Abstract: Longitudinal cognitive diagnostics can assess students’ strengths and weaknesses over time, profile students’ developmental trajectories, and can be used to evaluate the effectiveness of teaching methods and optimize the teaching process. Researchers have proposed different longitudinal diagnostic classification models, which provide methodological support for the analysis of longitudinal cognitive diagnostic data. Although these parametric longitudinal cognitive diagnostic models can effectively assess students’ growth trajectories, their requirements for coding ability and sample size hinder their application among frontline educators, and they are time-consuming and not conducive to providing timely feedback. On the one hand, the nonparametric approach is easy to calculate, efficient to apply, and provides timely feedback; on the other hand, it is free from the dependence on sample size and is particularly suitable for analyzing assessment data at the classroom or school level. Therefore, this study attempts to apply nonparametric method to longitudinal cognitive diagnostic assessments for tracking student’s learning trajectories. This study extended the longitudinal Hamming distance discriminant (Long-HDD) based on the Hamming distance discriminant (HDD), which uses the Hamming distance to represent the dependence between attribute mastery patterns of the same student at adjacent time points. To explore the performance of Long-HDD in longitudinal cognitive diagnostic data, we conducted three simulation studies and an empirical study and compared the classification accuracy of the HDD, Long-HDD, and Long-DINA models. The purpose of simulation study 1 was to compare the performance of performance of three methods under different simulation conditions. Simulation study 2 focused on the classification accuracy of the three methods at moderate attributes transfer probability level (p(0→1)=0.5, p(1→0)=0.05) and high attributes transfer probability level (p(0→1)=0.8, p(1→0)=0.05). To further highlight the advantages of the Long-HDD in small-scale assessments, the Long-DINA model was used as the data generation model in Study 3. At this point, if Long-HDD still outperforms or does not lose out to Long-DINA model's, the relative advantage of using Long-HDD in a small-scale assessments can be further highlighted. Furthermore, an empirical study was conducted to illustrate the application of the Long-HDD. Under the comparison of the three methods, the results of the simulation studies showed that (1) Long-HDD had higher classification accuracy in longitudinal diagnostic data analysis; (2) Long-HDD performed almost independently of sample size and performed better with a smaller sample size compared to Long-DINA; and (3) Long-HDD consumed much less computational time than Long-DINA. In addition, the results of the empirical study showed that there was good consistency between the results of the Long-HDD and the Long-DINA model?in tracking changes in attribute development. The percentage of mastery of each attribute increased with the increase of time points. In summary, the long-HDD proposed in this study extends the application of nonparametric methods to longitudinal cognitive diagnostic data and can provide high classification accuracy. Compared with parameterized longitudinal DCM, it can provide timely diagnostic feedback due to the fact that it is not affected by sample size, simple calculation, and less time-consuming. It is more suitable for small-scale longitudinal assessments such as class and school level.

Key words: cognitive diagnosis, nonparametric classification, longitudinal data analysis, Hamming distance

摘要： 研究通过在纵向诊断数据分析中引入计算简单、耗时少的汉明距离判别法(HDD)，提出了纵向HDD (Long-HDD)。与HDD相比，Long-HDD额外使用汉明距离刻画个体在相邻时间点上对属性掌握的相依性，以利用前一时间点信息提高当前时间点的分类准确性。三个模拟研究的结果主要表明：在分析纵向诊断数据时，与参数化模型相比，Long-HDD的分类准确性几乎不受样本量影响，在样本量较小时表现更优；且其计算耗时更少，更有利于提供及时性诊断反馈。实证研究结果表明Long-HDD可用于分析实践测评数据，且其追踪诊断结果与参数化模型的存在一致性。

关键词: 认知诊断, 非参数分类法, 纵向数据分析, 汉明距离