058 .784 .216#> recR_6 -.067 .510 -.020 .053 .212 .154 .534 .466#> socR_1 .025 .029 .016 .679 -.137 -.162 .400 .600#> socR_2 .103 -.035 .057 .602 .072 -.020 .401 .599#> socR_3 .029 .061 -.084 .559 -.199 .110 .348 .652#> socR_4 .023 -.082 -.090 .583 .197 -.057 .372 .628#> socR_5 -.086 .032 .097 .370 .077 .117 .234 .766#> socR_6 -.072 -.035 .134 .568 -.079 .142 .363 .637#>#> ── Factor Intercorrelations ────────────────────────────────────────────────────#>#> F1 F2 F3 F4 F5 F6#> F1 1.000 0.349 0.455 0.046 0.502 0.463#> F2 0.349 1.000 0.357 0.316 0.594 0.441#> F3 0.455 0.357 1.000 0.031 0.567 0.329#> F4 0.046 0.316 0.031 1.000 0.287 0.324#> F5 0.502 0.594 0.567 0.287 1.000 0.486#> F6 0.463 0.441 0.329 0.324 0.486 1.000#>#> ── Variances Accounted for ─────────────────────────────────────────────────────#>#> F3 F4 F1 F2 F6 F5#> SS loadings 2.457 2.518 2.565 2.213 2.225 1.717#> Prop Tot Var 0.082 0.084 0.086 0.074 0.074 0.057#> Cum Prop Tot Var 0.082 0.166 0.251 0.325 0.399 0.457#> Prop Comm Var 0.179 0.184 0.187 0.162 0.162 0.125#> Cum Prop Comm Var 0.179 0.363 0.551 0.712 0.875 1.000#>#> ── Model Fit ───────────────────────────────────────────────────────────────────#>#> CAF: .48#> df: 270
The indicator names in the output (i.e., the rownames of the rotatedloadings section) tell us which domain (out of ethical, financial,health, recreational, and social risks) an indicator stems from. Fromthe pattern coefficients it can be seen that these theoretical domainsare recovered relatively well in the six factor solution, that is,usually, the indicators from the same domain load onto the same factor.When we take a look at the factor intercorrelations, we can see thatthere are some strong and some weak correlations. It might be worthwhileto explore whether a general factor can be obtained, and which factorsload more strongly on it. To this end, we will use a Schmid-Leiman (SL)transformation.
McDonald’s Omegas
Finally, we can compute omega estimates and additional indices ofinterpretive relevance based on the SL solution. To this end, we caneither specify the variable-to-factor correspondences, or let them bedetermined automatically (in which case the highest factor loading willbe taken, which might lead to a different solution than what is desired,in the presence of cross-loadings). Given that no cross-loadings arepresent here, it is easiest to let the function automatically determinethe variable-to-factor correspondence. To this end, we will set thetype argument to"psych".
OMEGA(sl_dospert,type ="psych")#> Omega total, omega hierarchical, omega subscale, H index, explained common variance (ECV), and percent of uncontaminated correlations (PUC) for the general factor (top row) and omegas and H index for the group factors:#>#> tot hier sub H ECV PUC#> g 0.922 0.718 0.177 0.886 0.441 0.828#> F1 0.791 0.377 0.415 0.652#> F2 0.739 0.124 0.615 0.726#> F3 0.912 0.344 0.568 0.739#> F4 0.867 0.474 0.393 0.683#> F5 0.745 0.295 0.451 0.571#> F6 0.709 0.531 0.177 0.305
If we wanted to specify the variable to factor correspondencesexplicitly (for example, according to theoretical expectations), wecould do it in the following way:
OMEGA(sl_dospert,factor_corres =matrix(c(rep(0,18),rep(1,6),rep(0,30),rep(1,6),rep(0,6),1,0,1,0,1,rep(0,19),rep(1,6),rep(0,31),1,0,1,0,1,rep(0,30),rep(1,6),rep(0,12)),ncol =6,byrow =FALSE))#> Omega total, omega hierarchical, omega subscale, H index, explained common variance (ECV), and percent of uncontaminated correlations (PUC) for the general factor (top row) and omegas and H index for the group factors:#>#> tot hier sub H ECV PUC#> g 0.922 0.718 0.089 0.886 0.561 0.848#> F1 0.536 0.535 0.001 0.026#> F2 0.742 0.082 0.660 0.722#> F3 0.912 0.344 0.568 0.739#> F4 0.318 0.317 0.001 0.022#> F5 0.745 0.295 0.451 0.571#> F6 0.567 0.479 0.088 0.262