We conducted an online-questionnaire study with a battery of inventories that were used for two distinct projects. Some of the data were also used for illustration purposes within a statistics class of the Master of Science program in Psychology at the University of Bonn. The study was approved by the institutional ethics board of the University of Bonn and conducted in line with the Declaration of Helsinki. Participants were recruited by staff members and students from the Institute of Psychology. All participants volunteered to take part in the study, which took ~45–60 min. Participants received course credit for participation if needed for their degree. A total of 999 participants took part in the study. For the final data, we included only those participants who filled out all items of the relevant measures of this study. The final dataset consisted of 822 participants (516 female, 306 male), who were 16 to 89 years old (M = 32.68, SD = 14.79).

We followed Wilson et al. (2013) and used a bi-factor model, comprising three orthogonal factors: Distress as a general factor for interpersonal problems and Dominance and Love as the two circumplex factors. For MCFA, we used maximum likelihood estimation with robust standard errors to counteract effects of violated multivariate normality. We z-standardized all IIP scales before entering them into the analysis to level out possible differences in circumplex loadings due to different variances of the IIP scales. In MCFA, loadings on Dominance and Love were fixed according to the circumplex model (Table 2). The same loadings were entered as the mean of normally distributed priors in BCFA with a prior variance of σ² = 0.01. It has been recommended to start with smaller prior variances and to increase the prior variance in a second step (Asparouhov et al., 2015). We therefore performed the second BCFA with a prior variance of σ² = 0.1. In both analyses, the loadings of PA on Dominance and of LM on Love were fixed to 1 for scaling adjustments in BCFA. In both MCFA and BCFA, loadings for Distress were estimated freely, whereby the variance of Distress was fixed to σ² = 1 to obtain an identified model. All factors were set to be uncorrelated. Zitzmann and Hecht (2019) noted that the precision of the estimation should be controlled for by means of the potential scale reduction (PSR). We therefore provide the trace plot of the parameter with the greatest PSR value (see Supplementary Figure S1). For both BCFA models, the maximum PSR converged to 1.001 indicating a high precision of the estimation.

Model fit of the bi-factor circumplex model by MCFA was χ²(18) = 207.13, p < 0.001, CFI = 0.94, SRMR = 0.09, RMSEA = 0.11, 90% CI [0.10, 0.13]. Model fit of the bi-factor circumplex BCFA model based on a prior variance of σ² = 0.01 was 95% CI [34.10, 86.48] for χ²(40), the χ²-based posterior predictive p-value was p < 0.001, CFI = 0.98, RMSEA = 0.09, 90% CI [0.08, 0.10]. For a prior variance of σ² = 0.1, model fit was 95% CI [29.59, 80.05] for χ²(40), the χ²-based posterior predictive p-value was p < 0.001, CFI = 0.98, RMSEA = 0.09, 90% CI [0.08, 0.11]. Posterior predictive p-values were the likelihood-ratio χ²-statistic as the Mplus default. Since the fit of the two BCFA models was rather similar, we also computed Tucker's congruence coefficient c (Lorenzo-Seva and ten Berge, 2006) of the loadings on Dominance and Love with the ideal circumplex loadings for both BCFA models. For a prior variance of σ² = 0.01, congruences were c = 0.984 for Dominance and c = 0.984 for Love. For a prior variance of σ² = 0.1, congruences were c = 0.976 for Dominance and c = 0.977 for Love. As one would expect, the congruence with the ideal circumplex was a bit higher for the BCFA model based on a prior variance of σ² = 0.01. We therefore used this model for further analyses.

We saved MCFA scores estimated from the parameters of the MCFA model. By Mplus default, these scores are regression scores (McDonald, 2011), that is, these scores have the maximal correlation of a linear combination of the measured variables with the corresponding factor. The weights that were multiplied with the IIP scales' scores to obtain MCFA regression scores for the present dataset can be found in the Supplement (Supplementary Table S1).

Maximum likelihood based TEFA resulted in an acceptable model fit χ²(5) = 62.03, p < 0.001, CFI = 0.99, SRMR = 0.01, RMSEA = 0.12, 90% CI [0.09, 0.15]. The ideal circumplex loadings of Dominance and Love were used as target loadings for orthogonal target rotation. No target rotation was performed for the Distress factor. The congruence of the TEFA loadings with the ideal circumplex loadings was c = 0.798 for Dominance and c = 0.735 for Love. When congruence scores were adjusted for differences in communalities (Kaiser normalization), they were c = 0.812 for Dominance and c = 0.714 for Love. As congruences were substantially smaller for TEFA than for BCFA, we did not conduct follow-up TEFA analyses or compute TEFA factor scores.

Table 2 shows the ideal circumplex loadings as specified in MCFA as well as the BCFA loadings, so that one can see their deviations from perfect circumplexity. In BCFA, deviations from the perfect circumplex were largest (|>0.20|) for two loadings that were fixed to 0 in MCFA: The loading of the LM/Overly nurturant scale on Dominance and the loading of the PA/Domineering scale on Love, hence, for scales that were supposed to each be perfectly aligned on the opposite factor (see Figure 1). We also conducted an additional misspecification analysis of MCFA circumplex loadings (Saris et al., 1987, 2009), which indicated large expected parameter changes for the two respective MCFA loadings in the direction of the BCFA loadings. Loadings on the general factor Distress were positive and significant with p < 0.001 for all IIP octants for both MCFA and BCFA. Congruence between the corresponding MCFA/BCFA Distress factors was c = 0.999.

Additionally, we investigated shifts in model fit for MCFA and BCFA when error terms were allowed to be correlated. We selected those correlations with a modification index >10 in the first MCFA to be freed in a follow-up MCFA, resulting in 12 correlated error terms, seven of which were significant at p < 0.05 (MCFA model fit: χ²(6) = 92.70, p < 0.001, CFI = 0.97, SRMR = 0.08, RMSEA = 0.14, 90% CI [0.12, 0.17]). These 12 error terms were freely estimated in BCFA with a mean of zero and a prior variance of σ² = 0.01 (BCFA model fit: 95% CI [−23.86, 29.86] for χ²(52), χ²-based posterior predictive p = 0.412, CFI > 0.99, RMSEA = 0.04, 90% CI [0.00, 0.12]). Only two correlated error terms were significant in BCFA, indicating that the increased flexibility in the estimation of circumplex loadings tends to reduce the size of correlated errors.

As BCFA resulted in fewer significant error correlations than MCFA, the enhanced flexibility of BCFA estimation could also result in a more robust estimation of model fit. This was investigated by splitting the sample in two equally sized subsamples. The models were specified as before but without correlated error terms. In addition to the previously reported fit indices, we also report the Bayesian Information Criterion (BIC) to assess the variation of model fit (see Supplementary Table S2). The difference in model fit between the MCFA models in the two subsamples was slightly larger for the CFI, RMSEA, and BIC than for the BCFA models, indicating that BCFA model fit was less affected by variations in the two samples.

All further analysis (external validity of the IIP's structural model and the computation of factor scores) was based on the orthogonal three-factor model without correlated errors. In this model, loadings on Dominance and Love were fixed to the perfect circumplex in MCFA and entered as priors in BCFA with a prior variance of σ² = 0.01 (see Table 2). The number of iterations in BCFA was adjusted to result in PSR ≤ 1.001. The circumplex pattern of the BCFA posterior loadings can be inspected in Figure 3.

In addition to the analysis of internal validity, we embedded external criteria in MCFA and BCFA to support the external validity of the IIP's structural model (Table 3). External measures were entered in the CFA as z-standardized sum scores like the IIP scales. We included Extraversion, Agreeableness, and Neuroticism scores from the IPIP40 (Hartig et al., 2003) and NPI-17 scores measuring subclinical narcissism (von Collani, 2014). Model fit was similar to model fit without the external measures. Model fit for the MCFA model with the Big Five measures was χ²(36) = 335.00, p < 0.001, CFI = 0.94, SRMR = 0.09, RMSEA = 0.10, 90% CI [0.09, 0.11]. MCFA model fit for the inclusion of subclinical grandiose narcissism was χ²(23) = 248.41, p < 0.001, CFI = 0.94, SRMR = 0.09, RMSEA = 0.11, 90% CI [0.10, 0.12]. Model fit for the BCFA model was 95% CI [113.73, 179.30] for χ²(55), χ²-based posterior predictive p < 0.001, CFI = 0.97, RMSEA = 0.08, 90% CI [0.08, 0.09] with Extraversion, Agreeableness, and Neuroticism. The BCFA model with grandiose narcissism resulted in a model fit of 95% CI [67.24, 124.26] for χ²(45), χ²-based posterior predictive p < 0.001, CFI = 0.98, RMSEA = 0.09, 90% CI [0.08, 0.10].

Results followed a similar pattern for MCFA and BCFA (Table 3) and will be reported for those loadings that are most relevant within the theoretical framework of the IIP (see Introduction). Extraversion loaded positively on Dominance and Love, whereby the loading on Dominance was greater than the loading on Love. Agreeableness had a negative loading on Dominance and a positive loading on Love, however, the latter to a smaller degree. Neuroticism showed large negative loadings on Distress as an overall measure of interpersonal problems. Lastly, subclinical narcissism showed large and positive loadings on Dominance and zero loadings on Love and Distress.

We analyzed higher-level scores for Dominance, Love, and Distress using regression factor scores from MCFA and weighted sum scores. Weighted sum scores were computed according to Equation 1. The weights to obtain MCFA regression scores from the IIP scales' scores can be seen in the (Supplementary Table S1). The MCFA regression scores were perfectly predicted by the IIP octant scales (R² = 1). As can be seen in Table 4, correlations between corresponding IIP higher-level scores from the two scoring methods were very large with r(820) > 0.99, p < 0.001, for all scores Dominance, Love, and Distress. With regard to intercorrelations of higher-level scores within each scoring method, the largest absolute correlation was found between the MCFA regression scores for Love and Dominance (see Table 4). The two circumplex scores Love and Dominance were uncorrelated with Distress for MCFA regression scores, whereas the weighted sum score for Dominance showed a slight negative correlation with Distress.

For all following analyses of reliability and validity of IIP higher-level scores, we used the scoring model implied by the two scoring methods. We computed regression-component loadings of the scoring models (Schönemann and Steiger, 1976; Beauducel, 2005) as

where Λwere the standardized loadings of the IIP octants on the higher-level scores, Λ^* were the Pearson product-moment correlations between the octants and higher-level scores, and C were the intercorrelations between the higher-level scores. The corresponding reproduced correlation matrix between the IIP octants Σ_r was computed by

The coefficients of congruence of the regression component loadings with the factor loadings of the models were very high (c ≥ 0.99) for all types of scores and factor models, including congruences between higher-level scores with BCFA factor loadings.

We computed reliability estimates for the higher-level scores by the following formula, originally designed for two sets of parallel observed variables (Cliff, 1988; p. 277, Eq. 4):

where B contained the weights obtained from a multiple linear regression in which the higher-level scores were predicted by the IIP octant scales (see Supplementary Table S1). Originally, Σ₁₁ should represent the intercorrelations of the first variable set, Σ₂₂ of the second variable set, and Σ₁₂ the correlations between the first and the second variable set. We assumed that Σ₁₁ = Σ₂₂ (Beauducel et al., 2016) and used intercorrelations of the octant scales as estimators for Σ₁₁ as well as Σ₂₂ and the intercorrelations of the octants reproduced by the scoring model, Σ_r, as estimators for Σ₁₂. Reliability estimates were very high and very similar for the two scoring methods (Table 4). They were largest for the general problem factor Distress with r_tts = 0.95, followed by Love (r_tts = 0.91), and lastly Dominance with r_tts = 0.90 for MCFA regression scores and r_tts = 0.91 for weighted sum scores.

We analyzed the circumplexity of the reproduced correlations Σ_r implied by the two higher-level scoring models by the SPMC and compared them to the results for the original octant scales (Table 1). The best approximation to perfect circumplexity was found for reproduced correlations from the weighted sum scores for Dominance, Love, and Distress for all combinations of equality constraints on spacing and communalities. MCFA regression scores improved model fit as compared to the original octant scales for some, but not all SPMC conditions and fit indices.

For the unequal-spacing/equal-communalities SPMC model, we also inspected the estimated angles for the different correlation matrices (Table 5) in addition to overall model fit. The angle of the LM/Overly nurturant scale was set to zero, from which the other angles were estimated. We quantified the proximity to the ideal circumplex of the estimated angles by the gap difference test (GDIFF; Upton and Fingleton, 1989), with smaller values indicating greater circumplexity of the scores. The GDIFF test comprises all squared differences between ideal and observed angles. We computed GDIFF with degrees rather than radians and used its square root to facilitate interpretation of results as

where θ_i was the observed angle of each of the eight IIP scales and θ_id was the ideal circumplex angle for the respective scale. Values for GDIFF representing overall deviation from ideal circumplex angles were greater for original than score-reproduced correlations. In descending order, we found GDIFF = 29.26 for original correlations, GDIFF = 23.26 for weighted sum scores, and GDIFF = 18.61 for MCFA regression scores. As can be seen in Table 5, the variation of GDIFF was mainly driven by spacing between NO/Intrusive, PA/Domineering, and BC/Vindictive, which was closest to constant 45° spacing for MCFA regression scores.

Parallel to the analysis of external validity for the three-factor model of the IIP, we investigated the correlational pattern of the IIP higher-level scores with Extraversion, Agreeableness, and Neuroticism scores from the IPIP40 (Hartig et al., 2003) and NPI-17 scores (von Collani, 2014). Results were in line with the results for the structural model and followed a similar pattern for MCFA regression scores and weighted sum scores (see Table 6). Extraversion correlated positively with Dominance and Love, whereby the correlation with Dominance was greater than the correlation with Love. Agreeableness showed a negative correlation with Dominance and a positive correlation with Love, however, the latter to a smaller degree. Neuroticism showed large negative correlations with Distress. Lastly, subclinical narcissism showed large and positive correlations with Dominance, slight negative correlations with Love, and zero correlations with Distress.

The flexibility of BCFA parameter estimates makes it particularly applicable in circumplex settings, where a whole pattern of exact loadings is specified. By making use of priors for the loading variance in BCFA, the specified circumplex loadings are adjusted to better represent the data. The posterior loading pattern of the two circumplex factors can then be compared to the ideal circumplex by Tucker's congruence coefficient (Lorenzo-Seva and ten Berge, 2006). This procedure of comparing observed loading patterns with the ideal circumplex is also possible within exploratory analysis (Jacobs and Scholl, 2005). However, we compared the congruences of TEFA loadings with the ideal circumplex loadings and found that they were substantially smaller than the congruences of the BCFA loadings. The present study therefore indicates that BCFA is a flexible approach that allows for a closer approximation of the circumplex model of the IIP than TEFA. Hence, BCFA combines the advantages of exploratory and confirmatory analysis. Furthermore, the procedure of comparing BCFA loadings of Dominance and Love to the ideal circumplex is especially relevant when dealing with interpersonal problems, where a third general factor is involved. CFA fit indices, which measure overall model fit, are mainly driven by loadings on Distress (see also Wilson et al., 2013), whereby overall adequacy of the hypothesized circumplex pattern cannot be disentangled from the overarching three-factor structure. Regarding the overall three-factor model of the IIP, we found that model fit was acceptable for conventional MCFA and increased when BCFA was conducted, hence, when circumplex constraints were loosened by specifying priors. This effect was mainly driven by those loadings that were fixed to zero in MCFA, indicating that zero loadings could be too restrictive in empirical circumplex settings. However, the exceedingly high congruence between posterior BCFA and MCFA loadings reveals that the factors can be considered equal (Lorenzo-Seva and ten Berge, 2006). Hence, BCFA produces loadings on Dominance and Love that fulfill nearly perfect circumplex structure whilst being flexible enough to loosen empirically unrealistic constraints and thereby improving model fit. More flexible estimation of model parameters also resulted in a smaller number of substantial correlated error terms for BCFA than MCFA and in a smaller variation of model fit indices across two subsamples. Thus, BCFA seems to be less sensitive toward correlated errors and yields more robust model fit estimates than MCFA. Considering the rather large RMSEA values found for both MCFA and BCFA, it has been argued that RMSEA tends to get biased and is susceptible to falsely reject adequate models in the case of large main loadings and circumplex structure in particular (Saris et al., 1987, 2009; Rogoza et al., 2021). For example, Gurtman and Pincus (2003) used a threshold of RMSEA < 0.13 when assessing circumplex models, which would indicate acceptable fit for our MCFA and BCFA model.

The results of circumplexity analysis to test the SPMC (Browne, 1992; Grassi et al., 2010) suggest that strict 45° equal spacing might not be a realistic constraint for the IIP octants. SPMC model fit improved greatly when spacing between the octants was freed. Estimated angles from the unequal-spacing/equal-communalities condition suggest that spacing between LM/Overly nurturant, NO/Intrusive, and PA/Domineering is better approximated by angles larger than 45°, while spacing between the other IIP scales could be smaller than 45°. Horowitz et al. (2017) report similar results for principal component angles of the original IIP scales in the validation study of the IIP. Theoretically, PA and LM should be perfectly aligned on Dominance and Love (Gurtman, 1993; Gurtman and Pincus, 2000) and thus be unrelated with a 90° distance. However, our results suggest that individuals who are more domineering according to the IIP scales are slightly less affectionate and vice versa. Taking a closer look at the items of the scales reveals that the wording of items from the PA/Domineering scale could be responsible for the angular deviations. For example, one item says, “I try to exert influence on others to get what I want,” which could be understood as manipulative behavior and overlap with the BC/Vindictive scale. Another one says, “I am too aggressive toward others,” which can hardly be independent from content of the LM/Overly nurturant scale. While items from the NO/Intrusive scale are worded more clearly as intrusive behavior, it seems like being caring (i.e., scoring high in the LM/Overly nurturant scale) is not as close to being overinvolved in other people's lives as one could assume.

The results from MCFA and BCFA with the inclusion of external measures were in line with the hypothesized pattern and support the validity of the structural model of the IIP. Results did not differ between MCFA and BCFA, which indicates that differences between the CFA methods are most relevant when the internal validity, that is, the validity of the circumplex pattern, is concerned. Previous research has linked Extraversion and Agreeableness to interpersonal behavior in general and not interpersonal problems in particular (McCrae and Costa, 1989; DeYoung et al., 2013; Barford et al., 2015). The results of the present study indicate that the expected pattern for Dominance and Love can be transferred to interpersonal problems if the circumplex factors are separated from the general problem factor Distress. Only the positive loadings of Agreeableness on Love were smaller in size than the negative loadings of Agreeableness on Dominance, which was expected to be the other way around. However, the exact size of loadings and correlations also depends on the reliability of the scales that are administered, and we found greater Cronbach's alpha for the Extraversion scale than for the Agreeableness scale of the IPIP40 (Hartig et al., 2003). The large loadings of Neuroticism on Distress highlight that the third factor of the IIP is not only a response factor but has relevant meaning as a general factor associated with emotional and interpersonal problems (see also Tracey et al., 1996).

Our findings on grandiose narcissism and IIP higher-level factors were in line with our hypotheses and previous findings. The positive loading on Dominance is in line with previously found patterns (Dickinson and Pincus, 2003; Miller et al., 2012). While clinical narcissism seems to be associated with an overall impairment in interpersonal behavior (Ogrodniczuk et al., 2009), our results support the findings by Dickinson and Pincus (2003) and suggest that subclinical grandiose narcissism is not linked to general interpersonal distress. This finding could also be attributed to the fact that subjects with high grandiose narcissistic tendencies present themselves more favorably–that is, with fewer interpersonal problems–than what would be experienced by their surroundings.

Regarding the higher-level scores for Dominance, Love, and Distress, our results suggest that both regression scores from MCFA and weighted sum scores measure interpersonal problems with excellent reliability and good validity. Reliability estimates for Dominance, Love, and Distress scores even exceeded those of the original IIP octants, which strongly supports the use of higher-level scores. We did not include mean plausible values from BCFA into the analysis because of their susceptibility to bias in individual scoring settings (Lüdtke and Robitzsch, 2017). However, the excellent congruences of the MCFA regression scores and weighted sum scores with BCFA factor loadings indicate that they are a good representation of the IIP's higher-level structure and a feasible alternative to BCFA mean plausible values.

The results of circumplexity analysis by the SPMC (Browne, 1992; Grassi et al., 2010) for reproduced correlations from the higher-level scores indicate that correlations between the IIP octants fulfill circumplexity to a greater degree when they are reproduced from higher-level scores as opposed to the original IIP correlations. Although this is not surprising given the circumplex specifications of the underlying models, the results support the validity of IIP higher-level scores for their use in research and diagnostics. Across all SPMC conditions, weighted sum scores showed best circumplex validity. This can be attributed to the fact that weighted sum scores are based on fixed circumplex weights without any data-driven adjustments. However, all higher-level scores seem to be well-fit with respect to circumplex properties of reproduced correlations. The estimated angles suggest that higher-level scores improve circumplex spacing of the IIP octants, whereby MCFA regression scores seem to best approximate ideal 45° spacing.

Correlations between higher-level scores and external measures were in line with the hypothesized pattern and support the validity of Dominance, Love, and Distress scores for MCFA regression scores and weighted sum scores. The similarity to the findings for the factor model supports the notion that the IIP higher-level scores are a good representation of the underlying factors, also with regard to the relationship with external criteria. The findings for the correlations only differed slightly from the MCFA/BCFA findings for subclinical grandiose narcissism. While only the positive loading on Dominance was significant in the MCFA and BCFA, we also found slight positive correlations of NPI-17 scores with Love scores. These correlations suggest that subclinical grandiose narcissism could be positioned at 90–135° within the interpersonal circumplex. This hypothesis is in line with results by Nagy et al. (2019), who found the admiration aspect of narcissism to be positioned within this angular range when modeling the interpersonal circumplex by the SPMC.

Taken together, our results reveal convergence of different modeling and scoring procedures for the IIP. SPMC analysis, MCFA, BCFA, and TEFA reveal the circumplex structure in a similar fashion. Moreover, we found high convergence for the higher-level scores computed by different methods. Therefore, our results demonstrate that methodological pluralism, as it has been advocated by Zitzmann and Loreth (2021), does not preclude convergence of results. Although the results reveal that the IIP circumplex can be modeled by means of MCFA, BCFA, TEFA, and SPMC, there were nevertheless relevant differences. As in MCFA the ideal loadings are directly fixed, there is no information on the departure of single items from the circumplex unless modification indices are investigated. In contrast, BCFA, TEFA, and SPMC allow for a description of the conformity of each measured variable to the circumplex. Whereas, the circumplexity does not affect model fit of TEFA, it affects the model fit of MCFA, BCFA, and SPMC. Accordingly, MCFA, BCFA, TEFA, and SPMC must not necessarily provide converging results. Therefore, the convergence of the methods found in the present study corroborates the circumplex model of the IIP. Regarding the comparison of MCFA and BCFA, we note that MCFA resulted in more significant correlated error terms and in a more substantial variation of model fit across subsamples than BCFA. Thus, we found some advantages in using prior variances in BCFA in order to avoid over-specification of circumplex loadings. However, it follows from the perspective of Zitzmann and Loreth (2021) that researchers using BCFA for circumplex modeling should tolerate those who use another approach and should be aware of the advantages and disadvantages of their methodological choices.

There are three main limitations of the present study. First, we used rather short and less differentiated scales for the external measures, which could affect the reliability of results. Future research on the topic could use measures like the Revised NEO Personality Inventory (Costa and McCrae, 2008) for the Big Five as well as more exhaustive scales for narcissism. The items of the NPI-17 only measure grandiose narcissism, that is, a tendency to overemphasize one's strengths and power whilst downplaying one's weaknesses (von Collani, 2014; Foster et al., 2015). Future research could include vulnerable narcissism, which is likely to produce different results. For example, we would expect positive correlations of vulnerable narcissism with general interpersonal distress because vulnerable narcissists are more centered around their own suffering than grandiose narcissists (Dickinson and Pincus, 2003).

The second limitation concerns the sample of our study. We did not use a random sample for the present investigation, which complicates interpretation of results for the general population, especially regarding the exact size of effects and parameters. However, our results are in line with theoretical expectations concerning the circumplex structure of the IIP, which supports the validity of the present findings. Furthermore, results remained stable when the data set was split and when different methodological approaches were applied. Moreover, we used a non-clinical sample. While research on non-clinical samples is important to set a baseline for the majority of the population, the IIP as a measure of problematic behavior is also designed for clinical use (Horowitz et al., 2017). Therefore, it should be investigated whether the results can be replicated in a clinical sample. Furthermore, one could examine whether IIP higher-level scores can differentiate between patient groups or predict treatment outcome, as previously shown for the corresponding octants (Alden and Phillips, 1990; Pincus and Wiggins, 1990; Ruiz et al., 2004; Tilden et al., 2010). The IIP higher-level scores could be particularly interesting when personality disorders are investigated because personality disorders are greatly concerned with impairments in the interpersonal domain (Pincus and Wiggins, 1990). Thereby, it would be important to include clinical interviews and ratings rather than relying solely on self-report measures because individuals with personality disorders might reach and report different conclusions about themselves than an external judge. In a clinical setting, it could also be examined which of the three higher-level scores Dominance, Love, and Distress has the greatest predictive validity for which psychological disorder and compare factor score estimates and weighted sum scores as well.

Thirdly, we used only a subset of possible models for the IIP higher-level and circumplex structure. The factor models in this study ranged from strictly constrained to the perfect circumplex in MCFA, partially constrained by priors in BCFA, to completely free and only matched to the circumplex by target rotation in TEFA. However, it would be possible to specify various factor models with different levels of constraints for both MCFA and BCFA. We also applied the SPMC to our data, which can be viewed as a specific type of CFA model that can be estimated by structural equation modeling (Browne, 1992; Nagy et al., 2019). Variants of the SPMC could be examined in greater detail for the IIP in future research. The SPMC is a very flexible approach for circumplex structure and it is applicable to a variety of structural models. Typically, the SPMC is analyzed by using maximum-likelihood estimation, but it would be possible to analyze it by Bayesian estimation with the appropriate constraints as well (see Lenk et al., 2006). Moreover, Nagy et al. (2019) provide an extended version of the SPMC, which makes it possible to include external variables to investigate the external validity of the circumplex structure (see also Etzel et al., 2021). For example, it would be interesting to enter different measures of narcissism and test their angular position within the circumplex pattern of the IIP using the SPMC.

In summary, the present study supports the three-factor model of the IIP and the use of Bayesian modeling (Muthén and Asparouhov, 2012) in circumplex settings. The possibility to specify exact loading hypotheses whilst simultaneously allowing for enough flexibility to loosen unrealistic constraints (e.g., strict zero loadings) makes it particularly suitable for scales that comprise a circumplex structure. We also encourage researchers and practitioners to use the IIP higher-level scores Love, Dominance, and Distress. They are reliable, valid, and economic measures of interpersonal problems. We can recommend both MCFA regression scores and weighted sum scores. Which of these options should be preferred may depend on sample size and on the context of research or application. Weighted sum scores could be particularly relevant in individual diagnostic settings. They are of similar reliability and validity as CFA-based scores and can be computed easily for each individual based on fixed circumplex weights without requiring a large sample size for preliminary CFA.