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.”

“Exploratory analyses allow researchers to identify potential modeling problems directly rather than indirectly through post hoc inspection of fit and modification indices after estimating a confirmatory model (see Browne, 2001, pp. 124–125 for additional commentary).”

“An elegant and simple bifactor estimation method—called the Schmid-Leiman orthogonalization (SL; Schmid & Leiman, 1957)—was introduced 20 years later (see also Schmid, 1957; Wherry, 1959).”

“One approach to implementing this model simply is to replace Pearson correlations with tetrachoric or polychoric correlations and then conduct a limitedinformation factor analysis (e.g., weighted least squares; see Wirth & Edwards, 2007).”

“This topic was covered extensively in F. F. Chen et al. (2006) and will not be reviewed here.”

“IRTPRO also shows indices that reflect the degree of local dependence (Chen & Thissen, 1997) with larger values indicating higher residual correlations between item pairs after controlling for the latent trait(s).”

“Although originally described over 70 years ago (Holzinger & Harman, 1938; Holzinger & Swineford, 1937), bifactor modeling has spent the last 50 years overshadowed by the numerous applications of Thurstone’s correlated-factors model.”

“For example, Simms, Gros, Watson, and O’Hara (2008) used confirmatory bifactor modeling to explore the relative contribution of general and group factors in affecting responses to the Inventory of Depression and Anxiety Symptoms (Watson et al., 2007).”

“In IRTPRO the fit of each item is judged using adjusted chi-square statistics developed by Orlando and Thissen (2000, 2003).”

“This method often is referred to as “full-information” item factor analysis because it uses the entire item response matrix as part of the calibration (Gibbons & Hedeker, 1992).”

“Partitioning Item Variance Psychological constructs, and item response data resulting from measures designed to assess such constructs, often are proposed to have a “hierarchical” or “multifaceted” structure (Brunner et al., 2012; Chen et al., 2012).”

“Partitioning Item Variance Psychological constructs, and item response data resulting from measures designed to assess such constructs, often are proposed to have a “hierarchical” or “multifaceted” structure (Brunner et al., 2012; Chen et al., 2012).”

“In IRTPRO the fit of each item is judged using adjusted chi-square statistics developed by Orlando and Thissen (2000, 2003).”

“After describing limitations of the Schmid-Leiman, 2 newer methods of exploratory bifactor modeling are considered, namely, analytic bifactor (Jennrich & Bentler, 2011) and target bifactor rotations (Reise, Moore, & Maydeu-Olivares, 2011).”

“Finally, three goodness-of-fit statistics based on the overall contingency table— chi-square, Pearson (both not computable here due to a sparse contingency table), and M2 (Maydeu-Olivares & Joe, 2005)—are provided.”

“Only recently have highly efficient dimensionality reduction techniques become available (Cai, 2010a, 2010b, 2010c) that greatly expand the Note.”

“Only recently have highly efficient dimensionality reduction techniques become available (Cai, 2010a, 2010b, 2010c) that greatly expand the Note.”

“…distinguishes broad and narrow factors” (p. 107). men, 2009); demonstrate important extensions of bifactor modeling to computerized adaptive testing (Gibbons et al., 2008), vertical scaling (Li & Lissitz, 2012), and assessing differential item functioning (Fukuhara & Kamata, 2011; Jeon, Rijmen, &…”

“However, it has long been recognized that the two-parameter normalogive IRT model and the factor-analytic model for ordinal item responses are equivalent for either dichotomous or polytomous data (Kamata & Bauer, 2008; Takane & de Leeuw, 1987).”

“Several scholars have advised that only if the least restricted model (in this case the bifactor) is judged to fit the data (Yuan & Bentler, 2004) is it appropriate to consider whether applying a more restricted, nested model significantly degrades that fit.”

“…when multidimensional item response data are “unidimensional enough” (e.g., ratio of 1st to 2nd eigenvalue, residuals after extracting a single factor) and several proposed methods of evaluating the degree of unidimensionality (see, e.g., research on the DETECT index; Zhang & Stout, 1999).”

“For example, consider a model for “realistic” data generation proposed by MacCallum and Tucker (1991) where items may be influenced by one or more “dominant” factors but also by dozens of smaller common factors.”

“Knol and Berger (1991), for example, demonstrated that an ordinary least squares factor analysis of tetrachorics often can recover known item parameters just as well, if not better than, more complicated estimation methods (see also Finch, 2010, 2011).”

“Published psychometric articles now compare the bifactor with competing structural representations (Chen, West, & Sousa, 2006; Rijmen, 2010); provide solutions to challenging estimation problems (Cai, 2010c; Rij-”

“An elegant and simple bifactor estimation method—called the Schmid-Leiman orthogonalization (SL; Schmid & Leiman, 1957)—was introduced 20 years later (see also Schmid, 1957; Wherry, 1959).”

“On the other hand, if the group factors are allowed to correlate with each other (see Jennrich & Bentler, 2012), this suggests the presence of additional and unmodeled general factors.”

“This value can be contrasted to omega ¨ (Lucke, 2005) shown in 7Note that researchers who use a variance-covariance matrix in place of a tetrachoric or polychoric correlation matrix in their analyses must also make sure that the factor loadings in Equation 10 are unstandardized versions.”

“As such, the mathematics of coefficient omega hierarchical can be an invaluable tool in judging whether it is reasonable to report subscale scores (see also Gignac et al., 2007).”

“In a Monte Carlo simulation, Reise et al. (2011) generated dichotomous item response data from populations with known bifactor loading patterns.”

“Although originally described over 70 years ago (Holzinger & Harman, 1938; Holzinger & Swineford, 1937), bifactor modeling has spent the last 50 years overshadowed by the numerous applications of Thurstone’s correlated-factors model.”

“…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.”

“Knol and Berger (1991), for example, demonstrated that an ordinary least squares factor analysis of tetrachorics often can recover known item parameters just as well, if not better than, more complicated estimation methods (see also Finch, 2010, 2011).”

“…(p. 107). men, 2009); demonstrate important extensions of bifactor modeling to computerized adaptive testing (Gibbons et al., 2008), vertical scaling (Li & Lissitz, 2012), and assessing differential item functioning (Fukuhara & Kamata, 2011; Jeon, Rijmen, & Rabe-Hesketh, in press); and, it is…”

“However, the potential parameter distorting effects of forcing small cross-loadings to zero, and accommodating items with substantial cross-loadings on group factors, remains troublesome (see also Finch, 2011).”

“An elegant and simple bifactor estimation method—called the Schmid-Leiman orthogonalization (SL; Schmid & Leiman, 1957)—was introduced 20 years later (see also Schmid, 1957; Wherry, 1959).”

“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).”