On the Detection of the Correct Number of Factors in Two-Facet Models by Means of Parallel Analysis

Abstract

Methods for optimal factor rotation of two-facet loading matrices have recently been proposed. However, the problem of the correct number of factors to retain for rotation of two-facet loading matrices has rarely been addressed in the context of exploratory factor analysis. Most previous studies were based on the observation that two-facet loading matrices may be rank deficient when the salient loadings of each factor have the same sign. It was shown here that full-rank two-facet loading matrices are, in principle, possible, when some factors have positive and negative salient loadings. Accordingly, the current simulation study on the number of factors to extract for two-facet models was based on rank-deficient and full-rank two-facet population models. The number of factors to extract was estimated from traditional parallel analysis based on the mean of the unreduced eigenvalues as well as from nine other rather traditional versions of parallel analysis (based on the 95th percentile of eigenvalues, based on reduced eigenvalues, based on eigenvalue differences). Parallel analysis based on the mean eigenvalues of the correlation matrix with the squared multiple correlations of each variable with the remaining variables inserted in the main diagonal had the highest detection rates for most of the two-facet factor models. Recommendations for the identification of the correct number of factors are based on the simulation results, on the results of an empirical example data set, and on the conditions for approximately rank-deficient and full-rank two-facet models.

Facet models have been discussed since L. Guttman (1954), who demonstrated their utility for intelligence research (L. Guttman & Levy, 1991; Schlesinger & Guttman, 1969). Other models of intelligence structure were also facet models (Guilford, 1967, 1975, 1988; Jäger, 1982, 1984; Jäger et al., 1997). Thus, facet models have been shown to be useful for intelligence research because they allow for a flexible representation of overlapping variances (Sternberg, 1981). Even when facet models have first been developed for models of intelligence structure, there have been attempts to use facet models for research on personality (Beauducel et al., 2005; Leue & Beauducel, in press) and in organizational psychology (Cronshaw & Jethmalani, 2005; Wheeler, 1993). Moreover, the similarity of two-facet models with multitrait–multimethod (MTMM) models (Campbell & Fiske, 1959) has also been noted (Süß & Beauducel, 2005). To sum up, facet models have been proposed in several fields of psychology (R. Guttman & Greenbaum, 1998).

According to L. Guttman (1954), a facet is a set that is a component of a cartesian product (Shye, 1998). For example, the cartesian product of facet A with the elements (1, 2, 3) and facet B with the elements (a, b) is A × B = C with the elements (1a, 2a, 3a, 1b, 2b, 3b). When applied to factor analysis, the elements of the facets are factors so that a two-facet loading pattern comprises two sets of factors with each variable loading on one factor of each facet. An example for a two-facet model is Jäger’s (1982, 1984) Berlin intelligence structure (BIS) model comprising an operation facet with four factors (Speed, Memory, Creativity, Reasoning), a content facet with three factors (Figural, Verbal, and Numerical Intelligence), and a general intelligence “g” factor (see Figure 1). The cartesian product of the operation and content facet yields the “cells” (or “structupels”) of the BIS. Each performance task can be classified into one “cell” of the BIS. Accordingly, each task is expected to load on one operation factor and on one content factor. As a general intelligence factor is assumed, positive intercorrelations of the primary factors are expected. The example demonstrates that the loading patterns resulting from facet models are more complex than conventional simple structure loading patterns, where each variable has only a substantial loading on one factor. Probably, exploratory factor analysis (EFA) with subsequent factor rotation has rarely been used for the investigation of facet models because of the complexity of their loading patterns. Facet models have more often been investigated by means of confirmatory factor analyses or parcel-based EFA (Bucik & Neubauer, 1996; Jäger, 1984; Süß & Beauducel, 2015). In order to close this gap, Beauducel and Kersting (2020) investigated by means of a simulation study how well two-facet models can be identified by means of EFA with subsequent factor rotation. They investigated a large number of rotation methods and they found that two-facet factor loading patterns can successfully be identified by means of EFA with subsequent Tandem II rotation (Comrey, 1967) for orthogonal loading patterns, and a Promax-based version of Tandem II rotation, or Geomin rotation (Yates, 1987) with Epsilon of .01 for oblique loading patterns.

Figure 1.

Example for a two-facet model: The Berlin intelligence structure model.

Although Beauducel and Kersting (2020) showed that two-facet models can be found by means of EFA with appropriate factor rotation, they rotated the correct number of factors and did not investigate methods for the identification of the correct number of factors in two-facet models. The aim of the present study was therefore to investigate methods for the identification of the correct number of factors in two-facet models. However, the identification of the correct number of factors is a more complex issue for facet models than for conventional simple structure models because facet models may —or may not—be rank deficient. Deficient rank means that the number of independent columns of the unrotated factor loading matrix is q−1 although the corresponding two-facet loading matrix has q interpretable columns (factors). Beauducel and Kersting (2020) referred to studies showing the deficient rank of the loading matrices from MTMM models or from facet models (Eid, 2000; Eid et al., 2003; Grayson & Marsh, 1994). In order to set up an unrotated loading matrix for subsequent rotation toward a rank-deficient two-facet model, they proposed to extract two loading matrices by means of EFA: One with q−1 factors and one with q factors. Then, they performed orthogonal Procrustes/target rotation (Schönemann, 1966) of the unrotated loading matrix with q−1 factors toward the unrotated loading matrix with q factors. This yields a rank-deficient unrotated loading matrix that can be rotated in order to find a rank-deficient two-facet loading matrix with two salient loadings of each variable (Beauducel & Kersting, 2020). However, adding a further factor (by means of orthogonal target rotation, as described above) will only result in q factors when methods for the number of factors to extract indicate that q−1 factors should be retained for rotation. It will be shown below that two-facet models are rank deficient only when all salient loadings have the same magnitude and sign. When the sign and magnitude of the salient loadings is unknown (as is typical for EFA), the rank of two-facet loading matrices is also unknown. This is an additional challenge for the identification of the number of factors to extract. In a completely exploratory setting, it is unknown whether a rank-deficient two-facet loading matrix, a full-rank two-facet loading matrix, a full-rank conventional simple structure loading matrix or any other loading matrix might be expected. However, it follows from the setup of the unrotated loading matrix that is needed to find a rank-deficient two-facet loading matrix, that conventional EFA with the extraction and rotation of full-rank loading matrices is biased against the identification of rank-deficient two-facet loading matrices. Accordingly, an exact identification of the number of factors to extract is needed as a basis for the aforementioned procedure that allows to explore rank-deficient two-facet loading matrices by means of EFA with subsequent factor rotation.

A method that has been shown to be useful for the identification of the number of factors to extract is Horn’s (1965) parallel analysis (PA). PA is based on the comparison of the eigenvalues of an empirical correlation matrix with the eigenvalues of a correlation matrix based on random numbers with the same number of observed variables and the same number of cases as in the empirical data set. Traditional PA and several variations of PA have been investigated in several different contexts (Achim, 2017; Auerswald & Moshagen, 2019; Beauducel, 2001; Buja & Eyuboglu, 1992; Crawford et al., 2010; Cho et al., 2009; Green et al., 2012; Green et al., 2016; Lim & Jahng, 2019; Zwick, & Velicer, 1986) and PA can still be regarded as one of the most promising methods when compared with other methods. As PA is one of the best methods for the identification of the number of factors to extract, the focus of the present study was on relevant PA versions. A further reason for the focus on PA was that it is based on eigenvalues. Even when the relationship between eigenvalues and the rank of matrices is complex, eigenvectors with distinct eigenvalues are linearly independent (Magnus & Neudecker, 2007). This implies that the rank of a loading matrix is at least as large as the number of distinct nonzero eigenvalues. Therefore, the inspection of the eigenvalues, as it is performed with PA, is helpful in order to find the (approximate) rank of a loading matrix (see next section). Finally, an advantage of the focus on PA is that a large number of studies on PA is available. For example, Cho et al. (2009) have shown that the use of Pearson correlations for the random data can be justified even when the empirical data are categorical. However, only continuous data will be considered here because the identification of two-facet models is already quite a challenge due to the possible rank deficiency of the loading matrices. As several PA versions have been proposed, Table 1 contains an overview of the PA versions investigated in the present study. Some more detailed descriptions of relevant PA versions are given in the following.

Table 1.

Versions of Parallel Analysis Considered in the Present Study.

Label	Description	Authors
mPCA	Based on the mean eigenvalues of the unreduced correlation matrix	Horn (1965) a
95thPCA	Based on the 95th percentile of the unreduced correlation matrix	Glorfeld (1995)
m1PCA	Based on mean eigenvalues of the unreduced correlation matrix when the first eigenvalue is eliminated	Turner (1998)
95th1PCA	Based on the 95th percentile of eigenvalues of the unreduced correlation matrix when the first eigenvalue is eliminated	Glorfeld (1995), Turner (1998)
mFA	Based on the mean eigenvalues of the correlation matrix with squared multiple correlations in the main diagonal	Humphreys & Ilgen (1969)
95thFA	Based on the 95th percentile of eigenvalues of the correlation matrix with squared multiple correlations in the main diagonal	Humphreys & Ilgen (1969); Glorfeld (1995)
m1FA	Based on the mean eigenvalues of the correlation matrix with squared multiple correlations in the main diagonal when the first eigenvalue is eliminated	Humphreys & Ilgen (1969); Turner (1998)
95th1FA	Based on the 95th percentile of the correlation matrix with squared multiple correlations in the main diagonal when the first eigenvalue is eliminated	Humphreys & Ilgen (1969); Glorfeld (1995); Turner (1998)
slope-PCA	Based on the mean of the slopes of three subsequent eigenvalues of the unreduced correlation matrix	Gorsuch & Nelson (1981)
slope-FA	Based on the mean of the slopes of three subsequent eigenvalues of the correlation matrix with squared multiple correlations in the main diagonal	Humphreys & Ilgen (1969); Gorsuch & Nelson (1981)

^aAs Horn (1965) initially proposed parallel analysis, this author might be listed in all lines.

It has been proposed to insert the squared multiple correlation of a variable with all other variables as a communality estimate for PA (Humphreys & Ilgen, 1969). Buja and Eyuboglu (1992) found problems with this PA version whereas Crawford et al. (2010) found that this PA version does not perform consistently worse than PA based on the eigenvalues of the unreduced correlation matrix. In the following, PA based on the squared multiple correlation inserted into the main diagonal of the correlation matrix as a communality estimate will be referred to as PA based on factor analysis (FA), whereas PA based on the unreduced correlation matrix will be referred to as PA based on principal components analysis (PCA; see Table 1). Another aspect that remains to be investigated is whether PA with facet data should be based on the mean or on the 95th percentile (Glorfeld, 1995) of the random eigenvalues. When mean-based PA is performed empirical factors are extracted when their eigenvalue is greater than the mean of the simulated number of random eigenvalues. When 95th percentile PA is performed only empirical factors with an eigenvalue greater than the 95th percentile of the simulated eigenvalues are retained. Crawford et al. (2010) found that the optimal PA method depends on the properties of the population data. They report specific results for each PA method, but they did not investigate faceted models. It can therefore not a priori be known whether PA based on PCA or PA based on FA combined with mean eigenvalues or with the 95th percentile of the eigenvalues is more appropriate for identifying the relevant number of factors in data based on two-facet models. It should also be noted that some problems with PA may occur when factors or components are correlated, which typically implies that the first eigenvalue is very large compared with the other eigenvalues (Beauducel, 2001; Turner, 1998). The elimination of the first eigenvalue from the empirical eigenvalues has been proposed as a method to overcome these problems (Turner, 1998) and will also be considered here (see Table 1).

Moreover, a method for the identification of the relevant number of factors can also be based on the relative size of the eigenvalues within one empirical data set. Zoski and Jurs (1993, 1996), Gorsuch and Nelson (1981), and Gorsuch (1983) proposed more objective versions of Cattell’s (1966) scree test. According to Gorsuch (1983), the series of mean slopes across three subsequent eigenvalues of the empirical data set is computed with the conventional objective scree test. Although this method could be promising, it was considered to remain close to the framework of PA in that the mean slopes of successive empirical eigenvalues were compared with the mean slopes of the corresponding noise eigenvalues in order to extract components or factors when the mean slope of empirical eigenvalues is larger than the mean slope of the corresponding noise eigenvalues. Let ${\gamma }_i$ be the ith empirical eigenvalue, the slope of empirical eigenvalues is

$$\mathrm{\Delta }{\gamma }_i=({\gamma }_{i-1}-{\gamma }_{i+1})/2,\mathrm{for}i=2,...,p-1.$$ (1)

Let ${\theta }_i$ be the ith noise eigenvalue, the noise-eigenvalues slope is computed as

$$\mathrm{\Delta }{\theta }_i=({\theta }_{i-1}-{\theta }_{i+1})/2,\mathrm{for}i=2,...,p-1.$$ (2)

Factors or components are retained for rotation when $\mathrm{\Delta }{\gamma }_i$ is greater than the mean of $\mathrm{\Delta }{\theta }_i$ from 100 runs with noise eigenvalues. Since the empirical slope will only be as flat as the mean noise slope when there are no substantial empirical eigenvalues, the slope of q+ 1 factors will be larger than the mean noise slopes when there are q substantial eigenvalues. The slope-based PA can only be performed for the eigenvalues 2 to p−1, which is a minor limitation in the present context because two-facet models will always comprise more than one factor or component. Moreover, the slope-based PA as well as the other PA versions can be computed for unreduced eigenvalues from PCA as well as for the reduced eigenvalues from FA. As in Crawford et al. (2010), the reduced FA eigenvalues were based on communalities estimated from the squared multiple correlation between a variable and all remaining variables in a matrix (Humphreys & Ilgen, 1969). As the eigenvalues of the reduced correlation matrix can be negative, the additional condition was introduced that the reduced eigenvalue slope was only based on positive empirical eigenvalues.

When the PA versions described above are combined, this leads to the following 10 PA versions (see Table 1): Mean PCA eigenvalue (mPCA), 95th percentile of PCA eigenvalue (95^thPCA), mean PCA eigenvalue without first empirical eigenvalue (m1PCA), 95th percentile of PCA eigenvalue when the first empirical component is eliminated (95^th1PCA), mean FA eigenvalue (mFA), 95th percentile of FA eigenvalues (95^thFA), mean FA eigenvalue when the first empirical factor is eliminated (m1FA), 95th percentile of FA eigenvalues when the first empirical factor is eliminated (95^th1FA), mean slope of successive PCA eigenvalues (slope-PCA), mean slope of successive FA eigenvalues (slope-FA). Although further PA refinements (e.g., Achim, 2017; Dobriban & Owen, 2019; Green et al., 2012) and analyses of PA with binary and ordered polytomous items have meanwhile been published (Timmerman & Lorenzo-Seva, 2011; Weng & Cheng, 2005), the present selection already covers a relevant number of PA versions. As noted by Lim and Jahng (2019), the traditional PA has not been systematically outperformed by the newer PA versions and the new PA versions need to be investigated in several contexts. Since the detection of q−1 or q factors of two-facet models are a rather new context, even for traditional PA, the focus of the present study is not on the newest PA versions but on PA versions that have already been widely investigated. Even when the investigation of additional new PA versions may be interesting, the performance of more traditional PA versions is regarded as an important first step. Therefore, the selection of PA versions of the present study is close to the PA versions investigated by Crawford et al. (2010) with the only exception that a PA version based on eigenvalue differences that is close to the objective scree-test (Gorsuch, 1983) was also included. Moreover, only PA versions based on normally distributed variables were investigated.

First, ideal two-facet loading patterns are described and examples for rank-deficient and full-rank two-facet models are presented as a basis for the population models of the simulation study. Second, it is shown that two-facet loading matrices can also be approximately rank deficient and that the eigenvalues may indicate the approximate rank of the two-facet loading matrices. Finally, the specifications of the simulation study for two-facet models and the corresponding results are reported.

The Ideal Two-Facet Loading Patterns

An ideal two-facet loading pattern ${\mathbf{\Lambda }}_{\mathrm{f}}$ can be defined as

$${\mathbf{\Lambda }}_{\mathrm{f}}=\left[{\mathbf{\Lambda }}_{\mathrm{A}}|{\mathbf{\Lambda }}_{\mathrm{B}}\right],\mathrm{with}{\mathbf{\Lambda }}_{\mathrm{A}}={\mathbf{I}}_{q_{\mathrm{A}}}\otimes 1_{p_{\mathrm{A}}}0.5^{1/2}\mathrm{and}{\mathbf{\Lambda }}_{\mathrm{B}}=0.5^{1/2}{1}_{p_{\mathrm{B}}}\otimes {\mathbf{I}}_{q_{\mathrm{B}}}.$$ (3)

The overall number of variables is p = p_Aq_A = p_Bq_B and the overall number of factors is q = q_A+q_B, ${\mathbf{I}}_{q_{\mathrm{A}}}$ is a q_A×q_A identity matrix, $1_{p_{\mathrm{A}}}$ is a p_A× 1 unit-vector, ${\mathbf{I}}_{q_{\mathrm{B}}}$ is a q_B×q_B identity matrix, $1_{p_{\mathrm{B}}}$ is a p_B× 1 unit-vector, and “$\otimes$” denotes the Kronecker-product. Since each variable has two nonzero elements, the maximal size of the nonzero elements is 0.5^1/2 when all nonzero loadings are at maximum. For p_A = p_B = 2, p = 4, q_A = q_B = 2, and q = 4, Equation (3) yields

$${\mathbf{\Lambda }}_{\mathrm{f}}=0.5^{1/2}[\left(\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right)\otimes \left(\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{1}\end{array}\right)|\left(\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{1}\end{array}\right)\otimes \left(\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right)]=0.5^{1/2}\left[\left(\begin{array}{cc}\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{1}\end{array}\hfill & \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}}& \begin{array}{c}1\hfill \\ \multicolumn{1}{c}{1}\end{array}\hfill \end{array}\right)|\left(\begin{array}{cc}\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill & \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{0}\end{array}}& \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\end{array}\hfill \end{array}\right)\right],$$ (4)

which is the smallest version of a two-facet model. However, in ideal two-facet models all facets contain the same number of factors, all factors are represented by at minimum four nonzero variables (Fabrigar et al., 1999), so that all variables have substantial loadings on two factors and all factors have the same good chance to occur in EFA. Accordingly, for p_A = p_B = 4, p = 4, q_A = q_B = 2, and q = 4, Equation (3) yields

$${\mathbf{\Lambda }}_{\mathrm{f}}=0.5^{1/2}[\left(\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right)\otimes \left(\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{1}\end{array}\right)|\left(\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{1}\end{array}\right)\otimes \left(\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right)]=0.5^{1/2}\left[\left(\begin{array}{cc}\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{1}\end{array}\hfill & \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}}& \begin{array}{c}1\hfill \\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{1}\end{array}\hfill \end{array}\right)|\left(\begin{array}{cc}\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill & \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{0}\end{array}}& \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{1}\end{array}\hfill \end{array}\right)\right].$$ (5)

There is a linear dependency of Columns 1 and 2 with Columns 3 and 4 so that the rank is q−1 as can be seen from the corresponding row echelon form (see the appendix). Two-facet models with an asymmetric number of variables per factor are also possible. For example, for p_A = 2, q_A = 4, p_B = 4, q_B = 2, resulting in p = 8 and q = 6, Equation (3) yields

$${\mathbf{\Lambda }}_{\mathrm{f}}=0.5^{1/2}[\left(\begin{array}{c}\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}}\end{array}\right)\otimes \left(\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{1}\end{array}\right)|\left(\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{1}\end{array}\right)\otimes \left(\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right)]=0.5^{1/2}\left[\left(\begin{array}{c}\begin{array}{cc}\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{1}\end{array}\hfill & \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}}& \begin{array}{c}1\hfill \\ \multicolumn{1}{c}{1}\end{array}\hfill \end{array}\begin{array}{cc}\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill & \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}}& \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill & \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}}& \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill \end{array}\begin{array}{cc}\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{1}\end{array}\hfill & \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}}& \begin{array}{c}1\hfill \\ \multicolumn{1}{c}{1}\end{array}\hfill \end{array}}\end{array}\right)|\left(\begin{array}{c}\begin{array}{cc}\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill & \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{0}\end{array}}& \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\end{array}\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill & \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{0}\end{array}}& \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\end{array}\hfill \end{array}}\end{array}\right)\right].$$ (6)

In the example given in Equation (6) the chances for the identification of the facet-B factors 5 and 6 is larger than the chances for the identification of the facet-A factors 1 to 4. As can be seen in Equations (3) to (5), each variable has a substantial loading on two factors in a two-facet loading pattern. Of course, there is a linear dependency of Columns 1 to 4 with Columns 5 and 6. Accordingly, again, the rank of ${\mathbf{\Lambda }}_{\mathrm{f}}$ is q−1 (see the appendix). Although the exact rank deficiency of the loading matrices in Equations (5) and (6) also depends on the equal size of the salient loadings, it follows from these examples that loading matrices with a slight variation of the relative size of salient loadings can be approximately rank deficient, which may have consequences for the number of substantial eigenvalues of the correlation matrix reproduced from the loading matrix. In order to give an account of the relationship between rank, variation of salient loading size, and the eigenvalues of the reproduced correlation matrix, the salient loadings of Equation (5) were multiplied by .5 in order to provide a start loading matrix ${\mathbf{\Lambda }}_{\mathrm{f}1}$ with rank(${\mathbf{\Lambda }}_{\mathrm{f}1}$) = q−1. Then, in 20 steps an increased amount of random noise was added to the salient loadings so that the variability of the salient loadings increased. For the resulting loading matrices ${\mathbf{\Lambda }}_{\mathrm{f}2}$ to ${\mathbf{\Lambda }}_{\mathrm{f}20}$ the rank is q. Equation (7) represents the start loading matrix ${\mathbf{\Lambda }}_{\mathrm{f}1}$ and the final loading matrix ${\mathbf{\Lambda }}_{\mathrm{f}20}$ resulting from this procedure:

$${\mathbf{\Lambda }}_{\mathrm{f}1},...,{\mathbf{\Lambda }}_{\mathrm{f}20}=\left[\begin{array}{cc}\begin{array}{l}.50\\ .50\\ .50\\ .50\end{array}& \begin{array}{l}.00\\ .00\\ .00\\ .00\end{array}\\ \begin{array}{l}.00\\ .00\\ .00\\ .00\end{array}& \begin{array}{l}.50\\ .50\\ .50\\ .50\end{array}\end{array}\begin{array}{cc}\begin{array}{l}.50\\ .00\\ .50\\ .00\end{array}& \begin{array}{l}.00\\ .50\\ .00\\ .50\end{array}\\ \begin{array}{l}.50\\ .00\\ .50\\ .00\end{array}& \begin{array}{l}.00\\ .50\\ .00\\ .50\end{array}\end{array}\right],...,\left[\begin{array}{cc}\begin{array}{l}.58\\ .43\\ .41\\ .55\end{array}& \begin{array}{l}.00\\ .00\\ .00\\ .00\end{array}\\ \begin{array}{l}.00\\ .00\\ .00\\ .00\end{array}& \begin{array}{l}.39\\ .44\\ .50\\ .62\end{array}\end{array}\begin{array}{cc}\begin{array}{l}.41\\ .00\\ .53\\ .00\end{array}& \begin{array}{l}.00\\ .49\\ .00\\ .40\end{array}\\ \begin{array}{l}.45\\ .00\\ .51\\ .00\end{array}& \begin{array}{l}.00\\ .56\\ .00\\ .58\end{array}\end{array}\right]$$ (7)

Although the rank of ${\mathbf{\Lambda }}_{\mathrm{f}2}$ to ${\mathbf{\Lambda }}_{\mathrm{f}20}$ is q, the number of substantial eigenvalues of the correlation matrices reproduced from these loading matrices is q−1 (see Figure 2A).

Figure 2.

(A) Approximately rank deficient loading matrices: Eigenvalues of the factors 1 to 8 for ${\mathbf{\Lambda }}_{\mathrm{f}1}$ to ${\mathbf{\Lambda }}_{\mathrm{f}20}$. (B) Full rank loading matrices: Eigenvalues of the factors 1 to 8 for ${\mathbf{\Lambda }}_{\mathrm{f}1}$ to ${\mathbf{\Lambda }}_{\mathrm{f}20}$

As can be seen from this example, several two-facet loading matrices would be rank deficient when the salient loadings are of equal size and have full column rank only because of a variable size of the salient loadings. These loading matrices are regarded as approximately rank deficient. Beauducel and Kersting (2020) based their rotational procedure on approximately rank-deficient loading matrices that might be typical for several two-facet models. Moreover, rank(${\mathbf{\Lambda }}_{\mathrm{f}}$) = q−1 and a number of q−1 substantial eigenvalues of the reproduced correlation matrix is not true for two-facet loading matrices based on equal absolute salient loadings with variable sign. For example, rank(${\mathbf{\Lambda }}_{\mathrm{f}}$) = q for

$${\mathbf{\Lambda }}_{\mathrm{f}}=0.5^{1/2}[\left(\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -1\hfill \end{array}\right)\otimes \left(\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{-1}\\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{-1}\end{array}\right)|\left(\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{-1}\\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{-1}\end{array}\right)\otimes \left(\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -1\hfill \end{array}\right)]=0.5^{1/2}\left[\left(\begin{array}{cc}\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{-1}\\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{-1}\end{array}\hfill & \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}}& \begin{array}{c}-1\hfill \\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{-1}\\ \multicolumn{1}{c}{1}\end{array}\hfill \end{array}\right)|\left(\begin{array}{cc}\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill & \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{-1}\end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}-1\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{-1}\\ \multicolumn{1}{c}{0}\end{array}}& \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{-1}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{1}\end{array}\hfill \end{array}\right)\right],$$ (8)

as can be seen from the row echelon form (see the appendix). In order to explore the effect of the salient loading sign together with the effect of salient loading variability, the eigenvalues reproduced from the correlation matrices of the following 20 loading matrices with variable loading sign and rank = q were computed (Equation 9). The variability of the absolute salient loading size was the same for ${\mathbf{\Lambda }}_{\mathrm{f}21}$ to ${\mathbf{\Lambda }}_{\mathrm{f}40}$ as for ${\mathbf{\Lambda }}_{\mathrm{f}1}$ to ${\mathbf{\Lambda }}_{\mathrm{f}20}$:

$${\mathbf{\Lambda }}_{\mathrm{f}21},...,{\mathbf{\Lambda }}_{\mathrm{f}40}=\left[\begin{array}{cc}\begin{array}{l}.50\\ .50\\ .50\\ .50\end{array}& \begin{array}{l}.00\\ .00\\ .00\\ .00\end{array}\\ \begin{array}{l}.00\\ .00\\ .00\\ .00\end{array}& \begin{array}{l}.50\\ .50\\ .50\\ .50\end{array}\end{array}\begin{array}{cc}\begin{array}{l}.50\\ .00\\ -.50\\ .00\end{array}& \begin{array}{l}.00\\ -.50\\ .00\\ .50\end{array}\\ \begin{array}{l}.50\\ .00\\ -.50\\ .00\end{array}& \begin{array}{l}.00\\ -.50\\ .00\\ .50\end{array}\end{array}\right],...,\left[\begin{array}{cc}\begin{array}{l}.58\\ .43\\ .41\\ .55\end{array}& \begin{array}{l}.00\\ .00\\ .00\\ .00\end{array}\\ \begin{array}{l}.00\\ .00\\ .00\\ .00\end{array}& \begin{array}{l}.39\\ .44\\ .50\\ .62\end{array}\end{array}\begin{array}{cc}\begin{array}{l}.41\\ .00\\ -.47\\ .00\end{array}& \begin{array}{l}.00\\ -.51\\ .00\\ .40\end{array}\\ \begin{array}{l}.45\\ .00\\ -.49\\ .00\end{array}& \begin{array}{l}.00\\ -.44\\ .00\\ .58\end{array}\end{array}\right]$$ (9)

The correlation matrices reproduced from these loading matrices have q substantial eigenvalues (Figure 2B), which indicates that the loading matrices ${\mathbf{\Lambda }}_{\mathrm{f}21}$ to ${\mathbf{\Lambda }}_{\mathrm{f}40}$ are not approximately rank deficient. There are several negative loadings in ${\mathbf{\Lambda }}_{\mathrm{f}21}$ to ${\mathbf{\Lambda }}_{\mathrm{f}40}$, but even with rather few negative loadings, it is possible to get rank(${\mathbf{\Lambda }}_{\mathrm{f}}$) = q, as for example, for

$${\mathbf{\Lambda }}_{\mathrm{f}}={0.5}^{1/2}\left[\begin{array}{cc}\begin{array}{l}1\\ 1\\ 1\\ 1\end{array}& \begin{array}{l}0\\ 0\\ 0\\ 0\end{array}\\ \begin{array}{l}0\\ 0\\ 0\\ 0\end{array}& \begin{array}{l}1\\ 1\\ 1\\ 1\end{array}\end{array}\begin{array}{cc}\begin{array}{l}1\\ 0\end{array}& \begin{array}{l}0\\ -1\end{array}\\ \begin{array}{l}1\\ 0\\ 1\\ 0\\ 1\\ 0\end{array}& \begin{array}{l}0\\ 1\\ 0\\ 1\\ 0\\ 1\end{array}\end{array}\right].$$ (10)

The examples show that full-rank two-facet loading matrices may be approximately rank deficient when there are only salient loadings with the same sign (Figure 2A) and that they are not approximately rank deficient when some factors have positive as well as negative loadings (Figure 2B). Negative salient loadings, as they occur in Equations (8) to 10, are not unusual in factor analyses of real data.

The linear dependency between two columns j and k of ${\mathbf{\Lambda }}_{\mathrm{f}}$ results in nonzero covariances. As any two-facet model implies that each observed variable i to p has at least two salient loadings, it follows that

$$\sum _{j=1}^{q_A}\sum _{k=1}^{q_B}\sum _{i=1}^p\left|{\lambda }_{\mathrm{ij}}\right|\left|{\lambda }_{\mathrm{ik}}\right|>0,$$ (11)

for any two-facet loading matrix, where “$\left|\right|$” denotes the absolute value. However, if there is some variability of salient loading size and if one or more salient loadings of ${\mathbf{\Lambda }}_{\mathrm{f}}$ are negative and all other salient loadings are positive, the condition of Equation (11) still holds, although one cannot exclude that

$$\sum _{j=1}^{q_A}\sum _{k=1}^{q_B}\left|\sum _{i=1}^p{\lambda }_{\mathrm{ij}}{\lambda }_{\mathrm{ik}}\right|=0,$$ (12)

because some ${\lambda }_{\mathrm{ij}}{\lambda }_{\mathrm{ik}}$ may be positive and one ${\lambda }_{\mathrm{ij}}{\lambda }_{\mathrm{ik}}$ may be negative so that the sum across all p elements may be zero. This implies that the covariance between all loading columns of ${\mathbf{\Lambda }}_{\mathrm{f}}$ can be zero. It follows that two-facet loading matrices are not necessarily rank deficient so that the rank of these matrices can be q or q−1. Accordingly, it might be reasonable to compute rotated solutions for one factor more than proposed by PA. Note that the rank uncertainty of two-facet loading matrices demonstrated here is not related to sampling error.

Simulation Study

Method

A simulation study based on n = 400 and 1,000 cases, nine rank-deficient population two-facet factor models and nine full-rank population two-facet models with q = 4 to 12 factors in two facets with salient loadings l = .40, .50, and .60 was performed in order to investigate the detection of the correct number of population factors (q−1 or q, depending on the rank of the population matrix) by means of the 10 aforementioned PA versions (mPCA, 95^thPCA, m1PCA, 95^th1PCA, mFA, 95^thFA, m1FA, 95^th1FA, slope-PCA, slope-FA). All nonsalient population loadings were zero. Two levels of intercorrelation (ρ = .00, .30) between all factors were specified for l = .40, .50, and .60. A higher intercorrelation (ρ = .60) between all factors could only be specified for l = .40 and .50 because the communalities for l = .60 and ρ = .60 would have exceeded one. An example for the specification of salient loadings and factor intercorrelations is given in Figure 3. Further details how the sample correlations were computed from the population model parameters are given in the syntax example (Supplement, Part 1). As recommended from previous research (Fabrigar et al., 1999; Floyd & Widaman, 1995), each population factor was measured by at least four variables with salient loadings. This results in the following number of factors in the two facets and the following number of variables (p): 2 + 2 (p = 8), 2 + 3 (p = 18), 3 + 3 (p = 18), 3 + 4 (p = 24), 4 + 4 (p = 16), 4 + 5 (p = 20), 5 + 5 (p = 25), 5 + 6 (p = 30), and 6 + 6 (p = 36). The distribution of salient loadings on the factors was identical for the full-rank models and for the rank-deficient models. Therefore, all loading matrices were presented for the rank-deficient two-facet uncorrelated factor models with l = .50 in the Supplement (Part 2). Nine two-facet models × 2 model types (rank deficient vs. full rank) × 3 loadings sizes (l = .40, .50, .60) × 2 factor-intercorrelations (ρ = .00, .30) × 2 sample sizes (n = 400, 1,000) result in 216 conditions. For ρ = .60, nine two-facet models × 2 model types × 2 loadings sizes (l = .40, .50) × 2 sample sizes result in 72 additional conditions. For each condition 1,000 samples were drawn from the population. The dependent variable was the percentage of q−1 (for rank-deficient models) or q (for full-rank models) population factors detected by means of the PA versions. The percentage of over- and underfactorization of factors was also computed (see, Supplement, Part 3, Tables S1-S16). The simulation study was performed with IBM SPSS Version 26. An example for the syntax can be found in the Supplement (Part 1).

Figure 3.

Conceptual graphic of the population two-facet correlated factor model with q = 4 and p = 8; simple arrows represent salient loadings, double arrows represent factor intercorrelations.

Results

The results for the rank-deficient population models are given in Tables 2 to 4. The percentage of q−1 sample eigenvalues based on two-facet factor models with salient population loadings of l = .40 larger than the respective noise eigenvalues is presented in Table 2. The traditional PA (mPCA) performs well for the orthogonal models with n = 400 and n = 1,000. Alternative PA versions like 95^thPCA, 95^thFA, m1PCA, m1FA, 95^th1PCA, and 95^th1FA did not lead to substantial improvements of detection rates. Since the detection rates were high for these PA versions, the percentage of over- and underfactorizations was very low (Supplement, Table S1). However, mFA and 95^thFA resulted in acceptable detection rates for n = 400 and perfect detection rates for n = 1,000. This performance ranking of PA versions is not altered when uncorrelated factor models based on l = .50 (Table 3) and l = .60 (Table 4) are considered. To sum up, when rank-deficient two-facet uncorrelated factor models are to be expected, mPCA, 95^thPCA, m1PCA, 95^th1PCA, mFA, 95^thFA, and 95^th1FA can be recommended.

Table 2.

Percent of Correct Detection of q−1 Factors for Rank-Deficient Two-Facet Models for l = .40.

		ρ = .00
n	q	mPCA	95thPCA	m1PCA	95th1PCA	mFA	95th FA	m1FA	95th1FA	slope-PCA	slope-FA
	4	100	99	34	76	100	100	0	99	43	96
	5	100	100	99	100	98	100	94	100	33	38
	6	100	100	99	100	97	100	91	99	28	29
	7	100	100	100	100	95	99	92	99	23	26
400	8	96	86	93	99	98	100	87	99	18	62
	9	93	80	96	99	94	99	87	98	16	21
	10	98	94	99	100	91	99	83	97	14	15
	11	97	93	99	100	83	97	78	95	13	14
	12	100	98	100	100	79	95	74	94	10	10
	4	100	100	81	97	100	100	0	1	45	97
	5	100	100	100	100	100	100	100	100	37	40
	6	100	100	100	100	100	100	100	100	29	54
	7	100	100	100	100	100	100	100	100	28	31
1,000	8	100	100	100	100	100	100	0	100	21	79
	9	100	100	100	100	100	100	100	100	19	90
	10	100	100	100	100	100	100	100	100	14	60
	11	100	100	100	100	100	100	100	100	16	17
	12	100	100	100	100	100	100	100	100	12	13
		ρ = .30
n	q	mPCA	95thPCA	m1PCA	95th1PCA	mFA	95th FA	m1FA	95th1FA	slope-PCA	slope-FA
	4	5	1	18	68	100	99	0	100	60	99
	5	50	28	99	100	100	100	100	100	48	53
	6	28	11	98	100	99	100	99	100	40	43
	7	17	6	100	100	99	100	99	100	37	39
400	8	0	0	74	95	97	88	94	96	20	39
	9	0	0	87	95	94	87	93	92	18	20
	10	0	0	95	98	96	95	96	95	14	15
	11	0	0	97	93	95	93	96	92	14	13
	12	0	0	98	96	96	97	98	94	10	11
	4	6	1	49	88	100	100	0	0	60	100
	5	97	92	100	100	100	100	100	100	51	58
	6	95	86	100	100	100	100	100	100	41	54
	7	90	78	100	100	100	100	100	100	40	44
1,000	8	0	0	100	100	100	100	8	100	23	56
	9	0	0	100	100	100	100	100	100	26	70
	10	1	0	100	100	100	100	100	100	16	36
	11	1	0	100	100	100	100	100	100	18	20
	12	15	5	100	100	100	100	100	100	15	16
		ρ = .60
n	q	mPCA	95thPCA	m1PCA	95th1PCA	mFA	95thFA	m1FA	95th1FA	slope-PCA	slope-FA
	4	0	0	1	36	67	19	0	67	79	97
	5	0	0	98	95	91	62	94	68	67	72
	6	0	0	92	93	83	37	89	46	45	46
	7	0	0	91	64	59	21	55	17	44	45
400	8	0	0	13	27	6	0	21	0	4	4
	9	0	0	33	9	4	0	5	0	2	3
	10	0	0	42	2	3	0	1	0	2	2
	11	0	0	21	0	1	0	0	0	1	1
	12	0	0	7	0	0	0	0	0	0	0
	4	0	0	2	37	99	90	0	2	81	100
	5	0	0	100	100	100	100	100	100	72	77
	6	0	0	100	100	100	100	100	100	49	51
	7	0	0	100	100	100	100	100	100	53	58
1,000	8	0	0	70	97	96	55	62	94	10	14
	9	0	0	93	99	96	72	99	91	12	17
	10	0	0	100	100	99	94	100	97	5	4
	11	0	0	100	99	99	90	99	90	6	7
	12	0	0	100	99	99	96	99	93	3	3

Note. mPCA = mean PCA eigenvalue; 95^th PCA = 95th percentile of PCA eigenvalue; m1PCA = mean PCA eigenvalue without first empirical eigenvalue; 95^th1PCA = 95th percentile of PCA eigenvalue without first empirical component; mFA = mean FA eigenvalue; 95^thFA = 95th percentile of FA eigenvalues; m1FA = mean FA eigenvalue without first empirical factor; 95^th1FA = 95th percentile of FA eigenvalues without first empirical factor; slope-PCA = mean slope- of three successive PCA eigenvalues; slope-FA = mean slope of three successive FA eigenvalues.

Table 4.

Percent of Correct Detection of q− 1 Factors for Rank-Deficient Two-Facet Models for l = .60.

		ρ = .00
n	q	mPCA	95thPCA	m1PCA	95th1PCA	mFA	95thFA	m1FA	95th1FA	slope-PCA	slope-FA
	4	100	100	100	100	100	100	100	100	100	100
	5	100	100	100	100	100	100	100	100	99	100
	6	100	100	100	100	100	100	100	100	97	98
	7	100	100	100	100	100	100	100	100	96	99
400	8	100	100	100	100	100	100	100	100	85	89
	9	100	100	100	100	100	100	100	100	92	98
	10	100	100	100	100	100	100	100	100	86	89
	11	100	100	100	100	100	100	100	100	90	92
	12	100	100	100	100	100	100	100	100	85	88
	4	100	100	100	100	100	100	100	100	100	100
	5	100	100	100	100	100	100	100	100	100	100
	6	100	100	100	100	100	100	100	100	98	99
	7	100	100	100	100	100	100	100	100	98	99
1,000	8	100	100	100	100	100	100	100	100	89	92
	9	100	100	100	100	100	100	100	100	94	98
	10	100	100	100	100	100	100	100	100	91	95
	11	100	100	100	100	100	100	100	100	93	99
	12	100	100	100	100	100	100	100	100	88	93
		ρ = .30
n	q	mPCA	95thPCA	m1PCA	95th1PCA	mFA	95thFA	m1FA	95th1FA	slope-PCA	slope-FA
	4	18	7	100	100	100	100	100	100	100	100
	5	100	99	100	100	100	100	100	100	100	100
	6	100	99	100	100	100	100	100	100	97	96
	7	98	96	100	100	100	100	100	100	99	99
400	8	0	0	100	100	100	100	100	100	56	52
	9	0	0	100	100	100	100	100	100	83	81
	10	19	8	100	100	100	100	100	100	64	61
	11	16	7	100	100	100	100	100	100	83	82
	12	61	45	100	100	100	100	100	100	74	71
	4	35	17	100	100	100	100	100	100	100	100
	5	100	100	100	100	100	100	100	100	100	100
	6	100	100	100	100	100	100	100	100	97	97
	7	100	100	100	100	100	100	100	100	100	100
1,000	8	1	0	100	100	100	100	100	100	58	56
	9	2	1	100	100	100	100	100	100	94	93
	10	95	90	100	100	100	100	100	100	69	68
	11	97	93	100	100	100	100	100	100	93	93
	12	100	100	100	100	100	100	100	100	80	79

Note. mPCA = mean PCA eigenvalue; 95^thPCA = 95th percentile of PCA eigenvalue; m1PCA = mean PCA eigenvalue without first empirical eigenvalue; 95^th1PCA = 95th percentile of PCA eigenvalue without first empirical component; mFA = mean FA eigenvalue; 95^thFA = 95th percentile of FA eigenvalues; m1FA = mean FA eigenvalue without first empirical factor; 95^th1FA = 95th percentile of FA eigenvalues without first empirical factor; slope-PCA = mean slope-of three successive PCA eigenvalues; slope-FA = mean slope-of three successive FA eigenvalues.

Table 3.

Percent of Correct Detection of q− 1 Factors for Rank-Deficient Two-Facet Models for l = .50.

		ρ = .00
n	q	mPCA	95thPCA	m1PCA	95th1PCA	mFA	95thFA	M1FA	95th1FA	slope-PCA	slope-FA
	4	100	100	100	100	100	100	58	100	75	100
	5	100	100	100	100	100	100	100	100	63	70
	6	100	100	100	100	100	100	100	100	56	64
	7	100	100	100	100	100	100	100	100	52	58
400	8	100	100	100	100	100	100	100	100	42	85
	9	100	100	100	100	100	100	100	100	39	84
	10	100	100	100	100	100	100	100	100	38	42
	11	100	100	100	100	100	100	100	100	32	33
	12	100	100	100	100	100	100	100	100	30	32
	4	100	100	100	100	100	100	0	99	74	100
	5	100	100	100	100	100	100	100	100	68	75
	6	100	100	100	100	100	100	100	100	59	98
	7	100	100	100	100	100	100	100	100	53	63
1,000	8	100	100	100	100	100	100	100	100	49	88
	9	100	100	100	100	100	100	100	100	46	97
	10	100	100	100	100	100	100	100	100	42	90
	11	100	100	100	100	100	100	100	100	39	91
	12	100	100	100	100	100	100	100	100	34	38
		ρ = .30
n	q	mPCA	95thPCA	m1PCA	95th1PCA	mFA	95thFA	M1FA	95th1FA	slope-PCA	slope-FA
	4	9	3	100	100	100	100	100	100	96	100
	5	93	84	100	100	100	100	100	100	94	96
	6	86	73	100	100	100	100	100	100	84	88
	7	76	58	100	100	100	100	100	100	86	89
400	8	0	0	100	100	100	100	100	100	43	49
	9	0	0	100	100	100	100	100	100	59	69
	10	1	0	100	100	100	100	100	100	43	45
	11	1	0	100	100	100	100	100	100	51	53
	12	3	1	100	100	100	100	100	100	44	45
	4	16	5	100	100	100	100	100	100	98	100
	5	100	100	100	100	100	100	100	100	96	97
	6	100	100	100	100	100	100	100	100	87	94
	7	100	100	100	100	100	100	100	100	89	92
1,000	8	0	0	100	100	100	100	100	100	52	57
	9	0	0	100	100	100	100	100	100	73	87
	10	42	23	100	100	100	100	100	100	53	62
	11	43	26	100	100	100	100	100	100	66	81
	12	97	94	100	100	100	100	100	100	52	57
		ρ = .60
n	q	mPCA	95thPCA	m1PCA	95th1PCA	mFA	95thFA	m1FA	95th1FA	slope-PCA	slope-FA
	4	0	0	100	100	100	100	100	100	100	100
	5	0	0	100	100	100	100	100	100	100	100
	6	0	0	100	100	100	100	100	100	64	60
	7	0	0	100	100	100	100	100	100	86	84
400	8	0	0	100	89	100	94	98	58	2	2
	9	0	0	100	95	100	98	99	71	5	4
	10	0	0	100	91	100	98	98	64	2	2
	11	0	0	100	91	100	98	95	56	3	3
	12	0	0	100	92	100	98	92	51	1	1
	4	0	0	100	100	100	100	100	100	100	100
	5	0	0	100	100	100	100	100	100	100	100
	6	0	0	100	100	100	100	100	100	68	67
	7	0	0	100	100	100	100	100	100	95	94
1,000	8	0	0	100	100	100	100	100	100	3	2
	9	0	0	100	100	100	100	100	100	28	24
	10	0	0	100	100	100	100	100	100	2	2
	11	0	0	100	100	100	100	100	100	18	16
	12	0	0	100	100	100	100	100	100	3	3

Note. mPCA = mean PCA eigenvalue; 95^thPCA = 95th percentile of PCA eigenvalue; m1PCA = mean PCA eigenvalue without first empirical eigenvalue; 95^th1PCA = 95th percentile of PCA eigenvalue without first empirical component; mFA = mean FA eigenvalue; 95^thFA = 95th percentile of FA eigenvalues; m1FA = mean FA eigenvalue without first empirical factor; 95^th1FA = 95th percentile of FA eigenvalues without first empirical factor; slope-PCA = mean slope of three successive PCA eigenvalues; slope-FA = mean slope of three successive FA eigenvalues.

In contrast, mPCA does not work consistently well for rank-deficient two-facet correlated factor models based on l = .40 and ρ = .30. It does not detect q−1 for l = .40 and ρ = .60. The percentage of over- and underfactorization of the factors for l = .40 and ρ = .60 reveals that mPCA tends to underfactorization for correlated factor models (Supplement Table S3). Relevant improvements of the detection rates were obtained for m1PCA and especially for 95^th1PCA (see Table 2). For l = .40 and ρ = .30 high detection rates were found for mFA, 95^thFA, 95^th1FA, m1FA. For l = .40, ρ = .60, and n = 400, no PA version had consistently high detection rates. All PA versions tend to underfactorization of factors in this condition (Supplementary Table S3; available online). For l = .40, ρ = .60, and n = 1,000 mFA had consistently high detection rates. For this condition, the mFA detection rates were higher across the models than for any other PA version. Moreover, perfect detection rates were found for m1PCA, 95^th1PCA, mFA, 95^thFA, m1FA, and 95^th1FA for rank-deficient two-facet correlated factor models with l = .50 and l = .60 and ρ = .00 and ρ = .30 (Tables 2 and 3). As l = .40 is the most critical loading condition and as mFA resulted in the highest overall detection rates across samples sizes and factor intercorrelation sizes for this critical condition as well as in the other conditions, mFA can be recommended when two-facet correlated factor models tend to be rank deficient. It should, however, be noted that for factor intercorrelations of about .60, a large sample size (n = 1,000) or large salient loadings (≥.50) are necessary in order to get high detection rates with mFA.

The results for the full-rank population two-facet factor models are given in Tables 5 to 7. The percentage of q sample eigenvalues based on factor models with salient population loadings of l = .40 larger than the respective noise eigenvalues is presented in Table 5. Overall, the detection rates of the q population factors were considerably lower for full-rank population models. In sum, only mFA had acceptable detection rates for uncorrelated and correlated factor models up to q = 6 for ρ = .00 and ρ = .30. For q > 6 mFA tends to underfactorization of factors in this condition (Supplement Tables S9 and S10). For ρ = .60 and l = .40 no method had acceptable detection rates for a relevant subset of models (Table 5). All PA versions except the slope-based PA versions (which had overall low detection rates) tend to underfactorization of factors in this condition (Supplement Table S11). Detection rates increase for l = .50 with mFA and 95^thFA resulting in acceptable to perfect detection rates for factor models up to q = 7 for n = 1,000, ρ = .00 and ρ = .30 (Table 6). Again, all PA versions that were not based on slopes tend to underfactorization of factors for models with large q (Supplement Table S12). Moreover, slope-FA had acceptable detection rates for full-rank population models based on uncorrelated factors with n = 1,000 for q = 4 to 6 and q = 8 to 11. For l = .50, ρ = .60, n = 1,000, and q = 4 to 6 mFA and 95^thFA had nearly perfect detection rates (Table 6). For l = .60 and full-rank two-facet factor models based on n = 400, the detection rates for mFA and 95^thFA were nearly perfect for q = 4 to 10 (Table 7). For l = .60 and n = 1,000, only mFA and 95^thFA had perfect detection rates for all factor models. The detection rates for slope-FA were nearly perfect for l = .60, n = 1,000 and uncorrelated factors and acceptable for q = 4 to 6 and q = 8 to 11. No PA version had consistently high detection rates across all conditions for q > 6 full-rank two-facet models. All PA versions that were not based on slopes tend to underfactorization of factors. However, compared with the other PA versions mFA had the overall highest detection rates across conditions for the full-rank models. Although the overall detection rates were lower for the full-rank models with q > 6 than for the rank-deficient models, the relative performance of the PA versions was not very different for the full-rank models than for the rank-deficient models.

Table 5.

Percent of Correct Detection of q Factors for Full-Rank Two-Facet Models for l = .40.

		ρ = .00
n	q	mPCA	95thPCA	m1PCA	95th1PCA	mFA	95thFA	m1FA	95th1FA	slope-PCA	slope-FA
	4	100	100	19	43	100	100	0	0	31	68
	5	62	42	74	93	97	99	87	98	31	33
	6	42	24	65	80	93	93	82	93	26	28
	7	0	0	17	7	56	32	60	40	24	25
400	8	0	0	22	6	15	3	9	2	23	48
	9	0	0	11	2	18	4	29	10	23	57
	10	0	0	7	1	28	10	35	15	19	21
	11	0	0	3	1	29	10	35	14	16	19
	12	0	0	1	0	31	13	33	15	18	17
	4	100	100	30	50	100	100	0	0	31	72
	5	91	82	81	94	100	100	100	100	32	44
	6	63	46	87	98	100	100	100	100	26	88
	7	0	0	12	5	89	72	93	81	25	26
1,000	8	0	0	3	1	6	1	34	14	26	21
	9	0	0	1	0	9	3	1	8	22	55
	10	0	0	0	0	18	7	26	12	19	87
	11	0	0	0	0	8	2	12	3	18	36
	12	0	0	0	0	12	4	15	5	16	16
		ρ = .30
n	q	mPCA	95thPCA	m1PCA	95th1PCA	mFA	95thFA	m1FA	95th1FA	slope-PCA	slope-FA
	4	87	65	3	20	100	97	0	0	8	76
	5	1	0	84	79	90	72	88	81	16	37
	6	1	0	68	65	81	61	81	68	11	31
	7	0	0	11	3	19	4	23	6	21	30
400	8	0	0	25	7	4	0	0	1	22	30
	9	0	0	13	2	6	0	10	1	19	37
	10	0	0	8	1	10	1	10	1	16	21
	11	0	0	4	0	10	1	7	1	17	22
	12	0	0	2	0	9	1	4	0	14	17
	4	100	98	1	10	100	100	0	0	1	70
	5	0	0	99	99	100	99	100	100	8	41
	6	0	0	96	91	99	93	99	97	4	75
	7	0	0	12	4	33	14	51	25	13	30
1,000	8	0	0	8	1	0	0	24	2	8	5
	9	0	0	1	0	1	0	0	0	6	34
	10	0	0	0	0	6	1	6	1	5	74
	11	0	0	0	0	2	0	2	0	7	28
	12	0	0	0	0	2	0	1	0	7	21
		ρ = .60
n	q	mPCA	95thPCA	m1PCA	95th1PCA	mFA	95thFA	m1FA	95th1FA	slope-PCA	slope-FA
	4	9	0	0	3	67	25	0	3	0	62
	5	0	0	46	27	24	5	30	7	4	37
	6	0	0	34	30	26	7	33	9	2	27
	7	0	0	6	1	2	0	2	0	9	27
400	8	0	0	3	8	1	0	0	0	13	5
	9	0	0	8	1	0	0	0	0	10	3
	10	0	0	9	0	0	0	0	0	6	3
	11	0	0	2	0	0	0	0	0	5	2
	12	0	0	0	0	0	0	0	0	4	1
	4	29	4	0	0	97	83	0	0	0	67
	5	0	0	84	73	66	36	82	56	0	40
	6	0	0	70	61	53	26	71	42	0	33
	7	0	0	6	1	2	0	3	0	5	31
1,000	8	0	0	19	10	0	0	10	0	6	3
	9	0	0	13	2	0	0	1	0	2	26
	10	0	0	6	0	1	0	1	0	3	23
	11	0	0	1	0	0	0	0	0	2	20
	12	0	0	0	0	0	0	0	0	6	16

Note. mPCA = mean PCA eigenvalue; 95^thPCA = 95th percentile of PCA eigenvalue; m1PCA = mean PCA eigenvalue without first empirical eigenvalue; 95^th1PCA = 95th percentile of PCA eigenvalue without first empirical component; mFA = mean FA eigenvalue; 95^thFA = 95th percentile of FA eigenvalues; m1FA = mean FA eigenvalue without first empirical factor; 95^th1FA = 95th percentile of FA eigenvalues without first empirical factor; slope-PCA = mean slope of three successive PCA eigenvalues; slope-FA = mean slope of three successive FA eigenvalues.

Table 7.

Percent of Correct Detection of q Factors for Full-Rank Two-Facet Models for l = .60.

		ρ = .00
n	q	mPCA	95thPCA	m1PCA	95th1PCA	mFA	95thFA	m1FA	95th1FA	slope-PCA	slope-FA
	4	100	100	98	100	100	100	0	16	86	88
	5	98	96	100	100	100	100	100	100	98	99
	6	79	66	100	100	100	100	100	100	96	97
	7	0	0	5	2	100	100	100	100	95	98
400	8	0	0	0	0	100	100	100	100	94	98
	9	0	0	0	0	100	99	100	95	92	99
	10	0	0	0	0	100	98	99	94	92	98
	11	0	0	0	0	94	77	81	54	91	94
	12	0	0	0	0	89	70	72	44	88	90
	4	100	100	99	99	100	100	0	8	87	90
	5	100	100	100	100	100	100	100	100	99	100
	6	99	96	100	100	100	100	100	100	98	100
	7	0	0	3	3	100	100	100	100	97	100
1,000	8	0	0	0	0	100	100	98	100	96	98
	9	0	0	0	0	100	100	100	100	95	100
	10	0	0	0	0	100	100	100	100	94	100
	11	0	0	0	0	100	100	100	100	94	100
	12	0	0	0	0	100	100	100	100	89	98
		ρ = .30
n	q	mPCA	95thPCA	m1PCA	95th1PCA	mFA	95thFA	m1FA	95th1FA	slope-PCA	slope-FA
	4	100	100	100	100	100	100	94	100	0	95
	5	0	0	100	100	100	100	100	100	24	100
	6	0	0	98	95	100	100	100	100	0	99
	7	0	0	0	0	100	100	99	91	27	91
400	8	0	0	0	0	100	100	82	39	1	94
	9	0	0	0	0	100	98	20	5	0	96
	10	0	0	0	0	98	90	9	2	2	96
	11	0	0	0	0	68	36	0	0	0	97
	12	0	0	0	0	38	12	0	0	18	88
	4	100	100	100	100	100	100	100	100	0	95
	5	0	0	100	100	100	100	100	100	23	100
	6	0	0	100	100	100	100	100	100	0	99
	7	0	0	0	0	100	100	100	100	25	98
1,000	8	0	0	0	0	100	100	100	95	0	95
	9	0	0	0	0	100	100	64	30	0	99
	10	0	0	0	0	100	100	58	32	0	98
	11	0	0	0	0	100	100	8	2	0	100
	12	0	0	0	0	100	100	6	1	13	96

Note. mPCA = mean PCA eigenvalue; 95^thPCA = 95th percentile of PCA eigenvalue; m1PCA = mean PCA eigenvalue without first empirical eigenvalue; 95^th1PCA = 95th percentile of PCA eigenvalue without first empirical component; mFA = mean FA eigenvalue; 95^thFA = 95th percentile of FA eigenvalues; m1FA = mean FA eigenvalue without first empirical factor; 95^th1FA = 95th percentile of FA eigenvalues without first empirical factor; slope-PCA = mean slope of three successive PCA eigenvalues; slope-FA = mean slope of three successive FA eigenvalues.

Table 6.

Percent of Correct Detection of q Factors for Full-Rank Two-Facet Models for l = .50.

		ρ = .00
n	q	mPCA	95thPCA	m1PCA	95th1PCA	mFA	95thFA	m1FA	95th1FA	slope-PCA	slope-FA
	4	100	100	89	95	100	100	0	0	58	85
	5	89	76	100	100	100	100	100	100	57	66
	6	59	42	99	97	100	100	100	100	52	63
	7	0	0	6	2	98	91	97	90	46	53
400	8	0	0	0	0	58	28	0	35	46	89
	9	0	0	0	0	48	23	53	26	44	94
	10	0	0	0	0	52	28	51	27	37	55
	11	0	0	0	0	26	11	23	9	34	39
	12	0	0	0	0	25	9	19	6	33	37
	4	100	100	92	95	100	100	0	0	59	87
	5	100	99	100	100	100	100	100	100	63	95
	6	88	78	100	100	100	100	100	100	54	99
	7	0	0	5	3	100	100	100	100	50	73
1,000	8	0	0	0	0	84	63	1	1	51	94
	9	0	0	0	0	69	46	64	51	46	97
	10	0	0	0	0	79	60	77	58	42	99
	11	0	0	0	0	46	25	42	20	39	98
	12	0	0	0	0	50	31	44	27	34	54
		ρ = .30
n	q	mPCA	95thPCA	m1PCA	95th1PCA	mFA	95thFA	m1FA	95th1FA	slope-PCA	slope-FA
	4	99	97	99	100	100	100	0	0	1	95
	5	1	0	99	97	100	100	100	100	13	72
	6	0	0	91	77	100	99	100	98	4	68
	7	0	0	1	0	61	29	45	17	20	58
400	8	0	0	0	0	23	4	10	1	11	69
	9	0	0	0	0	17	3	4	1	5	80
	10	0	0	0	0	21	6	3	0	5	59
	11	0	0	0	0	6	2	1	0	4	52
	12	0	0	0	0	3	1	0	0	11	44
	4	100	100	100	100	100	100	0	0	0	95
	5	0	0	100	100	100	100	100	100	10	90
	6	0	0	100	99	100	100	100	100	0	98
	7	0	0	2	1	99	96	99	91	15	62
1,000	8	0	0	0	0	38	10	0	4	1	75
	9	0	0	0	0	25	8	5	1	0	82
	10	0	0	0	0	41	20	7	2	1	96
	11	0	0	0	0	12	3	0	0	0	95
	12	0	0	0	0	11	2	0	0	4	54
		ρ = .60
n	q	MPCA	95thPCA	m1PCA	95th1PCA	mFA	95thFA	m1FA	95th1FA	slope-PCA	slope-FA
	4	28	3	79	99	99	99	0	0	0	87
	5	0	0	67	38	88	65	73	41	0	67
	6	0	0	53	25	85	57	60	27	0	58
	7	0	0	0	0	1	0	0	0	8	39
400	8	0	0	1	0	4	0	0	0	1	29
	9	0	0	0	0	2	0	0	0	0	27
	10	0	0	0	0	2	0	0	0	5	29
	11	0	0	0	0	0	0	0	0	1	23
	12	0	0	0	0	0	0	0	0	9	13
	4	61	29	100	100	100	100	0	0	0	89
	5	0	0	99	95	100	100	100	99	0	79
	6	0	0	86	67	100	99	98	91	0	71
	7	0	0	0	0	26	6	4	0	6	47
1,000	8	0	0	0	0	8	1	0	0	0	43
	9	0	0	0	0	5	0	0	0	0	61
	10	0	0	0	0	10	2	0	0	0	64
	11	0	0	0	0	1	0	0	0	0	54
	12	0	0	0	0	0	0	0	0	7	30

Note. mPCA = mean PCA eigenvalue; 95^thPCA = 95th percentile of PCA eigenvalue; m1PCA = mean PCA eigenvalue without first empirical eigenvalue; 95^th1PCA = 95th percentile of PCA eigenvalue without first empirical component; mFA = mean FA eigenvalue; 95^thFA = 95th percentile of FA eigenvalues; m1FA = mean FA eigenvalue without first empirical factor; 95^th1FA = 95th percentile of FA eigenvalues without first empirical factor; slope-PCA = mean slope of three successive PCA eigenvalues; slope-FA = mean slope of three successive FA eigenvalues.

Empirical Example

The purpose of the empirical example was to investigate relevant PA versions in a data set for which a two-facet factor structure can be expected from previous research. The BIS model is a two-facet model comprising four operation factors and three content factors (see Figure 1) that has been replicated in several empirical data sets until now (e.g., Bucik & Neubauer, 1996; Beauducel & Kersting, 2002; Jäger et al., 1997; Süß & Beauducel, 2015; Süß et al., 2002). A two-facet loading pattern with positive loadings corresponding to the BIS model was also reported in Beauducel and Kersting (2020), who analyzed data based on a test constructed for the measurement of the BIS model (Jäger et al., 1997). It was therefore expected that the data presented by Beauducel and Kersting (2020) correspond to the BIS model. Accordingly, the intercorrelation matrix of 12 intelligence task aggregates provided in the Supplement of Beauducel and Kersting (2020) was used. The respective sample of 393 participants that has completed the BIS test is described in Beauducel and Kersting (2020). One task aggregate was formed for each of the 4 × 3 combinations (=12 cells) of operation and content factors. Beauducel and Kersting (2020) retained six factors and added a seventh loading column for factor rotation and report a seven-factor two-facet loading matrix corresponding to the BIS model. It follows from this result that PA should indicate that six factors are to be extracted in this data set. Only PA versions that were relatively successful in the simulation study were considered for the empirical example (Table 8). The most successful PA version, that is, mFA indicates that six factors should be extracted. Since Beauducel and Kersting (2020) have shown that six factors are the basis for the (rank-deficient) loading matrix of the BIS model, this result is compatible with expectations from previous research.

Table 8.

Empirical Eigenvalues Based on the Correlation Matrix of 12 Aggregates of Intelligence Tasks in Beauducel and Kersting (2020, Supplement, Table S2) and Corresponding Noise Eigenvalues of Relevant PA Versions.

	Number of factors
Eigenvalues of correlation matrix:	1	2	3	4	5	6	7	8	9	10	11	12	emp. eigenvalues > mean noise eigenvalues
Empirical eigenvalues	4.66	1.40	1.20	1.00	.73	.69	.51	.45	.41	.37	.31	.28	mPCA: 3
Empirical eigenvalues, first eliminated	—	1.74	1.59	1.42	1.12	1.04	.90	.85	.82	.77	.72	.64	m1PCA: 6
Mean noise eigenvalues	1.29	1.21	1.15	1.10	1.06	1.02	.97	.93	.89	.84	.80	.74
95th percentile noise eigenvalues	1.35	1.26	1.20	1.14	1.09	1.05	1.01	.96	.92	.88	.83	.79	95thPCA: 2
													95th1PCA: 5
Number of factors
Eigenvalues of reduced correlation matrix:	1	2	3	4	5	6	7	8	9	10	11	12
Empirical eigenvalues	4.12	.83	.66	.46	.15	.06	−.07	−.10	−.15	−.17	−.21	−.27	mFA: 6
Empirical eigenvalues, first eliminated	—	.81	.65	.46	.15	.06	.00	−.07	−.11	−.14	−.18	−.21	m1FA: 6
Mean noise eigenvalues	.32	.24	.18	.13	.08	.04	−.01	−.05	−.09	−.13	−.17	−.23
95th percentile noise eigenvalues	.39	.29	.23	.17	.11	.07	.03	−.02	−.06	−.10	−.14	−.19	95thFA: 5
													95th1FA: 5

Note. PA = parallel analysis; mPCA = mean PCA eigenvalue; 95^thPCA = 95th percentile of PCA eigenvalue; m1PCA = mean PCA eigenvalue without first empirical eigenvalue; 95^th1PCA = 95th percentile of PCA eigenvalue without first empirical component; mFA = mean FA eigenvalue; 95^thFA = 95th percentile of FA eigenvalues; m1FA = mean FA eigenvalue without first empirical factor; 95^th1FA = 95th percentile of FA eigenvalues without first empirical factor.

Discussion

This research was motivated by the finding that two-facet factor models can be identified by means of EFA when appropriate methods of exploratory factor rotation are used (Beauducel & Kersting, 2020). This is interesting because previous attempts to develop facet models were based on other methods as, for example, EFA with subsequent Procrustes-rotation (Guilford, 1967), EFA based on parceling (Jäger, 1982), smallest space analysis (L. Guttman & Levy, 1991), or confirmatory factor analysis (Bucik & Neubauer, 1996; Süß & Beauducel, 2015). The possibility to use EFA with exploratory factor rotation for the development of two-facet models and instruments may facilitate the exploration of faceted structures and may therefore allow for more conventional forms of model development and test construction in this area.

However, it was shown here that ideal two-facet loading matrices based on equal salient loading size can—but must not—be rank deficient. It was also shown that even full-rank two-facet loading matrices can be approximately rank deficient, with q−1 substantial eigenvalues of the reproduced correlation matrix, when the salient loadings on each factor have the same sign. In contrast, full-rank two-facet loading matrices with positive as well as negative salient loadings on some factors are typically not approximately rank deficient, that is, they result in q substantial eigenvalues. It follows that the exact rank and the approximate rank of two-facet loading matrices is not a priori known. Accordingly, the present simulation study investigates how well the correct number of factors can be detected by rather traditional PA versions when approximately rank-deficient and full-rank two-facet population loading matrices are given.

We investigated the detection rate of the correct number of factors in nine rank-deficient two-facet population models (rank: q−1) and nine full-rank two-facet population models (rank: q) by means of traditional PA (based on the mean of the unreduced eigenvalues; mPCA) as well as nine alternative PA versions. The dependent variable was the percentage of correctly detected q−1 factors for the rank-deficient population loading matrices and the percentage of correctly detected q factors for the full-rank population loading matrices.

It turns out that mPCA can only be recommended for orthogonal rank-deficient two-facet models and that mFA performed best across all conditions of the rank-deficient two-facet models. For correlated rank-deficient two-facet models mPCA tends to underfactorization, which might partially be compensated by the elimination of the first factor (m1PCA). However, mFA had higher detection rates as the other PA versions for the rank-deficient facet models. As it is not necessarily known in advance whether a two-facet model is based on uncorrelated or correlated factors, mFA can be recommended whenever a rank-deficient two-facet model might be expected. Overall, the detection rates for the full-rank two-facet models were considerably lower than for the rank-deficient two-facet models. The detection rates were especially low for full-rank two-facet correlated factor models, where all methods tend to underfactorization. Overall, mFA had the highest detection rate for full-rank two-facet models. For salient loadings of .50 and greater and interfactor correlations of .30 or less, mFA had acceptable detection rates. Thus, small salient loadings in combination with high factor intercorrelations may cause problems for mFA and for all other PA versions investigated here. The PA versions tend to underfactorization under these conditions. Other methods should be investigated in order to provide further recommendations for full-rank two-facet correlated factor models.

An empirical data set based on a test for the BIS model, a two-facet intelligence model that has been replicated several times (Bucik & Neubeuer, 1996; Jäger et al., 1997; Süß & Beauducel, 2015), was used in order to compare the PA versions with the highest detection rates. The most suitable PA version from the simulation study, that is, mFA, indicated that six factors should be extracted. This is compatible with the BIS model because a seventh loading column should be added in order to find an approximately rank-deficient two-facet model (Beauducel & Kersting, 2020). Overall, the results of the simulation study as well as results from the empirical data set underline that EFA based on PA is in principle suitable for the analysis of two-facet models but that—as in Beauducel and Kersting (2020)—rather large sample sizes and rather large salient loadings are necessary.

Some recommendations for EFA of two-facet models follow from the present research. As it is a priori unknown whether an expected two-facet population loading matrix tends to be rank deficient or tends to be of full rank, whether it is based on uncorrelated or correlated factors, a researcher has to perform EFA for at least two different numbers of factors and for the orthogonal and oblique rotations proposed in Beauducel and Kersting (2020). First, researchers may recode items in order to have rather clear expectations regarding the sign of salient loadings. This may enhance the probability of a conventional approximately rank-deficient two-facet model. Second, mFA with subsequent orthogonal rotation should be performed with one factor more as indicated (see, Beauducel & Kersting, 2020). If the resulting salient loadings of each factor have the same sign and a similar magnitude, an orthogonal approximately rank-deficient two-facet loading matrix was probably found. Inspection of cross-loadings may help to decide whether subsequent oblique rotation may be performed. However, if factors with positive and negative salient loadings or a considerable variation of the salient loadings occurs, researchers may not add a further factor. These recommendations, together with a sample size of at least 400 cases, may help to successfully explore two-facet models by means of EFA.

When a two-facet loading pattern has successfully been identified through mFA and appropriate subsequent rotation, it may be helpful to consider whether the loading pattern represents an MTMM model. This would be the case when the factors of one facet represent more formal aspects of the measurements. For example, when the data are based on questionnaire items, there might be a facet comprising factors for acquiescence or social desired responding. Another question is whether the factors of the two facets are reasonably well represented by measured variables or whether the factors of one facet are more completely represented than the factors of the other facet. Finally, it could also be that the two-facet loading pattern indicates that some combination of a factor of one facet with a factor of the other facet is not at all represented by measured variables. In order to detect possible underrepresentations of measured variables it might be helpful to construct a mapping sentence (R. Guttman & Greenbaum, 1998) from the factors and to develop hypotheses on the universe of measurements.

It should be acknowledged as a limitation of the present simulation study that some of the newest PA versions provided by Achim (2017), Green et al. (2012), and Dobriban and Owen (2019) were not investigated. Further studies on a large number of additional methods for the detection of the number of factors to retain for rotation should be performed in order to reach a more complete knowledge on this issue for two-facet models. This would be of special importance for full-rank two-facet models based on correlated factors because the current study shows that none of the rather traditional PA based methods investigated here had acceptable detection rates for the number of factors of these models. A further limitation of the present study is that only two-facet models and no three- or four-facet models were investigated. It should, however, be noted that the problem of unknown rank deficiency is even more pronounced for models with three or more facets. For example, three-facet models could be of full rank, could have a rank of q−1 or even q− 2. As the complexities resulting from the investigation of rank-deficient, approximately rank-deficient, and full-rank two-facet models are already considerable, a stepwise exploration of even more complex models may only be successful when a more complete knowledge on optimal procedures for EFA for two-facet models is available.

Appendix

Results of transformation to the reduced row echelon form according to the Linear Algebra Toolkit (v1.25, Bogacki, 2000-2019).

For loading matrix in Equation (5), Rank = 3:

$$\left[\begin{array}{llll}1& 0& 0& 1\\ 0& 1& 0& 1\\ 0& 0& 1& -1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$$

For loading matrix in Equation (6), Rank = 5:

$$\left[\begin{array}{cc}\begin{array}{l}1\\ 0\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 1\\ 0\\ 0\end{array}\\ \begin{array}{l}0\\ 0\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 0\\ 0\\ 0\end{array}\end{array}\begin{array}{cc}\begin{array}{l}0\\ 0\\ 1\\ 0\end{array}& \begin{array}{l}0\\ 0\\ 0\\ 1\end{array}\\ \begin{array}{l}0\\ 0\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 0\\ 0\\ 0\end{array}\end{array}\begin{array}{cc}\begin{array}{l}0\\ 0\end{array}& \begin{array}{l}1\\ 1\end{array}\\ \begin{array}{l}0\\ 0\\ 1\\ 0\\ 0\\ 0\end{array}& \begin{array}{l}1\\ 1\\ -1\\ 0\\ 0\\ 0\end{array}\end{array}\right]$$

For loading matrices in Equations (7) and (8), Rank = 4:

$$\left[\begin{array}{cc}\begin{array}{l}1\\ 0\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 1\\ 0\\ 0\end{array}\\ \begin{array}{l}0\\ 0\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 0\\ 0\\ 0\end{array}\end{array}\begin{array}{cc}\begin{array}{l}0\\ 0\end{array}& \begin{array}{l}0\\ 0\end{array}\\ \begin{array}{l}1\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 1\\ 0\\ 0\\ 0\\ 0\end{array}\end{array}\right]$$