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.”

“KEYWORDS Bi-factor rotation; exploratory factor analysis; Schmid-Leiman; target rotation The use of bi-factor analysis has dramatically increased in the last decade (e.g. Chen, West, & Sousa, 2006; Reise, 2012).”

“Reise and his colleagues (2013a, p. 137) tentatively proposed a rather strict advice to evaluate omega-hierarchical values for specific factors, namely the minimum value would be greater than 0.50 and the preferred value would be closer to 0.75 (see also Reise, 2012).”

“Though these are captured by the added covariance parameters, future applications of this approach may choose to more systematically account for these dependencies with extra path estimates, or possibly a bi-factor CFA model that models a general factor, and sub-factors within this factor (Reise, 2012; Reise et al., 2…”

“[50] First, we used coefficient omega hierarchical.”

“…(a) the existence of a general factor that accounts for the shared communality by all the items; and (b) the existence of several group factors, which contribute to a common variance shared within each cluster of items, beyond that of the general factor (Reise, 2012; Gu et al., 2015, 2017a).”

“One solution that has been proposed to account for substantial evidence of possible unidimmensionality between scales and observed multidimensionality among factor solutions is an exploratory bifactor approach [39].”

“This analytical strategy offers the advantage of distinguishing, within a single model, outcome variance that can be predicted by what is common to all dimensions (g-factor) from what is dimension-specific (s-factors; see Chen et al., 2006; Reise, 2012).”

“Recent bi-factor approaches to CFA also permit items to correlate with a strong general factor alongside residual sub-factors (Reise, 2012), wherein sub-factors and general factors are orthogonal, and compete for explaining item covariance (Beaujean, 2015).”

“The general factor assumes that all items in the measure explain one unidimensional factor as well as lower-level individual factors [21].”

“To further aid in the interpretation of each factor’s importance, and assessment of dimensionality, the explained common variance (ECV) was calculated as previously detailed by Reise (Reise, 2012).”

“The true variance of the score of each item in a multidimensional construct is related to the variance of the total score and the variance of the items’ specific scores (Reise, 2012).”

“The orthogonal rotation allows for easier interpretation of how S-factors relate to observed items above and beyond the G-factor (Reise, 2012; Reise et al., 2010).”

“Of the total sample of 11,878 youths, about three fourths of participants (9000 youths) were randomly selected for an exploratory structural equation modeling (ESEM) analysis (24) to determine what underlying factors may be present, and the remaining participants (2878 youths) were reserved for a confirmatory bifactor…”

“This condition is met for the one-factor and two-factor models, both nested in the bifactor model (Reise, 2012), and the two-factor model, which is nested in the one-factor model.”

“, the explained common variance [ECV]) and the proportion of correlations in the analysis that are uncontaminated by general and group factor variance (i.e., PUC; Reise, 2012) are used.”

“Before the simulation of the adaptive administration of this item bank could be carried out, the factor analytic estimates needed to be converted to IRT parameters by using the following formulas [18, 36]; for each item i = 1, .”

“Several studies have pointed out that bifactor models usually tend to fit better than other competing models (e.g., Reise, 2012).”

“En conjunto, estos resultados sugieren la presencia de un factor general junto con algunos factores específicos de naturaleza residual, lo cual corresponde a un modelo bifactor (Reise, 2012; Reise, Bonifay, & Haviland, 2013; Rodriguez, Reise, & Haviland, 2016a, 2016b).”

“Bifactor-ESEM models provide a way to achieve a complete disaggregation of participants’ ratings into one global component reflecting the variance shared across all items and into specific components reflecting the variance shared among items, forming a dimension but not already explained by the G-factor, while taking…”

“, 2006), the ease of interpretation and breadth of influence of the direct hierarchical model recommends that model over the indirect hierarchical model (Chen, Hayes, Carver, Laurenceau, & Zhang, 2012; Gignac, 2008; Reise, 2012).”

“As a follow-up, we fitted three CFA models to the data (Additional file 2): (1) a unidimensional model, where the 11 PS represented a single construct of global disability; (2) a two-dimensional model composed of two correlated factors that we loosely referred to as “physical disability” (mobility, hand, bladder/bowel…”

“Although commonly applied only to the general factor in a bifactor model, the concepts underlying hierarchical omega can be applied to the subscale reliability, controlling for the effects of the general factor (Reise, 2012).”

“…there is an accumulation of evidence that for multidimensional questionnaires in general, bifactor models often provide better fit than correlated factors models (Brunner, Nagy, & Wilhelm, 2012; Chen, West, & Sousa, 2006; Gignac, 2016; Markon, 2009; Reise, 2012; Yung, Thissen, & McLeod, 1999).”

“Previous studies have extensively demonstrated the benefits of using the bifactor model to analyze behavioral data (see Chen et al., 2006; Pomplun, 2007; Reise, 2012).”

“The optimal case for the validity of a factor is that the loadings of its indicators should be relatively consistent and moderate to high in magnitude (K. Bollen, 2011; K. A. Bollen, 2020; Fabrigar et al., 1999; Reise, 2012; Savalei & Reise, 2019; Yang & Green, 2010).”

“Although the bifactor model may not be appropriate for all data and psychological phenomena, the bifactor model has been found to fit psychological data well across different problem areas, sample types, and ages (e.g., Ebesutani, McLeish, Luberto, Young, & Maack, 2014; Reise, 2012; Reise et al., 2007).”

“As Reise (2012) stated A bifactor structural model specifies that the covariance among a set of item responses can be accounted for by a single general factor that reflects the common variance running among all scale items and group factors that reflect additional common variance among clusters of…”

“Thus, we used bifactor modeling (see Reise, 2012) to build one general context perception factor (from perceptions of interactional justice as well as perceptions of organizational politics) and one general job performance factor (from task performance and OCB).”

“As a second method of evaluating the model, Table 4 presents McDonald’s (1999) coefficient ω as a measure of reliability calculated for each of the factors at Years 9 and 10 (see also Reise, 2012; Zinbarg, Yovel, Revelle, & McDonald, 2006).”

“This bifactor model (Reise, 2012), which has also been called a direct hierarchical model (Gignac, 2008) and a nested factor model (Gustafsson & Balke, 1993), has been recommended for hierarchically structured constructs such as intelligence (Brunner, Nagy, & Wilhelm, 2012), has been applied to…”

“Liu et al. 619 Bifactor analysis has recently gained wide recognition in hierarchical modeling applications (Reise, 2012).”

“In the bifactor model the general and specific factors were specified as orthogonal to ensure interpretability (Reise, 2012).”

“For Aim 1, latent bifactor modeling (Reise, 2012) was used to estimate P, INT, and EXT factors using a combination of mother and partner reports of child psychopathology at age 3.”

“The latent variables were specified to be uncorrelated (Reise, 2012) and all unique variances were uncorrelated.”

“It is therefore conceivable that variation in responses stems in part from the foundation that the item taps into and in part from whether it is a judgment item or a relevance item, and this can be adequately modeled through a bifactor model (Reise, 2012).”

“Although this suggests similarities between higher-order and bifactor models, the SLP shows that hierarchical factor models implicitly rely on more stringent assumptions (Chen, West, & Sousa, 2006; Reise, 2012).”

“Because bifactor models are frequently applied in practice in recent years (Reise, 2012), it is a helpful heuristic rule for researchers to know to explain problems that may occur in conducting their analyses.”

“Bifactor structures have been used to model multidimensionality of measures when all items are expected to measure a common construct while a subset of items measure a subdomain construct in addition to the common trait (Reise, 2012).”

“…be addressed via methods such as utilising a bi-factor measurement model to separate variation common versus unique to these forms of aggression, in practice in the case of the measure used in the current sample, this leaves insufficient reliable variance in the unique factors (e.g. Reise, 2012).”

“The bifactor model is an increasingly popular approach to measurement (see Reise, 2012), wherein the items constituting a scale are hypothesized to reflect an overall latent dimension (or “general factor”) along with several clusters of dimensions (or “specific factors”) that have shared content.”

“Besides these statistical approaches, there are further analysis strategies that may profit from comparisons of the variances of latent variables, such as the multitrait-multimethod approach (Marsh and Grayson, 1995) and the bifactor approach (Reise, 2012).”

“Bifactor Model In recent years, bifactor models have been rediscovered by researchers to test multidimensional constructs where a general factor and a set of specific subdomains coexist under some basic rules: (1) each indicator shares variance with a general factor and with only one subscale at the same time; (2) the…”

“Bi-factor modelling is a structural equation approach used to parse variance explained by a general factor (e.g., an overall latent factor of self- and other-ratings) from factors that explain variance byhow they are clustered (e.g., latent factors for each rater-source; Reise, 2012).”

“, internalizing, conduct problems, and attention-deficit/ hyperactivity disorder [ADHD]) (15).”

“…the covariation among the 16 GRAT–RS items could be best accounted for by (a) a single general gratitude factor that reflects the common variance across all items, and (b) three specific factors that capture additional (i.e., unique) common variance among clusters of items (Reise, 2012, p. 688).”

“saturation of a multidimensional scale by a general factor [18,19].”

“On the other hand, bifactor models (Reise, 2012; see Figure 1 below) allow for the estimation of an overarching general construct without relying on the restrictive implicit assumption of the proportionality constraint, and for a separate and straightforward assessment of variance uniquely…”

“…the original analyses by Scalas et al. (2013) and consistent with recommendations by Marsh et al. (2010), here we also use a bifactor model (Reise, 2012; also see Marsh, Morin, Parker, & Kaur, 2014) in which all 10 items load on the global esteem factor, whereas positively worded items…”

“A bifactor model is a measurement model in which each indicator loads on two factors: a general factor common to all indicators and a group factor common to a subset of indicators (Reise, 2012).”

“Higher ECV values indicate a strong general factor, which may guide the decision to fit a unidimensional model even with data that are multidimensional (20).”

“Despite this, there was a strong single factor in the S-BAT, which is consistent with the idea that the scale measures only one construct (i.e., burnout syndrome) (Reise, 2012).”

“Some researchers have argued that this approach may best capture the effects of latent constructs on various outcomes since residual variance does not necessarily denote measurement error (e.g. Reise, 2012).”

“Speci fi cally, we used bifactor modeling (Reise, 2012; Reise et al., 2010), which allows items to load on one general factor (in this case, Schizotypy) and one speci fi c factor (depending on the sub-factor to which it belongs).”

“A prominent example of these transformed models is the bifactorMIRT discovered by Gibbons and Hedeker (1992), later improved by Cai, Yang, and Hansen (2011) and Reise (2012).”

“All factors were specified as orthogonal, consistent with standard procedures for bi-factor modeling (Reise 2012) and with the original model (Friedman et al.”

“Considering the work of Reise and colleagues (Reise, 2012; Reise, Moore, & Haviland, 2010) we made our first assessment of scale unidimensionality by comparing the standardized factor loadings for the general and specific factors of the championed bifactor model.”