# Unbiased Isotonic Regression Tree for Discovering Hidden Heterogeneity in Monotonicity Constraints
## 4 Investigating the Presence of Selection Bias in IRT
By adhering to this widely accepted convention, the simulation aligns with established practices in the field, enhancing its rigor and comparability to related research. To calculate the selection rate for six covariates, 1000 iterations were performed, each of which generated 1000 samples. Selection rates for the IRT-like method and the proposed variable selection method are reported in Figure 4. In Figure 4, the upper panels reflect Case 1, and the lower panels show Case 2 results. In each panel, the results of the IRT-like method are compared with those of the proposed method. According to Figure 4, the IRT-like method shows a clear preference for variables with a greater number of cutpoints. Specifically, the continuous variables \(Z_{1}\), \(Z_{2}\), and \(Z_{3}\) and categorical variables \(Z_{6}\) with a relatively higher number of cutpoints among categorical variables exhibit substantially higher selection rates compared to the others. In contrast, the proposed method achieves fair selection rates, approximately 1/6, across all the covariates. These results imply that the conventional IRT is susceptible to selection bias, but the proposed method is unbiased in split variable selection. \begin{table}
\begin{tabular}{c l l} \hline  & **Case 1 (Independent)** & **Case 2 (Dependent)** \\ \hline \(Z_{1}\) & Uniform [1; 5] & Uniform [1; 5] \\ \(Z_{2}\) & Exponential (2) & Exponential (2) \\ \(Z_{3}\) & Normal (0, 1) & \(Z_{2}\) + 0.2 Normal (0, 1) \\ \(Z_{4}\) & Multinomial (2) & Multinomial (2) depending on \(Z_{3}\) \\ \(Z_{5}\) & Multinomial (3) & Multinomial (3) \\ \(Z_{6}\) & Multinomial (6) & Multinomial (6) correlated with \(Z_{5}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Distribution of covariates in the simulation. The observed results underscore the limitations of IRT, which selects split variables by simultaneously optimizing both the variable and its cutpoint. This approach inherently favors variables with more potential cutpoints because they generate broader and finer optimization spaces [54]. For example, a continuous split variable \(Z_{\text{A}}\) with n\({}_{\text{1}}\) distinct values can have (n\({}_{\text{1}}-1\)) possible splits, while a categorical variable \(Z_{\text{B}}\) with n\({}_{\text{2}}\) categories can have (2\({}^{\text{n}_{\text{2}}-1}-1\)) possible splits. If (n\({}_{\text{1}}-1\)) > (2\({}^{\text{n}_{\text{2}}-1}-1\)), \(Z_{\text{A}}\) can create a larger solution space and, thus, is more likely to be selected; otherwise, \(Z_{\text{B}}\) is more likely to be selected. This bias is particularly pronounced for categorical variables with many categories. Consequently, variables with more split options are selected more frequently, regardless of their actual association with the target variable \(Y\)[24; 25; 26; 28]. To address this issue, it is essential to separate the split variable selection step from cutpoint optimization. The proposed UIRT achieves this by first selecting the split variable and then determining its optimal cutpoint. This decoupling removes the influence of the number of potential cutpoints, ensuring unbiased split variable selection. As shown in Figure 4, UIRT produces balanced selection rates across all covariates, effectively mitigating the selection bias present in IRT-like methods. To further evaluate the unbiasedness of the proposed model in a more practical and complex setting, a real-world dataset was utilized in addition to simulated data. The Abalone dataset was used in this analysis. This dataset is a widely recognized dataset for predicting the age of abalone based on physical measurements. For the evaluation, the variable 'Rings' is set as a response variable, and 'Shell weight' is chosen as the predictor because they have the strongest monotonically increasing relationship for each other based on Spearman's rank correlation. Five variables such as 'Shucked weight (SW)', 'Height (H)', 'Length (L)', 'Diameter (D)', and 'Gender (G)' are used as split variables.