Factor Retention Methods
As the goal of EFA is to determine the underlying factors from a set of multiple variables, one of the most important decisions is how many factors can or should be extracted. There exists a plethora of factor retention methods to use for this decision. The problem is that there is no method that consistently outperforms all other methods. Rather, which factor retention method to use depends on the structure of the data: are there few or many indicators, are factors strong or weak, are the factor intercorrelations weak or strong. For rules on which methods to use, see, for example, Auerswald and Moshagen, (2019).
There are multiple factor retention methods implemented in the
EFAtools package. They can either be called with separate
functions, or all (or a selection) of them using the
N_FACTORS() function.
Calling Separate Functions
Let’s first look at how to determine the number of factors to retain by calling separate functions. For example, if you would like to perform a parallel analysis based on squared multiple correlations (SMC; sometimes also called a parallel analysis with principal factors), you can do the following:
# determine the number of factors to retain using parallel analysis
PARALLEL(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 printed explicitly:
# determine the number of factors to retain using parallel analysis
print(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: 10Other factor retention methods can be used accordingly. For example,
to use the empirical Kaiser criterion, use the EKC
function:
# determine the number of factors to retain using parallel analysis
print(EKC(DOSPERT_sub), plot = FALSE)
#> i 'x' was not a correlation matrix. Correlations are found from entered raw data.
#>
#> Empirical Kaiser criterion suggests 4 factors.The following factor retention methods are currently implemented:
comparison data (CD()), empirical Kaiser criterion
(EKC()), the hull method (HULL()), the
Kaiser-Guttman criterion (KGC()), parallel analysis
(PARALLEL()), scree test (SCREE()), and
sequential model tests (SMT()). Many of these functions
have multiple versions of the respective factor retention method
implemented, for example, the parallel analysis can be done based on
eigenvalues found using unity (principal components) or SMCs, or on an
EFA procedure. Another example is the hull method, which can be used
with different fitting methods (principal axis factoring [PAF], maximum
likelihood [ML], or unweighted least squares [ULS]), and different
goodness of fit indices. Please see the respective function
documentations for details.
Run Multiple Factor Retention Methods With
N_FACTORS()
If you want to use multiple factor retention methods, for example, to
compare whether different methods suggest the same number of factors, it
is easier to use the N_FACTORS() function. This is a
wrapper around all the implemented factor retention methods. Moreover,
it also enables to run the Bartlett’s test of sphericity and compute the
KMO criterion.
For example, to test the suitability of the data for factor analysis and to determine the number of factors to retain based on parallel analysis (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"))
#>
<U+0001F3C3> ( ) ( ) Running EKC
(*) <U+0001F3C3> ( ) Running PARALLEL
(*) (*) <U+0001F3C3> Running SMT
(*) (*) (*) Done!
#>
#> -- Tests for the suitability of the data for factor analysis -------------------
#>
#> Bartlett's test of sphericity
#>
#> v The Bartlett's test of sphericity was significant at an alpha level of .05.
#> These data are probably suitable for factor analysis.
#>
#> <U+0001D712>²(435) = 5843.05, p < .001
#>
#> Kaiser-Meyer-Olkin criterion (KMO)
#>
#> v 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 ------
#>
#> ( ) Empirical Kaiser criterion: 4
#> ( ) Parallel analysis with PCA: 5
#> ( ) Parallel analysis with SMC: 10
#> ( ) Sequential <U+0001D712>² model tests: 14
#> ( ) Lower bound of RMSEA 90% confidence interval: 6
#> ( ) Akaike Information Criterion: 13If all possible factor retention methods should be used, it is sufficient to provide the data object (note that this takes a while, as the comparison data is computationally expensive and therefore relatively slow method, especially if larger datasets are used). We additionally specify the method argument to use unweighted least squares (ULS) estimation. This is a bit faster than using principle axis factoring (PAF) and it enables the computation of more goodness of fit indices:
N_FACTORS(DOSPERT_sub, method = "ULS")
#>
<U+0001F3C3> ( ) ( ) ( ) ( ) ( ) ( ) Running CD
(*) <U+0001F3C3> ( ) ( ) ( ) ( ) ( ) Running EKC
(*) (*) <U+0001F3C3> ( ) ( ) ( ) ( ) Running HULL
(*) (*) (*) <U+0001F3C3> ( ) ( ) ( ) Running KGC
(*) (*) (*) (*) <U+0001F3C3> ( ) ( ) Running PARALLEL
(*) (*) (*) (*) (*) <U+0001F3C3> ( ) Running SCREE
(*) (*) (*) (*) (*) (*) <U+0001F3C3> Running SMT
(*) (*) (*) (*) (*) (*) (*) Done!
#>
#> -- Tests for the suitability of the data for factor analysis -------------------
#>
#> Bartlett's test of sphericity
#>
#> v The Bartlett's test of sphericity was significant at an alpha level of .05.
#> These data are probably suitable for factor analysis.
#>
#> <U+0001D712>²(435) = 5843.05, p < .001
#>
#> Kaiser-Meyer-Olkin criterion (KMO)
#>
#> v 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: 8
#> ( ) Empirical Kaiser criterion: 4
#> ( ) Hull method with CAF: 6
#> ( ) Hull method with CFI: 6
#> ( ) Hull method with RMSEA: 6
#> ( ) Kaiser-Guttman criterion with PCA: 8
#> ( ) Kaiser-Guttman criterion with SMC: 4
#> ( ) Kaiser-Guttman criterion with EFA: 3
#> ( ) Parallel analysis with PCA: 5
#> ( ) Parallel analysis with SMC: 10
#> ( ) Parallel analysis with EFA: 7
#> ( ) Sequential <U+0001D712>² model tests: 14
#> ( ) Lower bound of RMSEA 90% confidence interval: 6
#> ( ) Akaike Information Criterion: 13
Now, this is not the scenario one is happy about, but it still does happen: There is no obvious convergence between the methods and thus the choice of the number of factors to retain becomes rather difficult (and to some extend arbitrary). We will proceed with 6 factors, as it is what is typically used with DOSPERT data, but this does not mean that other number 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 use
method = "ULS" and
eigen_type_other = c("SMC", "PCA") to skip the slower
criteria. 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.
#>
(*) <U+0001F3C3> ( ) ( ) ( ) ( ) ( ) Running EKC
(*) (*) <U+0001F3C3> ( ) ( ) ( ) ( ) Running HULL
(*) (*) (*) <U+0001F3C3> ( ) ( ) ( ) Running KGC
(*) (*) (*) (*) <U+0001F3C3> ( ) ( ) Running PARALLEL
(*) (*) (*) (*) (*) <U+0001F3C3> ( ) Running SCREE
(*) (*) (*) (*) (*) (*) <U+0001F3C3> Running SMT
(*) (*) (*) (*) (*) (*) (*) Done!
#>
#> -- Tests for the suitability of the data for factor analysis -------------------
#>
#> Bartlett's test of sphericity
#>
#> v The Bartlett's test of sphericity was significant at an alpha level of .05.
#> These data are probably suitable for factor analysis.
#>
#> <U+0001D712>²(153) = 2173.28, p < .001
#>
#> Kaiser-Meyer-Olkin criterion (KMO)
#>
#> v 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
#> ( ) Empirical Kaiser criterion: 2
#> ( ) Hull method with CAF: 3
#> ( ) Hull method with CFI: 3
#> ( ) Hull method with RMSEA: 3
#> ( ) Kaiser-Guttman criterion with PCA: 3
#> ( ) Kaiser-Guttman criterion with SMC: 1
#> ( ) Parallel analysis with PCA: 3
#> ( ) Parallel analysis with SMC: 3
#> ( ) Sequential <U+0001D712>² model tests: 3
#> ( ) Lower bound of RMSEA 90% confidence interval: 2
#> ( ) Akaike Information Criterion: 3
