ISSN 0439-755X
CN 11-1911/B

Acta Psychologica Sinica ›› 2017, Vol. 49 ›› Issue (8): 1125-1136.doi: 10.3724/SP.J.1041.2017.01125

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 Simulated data comparison of the predictive validity between bi-factor and high-order models

 XU Shuangxue; YU Zonghuo; LI Yuemei   

  1.  (School of Psychology, Jiangxi Normal University, Nanchang 330022, China)
  • Received:2016-08-17 Published:2017-08-25 Online:2017-06-25
  • Contact: YU Zonghuo, E-mail: E-mail: E-mail:
  • Supported by:

Abstract:  Psychological and educational researchers are often confronted with multifaceted constructs which are comprised of several related dimensions. Bi-factor and high order factor models, which were regarded as two important methods to measure the multifaceted constructs, had been widely applied in many research fields. Generally speaking, a high order factor model was nested in the respective bi-factor model. These two models were equivalent, however, only when the proportionality constraints were imposed. Many researchers compared these two models from the perspective of model-fit. Empirical or simulation studies which were based on empirical data almost overwhelmingly found that the bi-factor model was better than the high-order factor model in model fit. In contrast, however, several simulation studies found that there was no significant difference between these two models in model fit indices. Hence, it is rather difficult to conclude which model is the best one. In addition to model fit, predictive validity is also an important aspect of models’ construct validity with no researcher comparing the predictive validity of these two models so far. It is thus important to compare these two models from the perspective of predictive validity, and to find out which of these two models is the better one. Based on the comparison of model fit between bi-factor and high-order factor models, this research focuses on the differences in the predictive accuracy of these two models in two studies using the Monte Carlo simulation method. Study One compared the difference on model fit indices between the two models under different levels of factor loadings (0.4, 0.5, 0.55, 0.6, 0.7). Their difference in predictive accuracy when the criterion was an exogenous variable was compared. To ensure the conclusions have wider generalizability, Study Two focused on the comparison of the predictive validity between the two models under the same experimental conditions when the criterion was an endogenous variable. The sample size was fixed at 1000 in all condition. The results of the simulation study suggested that, the bi-factor model and high order factor model both perfectly fitted these generated data under any conditions, and there was no significant difference between two models in model fit. When the criterion was an exogenous variable, the biases of the structural coefficients for both bi-factor model and high order factor model were small. In both models, it was more than ten percent bias in one case only, with the probability 1/80. It can be concluded that the two models’ structural coefficients were unbiased estimates. When the criterion was an endogenous variable, the structural coefficient biases of the high order factor model were all within 10%. There were, however, about 50% of the structural coefficient of the bi-factor model had a bias greater than 10%. Generally speaking, for bi-factor models, they can determine domain factors’ effect size through its factor loadings. When compared to regular SEMs, such as high-order factor models, however, starting values for the models might have to be provided for convergence. For high order models, they are more compact and superior in parameter estimation. To conclude, compared with high order factor models, bi-factor models were superior in determining its effect size through domain factors’ loading but they had with no advantage in model fit. High-order factor models were better than bi-factor model when the general and domain factors were used to predict criterion.

Key words: bi-factor model, high-order factor model, model-fit, structural coefficient bias.

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