ISSN 0439-755X
CN 11-1911/B

心理学报 ›› 2023, Vol. 55 ›› Issue (10): 1712-1728.doi: 10.3724/SP.J.1041.2023.01712

• 研究报告 • 上一篇    下一篇


刘彦楼1,2(), 陈启山3,4, 王一鸣2, 姜晓彤2   

  1. 1曲阜师范大学教育大数据研究院, 山东 济宁 273165
    2曲阜师范大学心理学院, 山东 济宁 273165
    3“儿童青少年阅读与发展”教育部哲学社会科学实验室(华南师范大学), 广州 510631
    4华南师范大学心理学院, 广州 510631
  • 收稿日期:2023-03-02 发布日期:2023-07-26 出版日期:2023-10-25
  • 通讯作者: 刘彦楼, E-mail:
  • 基金资助:

On the reliability of point estimation of model parameters: Taking cognitive diagnostic models as an example

LIU Yanlou1,2(), CHEN Qishan3,4, WANG Yiming2, JIANG Xiaotong2   

  1. 1Academy of Big Data for Education
    2School of Psychology, Qufu Normal University, Jining 273165, China
    3Philosophy and Social Science Laboratory of Reading and Development in Children and Adolescents (South China Normal University), Ministry of Education
    4School of Psychology, South China Normal University, Guangzhou 510631, China
  • Received:2023-03-02 Online:2023-07-26 Published:2023-10-25


心理学研究中, 不恰当的模型参数估计框架或收敛准则严重影响模型参数点估计的可靠性, 进而影响到研究结论的可靠性。本研究提出了基于MLE-EM的CDM模型参数估计新框架, 以及新收敛判断方法。通过模拟研究与实证数据分析的方式, 探索了新参数估计框架和新收敛判断方法的表现, 并与已有模型参数估计框架及收敛判断方法进行了比较。结果显示, 新的模型参数估计框架及收敛准则的表现优于已有的模型参数估计框架及收敛准则, 能有效提高模型参数点估计的可靠性。

关键词: 参数估计, 点估计, 收敛准则, 认知诊断模型


Cognitive diagnostic models (CDMs) are psychometric models that have received increasing attention within fields such as psychology, education, sociology, and biology. It has been argued that an inappropriate convergence criterion for a maximum likelihood estimation using the expectation maximization (MLE-EM) algorithm could result in unpredictable and inaccurate model parameter estimates. Thus, inappropriate convergence criteria may yield unstable and misleading conclusions from the fitted CDMs. Although several convergence criteria have been developed, it remains an unexplored question, how to specify the appropriate convergence criterion for fitted CDMs.

A comprehensive method for assessing convergence is proposed in this study. To minimize the influence of the model parameter estimation framework, a new framework adopting the multiple starting values strategy (mCDM) is introduced. To examine the performance of the convergence criterion for MLE-EM in CDMs, a simulation study under various conditions was conducted. Five convergence assessment methods were examined: the maximum absolute change in model parameters, the maximum absolute change in item endorsement probabilities and structural parameters, the absolute change in log-likelihood, the relative log-likelihood, and the comprehensive method. The data generating models were the saturated CDM and the hierarchical CDM. The number of items was set to J = 16 and 32. Three levels of sample sizes were considered: 500, 1000, and 4000. The three convergence tolerance value conditions were 10-4, 10-6, and 10-8. The simulated response data were fitted by the saturated CDM using the mCDM and the R package GDINA. The maximum number of iterations was set to 50000.

The simulation results suggest the following.

(1) The saturated CDM converged under all conditions. However, the actual number of iterations exceeded 30000 under some conditions, implying that when the predefined maximum iteration number is less than 30000, the MLE-EM algorithm might inadvertently stop.

(2) The model parameter estimation framework affected the performance of the convergence criteria. The performance of the convergence criteria under the mCDM framework was comparable or superior to that of the GDINA framework.

(3) Regarding the convergence tolerance values considered in this study, 10-8 consistently had the best performance in providing the maximum value of the log-likelihood and 10-4 had the worst performance. Compared to all other convergence assessment methods, the comprehensive method in general had the best performance, especially under the mCDM framework. The performance of the maximum absolute change in model parameters was similar to the comprehensive method, but this good performance was not consistent. On the contrary, the relative log-likelihood had the worst performance under the mCDM and GDINA frameworks.

The simulation results showed that the most appropriate convergence criterion for MLE-EM in CDMs was the comprehensive method with tolerance 10-8 under the mCDM framework. The results from the real data analysis also demonstrated that the proposed comprehensive method and mCDM framework had good performance.

Key words: model parameter estimation, point estimation, convergence criterion, cognitive diagnostic model