In learning environments, understanding the longitudinal path of learning is one of the main goals. Cognitive diagnostic models (CDMs) for measurement combined with a transition model for mastery may be beneficial for providing fine-grained information about students’ knowledge profiles over time. An efficient algorithm to estimate model parameters would augment the practicality of this combination. In this study, the Expectation–Maximization (EM) algorithm is presented for the estimation of student learning trajectories with the GDINA (generalized deterministic inputs, noisy, “and” gate) and some of its submodels for the measurement component, and a first-order Markov model for learning transitions is implemented. A simulation study is conducted to investigate the efficiency of the algorithm in estimation accuracy of student and model parameters under several factors—sample size, number of attributes, number of time points in a test, and complexity of the measurement model. Attribute- and vector-level agreement rates as well as the root mean square error rates of the model parameters are investigated. In addition, the computer run times for converging are recorded. The result shows that for a majority of the conditions, the accuracy rates of the parameters are quite promising in conjunction with relatively short computation times. Only for the conditions with relatively low sample sizes and high numbers of attributes, the computation time increases with a reduction parameter recovery rate. An application using spatial reasoning data is given. Based on the Bayesian information criterion (BIC), the model fit analysis shows that the DINA (deterministic inputs, noisy, “and” gate) model is preferable to the GDINA with these data.

在学习环境中，了解学习的纵向路径是主要目标之一。用于测量的认知诊断模型 (CDM) 与用于掌握的过渡模型相结合，可能有助于提供有关学生知识概况的细粒度信息。估计模型参数的有效算法将增强这种组合的实用性。在这项研究中，提出了期望最大化 (EM) 算法，用于使用 GDINA（广义确定性输入、噪声、“和”门）及其测量组件的一些子模型来估计学生的学习轨迹，以及第一个实现了学习转换的阶马尔可夫模型。进行了仿真研究，以研究该算法在样本大小、属性数量、测试中的时间点数量和测量模型的复杂性等几个因素下估计学生和模型参数的准确性的效率。研究了模型参数的属性和向量级一致性率以及均方根误差率。此外，还会记录收敛的计算机运行时间。结果表明，在大多数情况下，参数的准确率与相对较短的计算时间相结合是非常有希望的。只有在样本量相对较小和属性数量较多的情况下，计算时间才会随着缩减参数恢复率的增加而增加。给出了一个使用空间推理数据的应用程序。