Psychopathy might well be the major trait of the Dark Triad (Muris et al., Reference Muris, Merckelbach, Otgaar and Meijer2017, p. 189); its locus within Eysenckian personality space (the Big Three) being in the octant of high Extraversion, high Neuroticism, and high Psychoticism (Eysenck, Reference Eysenck1995). Its sister dark trait of Machiavellianism lies in the high Psychoticism, high Extraversion quadrant (Allsopp et al., Reference Allsopp, Eysenck and Eysenck1991). Narcissism completes the Dark Triad, and is aligned with Neuroticism (Jang et al., Reference Jang, Livesley and Vernon1998). The Big Three, however, is not the dominant paradigm for personality research at the present, and various models have been proposed suggesting more than three central dimensions of personality. Given this, where would one locate the Dark Triad in a space of higher dimensionality? To answer this, we first have to address these candidate multidimensional personality spaces.
On making the transition from a focus on the Big Three (Eysenck & Eysenck, Reference Eysenck and Eysenck1985) to the Big Five (McCrae & Costa, Reference McCrae and Costa1997), a consensus was prescribed indicating that this was the working paradigm for personality research. Indeed, Ozer and Reise (Reference Ozer and Reise1994, p. 361) could proclaim that “Personality psychologists who continue to employ their preferred measure without locating it within the five-factor model can only be likened to geographers who issue reports of new lands but refuse to locate them on a map for others to find ….”. The Big Five (Neuroticism, Extraversion, Agreeableness, Conscientiousness, and Openness to Experience) not only describes normative personality, but also embeds personality disorders within the same personality space (Costa & McCrae, Reference Costa and McCrae1992).Nevertheless, within the space of ten years (from 1994), the paradigmatic status of the Big Five was beginning to be challenged, on three fronts.
First, an ‘alternative’ Big Five (Zuckerman, Reference Zuckerman2005) was promoted, one in which the Big Three could be embedded (Zuckerman & Glicksohn, Reference Zuckerman and Glicksohn2016). While this ‘alternative’ Big Five has not succeeded in generating a large literature, it nevertheless presents an alternative framework within which one can conduct personality research, as in the present paper. Second, a Big Six was advanced, namely that of HEXACO (Honesty-Humility, Emotionality, Extraversion, Agreeableness, Conscientiousness, and Openness to Experience; Lee & Ashton, Reference Lee and Ashton2004). While the important addition here is the dimension of Hostility-Humility (Saucier, Reference Saucier2019), it is not the case that this is a simple addition to the existing Big Five. In particular, HEXACO Emotionality is not comparable with Big Five Neuroticism (Howard & van Zandt, Reference Howard and van Zandt2020); and Honesty-Humility has no clear projection on Big Five Agreeableness and Conscientiousness (Ashton & Lee, Reference Ashton and Lee2020). Again, while this HEXACO framework has generated quite a large literature, this is still not as extensive as that generated by the Big Five. Nevertheless, this is also an alternative framework within which one can conduct personality research. Third, the Dark Triad made its appearance in the literature (Paulhus & Williams, Reference Paulhus and Williams2002). While this did not pose a challenge to the Big Five, it certainly suggested that the Big Five could be complemented by looking at other personality dimensions. Thus, what Lee and Ashton (Reference Lee and Ashton2014, p. 3) refer to as the “Big-Five plus-Dark-Triad” (B5–plus–D3 model; see, e.g., Kowalski et al., Reference Kowalski, Vernon and Schermer2019, for a recent example of this) can now be contrasted with HEXACO as two competing theories (Ashton & Lee, Reference Ashton and Lee2020; Howard & van Zandt, Reference Howard and van Zandt2020) whose predictive value are under scrutiny at the present.
Turning now to HEXACO space, where would the Dark Triad be located? One finds two prominent patterns in the literature. The first identifies the Dark Triad with Honesty-Humility (Ashton & Lee, Reference Ashton and Lee2009, Reference Ashton and Lee2020; Hodson et al., Reference Hodson, Book, Visser, Volk, Ashton and Lee2018; Paulhus, Reference Paulhus2014); the second places the Dark Triad within a subspace marked by Honesty-Humility, Agreeableness, and Conscientiousness (Watts et al., Reference Watts, Waldman, Smith, Poore and Lilienfeld2017). There is an ongoing debate regarding the degree of convergence of Honesty-Humility and the Dark Triad. While the two constructs do converge (Ashton & Lee, Reference Ashton and Lee2009, p. 344; Kowalski et al., Reference Kowalski, Rogoza, Saklofske and Schermer2021), one line of thought is that the ‘dark core’ of the Dark Triad (and of all such dark traits of personality) is much wider than the content expressed by Honesty-Humility (Moshagen et al., Reference Moshagen, Hillbig and Zettler2018). A second line of thought is that the two constructs, in fact, have maximal overlap (Hodson et al., Reference Hodson, Book, Visser, Volk, Ashton and Lee2018, p. 128). The most recent argument is that the two constructs are, in fact, functionally distinct, while still bearing a great degree of overlap (Horsten et al., Reference Horsten, Moshagen, Zettler and Hilbig2021). Hence, it is important to continue investigating where exactly the Dark Triad is located in HEXACO space, and our use of Smallest Space Analysis (see below) will comprise an important contribution here.
A second major goal is to study the space of the ‘alternative’ Big Five, the ZKA-PQ space (Extraversion, Neuroticism, Activity, Sensation Seeking, and Aggressiveness; Glicksohn et al., Reference Glicksohn, Golan-Smooha, Naor-Ziv, Aluja and Zuckerman2018; Rossier et al., Reference Rossier, Aluja, Blanch, Barry, Hansenne, Carvalho, Valdivia, Wang, Desrichard, Hyphantis, Suranyi, Glicksohn, De Pascalis, León-Mayer, Piskunov, Stivers, Morizot, Ostendorf, Cekrlija and Karagonlar2016). Note that this ‘alternative’ Big Five model also enables one to place both ‘normal’ and pathological traits within the same personality space (Aluja et al., Reference Aluja, Sayans-Jiménez, García and Gutierrez2021). Given that Psychoticism converges with Sensation Seeking, Aggressiveness and Activity (Zuckerman & Glicksohn, Reference Zuckerman and Glicksohn2016, p. 50), one option is for the Dark Triad to be embedded in ZKA-PQ space within the zone corresponding to the high end of these dimensions. Another would be to consider a subspace such as that of Sensation Seeking and Aggressiveness (Glicksohn et al., Reference Glicksohn, Golan-Smooha, Naor-Ziv, Aluja and Zuckerman2018, p. 21). We are particularly interested in seeing whether Machiavellianism and Psychopathy can be clearly differentiated in ZKA-PQ space. For as Kowalski et al. (Reference Kowalski, Vernon and Schermer2019, p. 3) have recently suggested, in contrast to the Big Five framework, a personality space that emphasizes such a dimension as impulsivity might be better suited to differentiate between Psychopathy and Machiavellianism. ZKA-PQ space would be a good candidate here.
In order to investigate the location of the Dark Triad within each of these multidimensional personality spaces, we employ multidimensional scaling (specifically, Smallest Space Analysis), to reduce the dimensionality of each space (for HEXACO, this is 6; for the ZKA-PQ, this is 5) to that of a 2D representation. Given that the reader may be unfamiliar with this technique, we outline below the rationale of the analysis, exemplify its use, and then present a working hypothesis for the structure of personality space that should be uncovered using Smallest Space Analysis (SSA).
SSA is a technique of data reduction, whereby a correlation matrix (or matrix of any suitable measure of similarity) is translated into a matrix of distances.Footnote 1 Consider, for example, the 3 × 3 correlation matrix for the Dark Triad: the 3 above-diagonal correlations are that between Psychopathy (P) and Narcissism (N; r PN), that between Psychopathy and Machiavellianism (M; r PM), and that between Narcissism and Machiavellianism (r NM). In the study reported here, these were found to be as follows: r PN = .31; r PM = .51; r NM = .35 (see Table 1). The correlation between P and M is translated into a distance between a point in a plane representing P and a point in the same plane representing M; this distance is d PM. In the same manner, r NM is translated into d NM and r PN into d PN. The three points P, N, and M must be situated such that inequalities between correlations (e.g., r PM > r NM) are reflected in the corresponding inequalities between distances (i.e., d PM < d NM); in addition, given that the three points form a triangle, these distances must preserve the triangle inequality, namely that the sum of any two sides must be greater than the third side. In a larger k × k correlation matrix, there are multiple constraints on these calculated distances. SSA preserves the order among the correlations in the smallest possible Euclidean space (Schlesinger & Guttman, Reference Schlesinger and Guttman1969, p. 95). In practice, the 2D space is the most useful one for exploring the structure of that space.
Table 1. Pearson Correlations* and Descriptive Statistics for the Five Dimensions of the ‘Alternative’ Big Five, Assessed Using the ZKA-PQ, the Six Dimensions of HEXACO, and the Three Dimensions of the Dark Triad, Assessed Using the SD3
Note. *significant at p < .0001.
SSA has been employed in the past to examine the 2D space representing structural brain asymmetry (Glicksohn & Myslobodsky, Reference Glicksohn and Myslobodsky1993; Myslobodsky et al., Reference Myslobodsky, Glicksohn, Coppola and Weinberger1991), mood (Glicksohn & Boikova, Reference Glicksohn and Boikova2018; Glicksohn et al., Reference Glicksohn, Gvirtsman and Offer1996), personality disorders (Kumar et al., Reference Kumar, Ryan and Wagner2012; Pukrop et al., Reference Pukrop, Herpertz, Saß and Steinmeyer1998; Turkheimer et al., Reference Turkheimer, Ford and Oltmanns2008), parenting styles (Alt, Reference Alt2016), and other varied topics. Of present interest is the use of SSA to study the structure of personality space, be this the Eysenckian Big Three (Hammond, Reference Hammond1987), the Big Five (Maraun, Reference Maraun1997), and as shown in this paper, to study the personality space underlying HEXACO in comparision with that underlying the ZKA-PQ. As opposed to studying the factor space underlying a personality inventory, with its inherent focus on the number of factors entailed (e.g., the Big Three, the Big Five, etc.), SSA is concerned with the structure of that personality space, how the space can be partitioned into regions, and what those regions represent.
Furthermore, as a nonlinear and nonmetric technique of data analysis, SSA comprises a major alternative to the linear type of data reduction incorporated in factor analysis (see the Appendix). Another approach to the depiction of the structure of personality space, which also presents an alternative to such linear data reduction, is that of a network representation of personality space (Cramer et al., Reference Cramer, van der Sluis, Noordhof, Wichers, Geschwind, Aggen, Kendler and Borsboom2012; Epskamp et al., Reference Epskamp, Maris, Waldorp, Borsboom, Irwing, Booth and Hughes2018). A benefit of a network analysis is to indicate the centrality of particular traits within the network, and to present their spatial interdependencies (for recent use of this with respect to the Dark Triad and the Big Five, see Jordan et al., Reference Jordan, Jonason, Zeigler-Hill, Winer, Fletcher and Underhill2021; Wehner et al., Reference Wehner, Ziegler, Gödeke and Lämmle2021), and this is also apparent in SSA. In addition to this, however, SSA also enables one to look for a particular structure in the data—and in the present context, a particular structure of the personality space, as we detail below.
Using SSA, Maraun (Reference Maraun1997) has uncovered a radex (Shepard, Reference Shepard and Shye1978, p. 57) for the Big Five, which entails the partitioning of the 2D SSA space into conical regions corresponding to each of the Big Five dimensions, with rays emanating from a common point of origin delineating these regions. The ordering of regions in a radex is circular, and Maraun (Reference Maraun1997) reported the following circular ordering, moving in a clockwise direction: Neuroticism—Agreeableness—Conscientiousness—Extraversion—Openness to Experience. Both Alt and Boniel-Nissim (Reference Alt and Boniel-Nissim2018) and de Souza et al. (Reference de Souza, de Souza, Roazzi, Lula, Roazzi, de Souza and Bilsky2015) also reported a radex structure for the Big Five—and in each study there was a slightly different circular ordering of the regions. We note, however, that in each of these studies, a different measure of the Big Five was used; hence, while the radex structure seems to be common, its particular version is dependent on the questionnaire employed to assess the Big Five.
Will we also find that both the HEXACO and the ZKA-PQ spaces conform to this structure of a radex? We know of no previous study that has addressed this question. Based on the radex structure that has been reported for the Big Five, this is a plausible hypothesis to be examined for these other personality spaces. Further, where will the Dark Triad traits be located in this space? Finally, on embedding the Dark Triad traits within these 2D SSA spaces, will there be a distortion of the space? This is an interesting possibility, and we can demonstrate this when looking at how SSA was used by Hammond (Reference Hammond1987) to examine the Eysenckian Big Three. In order to do so, we first reflect on the fourth factor proposed by Eysenck. For Eysenck incorporated a Lie Scale (L) in his personality questionnaires (Eysenck & Eysenck, Reference Eysenck and Eysenck1985), so as to assess the degree to which subjects attempt to ‘fake good.’ The negative correlation between P and L (Glicksohn & Abulafia, Reference Glicksohn and Abulafia1998) was the basis for profiling the antisocial risk taker (Glicksohn et al., Reference Glicksohn, Ben-Shalom and Lazar2004), and for mapping psychiatric patients onto a P-L plane (Glicksohn & Bar-El, Reference Glicksohn and Bar-El2004).
When excluding L from his analysis, Hammond (Reference Hammond1987) presented a clear partitioning of the 2D SSA space into three regions, one comprising items of the Extraversion scale, a second comprising items of the Neuroticism scale, and the third items of the Psychoticism scale, in support of the Big Three. A potential problem arises when L items are included in the analysis, because of the negative correlation between P and L (Hammond, Reference Hammond1987, p. 546). Thus, L could well distort the 2D SSA space for the Big Three. In fact, however, this was not the case, and when the L items were entered into his analysis, Hammond (Reference Hammond1987) reported a clear partitioning of the 2D SSA space into four regions, with the P and L regions extending successively and in separate along one diagonal of the plane. Note that in an SSA plot, when two regions such as those of P and L lie at 180 degrees to each other, this reflects the negative correlation between the two domains.
In the present study, a distortion of the 2D SSA space might occur for the HEXACO space, in comparison to the radex found for the Big Five, because of the very presence of the sixth dimension of Honesty-Humility in the matrix. In addition, for both the HEXACO and ZKA-PQ spaces, a distortion of the respective 2D SSA spaces might occur on entering the Dark Triad items into the analysis. Note that the surface pattern of intercorrelation is quite predictive of the 2D SSA structure. For example, the positive correlations between the Dark Triad and Aggressiveness (see Table 1) would ensure that the Dark Triad should appear in the subspace of the SSA structure for the ZKA-PQ pertaining to Aggressiveness, while the negative correlations between the Dark Triad and Honesty-Humility (see Table 1) would ensure that the Dark Triad should be diametrically opposite Honesty-Humility in the SSA structure for HEXACO. Nevertheless, given that these are not the only correlations observed between the Dark Triad and the various traits under investigation here, one has to take into consideration the whole pattern of intercorrelation—and that is exactly what the 2D SSA solution will do. The answers to these three questions should, therefore, be of interest for both the study of personality structure and that of personality assessment: (a) Do the HEXACO and the ZKA-PQ 2D SSA spaces conform to the structure of a radex, as does that 2D SSA space pertaining to the Big Five? (b) On embedding the Dark Triad traits within these 2D SSA spaces, will there be a distortion of the space, or will the space remain essentially unchanged in structure?Footnote 2 (c) Where will the Dark Triad traits be located in each of these 2D SSA spaces?
To conclude this section, what will be gained from embedding the Dark Triad within these two personality spaces? First, it should be clear from the results that we report whether the Dark Triad is, as Moshagen et al. (Reference Moshagen, Hillbig and Zettler2018, p. 27) put it, “well suited for inclusion in a more general model of personality dimensions akin to the … HEXACO model.” Second, whether the Dark Triad has, as Hodson et al. (Reference Hodson, Book, Visser, Volk, Ashton and Lee2018, p. 128) put it, any “incremental value … at least as a latent ‘constellation’ ” will surely be of interest to the literature. Thus, if the Dark Triad traits are clearly, and unambiguously, aligned only with Hostility-Humility in HEXACO space (Ashton & Lee, Reference Ashton and Lee2009, p. 344), then the “Big-Five plus-Dark-Triad” option can be safely exchanged with that of HEXACO (Ashton & Lee, Reference Ashton and Lee2020). Furthermore, if this is the case, then this alignment of the Dark Triad with Hostility-Humility, and not with Agreeableness in HEXACO space, will surely support the distinction of these two dimensions, in spite of their similarities (Howard & van Zandt, Reference Howard and van Zandt2020).
In turn, if the Dark Triad traits do lie in the quadrant of a higher-order circumplex, spanning Disharmony, Disinhibition, and Sensation Seeking (Rogoza et al., Reference Rogoza, Kowalski, Saklofske and Schermer2022, Figure 1), then this would suggest that they should be easily embedded within a lower-order ZKA-PQ space. This is because Sensation Seeking in this circumplex is defined, using HEXACO, as High Emotionality, High Extraversion, High Openness to Experience, Low Agreeableness, Low Conscientiousness, and Low Hostility-Humility (Rogoza et al., Reference Rogoza, Kowalski, Saklofske and Schermer2022, Figure 1), while in ZKA-PQ space, Sensation Seeking is itself one of the five dimensions defining this space. Thus, it is important to embed the Dark Triad within ZKA-PQ space, in order to further our understanding of these personality spaces, and their degree of compatibility.
Figure 1. The 2D Personality Space Obtained Via Smallest Space Analysis for HEXACO (a), and when Entering the Dark Triad into the Analysis (b)
Method
Participants
Two hundred and eighty-nine individuals (137 males, 145 females, 7 unspecified) participated in this study on a voluntary basis. Their age ranged between 18 and 87 (M = 43.3, SD = 17.47 years). We had hoped to achieve a sample size closer to 400, but given constraints (especially a deadline by which time we had to complete collecting our data, as part of a larger study—see Aluja et al., Reference Aluja, Rossier, Oumar, García, Bellaj, Ostendorf, Ruch, Wang, Kövi, Ścigała, Čekrlija, Stivers, Di Blas, Valdivia, Ben Jamaa, Atitsogbe, Hansenne and Glicksohn2020), we remained with the n reported here.Footnote 3
The study was approved by the university ethics board; all participants provided written informed consent. They completed the following measures in the same order as below.
Measures
Three questionnaires were employed, and each underwent a process of back-translation into Hebrew. The first author translated the items into Hebrew, and the second author translated the Hebrew version back into English, staying as closely as possible to the idiomatic language of the Hebrew. Both authors then reviewed these versions and jointly decided on the final Hebrew version. For the ZKA-PQ/SF, we could rely on our Hebrew version of the ZKA-PQ (Glicksohn et al., Reference Glicksohn, Golan-Smooha, Naor-Ziv, Aluja and Zuckerman2018) and only had to add a few items for this shortened form. At the time, we were unaware of the fact that a Hebrew version of both the HEXACO–60 and the SD3 had been prepared, independently of our own versions (Zeigler-Hill et al., Reference Zeigler-Hill, Besser and Marcus2017). The alpha reliabilities that we report here (see Table 1) closely match those reported by these colleagues.
The Hebrew version of the Zuckerman-Kuhlman-Aluja Personality Questionnaire, shortened form (ZKA-PQ/SF) comprises 80 items (Aluja et al., Reference Aluja, Lucas, Blanch, García and García2018; Aluja et al., Reference Aluja, Rossier, Oumar, García, Bellaj, Ostendorf, Ruch, Wang, Kövi, Ścigała, Čekrlija, Stivers, Di Blas, Valdivia, Ben Jamaa, Atitsogbe, Hansenne and Glicksohn2020), from which we derived scores for the five dimensions of the Alternative 5 (Zuckerman, Reference Zuckerman2005): Extraversion (EX), Neuroticism (NE), Sensation Seeking (SS), Aggressiveness (Ag) and Activity (Act). Participants selected their response to each item on a 4-point scale, ranging from strongly disagree (1) to strongly agree (4).
The Hebrew version of HEXACO–60 comprises 60 items (Ashton & Lee, Reference Ashton and Lee2009), from which we derived scores for the six dimensions of Honesty-Humility (H), Emotionality (E), Extraversion (X), Agreeableness (A), Conscientiousness (C), and Openness to Experience (O). Participants selected their response to each item on a 5-point scale, ranging from strongly disagree (1) to strongly agree (5).
The Hebrew version of the Short Dark Triad (SD3) comprises 27 items (Jones & Paulhus, Reference Jones and Paulhus2014), from which we derived scores for the three dimensions of Machiavellianism, Psychopathy, and Narcissism. Participants selected their response to each item on a 5-point scale, ranging from strongly disagree (1) to strongly agree (5).
Alpha reliabilities for the five dimensions of the Alternative 5 (ZKA-PQ), the six dimensions of HEXACO (HEXACO–60), and the three dimensions of the Dark Triad (SD3) are reported in Table 1. They range between .79 and .88 for the ZKA-PQ, between .68 and .78 for HEXACO, and between .72 and .78 for the SD3.
Procedure
This study was conducted as part of a large, cross-cultural project (Aluja et al., Reference Aluja, Rossier, Oumar, García, Bellaj, Ostendorf, Ruch, Wang, Kövi, Ścigała, Čekrlija, Stivers, Di Blas, Valdivia, Ben Jamaa, Atitsogbe, Hansenne and Glicksohn2020), and followed the same protocol. The three questionnaires were presented in one standard order to students studying with either the first author or the second author, who volunteered to participate in the study. These students were then invited to administer the questionnaires to family members, friends, associates and neighbors, with the following constraints: An equal number of men and women, within the following age ranges: (a) 18 to 30 years, (b) 31 to 45 years, (c) 46 to 60 years, and (d) more than 60 years old.
Data Analysis
The Smallest Space Analysis (SSA) in the present report is conducted on a k × k correlation matrix (here k is either 20 [ZKA-PQ] or 24 [HEXACO], increasing with the addition of the 3 Dark Triad traits to either 23 or 27), and our n-to-k ratio is roughly 10. This seems to be quite plausible.Footnote 4 The analysis is provided by the SSA module of HUDAP (Guttman & Greenbaum, Reference Guttman and Greenbaum1998).Footnote 5 The coefficient of alienation (a measure of ‘lack-of-fit’) for the 2D SSA space for HEXACO was found to be 0.272, and while this would not be considered to be adequate (for the 3D space, the coefficient was 0.165, which would be considered to be adequate), we follow the guidelines advanced by Borg and Groenen (Reference Borg and Groenen2005, p. 73) who, in discussing an example, stress that the interpretability of the space is more important than is the actual value of such a goodness-of-fit criterion. Furthermore, a rotation of this space is allowed, because the 2D SSA space can be rotated along any axis. This was reported by Tiliopoulos et al. (Reference Tiliopoulos, Pallier and Coxon2010, pp. 36–37) for the Big Five, and by Glicksohn et al. (Reference Glicksohn, Gvirtsman and Offer1996) for mood space (see also Glicksohn & Boikova, Reference Glicksohn and Boikova2018).
Results
Table 1 presents Pearson correlations and descriptive statistics for all the personality dimensions studied here. Note the following: (a) The positive correlation of .45 between Aggressiveness and Neuroticism; (b) the positive correlation between Psychopathy and Machiavellianism of .51; (c) the negative correlation between Aggressiveness and Agreeableness; (d) the positive correlation between Neuroticism and Emotionality, and that between the two dimensions of Extraversion, these tapping essentially the same personality dimensions; (e) the negative correlation between Aggressiveness and Honesty-Humility; (f) for Psychopathy, its positive correlation with Aggressiveness and its negative correlation with Agreeableness; (g) the negative correlations between each measure of the Dark Triad and Honesty-Humility; and (h) the positive correlations between each measure of the Dark Triad and Aggressiveness.
Embedding the Dark Triad in HEXACO Space
The 2D SSA space depicted in Figure 1a can be partitioned into five regions, four of which demark the HEXACO dimensions of Emotionality, Agreeableness, Honesty-Humility, and Openness to Experience, in that counterclockwise order. In the fifth region, a cluster indicative of Extraversion is embedded within a region that also spans the Conscientiousness dimension. Hence, these two dimensions are not differentiated in the 2D SSA space. On entering the Dark Triad scales into the analysis (Figure 1b), one notes the following: (a) The same circular ordering of dimensions (regions) is preserved; (b) the Dark Triad lie in the same region of Extraversion + Conscientiousness; and (c) the presence of the Dark Triad in this space impacts on the space, the other 4 regions being more squished together.
Embedding the Dark Triad in ZKA-PQ Space
Figure 2a presents the 2D SSA space (coefficient of alienation = 0.148), for the 20 facets of the ZKA-PQ, with the structure being clearly indicative of a radex: The space is evenly partitioned into regions corresponding to each of the dimensions, the circular counterclockwise ordering being Activity, Extraversion, Sensation Seeking, Aggressiveness, and Neuroticism. On entering the Dark Triad scales into the analysis (Figure 2b), one notes the following: (a) There is no major change in this space, hence this space retains the structure of a radex and, moreover, of an invariant gestalt; (b) the Dark Triad lie within this space, with Psychopathy and Machiavellianism being in closer proximity, in the Aggressiveness region, than either is with Narcissism; and (c) the common origin of the radex is also the locus for Narcissism.
Figure 2. The 2D Personality Space Obtained Via Smallest Space Analysis for the ZKA-PQ (a), and when Entering the Dark Triad into the Analysis (b)
Discussion
Turning first to the Pearson correlations, note the following. First, the positive correlation of .45 between Aggressiveness and Neuroticism is higher than the .34 correlation reported for these by Aluja et al. (Reference Aluja, Lucas, Blanch, García and García2018). Second, for HEXACO all intercorrelations were quite low, as one would expect given the orthogonality of these dimensions (Lee & Ashton, Reference Lee and Ashton2014); for the Dark Triad, the correlation between Psychopathy and Machiavellianism of .51 is comparable to the value of .47 reported by Jones and Paulhus (Reference Jones and Paulhus2014, Study 3). Third, the negative correlation between Aggressiveness and Agreeableness is expected (Aluja et al., Reference Aluja, García and García2002; Dinić & Smederevac, Reference Dinić and Smederevac2018; Zuckerman et al., Reference Zuckerman, Kuhlman, Joireman, Teta and Kraft1993). Fourth, the positive correlation of Psychopathy with Aggressiveness, and its negative correlation with Agreeableness, is internally consistent, given the negative correlation between Aggressiveness and Agreeableness, and places Psychopathy in both personality spaces. Fifth, the negative correlations between each measure of the Dark Triad and Honesty-Humility, clearly aligns the Dark Triad with this dimension of the HEXACO space. Sixth, the positive correlations between each measure of the Dark Triad and Aggressiveness, clearly aligns the Dark Triad with this dimension of the ZKA-PQ space; furthermore, this is internally consistent, given the negative correlation between Aggressiveness and Honesty-Humility.
Turning now to the 2D SSA spaces, why is it important to know where to locate the Dark Triad here? There are two considerations here. The first is that of redundancy (O’Boyle et al., Reference O’Boyle, Forsyth, Banks, Story and White2015), especially given the notion that in the HEXACO space, the dimension of Honesty-Humility might very well cover the same ground as that of the Dark Triad (e.g., Lee & Ashton, Reference Lee and Ashton2014; Schreiber & Marcus, Reference Schreiber and Marcus2020). The second is that of the internal structure of the Dark Triad, especially given the debate in the literature regarding the distinction (or lack of) between Psychopathy and Machiavellianism (e.g., Jones & Figueredo, Reference Jones and Figueredo2013; Rogoza & Cieciuch, Reference Rogoza and Cieciuch2020).
Embedding the Dark Triad in HEXACO Space
SSA space for HEXACO does not conform to a radex as has been reported for the Big Five (Maraun, Reference Maraun1997, p. 638). Why might this be so? The problem seems to lie in the extensive overlap between Conscientiousness and Extraversion. While partial overlap of these two dimensions has been reported recently (Schwaba et al., Reference Schwaba, Rhemtulla, Hopwood and Bleidorn2020, p. 17; Strus & Cieciuch, Reference Strus and Cieciuch2021, p. 3; Woods & Anderson, Reference Woods and Anderson2016, pp. 589–590), it is only when the data are analyzed using SSA that one can appreciate the problematic nature of this. Note that the second largest correlation in the HEXACO intercorrelation matrix (see appropriate submatrix in Table 1) is that between Conscientiousness and Extraversion, with a value of .15 which, while being modest at best, is still of the same size as that reported recently by both García et al. (Reference García, Aluja, Rossier, Ostendorf, Glicksohn, Oumar, Bellaj, Ruch, Wang, Kövi, Ścigała, Čekrlija, Stivers, Di Blas, Valdivia, Ben Jemaa, Atitsogbe and Hansenne2021) and Thielmann et al. (Reference Thielmann, Akrami, Babarović, Belloch, Bergh, Chirumbolo, Čolović, de Vries, Dostál, Egorova, Gnisci, Heydasch, Hilbig, Hsu, Izdebski, Leone, Marcus, Međedović, Nagy and Lee2020). This is not, however, the result, as an anonymous reviewer has suggested, of “metric partial non-invariance of the Hebrew version” of HEXACO–60, because García et al. (Reference García, Aluja, Rossier, Ostendorf, Glicksohn, Oumar, Bellaj, Ruch, Wang, Kövi, Ścigała, Čekrlija, Stivers, Di Blas, Valdivia, Ben Jemaa, Atitsogbe and Hansenne2021) reported metric invariance of the HEXACO–60 across eighteen cultures/languages, including that of our sample. Such cross-cultural metric invariance has also been reported by Thielmann et al. (Reference Thielmann, Akrami, Babarović, Belloch, Bergh, Chirumbolo, Čolović, de Vries, Dostál, Egorova, Gnisci, Heydasch, Hilbig, Hsu, Izdebski, Leone, Marcus, Međedović, Nagy and Lee2020). Furthermore, the presence of the Dark Triad within this 2D space distorts the space, while the presence of the Dark Triad in the 6D factorial space preserves that space (see the Appendix). What this means is that on entering the Dark Triad into the analysis, the same 6D factorial space defined by HEXACO remains invariant, with the Dark Triad loading (with negative sign) on the factor of Honesty-Humility, as expected (Paulhus, Reference Paulhus2014, p. 422), on the one hand, and with Narcissism loading in addition on Extraversion, and Psychopathy loading in addition on Agreeableness. Thus, the Dark Triad differentially load on a subspace of Honesty-Humility × Extraversion × Agreeableness, which is a subspace worthy of further investigation (Schreiber & Marcus, Reference Schreiber and Marcus2020, pp. 1036–1037); and further, all six HEXACO factors are not compromised by the presence of the Dark Triad traits in the factor analysis. It is when the 6D HEXACO space is reduced to a 2D space that the impact of the Dark Triad is realized. This ties in with some recent comments in the literature that, because the underlying D factor of the Dark Triad represents a blend of basic traits, it would be difficult to preserve orthogonality with other dimensions of the personality space (Moshagen et al., Reference Moshagen, Hillbig and Zettler2018, p. 27), and that the Dark Triad comprises a heterogeneous constellation (Watts et al., Reference Watts, Waldman, Smith, Poore and Lilienfeld2017, p. 952). We wonder whether a different type of geometric space is required for making a smooth transition to a space of lower dimensionality, one that would enable a better embedding of the Dark Triad in that space (Townsend et al., Reference Townsend, Solomon, Spencer-Smith, Wenger and Townsend2001; Townsend et al., Reference Townsend, Burns, Pei, Berglund, Townsend, Rossi and Pendrill2012).
We were interested in seeing whether Machiavellianism and Psychopathy are coincident in HEXACO space (Muris et al., Reference Muris, Merckelbach, Otgaar and Meijer2017, p. 188; Vize et al., Reference Vize, Lynam, Collison and Miller2018, p. 109). Note that the Dark Triad and Honesty-Humility lie at 180 degrees to each other in the 2D SSA space, and this is an interesting finding, given the discussions in the literature as to whether there is much overlap between Honesty-Humility and the Dark Triad (Ashton & Lee, Reference Ashton and Lee2020; Hodson et al., Reference Hodson, Book, Visser, Volk, Ashton and Lee2018; Moshagen et al., Reference Moshagen, Hillbig and Zettler2018). This finding reflects the fact that these measures are negatively correlated. Furthermore, in the 2D SSA space for HEXACO, Machiavellianism and Psychopathy lie in the same region, while being quite distinct. This is important to stress, given that their differentiation is still being debated in the literature (Kowalski et al., Reference Kowalski, Vernon and Schermer2019; Rogoza et al., Reference Rogoza, Kowalski, Saklofske and Schermer2022).
Embedding the Dark Triad in ZKA-PQ Space
We report a major finding: SSA space for the ZKA-PQ conforms to a radex; furthermore, this radex is preserved when embedding the Dark Triad in this space, and thus this is indicative of the fact that the ZKA-PQ radex is a gestalt (Glicksohn et al., Reference Glicksohn, Golan-Smooha, Naor-Ziv, Aluja and Zuckerman2018). In this radex, Machiavellianism and Psychopathy are close in proximity, as would be expected (Rogoza & Cieciuch, Reference Rogoza and Cieciuch2020), hence might well reflect the very similar personality constructs (Glenn & Sellbom, Reference Glenn and Sellbom2015, p. 363) in the SD3 instantiation of these traits (Jones & Paulhus, Reference Jones and Paulhus2014). The differential placement of the Dark Triad traits within the radex comes in support of the argument that Machiavellianism and Psychopathy might be combined into one factor, which is differentiated from Narcissism (Koehn et al., Reference Koehn, Okan and Jonason2019). Furthermore, given the fact that both Psychopathy and Machiavellianism lie in close proximity in the Aggressiveness region of the radex, together with the fact that Aggressiveness and Agreeableness are negatively correlated (Aluja, Reference Aluja2019, p. 187), we find support for the argument that it is not only Hostility-Humility which is related to the Dark Triad traits, but also Agreeableness (Vize et al., Reference Vize, Collison, Miller and Lynam2019, p. 97).
While Machiavellianism and Psychopathy are in close proximity in the radex, they are not coincident. The literature has stressed that these two measures from the SD3 poorly reflect the richness of Psychopathy (Muris et al., Reference Muris, Merckelbach, Otgaar and Meijer2017, p. 195); that the Psychopathy measure may not assess both primary and secondary Psychopathy (Koehn et al., Reference Koehn, Okan and Jonason2019); that there is a need to clarify the difference between Psychopathy and Machiavellianism (Rogoza et al., Reference Rogoza, Kowalski and Schermer2019, p. 174); and that the SD3 does not adequately assess the subdimensions of Psychopathy, which might be related differentially to external criteria (Lilienfeld, Reference Lilienfeld2018, p. 81). If what ‘binds’ the Dark Triad is primary psychopathy (Neumann et al., Reference Neumann, Hare and Pardini2015, p. 679), then what might distinguish Machiavellianism from Psychopathy in the radex? The primary-secondary distinction within Psychopathy might not be the key to differentiating Machiavellianism and Psychopathy, because while one group of researchers has suggested that Machiavellianism may reflect secondary Psychopathy (Vize et al., Reference Vize, Lynam, Collison and Miller2018), a second group has suggested that this might reflect primary Psychopathy (Kavish et al., Reference Kavish, Jones, Rock, Johnson and Anderson2019, p. 205).
Consider, therefore, a second distinction. If Lilienfeld’s (Reference Lilienfeld2018) Impulsive Antisociality (IA) subdimension of Psychopathy is more closely associated with the SD3 measure of Psychopathy (Dowgwillo & Pincus, Reference Dowgwillo and Pincus2017, p. 25), then Machiavellianism, which should be unrelated to impulsivity (Watts et al., Reference Watts, Waldman, Smith, Poore and Lilienfeld2017, p. 108) might, instead, be related to Lilienfeld’s Fearless Dominance (FD) subdimension of Psychopathy. In addition, psychopathy and antisocial personality have a strong biological component (García et al., Reference García, Aluja, Fibla, Cuevas and García2010) and have been associated with Aggressiveness and Sensation Seeking (Dickey, Reference Dickey2014; Wilson & Scarpa, Reference Wilson and Scarpa2011). If so, then one intercorrelated factor of IA, Impulsivity and Neuroticism (Ben-Yaacov & Glicksohn, Reference Ben-Yaacov and Glicksohn2020) might characterize Psychopathy, while a second intercorrelated factor of FD, Sensation Seeking and Extraversion (Ben-Yaacov & Glicksohn, Reference Ben-Yaacov and Glicksohn2020) might characterize Machiavellianism. Note that in the radex, Machiavellianism is located somewhat closer to the Sensation Seeking region, while Psychopathy is located somewhat closer to the Neuroticism region. Conceivably, more refined measures of Psychopathy and Machiavellianism (and here we support this call for such measures, raised by Miller et al., Reference Miller, Hyatt, Maples-Keller, Carter and Lynam2017, p. 450) might further stretch apart these two dimensions (both of which would still be located in the region of Aggressiveness), pulling Psychopathy more in the direction of the Neuroticism region. That is a hypothesis worthy of future investigation.
Two limitations of the present study should be stressed. First, our participants completed the three questionnaires in the same order (ZKA-PQ/SF, HEXACO–60, SD3)—following the prescribed protocol of the larger, cross-cultural study (Aluja et al., Reference Aluja, Rossier, Oumar, García, Bellaj, Ostendorf, Ruch, Wang, Kövi, Ścigała, Čekrlija, Stivers, Di Blas, Valdivia, Ben Jamaa, Atitsogbe, Hansenne and Glicksohn2020; García et al., Reference García, Aluja, Rossier, Ostendorf, Glicksohn, Oumar, Bellaj, Ruch, Wang, Kövi, Ścigała, Čekrlija, Stivers, Di Blas, Valdivia, Ben Jemaa, Atitsogbe and Hansenne2021). Whether this impacted on our results is an unknown. Second, the SD3, as noted above, does not lend itself to a refined analysis of the subcomponents of Psychopathy. The same point can be made with respect to Narcissism, which has a grandiose form, and a vulnerable form (Egan et al., Reference Egan, Chan and Shorter2014; Lilienfeld et al., Reference Lilienfeld, Watts, Murphy, Costello, Bowes, Smith, Latzman, Haslam and Tabb2019; Watts et al., Reference Watts, Waldman, Smith, Poore and Lilienfeld2017). As suggested above, the use of more refined measures of the Dark Triad, coupled with SSA, should provide a better understanding of the location of these traits, and their subcomponents, in a multidimensional personality space.
In conclusion, we address the implications of the results reported here. In both HEXACO and ZKA-PQ 2D spaces, the Dark Triad traits form a triangular structure, indicating a specific constellation. The traits do not overlap in location, hence are differentiated. Of particular importance, while Machiavellianism and Psychopathy are located in close proximity, their differentiation is clearly captured using SSA. Their location in HEXACO 2D space is diametrically opposite that of Honesty-Humility, and had they been inversely coded, they would have been located within this Honesty-Humility section of the space. They lie in the conjoint section of Extraversion and Conscientiousness, and that is an interesting finding, worthy of further exploration. Furthermore, this 2D SSA space for HEXACO is not as well organized as is that for the ZKA-PQ. This is the case irrespective of whether or not the Dark Triad traits are entered into the analyses. Indeed, the 2D SSA space for HEXACO has captured a fair degree of overlap between Extraversion and Conscientiousness, that is not widely discussed in the literature, and which requires attention. This overlap impinges on the radex structure that is expected to be found for HEXACO. In contrast, ZKA-PQ space exhibits this radex, and the Dark Triad is embedded in this radex without altering its basic organization. Their location in ZKA-PQ 2D space is such that Narcissism is found at the very center of the radex, while Psychopathy and Machiavellianism lie in the Aggressiveness section of the radex. Their presence in this radex does not distort this structure (hence the gestalt-like nature of the radex), which further suggests that the Dark Triad fit very well into ZKA-PQ personality space.
The data reported here will be made available, following such a request to the corresponding author.
Appendix: Factor Analyses
Embedding the Dark Triad in HEXACO Space
Summary scores for each of the four facets for each of the six scales of the HEXACO–60 constituted the twenty-four variables for exploratory factor analysis (inspection of the scree plot, subsequent factor extraction and rotation, a criterial loading of 0.4, and ‘good structure’ being elements here). A total of 270 participants had complete data for these, hence the ratio of participants to variables is well in excess of 10. Table A1 provides the correlations of each of the Dark Triad scales with each of these 24 HEXACO facets, to aid in interpreting this factor analysis. The scree plot indicated a break in support of a six-factor solution, the eigenvalues being 3.08, 2.61, 2.37, 2.14, 1.90, 1.43, 1.02, 0.93, 0.87, and 0.75 for the first 10 factors. We then ran a parallel analysis (PA) test as a converging operation, employing FACTOR, version 10.10.02 (Lorenzo-Seva & Ferrando, Reference Lorenzo-Seva and Ferrando2006). PA (using 500 random data matrices) indicated between 5 and 6 factors (depending on criterion: 95th percentile or mean). We therefore retained the 6-factor solution reported here. The correlations among the six extracted factors ranged between –0.20 and +0.33, hence an orthogonal rotation was adopted. On entering the Dark Triad scales into the analysis, the same six-factor space was preserved. We stress that exploratory factor analysis (EFA), as opposed to confirmatory factor analysis (CFA), is the more appropriate analysis here, given that we explore where the Dark Triad lie in each personality space. Furthermore, the use of CFA to ‘confirm’ a given factorial structure of a personality model—be this that of the Big Five or that of the ‘alternative’ Big Five—has not found much support in the literature (Aluja et al., Reference Aluja, García, García and Seisdedos2005, p. 1884; Ginns et al., Reference Ginns, Martin, Liem and Papworth2014, p. 183; Hopwood & Donnellan, Reference Hopwood and Donnellan2010, pp. 341–342; Surányi & Aluja, Reference Surányi and Aluja2014, p. 5).
Table A1. Pearson Correlations* of the Three Dimensions of the Dark Triad, Assessed Using the SD3, with the 24 Facets of HEXACO
Note. *significant at p < .0001.
One factor was indicative of Emotionality; a second was indicative of Honesty-Humility, and on this factor, the three Dark Triad scales loaded with negative sign (as expected); a third factor was identified as Extraversion, and on this factor, Narcissism also loaded with positive sign; a fourth factor was identified with Conscientiousness; a fifth factor was indicative of Openness to Experience; the sixth factor was indicative of Agreeableness, and on this factor, Psychopathy also loaded with negative sign. We compared this factor solution to that provided by an oblique rotation, and can report that a highly similar pattern matrix was found—both prior to and after entering the Dark Triad scales into the analysis. Specifically, for the orthogonal solution the loadings on Honesty-Humility of Machiavellianism (–.70), Narcissism (–.65), and Psychopathy (–.52) were similar to those for the oblique rotation: –.69, –.63, and –.44, respectively.
Embedding the Dark Triad in ZKA-PQ Space
Summary scores for each of the four facets of the five scales of the ZKA-PQ-SF constituted the twenty variables for exploratory factor analysis. A total of 249 participants had complete data for these, and rather than imputing scores to replace missing data, we preferred to work with the smaller sample, which still preserved a ratio of participants-to-variables in excess of 10. Table A2 provides the correlations of each of the Dark Triad scales with each of these 20 ZKA-PQ facets, to aid in interpreting this factor analysis. The scree plot indicated a five-factor solution, the eigenvalues being 4.66, 3.43, 1.96, 1.71, 1.38, 0.76, 0.74, 0.68, 0.60, and 0.59 for the first 10 factors. While PA indicated a 4-factor solution, this solution provided by FACTOR had no ‘simple structure’; in addition, one factor was comprised of Aggressiveness and Sensation Seeking. In contrast, in the 5-factor solution, these two dimensions were clearly separated. We therefore retained the 5-factor solution reported here. The correlations among the five extracted factors ranged between –0.22 and +0.45, hence an oblique rotation was adopted. On entering the Dark Triad scales into the analysis, the same five-factor space was preserved. One factor was identified as Neuroticism; a second was indicative of Activity; a third factor was identified as Sensation Seeking; the fourth factor was that of Extraversion, and on this factor, Narcissism also loaded with positive sign; the fifth factor was identified as Aggressiveness, and on this factor, the three Dark Triad scales loaded with positive sign.
Table A2. Pearson Correlations* of the Three Dimensions of the Dark Triad, Assessed Using the SD3, with the 20 Facets of the ZKA–PQ
Note. *significant at p < .0001.
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