# Enhancing Ability Estimation with Time-Sensitive IRT Models in Computerized Adaptive Testing
## 3 Methods
### Parameter Estimation
#### 3.3.1 First Approach Maximum Likelihood Estimation (MLE) with Fixed Item Difficulties  
In the first optimization approach, item difficulties (b) were precomputed using the logit transformation of correct response probabilities, ensuring numerical stability by clipping probabilities between 0.05 and 0.95:  
\[b_{j}=ln\left(\frac{p_{j}}{1-p_{j}}\right) \tag{8}\]  
where \(p_{j}\) represents the proportion of correct responses for item j. Once determined, b remained fixed, and only \(\theta\) and \(\lambda\) were optimized. This approach minimized computational complexity but introduced potential bias if initial b estimates were inaccurate.  
The L-BFGS-B optimization algorithm was used to estimate \(\theta\) and \(\lambda\), enforcing theoretical constraints:  
* \(\theta\in[-3,3]\): Student abilities were constrained to a realistic range, preventing extreme values that could distort model interpretability.
* b \(\in[-1,\ 1]\): Item difficulties were bounded to ensure balanced item calibration, avoiding excessively easy or difficult items.
* \(\lambda>0\) for multiplicative models: Time-effect coefficients were constrained to positive values to maintain interpretability in multiplicative interactions.  
The L-BFGS-B optimization algorithm was employed for parameter estimation due to its ability to handle bound constraints, ensuring that parameters such as student abilities (\(\theta\in[-3,3]\)), item difficulties (\(\mathrm{b}\in[-1,1]\)), and time-effect coefficients (\(\lambda>0\)) remain within theoretically valid ranges. Its gradient-based approach and memory efficiency make it particularly suitable for large-scale problems, such as those encountered in IRT models with multiple parameters. Additionally, L-BFGS-B's robustness against local minima and its integration into widely-used libraries like scipy.optimize facilitate reliable and efficient estimation of model parameters.  
The objective function was defined as the regularized negative log-likelihood:  
\[L=-\sum_{i=1}^{N}\sum_{j=1}^{M}\left[Y_{ij}\cdot ln^{P_{ij}}+\left(1-Y_{ij} \right)\cdot\left(ln^{(1-P_{ij})}\right)\right]+\lambda_{r}\sum_{k}\theta_{k} ^{2} \tag{9}\]  
where \(\lambda_{r}=0.01\) represents the L2 penalty to prevent overfitting, and \(P_{\bar{\eta}}=\sigma\) (\(\theta_{i}-b_{j}\)) represents the probability of correct response under the logistic function.  
3.2 Second Approach Full Parameter Optimization (\(\theta\), \(\mathrm{b}\), \(\lambda\) Joint Estimation)  
In the second approach, all parameters \(\theta\), \(\mathrm{b}\), and \(\lambda\) were optimized simultaneously rather than fixing b beforehand. This allowed greater flexibility in estimating item characteristics while preserving interpretability. However, this method required careful initializationto prevent local minima. The regularized log-likelihood function was expanded to include constraints for b and \(\lambda\), ensuring a stable estimation process:  
\[L=-\sum_{i=1}^{N}\sum_{j=1}^{M}\left[\Upsilon_{ij}\cdot ln^{P_{ij}}+\left(1- \gamma_{ij}\right)\cdot\left(ln^{(1-P_{ij})}\right)\right]+\lambda_{r}\left(\sum _{k}\theta_{k}^{2}+\sum_{j}\theta_{j}^{2}\right) \tag{10}\]  
where an additional L2 penalty on b prevents large variations across item difficulties. Gradient norms were monitored during optimization, and termination criteria were set at 500 iterations or a tolerance of \(10^{-6}\) for parameter updates. The optimization process minimized the regularized negative log-likelihood function, which incorporated an L2 penalty (\(\lambda=0.01\)) to mitigate overfitting.