The Rediscovery of Bifactor Measurement Models

“Thus, to judge the degree to which composite scale scores are interpretable as a measure of a single common factor, we need a related index called coefficient omega hierarchical (McDonald, 1999; Zinbarg, Revelle, Yovel, & Li, 2005).”

“In other words, if the data are truly unidimensional and a unidimensional latent variable measurement model is proposed to fit the data, content representativeness does not make any difference in defining the latent variable or in the value of the estimated item parameters (Bollen & Lennox, 1991).”

“Specifically, many applications of IRT are based on the marginal maximum likelihood (MML; Bock & Aitken, 1981) estimation method.”

“…resolving important problems in conceptualizing and measuring psychological constructs (Brunner, Nagy, & Wilhelm, 2012; Cai, Yang, & Hansen, 2011; Chen, Hayes, Carver, Laurenceau, & Zhang, 2012; Gustafsson & Aberg-Bengtsson, 2010; Reise et al., 2010; Reise, Morizot, & Hays, 2007; Thomas, 2012); 3.”

“Models for essential unidimensionality can be implemented in different ways, for example with socalled bi-factor models or hierarchical measurement models (Gustafsson & Åberg-Bengtsson, 2010; Reise, 2012).”

“…on this equation, it is clear that, as the proportion of variance explained by the general factor increases, ECV will increase, and, as noted by Reise (2012), higher ECV values indicate a strong general factor, which could guide researchers in deciding whether to fit a unidimensional model…”

“An interesting example of this comes from a study conducted by Warnock and Lalmas (2015) where user engagement with two versions of a web interface was tested.”

“…to carry out further research to evaluate the functioning of the various reliability coefficients with more complex multidimensional structures (Reise, 2012; Green and Yang, 2015) and in the presence of ordinal and/or categorical data in which non-compliance with the assumption of normality is…”

“Statistically, the strict proportionality constraints that are at play in hierarchical models limit their flexibility and practical applicability (Brunner et al., 2012; Chen et al., 2006; Jennrich & Bentler, 2011; Reise, 2012).”

“Group factors are also generally constrained to be orthogonal to or uncorrelated with each other for identification and interpretability (Reise, 2012).”

“The specification of a bifactor type measurement model (e.g., Reise, 2012) is particularly useful for determining essential unidimensionality.”

“Other approaches include using testlet models (Wang & Wilson, 2005; Wilson & Adams, 1995) or a bifactor model (Reise, 2012).”

“, Hancock & Mueller, 2000), but recently, Rodriguez, Reise, and Haviland (2016a, 2016b) have put forward a well-organized proposal in the context of bifactor FA solutions (Reise, 2012).”

“Latent variable modeling To test the hypothesis of a general risk-preference factor,we implemented a psychometric bifactor model, which has a long tradition in research on intelligence (84) and has recently been rediscovered as an effective method to model construct-relevant multidimensionality (27, 85, 86).”

“In particular, to test the notion of one single common core of dark traits, we resorted to a bifactor approach (e.g., Reise, 2012).”

“One method uses a bifactor solution (Holzinger & Swineford, 1937; Reise, 2012; Rodriguez, Reise, & Haviland, 2016) using structural equation modeling software (e.”

“The more specific subfactors are specified as orthogonal to (i.e., uncorrelated with) the general factor and reflect correlations among the subsets of dimensions after the correlations among all dimensions captured by the general factor have been taken into account (Reise, 2012).”

“The specific factors are considered “residualized factors” (Reise, 2012, p. 691).”

“For example, bifactor models (Brunner, Nagy, & Wilhelm, 2012; Reise, 2012) that account for general or shared variance across dimensions, but also the unique variance of dimensions, might be useful in the case of career adaptability to identify the specific roles played by the dimensions.”

“In the case of the bifactor models (4a, 4b, 5a, and 5b), group factors were specified as orthogonal to the general factor, as is standard (e.g., Reise, 2012; Reise, Moore, & Haviland, 2010) but the two general factors were specified as correlated1 (see also Tóth-Király, Morin, Bőthe, Orosz, &…”

“The appropriate ECV benchmarks are determined by the PUC (Percentage of Uncontaminated Correlations; Reise, 2012; Reise, Scheines, Widaman, & Haviland, 2013) or the percentage of correlations among items in different specific factors and thus are not inflated due to shared general and specific…”

“The answer is affirmative in many cases because a common construct is what scale developers originally intended to measure and what typical researchers are most interested in (Cho & Kim, 2015; Reise, 2012).”

“Another convenient development in methodology that will help with the exploration of validity is the recent adoption by many researchers of the bifactor model itself (See Reise, 2012).”

“Thus, the BFM allows us to more directly investigate the amount of shared variance in each item due to the general and domain-specific factors (for an in-depth discussion of BFM in the area of personality research, see Reise, 2012).”

“More precisely, a bifactor model (Holzinger & Swineford, 1937; Morin, Tran, & Caci, 2013; Reise, 2012) analyzes the total covariance among the indicators to extract a global G factor underlying all indicators, and models the residual covariance not explained by the G factor through the specific S…”

“The first source pertains to the assessment of coexisting global and specific constructs with bifactor modeling being able to model the associations of the global and specific factors on the questionnaire items (see also Reise 2012).”

“…by many applications in latent variable modeling, including how best to handle cross-loadings (Asparouhov, Muthén, & Morin, 2015), bifactor models (Reise, 2012), multi-trait multi-method analyses (Kenny & Kashy, 1992), and mitigating response biases (McLarnon, Goffin, Schneider, & Johnston, 2016).”

“As noted by Reise (2012), the presence of multidimensionality does not dictate the creation of subscales nor does it ruin the interpretability of a unit-weighted composite score.”

“In contrast, a robust ωhs coefficient suggests that most of the reliable variance of that subscale is independent of the general factor and clinical interpretation of an examinee’s strengths and weaknesses beyond the general factor can be conducted (Brunner et al., 2012; DeMars, 2013; Reise, 2012).”

“—we first fitted a CFA model to the Chilean sample.”

“However, higher-order factor models rely on highly restrictive implicit assumptions that may not hold in practice and may explain why they often fail to meet minimal requirements of adequate fit (Gignac, 2008; Morin et al., 2016; Reise, 2012).”

“In a bifactormodel, the general Gr factor and specific factors each predict the indicators (Chen et al., 2006, 2012; Reise, 2012).”

“7 Bi-factor models provide a more flexible alternative, a way of capturing the hierarchical and multidimensional nature of task value (Chen, West, & Sousa, 2006; Reise, 2012).”

“As initial exploratory factor analyses indicated correlated traits of psychopathology, we used a bifactor confirmatory model, which can produce orthogonal scores from correlated traits (28).”

“Neff, Whittaker and Karl (2017) have argued that a bifactor approach (Reise, 2012) – which models the association of SCS items with the six specific factors of self-compassion and a general factor simultaneously – is more consistent with a systems-view of self-compassion.”

“There was therefore statistical support for the presence of a strong “general alexithymia” factor, as well as statistical support for five narrow factors reflecting the subdomains (i.e., the subscales) of the multidimensional alexithymia construct (Reise, 2012).”

“…bifactor model appears to have become embraced in the area of psychometrics and personality in recent years (Chen, Hayes, Carver, Laurenceau, & Zhang, 2012; Gibbons et al., 2007; Gruhl, Erosheva, & Crane, 2013; Morin, Arens, & Marsh, in press; Reise, 2012; Rodriguez, Reise, &Haviland, in press).”

“Bifactor models should be tested when the researcher is investigating multifaceted concepts (Reise, 2012) or when investigating the presence of a single global factor (Howard, Gagn e, Morin, & Forest, 2016).”

“It is also important to test bifactor models (Holzinger & Swineford, 1937; Reise, 2012) in examining interpersonal behaviors.”

“Bifactor model solutions can be estimated within both exploratory factoranalytic and confirmatory factor-analytic frameworks (Jennrich & Bentler, 2011; Reise, 2012).”

“, corresponding to b/c), and unlikely to hold in real life (Reise 2012; Yung et al. 1999).”

“Bi-factor models have a long history in intelligence research (Gustafsson & Balke, 1993; Holzinger & Swineford, 1937) and have seen a recent resurgence in the context of modeling psychometric multidimensionality (Chen, West, & Sousa, 2006; Morin, Arens, & Marsh, 2016; Reise, 2012).”

“Combining elements of both primary hypotheses in a bifactor model (11, 12), we also investigated a third hypothesis (Fig.”

“That is, the group factors reflected the part of the items' variance not explained by the global factors, and the global factors reflected the proportion of the items' variance that is shared across all items (e.g., Chen, West, & Sousa, 2006; Reise, 2012).”

“…of Structural Equation Modelling that is able to model the shared and unique variance of latent traits the questionnaires capture, and assess their independent contribution in predicting an outcome (e.g., Chen, Hayes, Carver, Laurenceau, & Zhang, 2012; Reise, 2012, see Methods section for details).”

“specific first-order factor, and are unlikely to hold in many real life situations (Reise, 2012).”

“No cross-loadings were allowed and factors were orthogonal to one another as per typical bifactor specifications (Reise, 2012).”

“Thus, the correlated factors model can be derived from a bi-factor model by constraining the general factor loadings to zero and relaxing the orthogonality constraint on the first-order factors [25].”

“Estimating the bifactor measurement structure is proposed to be an effective approach to modeling construct-relevant multidimensionality (Reise, 2012).”

“…factors, and, most commonly, the specific factors are also parameterized to be orthogonal to one another (Brown, 2015; Holzinger & Swineford, 1937; Reise, 2012)—although bifactor models with correlated specific factors have sometimes been used (Carragher et al., 2016; Caspi, et al., 2014;…”

“The bifactor model has enjoyed a resurgence of late (Reise, 2012), partly because of its strong performance in a number of model comparison studies.”

“For example, we did not pursue hierarchical or bifactor models (for current guidelines, see Canivez ( in press ); Reise (2012)) currently, but given the observed factor correlations (see Supplement) this could be an interesting avenue for future investigation.”

“Bifactor models have been rediscovered as a viable alternative structural representation of multiple types of higher order and multidimensional factor structures (Reise, 2012).”

“Such findings are more consistent, rather, with the p factor reflecting an amalgam of differentially weighted and psychologically distinct sources of variance (Krueger et al., 2018; Reise, 2012).”

“Bifactor ESEM models were specified as their bifactor CFA counterparts, although all cross-loadings involving the S-factors were freely estimated but assigned a target value of zero using an orthogonal bifactor target rotation procedure (Reise, 2012).”

“In a confirmatory Frontiers in Psychology | www.frontiersin.org 2 April 2015 | Volume 6 | Article 404 bifactor analysis (CBFA), two types of latent factors are defined (Reise, 2012).”

“In this research, our working model of multidimensionality is a bifactor structure (Holzinger & Swineford, 1937)—a multidimensional structure wherein measured variables reflect a single common trait (typically the primary dimension of interest to the researcher) and additional orthogonal group factors that typically a…”

“It is well-suited for representing the “construct-relevant multidimensionality” arising when a broad construct includes conceptually narrow subdomain constructs (Reise, 2012).”

“Instead, whenmulticollinearity is present, a bifactor solution (Reise, 2012) should be pursued which might be able to explain the high associations between the factors.”

“Among them were several rarely examined bifactor structures (cf. Reise, 2012) that allowed for facet-specific residual variations beyond a general factor common to all 12 items.”

“Second, bifactor models with uncorrelated general and specific factors allow for a clear partitioning of variance in the presence of multidimensionality and a strong general factor (Reise, 2012), as is the case for the PID-5-BF, IPO-16, and OPD-SQS (Zimmermann et al., 2020).”

“, commitment to the organization, the workgroup, the supervisor), and all factors are set to be orthogonal (Gignac, 2016; Morin, Arens, & Marsh, 2016; Reise, 2012).”

“It is worth noting that bifactor models only have first-order factors, which is distinct from second-order factor models (Canivez, 2016; Reise, 2012).”

“Reliability was determined using both an index of variance associated with all common (non-error) factors (omega; ω), and an index of variance associated with the general wisdom factor (omega hierarchical; ωH) (McDonald, 1999; Reise, 2012; Reise et al., 2010).”

“…(ESEM) is a new modeling framework that specifically deals with cross-loadings (Marsh et al., 2009, 2014), bifactor modeling which is available both within the traditional CFA and the ESEM framework differentiates between general and specific factors of constructs (Chen et al., 2012; Reise, 2012).”

“Perceived Stress Scale, bifactor modeling, measurement invariance alternative for modeling multidimensionality (Chen, West, & Sousa, 2006; Reise, 2012).”