Performance of Estimation Methods in Bifactor Models with Ordered Categorical Data

Abstract

Simulation studies are needed to investigate how many score categories are sufficient to treat ordered categorical data as continuous, particularly for bifactor models. The current simulation study aims to address such needs by investigating the performance of estimation methods in the bifactor models with ordered categorical data. Results support the application of categorical estimators to the ordered categorical data rather than the continuous estimators when sample size is large (750). Otherwise, an applied researcher may have to use the continuous estimators due to the model non-convergence. In this circumstance, the number of response categories needs to be at least 6 to avoid the rejection of correctly specified bifactor models by the chi-square test and estimate the model parameters accurately. The robust maximum likelihood (MLR) may be chosen among two continuous estimators due to its smaller type I error rate for the chi-square test than the ML. Practical implications of study findings are discussed.

Keywords:

• Bifactor models
• estimation methods
• number of score categories
• ordered categorical data

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 A study condition was selected (sample size = 750, the number of group factors = 3 with 6 items per group factor and the group factor loadings between 0.4 and 0.5) to study the performance of estimators across the study conditions with symmetric and asymmetric thresholds. The threshold symmetry (symmetric, moderately asymmetric, and extremely asymmetric) and the number of categories (2, 3, 4, 5, 6, and 7) were manipulated in the data generation. Unlike the ULSMV and WLSMV, the performance of ML and robust ML was affected by the threshold symmetry with a less severe impact on the MLR under the conditions with a greater number of score categories (see Online Supplemental Material—Tables B and C). Specifically, the type I error rate was inflated, and the relative bias of factor loadings was larger for the ML and robust ML as the observed score distribution became more asymmetric. When using a cutoff of $\pm 0.$05 as an acceptable value for the relative bias, the number of score categories generally needs to be at least 5 for the study conditions with the symmetric and moderately asymmetric thresholds, but at least 6 for the conditions with the extremely asymmetric thresholds so that the robust ML can be considered for the analysis of ordered categorical data via the bifactor models.