## 3 Methods
### Data Analysis  
As explained above, we used a response dataset including item responses and RT values obtained from a computer-based math assessment. We identified RT thresholds for each item using our proposed approach based on the random search and genetic algorithms and the NT method [5]. For the NT method, we used NT10 (10% of the average RT value), NT15 (15% of the average RT value), NT20 (20% of the average RT value), and NT25 (25% of the average RT value). Our preliminary analysis indicated that we did not have to set a maximum threshold (e.g., 10 s for NT10), since the maximum response time was 30 s for all items. For each method, we identified the RT thresholds for the entire assessment (i.e., 40 items) and reported various results (e.g., model fit indices, correlations between estimated scores, and percent of students with rapid guessing behavior). The data analysis was conducted using the randomsearch, GA [38; 39] and mirt [40] packages in R [41]. The following section describes the details of our proposed approach with the random search and genetic algorithms. Setting RT Thresholds with Random Search  
Random search refers to a family of stochastic methods designed for global optimization problems with continuous or discrete variables [42]. Unlike grid search, which relies on an exhaustive search of a manually specified hyperparameter space, random search uses the random sampling of a sequence of points in the hyperparameter space of the targeted algorithm [43]. Pure random search begins with the identification of a sequence of independent and identically distributed points within a reasonable search space. Then, the algorithm tries to find the direction of descent or ascent by randomly selecting values within the search space while continuously evaluating the performance of the target function [44]. Multiple iterations are implemented until finding the best value for local or global minima or maxima. Compared with grid search, random search is computationally less intensive and finds an optimal solution with significantly less computational time, because it relies on function evaluations with randomly drawn values, rather than gradient or Hessian information [42; 45]. In this study, the target function for finding local maxima (i.e., RT threshold for each item) is based on the log-likelihood difference between the baseline IRT model (i.e., rapid guesses are ignored) and the adjusted IRT model (i.e., rapid guesses are removed based on the RT threshold selected from the search space). For an IRT model, the log-likelihood can be expressed as  
\[\mathcal{L}_{IRTModel}(\mathbf{X})=Prob(\mathbf{X}|IRT\ Model), \tag{1}\]  
where \(\mathbf{X}\) refers to observed data points, and the IRT model refers to a logistic model that defines the probability of answering an item correctly based on a person's ability level and a set of item parameters. For example, the two-parameter logistic (2 PL) IRT model can be formulated as follows:  
\[P(X_{ij}=1|\theta_{j},a_{i},b_{i})=\frac{e^{a_{i}(\theta_{j}-b_{i})}}{1+e^{a_{ i}(\theta_{j}-b_{i})}}, \tag{2}\]  
where \(\theta\) is person \(j\)'s ability level on a logistic scale ranging from \(-\infty\) to \(+\infty\), \(a_{i}\) is the discrimination parameter for item \(i\), \(b_{i}\) is the difficulty parameter for item \(i\), \(e\) is the exponential function, and \(P(X_{ij}=1|\theta_{j},a_{i},b_{i})\) refers to person \(j\)'s probability of answering item \(i\) correctly given his/her ability and the parameters of item \(i\). The overall fit of an IRT model can be measured based on the deviance statistic (\(-2\log\)-likelihood) [46]. Theoretically, the deviance of the IRT model reflects the discrepancy between the expected and observed responses in the items based on the target ability (\(\theta\)). The deviance statistic increases as the observed responses deviate from the expected responses due to aberrant response behaviors (e.g., rapid guessing, item preknowledge, and cheating).