# Enhancing Ability Estimation with Time-Sensitive IRT Models in Computerized Adaptive Testing
## 4 Model Performance and Results Analysis
### Lambda (\(\lambda\)) Estimates  
This section presents the optimized values of the response-time sensitivity parameter (\(\lambda\)) for each of the four time-aware IRT models. These estimates were obtained using both full-dataset optimization and five-fold cross-validation procedures to ensure robustness and generalizability. As shown in Table 4, TP-IRT consistently yielded negative \(\lambda\) values (\(-0.2688\) with cross-validation and \(-0.2679\) with full data), indicating a systematic penalization of longer response times. In contrast, TWD-IRT and ART-IRT produced small but positive \(\lambda\) values, reflecting only a mild reward for faster responses. The DTA-IRT model displayed a notably larger \(\lambda\) under cross-validation (0.1557) compared to full-dataset training (0.0146), which suggests a stronger regularization effect when generalization is prioritized. These \(\lambda\) estimates highlight how each model interprets the influence of response time differently, offering insight into their respective time-sensitivity dynamics. These values confirm that \(\lambda\) was not fixed but was treated as a free parameter and empirically optimized during model training. The negative \(\lambda\) observed in TP-IRT supports the subtractive logic of this model, where prolonged response times correspond to reduced performance estimates. In contrast, TWD-IRT exhibited a small positive \(\lambda\), reflecting a modest amplification effect whereby response speed slightly increases the effective ability-difficulty gap. The relatively higher \(\lambda\) values observed in DTA-IRT and ART-IRT under cross-validation compared to the full-data scenario suggest that response-time sensitivity may vary across subpopulations and testing conditions. This variation highlights the relevance of flexible time-aware modeling, particularly in adaptive environments where generalization is crucial. Furthermore, we acknowledge the nested nature of some models, as noted by the reviewer:  
* The STP-IRT model is a restricted case of DTA-IRT when \(\lambda\) = 1. * Similarly, the NRT-IRT model is a special case of ART-IRT under the same constraint. However, the empirically optimized \(\lambda\) values were all significantly lower than 1 in both the full-data and cross-validation results. This observation empirically justifies the use of the more general DTA-IRT and ART-IRT formulations, which allow for greater flexibility and better data fit compared to their fixed-lambda counterparts. Figure 1 compares the AIC values of different models under three different computational approaches. A lower AIC value indicates a better model fit to the data. DTA-IRT and 1 PL-IRT models exhibit the lowest AIC values, suggesting that these models achieve better alignment with the dataset. TWD-IRT and ART-IRT models have the highest AIC values, indicating poorer model fit compared to other models. Approach 3 consistently produces the lowest AIC values, demonstrating that this approach is more effective in improving model performance. These findings highlight that AIC should not be the sole metric for model selection; other criteria must also be considered to ensure comprehensive model evaluation. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & \(\lambda\) **(Cross-Validation)** & \(\lambda\) **(Full Dataset)** \\ \hline
**TP-IRT** & \(-0.2688\) & \(-0.2679\) \\
**TWD-IRT** & \(0.0574\) & \(0.0556\) \\
**DTA-IRT** & \(0.1557\) & \(0.0146\) \\
**ART-IRT** & \(0.0787\) & \(0.0146\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Optimized lambda (\(\lambda\)) estimates for time-aware IRT models. Figure 2 presents the Bayesian Information Criterion (BIC) values for different models. Lower BIC values suggest better model fit while avoiding excessive complexity. The 1 PL-IRT and DTA-IRT models have the lowest BIC values, making them the most compatible with the dataset. ART-IRT and TWD-IRT models exhibit the highest BIC values, indicating that their complexity reduces their overall efficiency.