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.”

“Typically, in this class of models a general latent factor associated with all observed variables is specified, along with narrower latent factors associated with only a subset of observed variables (see Reise [83] for more details).”

“In other words, in the bi-factor model, the general factor reflects the conceptually broad construct that an instrument was designed to measure, and the specific factors reflect the more conceptually narrow subdimensional constructs (Reise, 2012; Stefansson et al., 2016).”

“Interpretation of GRS scores can be misleading given that obfuscation of common variance occurs when scales or lower order factors, or both, are highly correlated (Reise, 2012).”

“…initially invented in intelligence research (Holzinger & Swineford, 1937), but have recently experienced a renaissance in various areas of assessment as an important structural representation of multidimensionality besides the often used correlated factor model (Reise et al., 2010; Reise, 2012).”

“Organizing cognitive vulnerabilities in a way that accommodates both their common and unique aspects simultaneously calls for the use of latent bifactor models (Holzinger & Swineford, 1937; Reise, 2012).”

“…analytic models is readily available (Bornovalova et al., 2020; Brown, 2015; Brunner et al., 2012; Chen et al., 2012; Chen & Zhang, 2018; Gorsuch, 1983; Gustafsson & Åberg-Bengtsson, 2010; Markon, 2019; Reise, 2012; Reise et al., 2013, 2018), Sellbom & Tellegen, 2019; Styck, 2019; Watkins, 2018).”

“Additionally, the role of the specific factors was explored independent of the general factor (Chen, West, & Sousa, 2006; Reise, 2012).”

“Additional statistics, coefficient omega hierarchical ( h) and omega-subscale ( s), were calculated for both sexes (Reise, 2012; Zinbarg, Revelle, Yovel, & Li, 2005).3 h estimates the proportion of total variance attributable to the general factor, whereas the s represents the proportion of…”

“(see Brunner, Nagy, & Wilhelm, 2012; Reise, 2012), which is supported for the extension to robust estimation procedures (Bentler, 2009).”

“We also test the fit of the data with a bifactor model, which affords an alternative multidimensional structure of parenting (Reise, 2012).”

“hierarchical (ωH), omega general (ωG) and omega specific (ωS) [39].”

“, total and hierarchical general and group specific omega coefficients) to evaluate psychometric properties of scores and explore interrelations of general and group factors across different variables (see, e.g., Gonzalez & MacKinnon, 2018; Reise, 2012; Rodriguez, Reise, & Haviland, 2016a, 2016b).”

“For most of these test batteries, only CFAs are reported in the manual.”

“It has been concluded that the strong reliability estimates for the FSIQ permit individual interpretation on the superordinate factor level, whereas reliability estimates for each single index were deemed too weak for individual interpretation on the subscale level (Reise, 2012; Reise et al., 2013).”

“Omega hierarchical displays the amount of variance in all items or item subsets accounted for by either the g-factor or an s-factor (cf. Brunner et al. 2012; Reise 2012).”

“general and specific factors that are in accordance with their construct (Reise, 2012).”

“…by applied researchers who are more focused on item characteristics and the optimization of measurement tools rather than the multidimensional structure of the data (ten Holt et al. 2010; Wang and Russel 2005; although see Reise et al. 2007, and Reise 2012 for the case of bifactor IRT models).”

“Bifactor modeling (Reise, 2012; Reise et al., 2010) involves allowing each item to load on exactly two factors, one specific factor (e.g., psychosis) and one general factor comprising all items.”

“…technique that can be used to determine the plausibility of measuring the OCPD construct with a general factor that encapsulates overall OCPD severity and dysfunction in addition to subfactors that capture unique domains or traits of OCPD psychopathology (Jennrich & Bentler, 2011; Reise, 2012).”

“…TRS scales contained sufficient unique true score variance for confident interpretation based on ωHS coefficients as all were less than the .50 minimum criterion (Reise, 2012; Reise, Bonifay, et al., 2013) and H coefficients which were below the .80 criterion (Rodriguez et al., 2016a, 2016b).”

“Exploratory bifactor analysis is an increasingly popular quantitative framework for item analysis (Eid, Geiser, Koch, & Heene, 2017; Reise, 2012).”

“Bifactor modeling assumes that the correlations between factors are explained by the presence of a general factor (Reise, 2012).”

“For a review of the history, see Reise (2012).”

“…partitioned into general and group factor effects that respectively represent systematic variance shared across all items and independent variance common to items within each subscale (see, e.g., Reise, 2012; Reise et al., 2013; Rodriquez et al., 2016a, 2016b; Vispoel, Lee et al., 2022a, 2022b).”

“…general factor to justify a unidimensional model, it can also provide direction on the appropriateness of using a total score, or whether presenting the scores of the domain-specific latent factors would provide a meaningful interpretation (Chen, et al., 2006; Reise, 2012; Rodriguez et al., 2016).”

“Lorenz, Hyland, Perkonigg, & Maercker (2017) applied confirmatory bifactor modelling (CBM; Reise, 2012) within a convenience sample of individuals affected by involuntary job loss in Switzerland in order to better understand the structure of AjD symptoms.”

“In line with the growing interest in bifactor structures for modeling the multidimensionality of psychological constructs (Reise, 2012), the bifactor model had a markedly better fit to our data.”

“Under a bifactor model, different indices can be computed: (a) Coefficient omega (Lucke’sω) (Revelle and Zinbarg, 2009; Reise, 2012; McDonald, 2013), which is a reliability estimate based on factorial model that estimates the proportion of the observed variance in the total score attributed to all…”

“While Reise (2012) referred to the SL procedure as an approximate exploratory BF model, the SL procedure technically represents a mathematical transformation of the higher-order model using an elegant procedure to apportion the variance to the higher- and lower-order factors (Carroll, 1993;…”

“52 Because the data is collected on a 5-point Likert scale and is considered ordinal, we adjusted parameter estimates using weighted least squares mean and variance which is a recommended approach for the analysis of ordinal data.”

“…to bifactor models, it is important to note bifactor (group and general) estimates are often weaker than first-order factor models, which is due to the disaggregation of indicator-level covariance into estimates of two separate factor sets (general and group) rather than a single one (Reise 2012).”

“In a bi-factor structure, all items typically load on 1 general factor and 1 or more distinct specific factors that represent residual clusters of common variance not captured by the general factor ( Reise, 2012 ).”

“Bifactor modeling is an empirical approach for examining the extent to which scale scores reflect a single common factor, even when the data are multidimensional (Reise, 2012) Table 4.”

“In this bi-factor model the general and the group factors are assumed to be orthogonal, in order to enable us to estimate the unique variance accounted for by any components [44].”

“The terminology used in a traditional PCA is exemplified in the following quote describing the bifactor model of Reise [68]: “ . . . it is assumed that the general and group factors all are orthogonal.”

“The bi-factor model is particularly useful for multidimensional constructs (Reise, 2012).”

“General and domain-specific factor correlations are set to zero (Reise, 2012).”

“The Schmid–Leiman approach (SL; Schmid & Leiman, 1957) and the Jennrich–Bentler approach (JB; Jennrich & Bentler, 2011) are two alternative methods that may be used for exploratory bifactor analysis (Reise, 2012).”

“Omega hierarchical (ωh) is a measure of reliability for the general factor and omega specific (ωs) is a measure of reliability for the specific factors (McDonald, 1999; Reise, 2012).”

“Bifactor analysis (Jennrich & Bentler, 2011; Reise, 2012) was added within the ESEM approach.”

“…or communication of sexual preferences) as well as part of a general component (i.e. verbal or nonverbal communication) we used a bi-geomin orthogonal rotation to examine the existence of a bifactor structural model (see Muth en & Muth en, 2007; Reise, 2012; Reise, Moore, &Haviland, 2010).”

“According to Reis (2012), these latter figures demonstrate that “if both total scores and subscales were to be formed, the interpretation of the subscales as precise indicators of unique constructs is extremely limited” (p. 691).”

“Although ELA and CT scores are not necessary to report due to their low !s values (see Reise, 2012), they are presented here for the use of future research.”

“Thus, the SL procedure will only duplicate the results of a bifactor model if the measured phenomena actually came from a higher-order process; otherwise, the SL procedure can provide misleading results (Jennrich & Bentler, 2011; Reise, 2012; Yung, Thissen, & McLeod, 1999).”

“indicators load on both general and specific factors (Chen et al., 2012; Reise, 2012).”

“A single bifactor model (Reise 2012) was fitted to the data to define the SA constructs which is constituted of three different facets (i.e. a physiological, a social, and a well-being facet).”

“…items of subjects making decision about the number of dimensions that should be assessed in language proficiency test is a controversial issue and necessitates a number of validation and investigatory studies for a comprehensive response (e.g. Cucina & Byle, 2017; Dunn & McCray, 2020; Reise, 2012).”

“To verify if a single “risk” construct encompassed the three Static-2002R dimensions, a bifactor model (Reise, 2012) was run in the R mirt package.”

“Whilst alternative factorial techniques for simultaneous testing for the presence of a general trait and multi factorial structure exist such as bifactor analysis (Reise 2012) it is unlikely they would have discovered the important element of misfit that has been revealed here.”

“Similar to daily smoker results, the ECV associated with the health expectancies (general) factor in the nondaily sample solution increased from 0.80 to 0.84.”

“We also estimated the omega hierarchical (ωH), as the proportion of reliable variance in total scores that can be attributed to the single general factor, and the omega hierarchical subscale (ωHS), as the proportion of reliable variance associated with each factor once partitioning out variance associated with the sin…”

“Lower ωs coefficients suggest that interpretation of the factor indices beyond the general factor may not be appropriate as little variance exists beyond the general factor (Reise, 2012).”

“In a bifactor model, the commonality f all scale items is shared by the general factor and, after that, the ontribution of each cluster of items to the distinct specific facors is calculated (Chen et al., 2012; Reise, 2012).”

“280 (VC) and likely contained too little true score variance for clinical interpretation (Reise, 2012; Reise, Bonifay, & Haviland, 2013).”

“…covariance among a set of observed variables can be accounted for by a single general factor, that reflects the common variance running among all indicators, and group factors, that reflect additional common variance among clusters of indicators, typically, with highly similar content (Reise 2012).”

“These models are for data in which the primary dimensions are more strongly represented relative to the secondary dimensions (DeMars, 2013; Reise, 2012; Toland, Sulis, Giambona, Porcu, & Campbell, 2017), and our choice of values reflects this because the mean of the hyperprior related to primary…”

“For model-based reliability, omega coefficients were assessed considering omega (ω), omega subscales (ωS) (Reise, 2012; Rodrigue, Reise, & Haviland, 2016a, b).”

“Detailed resources regarding bifactor models in psychometric work are provided elsewhere (Canivez, 2016; Reise, 2012).”

“Omega (ω) describes the amount of reliable variance accounted for by the g-factor and all s-factors taken together, whereas omega hierarchical (ωH) describes the amount of reliable variance accounted for by the g-factor (Brunner et al., 2012; Reise, 2012).”

“A bifactor approach specifies that the covariance among a set of items can be accounted for by a single, general factor that captures the common variance among all items in the set, while also allowing for subfactors to explain item subgroups (40).”

“These results indicated that the individualism – collectivism scale is multidimensional and the four specific dimensions were meaningful for interpreting the present data (e.g., Reise, 2012; Reise, Scheines, Widaman, & Haviland, 2012; Rodriguez, Reise, & Haviland, 2016).”

“It is suitable when “multiple and distinct domains of item content are included to increase content validity” (Reise, 2012, p. 668).”

“To evaluate the unidimensionality of the scale under bifactor modeling, explained common variance (ECV), percentage of uncontaminated correlations (PUC), and percentage of reliable variance (PRV) were used (Brunner et al., 2012; Reise, 2012; Ríos and Wells, 2014; Bonifay et al., 2015).”

“Such models have a latent structure that includes a general factor that accounts for the commonality among all indicators and independent group factors that represent systematic variance specific to nested and non-overlapping subsets of indicators ([8,12,16,18]).”