## 3 Materials and Methods
### Methodology
#### 3.5.1 Statistical Functions of the CAFT Software
The likelihood function of the responses to all items administered is computed as follows:  
\[\hat{\theta}_{u_{A_{k}}}^{ML}\equiv\arg max_{\theta}[L(\theta|u_{A},\ \xi_{A}): \theta\in(-\infty,\infty)], \tag{5}\]  
where a learner's ability = \(\theta\), a bank of items \(\equiv B\), a set of items administered after stage \(k\equiv A_{k}\), a set of items remaining at stage \(k\equiv R_{k}=B-\ A_{k-1}\), a response to item \(i\) (\(m_{i}\) possible categories): \(u_{i}\in\{1,\,2,\,3,\,\ldots,\ m_{i}\}\), a single item administered at stage \(k\equiv i_{k}\), a response to item \(i\) at stage \(k\equiv u_{i_{k}}\), and \(\xi\) refers to the item characteristics, such as item difficulty and item discrimination, depending on the IRT models. Meanwhile, the expected a posterior estimate (EAP) combines prior information about learners with likelihood information from the data [34]. EAP assumes that proper prior information about learners exists. The EAP can be computed by the following equation:  
\[\hat{\theta}_{u_{A_{k}}}^{EAP}\equiv\mathrm{E}[\theta]=\int\theta g\big{(} \theta\big{|}u_{A_{k}},\ \xi_{A_{k}}\big{)}d\theta \tag{6}\]  
Additionally, this software computes the standard error of measurement for selecting and stopping CAT, for which posterior variance is used. Lastly, the (F) area is for selecting data files or conducting a simulation. The data files, including the response data, item characteristics data, and examinees' abilities data, can be uploaded, and a simulated data file can be generated. 5.2 Dynamic Bayesian Network for Diagnostic Learning Progression under the Adaptive Formative Assessment System  
Since CAT is an adaptive system, a psychometric method should be developed in order to estimate each student's learning progression over multiple measurement time points. The estimation of a learner's learning progression over multiple time points in CAFT is produced by DBNs [35]. DBNs are a way to extend a static Bayesian network to model probability distributions over multiple time points [36]. DBNs offer a real-time updating method to estimate a learner's previous, current, and future states of a system over a specific period of time [37]. The DBNs for a formative assessment system contain prior information on the hidden state, \(\mathrm{P(X_{1})}\), a transition function of the hidden states over multiple time points, \(\mathrm{P(X_{t}|\ X_{1:t-1})}\), and an observation function given the hidden state, \(\mathrm{P(Y_{t}|\ X_{t})}\). The detailed expression about three probability matrices are as follows:Initial probability matrix of the hidden state at the first time point, P(X\({}_{1}\));  
Transition probability matrix, P(X\({}_{t}|\)X\({}_{1:t-1}\));  
Conditional probability matrix, P(Y\({}_{t}|\)X\({}_{t}\)). Another aspect of DBNs supports the monitoring of learning progression over a specific period of time in a formative assessment. Once an observation has been made on a subset of the variables in the network at a certain point in time, educators are able to make inferences about the remaining unobserved variables in the network at any given time point. In other words, the DBNs reflect the states at previous and future points in time, as well as the current state, because the states at the current point in time will impact the state in the future and are impacted by the states in the past. Therefore, there are three main inferences that can be performed using DBNs:  
_Smoothing_: The process of monitoring states at previous time t - 1 given evidence at time t;  
_Filtering_: The process of monitoring states at time t given evidence at time t;  
_Prediction_: The process of monitoring states at future time t + 1 given evidence at time t.