The bifactor model is a widely applied model to analyze general and specific abilities. Extensions of bifactor models additionally include criterion variables. In such extended bifactor models, the general and specific factors can be correlated with criterion variables. Moreover, the influence of general and specific factors on criterion variables can be scrutinized in latent multiple regression models that are built on bifactor measurement models. This study employs an extended bifactor model to predict mathematics and English grades by three facets of intelligence (number series, verbal analogies, and unfolding). We show that, if the observed variables do not differ in their loadings, extended bifactor models are not identified and not applicable. Moreover, we reveal that standard errors of regression weights in extended bifactor models can be very large and, thus, lead to invalid conclusions. A formal proof of the nonidentification is presented. Subsequently, we suggest alternative approaches for predicting criterion variables by general and specific factors. In particular, we illustrate how (1) composite ability factors can be defined in extended first-order factor models and (2) how bifactor(S-1) models can be applied. The differences between first-order factor models and bifactor(S-1) models for predicting criterion variables are discussed in detail and illustrated with the empirical example.

A new multiple indicator multilevel latent state-trait (LST) model for the analysis of multitrait–multimethod–multioccasion (MTMM-MO) data is proposed. The LST-COM model combines current CFA-MTMM modeling approaches of interchangeable and structurally different methods and LST modeling approaches. The model enables researchers to specify construct and method factors on the level of time-stable (trait) as well as time-variable (occasion-specific) latent variables and analyze the convergent and discriminant validity among different rater groups across time. The statistical performance of the model is scrutinized by a simulation study and guidelines for empirical applications are provided.

A doubly latent residual approach (DLRA) is presented to explain latent specific factors in multilevel bifactor-(S-1) models. The new approach overcomes some important limitations of the multiple indicators multiple causes (MIMIC) approach and allows researchers to predict latent specific factors at different levels. The DLRA is illustrated using real data from a large-scale assessment study. Furthermore, the statistical performance and power of the DLRA is examined in a Monte Carlo simulation study. The results show that the new DLRA performs well if more than 50 clusters and more than 10 observations per cluster are sampled. The power to test structural parameters at level 2 was lower than at level 1. To test a medium effect at level 2, we recommend to sample at least 100 clusters with a minimum cluster size of 10. The advantages and limitations of the new approach are discussed and guidelines for applied researchers are provided.