Resumen de Avances en Análisis Bi-factorial Exploratorio
Bi-factor modelling constitutes today one of the preferred statistical tools in psychological research. Due to its unique characteristics, the bi-factor model has been reintroduced in significant areas of interest such as psychopathology, personality or intelligence. Accordingly, our understanding of this model could determine the advancement of our knowledge of many psychological phenomena. Despite considerable efforts to appropriately estimate these models, available methods present stringent limitations that question their overall validity and usefulness. This doctoral dissertation aims to provide a detailed account of the historical development of the bi-factor model, as well as its main applications from both, a confirmatory and an exploratory approach. Notably, this dissertation will detail why and how exploratory bi-factor analysis has emerged as one of the most compelling solutions to approximate these complex structures under realistic settings. Moreover, the central role played by the target rotation will be scrutinised, emphasising the limitations and potential improvements of current bi-factor target-based rotation methods.
In this context, this doctoral dissertation was ultimately concerned with proposing new alternatives to conduct bi-factor rotation, understanding their benefits with regards to both, parameter estimation and secondary statistics of interest. Furthermore, this thesis dissertation intended to provide free, user-friendly tools aimed for the general public to apply bi-factor exploratory factor analysis. In detail, these objectives were developed in seven chapters:
In Chapter 2, the Iterative Target Rotation based on a Schmid-Leiman solution (i.e., SLi) algorithm was introduced. This algorithm improved the original proposal by Reise, Moore and Maydeu-Olivares (2011) of defining bi-factor target rotation by including Moore, Reise, Depaoli and Haviland (2015) iterative target rotation scheme. Results from a Monte Carlo simulation evidenced that SLi improved factor recovery when compared with widespread methods such as the bi-geomin and the bi-quartimin criteria, the Schmid-Leiman solution or the non-iterative version of the bi-factor target rotation.
In Chapter 3, the Empirical Iterative Target Rotation based on a Schmid-Leiman solution (i.e., SLiD) algorithm was presented. This algorithm aimed to improve the SLi algorithm by including a novel method for the computation of factor-specific, empirically defined cut-off points for distinguishing non-vanishing and vanishing entries in the target matrix. This strategy was based on finding relevant differences in the distribution of normalised, sorted factor loadings in each group factor. Results from a Monte Carlo simulation demonstrated the superiority of the SLiD algorithm under realistic conditions (i.e., structures presenting a mixture of group factors with different average factor loadings altogether with a high number of cross-loadings).
In Chapter 4, the strategy behind the SLiD algorithm was applied to the study of the properties of the scores from a novel, brief intelligence test: the Last Twelve matrices of the Standard Progressive Matrices (SPM-LS). This application demonstrated the usefulness of the previous strategies to evaluate the assumption of essential unidimensionality as well as the presence of relevant nuisance factors. The use of the methods developed in this thesis revealed that while the scores could be considered essentially unidimensional, the bi-factor model represented the most appropriate measurement model.
In Chapter 5, the consequences of the choice of an exploratory bi-factor rotation were explored with regards to the estimation of the omega hierarchical statistic. Moreover, in this Chapter, two new bi-factor rotation methods were studied: the Direct Bi-factor and Direct Schmid-Leiman algorithms. Yet again, results from three different Monte Carlo simulations evidenced that the SLiD algorithm provided the best results regardless structure under consideration (i.e., full, rank-deficient bi-factor models or structures without a general factor), and across a wide range of conditions. Furthermore, it was shown that the application of a partially or completely specified target rotation determined the quality of omega hierarchical estimation when examining target-based algorithms. Lastly, the functioning of each algorithm in eight classic examples was presented to provide a better depiction of the results previously discussed.
In Chapter 6, the integration of the SLiD in the context of exploratory structural equation modelling was discussed. A free, user-friendly Shiny application SLiDApp was developed so to facilitate the translation of the estimated SLiD target to Mplus. To illustrate the usefulness of this application, a step-by-step guide was introduced using a novel bi-factor examination of the Generic Conspiracionist Belief Scale and its relationship with the Big Five personality traits.
This doctoral dissertation is concluded with a reflective, critical discussion of the benefits and limitations of both, the exploratory bi-factor model and the presented methods for approximating its estimation. In the same spirit, this discussion is focused on offering future research directions as well as in providing clear advice to applied researchers and psychometricians alike. It is hoped that this dissertation would be helpful to disentangle the benefits and drawbacks of bi-factor modelling, inspiring further research endeavours in this area.
