## 3 Methods
### Data Analysis
Using this information, we propose to use the change in the deviance statistic as a function to be maximized. Specifically, we anticipate that the deviance difference would decrease as rapid guesses are removed from the response dataset, which can be shown as  
\[\mathit{Deviance}_{Diff}=-2(\mathcal{L}_{Baseline\ Model}-\mathcal{L}_{ Adjusted\ Model}), \tag{3}\]  
where \(\mathcal{L}_{Baseline\ Model}\) is the log-likelihood for the baseline IRT model including all item responses, \(\mathcal{L}_{Adjusted\ Model}\) is the log-likelihood for the adjusted IRT model excluding rapid guesses, and \(\mathit{Deviance}_{Diff}\) is the deviance difference between the two IRT models. Our random search process begins with the estimation of the baseline IRT model (i.e., the 2 PL IRT model based on the original response dataset). After saving the log-likelihood value for the baseline model, the algorithm randomly selects a particular RT threshold from the search space, including all possible RT values. In our study, we defined the search space based on the previous empirical studies using the NT method. Specifically, the search space for RT values was between 10% and 30% of the average RT value for each item based on the recommendations of [PERSON] and [PERSON] [5] and [PERSON] et al. [9]. The selected RT threshold is used to remove rapid guesses from the response dataset, yielding a cleaned dataset. Next,an adjusted IRT model (i.e., the 2 PL IRT model based on the cleaned response dataset) is estimated, and its log-likelihood value is extracted. The target function (see Equation (3)) computes the deviance difference and stores this value for each iteration to find the optimal RT threshold. Although we use the deviance difference as the main criterion to look for the optimal RT threshold, we also consider the probability of correct response (i.e., cumulative proportion-correct) for rapid guessers. To achieve this, we first compute the cumulative proportion for each item at time \(t\), denoted as \(CUMP(t)\), for students who spend less than \(t\) seconds on the item. Then, we divide the deviance difference by this \(CUMP(t)\) and look for the RT threshold that maximizes \(Adjusted\ Deviance_{Diff}\):  
\[Adjusted\ Deviance_{Diff}=\operatorname*{argmax}_{t\in R}( Deviance_{Diff}/CUMP(t)), \tag{4}\]  
where the value of \(CUMP(t)\) can be fixed to 0.001 (or another small value) if the original \(CUMP(t)\) value is exactly zero, in order to avoid undefined results due to a division by zero. We use the adjusted deviance difference produced by Equation (4), rather than the original deviance difference, to determine the most optimal RT threshold. This approach allows us to combine a psychometric approach with the NT and CUMP methods. It should be noted that the objective function defined in Equation (4) is suitable for assessment settings where the researcher does not have any priori hypothesis about the ideal value of \(CUMP(t)\) (e.g., using constructed-response items). Therefore, Equation (4) can be further revised to accommodate other types of assessment settings. For example, one can use  
\[Adjusted\ Deviance_{Diff}=\operatorname*{argmax}_{t\in R}(| Deviance_{Diff}*log(CUMP(t))|), \tag{5}\]  
where \(log(CUMP(t))\) is the \(CUMP(t)\) value after the log transformation. For multiple-choice items, Equation (5) could be more suitable than Equation (4), because the adjustment would be stronger within the range of \(CUMP(t)=0\) and \(CUMP(t)=0.25\) (i.e., the random guessing rate in a multiple-choice item with 4 response options), whereas the adjustment would get weaker as\(CUMP(t)\) increases beyond 0.25, which is not desirable since the rapid guessing rate is not expected to exceed the random guessing rate.