# Unbiased Isotonic Regression Tree for Discovering Hidden Heterogeneity in Monotonicity Constraints
## 4 Investigating the Presence of Selection Bias in IRT  
A simulation study was carried out to investigate the possibility of selection bias for split selection in existing IRT and validate the unbiasedness of the proposed split variable selection. The simulation process is given as follows: First, generate two variables \(X\) and \(Y\) that have a monotonic increasing or decreasing trend with some variability. Six different types of split covariates \(\{Z_{1},\ Z_{2},\ \ldots,\ Z_{6}\}\) are considered, and they are independent of two variables \(X\) and \(Y\). The proposed method is applied to the simulated data and identifies the split variable. This process is repeated across multiple iterations, and the selection rate for each covariate (i.e., the proportion of iterations in which the covariate is chosen) is calculated. Ideally, an unbiased split variable selection method can produce similar selection rates across all six covariates, irrespective of their distribution and types. A comparison is made with a split variable selection method used in the IRT. Since IRT finds a split variable similar to CART, IRT is vulnerable to selection bias. Here, this study calls this the IRT-like method for convenience. For two variables \(X\) and \(Y\) in the simulation, the monotonic constraint is imposed by the following equation:  
\[Y=0.6+1.2X+0.5\text{sin}(X)+\epsilon \tag{5}\]  
where the first term is a constant value of 0.6 to ensure a positive baseline, the second term (e.g., 1.2\(X\)) serves as the overall monotonic increasing trend, the third term introduces periodic deviation from increasing trends using the sinusoidal component, and the last term is a random noise that follows normal distribution with zero mean and unit variance for simulating real-world data imperfections, respectively. Regarding six different split variables in the simulation, both continuous and categorical types are considered. Given that varying numbers of potential cutpoints of split variables introduce selection bias, this setup is suitable for examining the presence of selec  
Figure 3: The framework for constructing the proposed unbiased isotonic regression tree. tion bias. The design used in the simulation adopts the simulation setting of the renowned GUIDE algorithm [28], which serves as a well-established benchmark in the field, as well as a related study that employs the same configuration [54]. The six covariates consist of three continuous and three categorical variables whose distribution setups are described in Table 1. Based on the two previous studies [28; 54], two scenarios were assessed as follows: Case 1, where all covariates are independent, and Case 2, where certain covariates are dependent. Each scenario represents practical data generation situations. In Case 1, \(Z_{1}\) is uniformly distributed from one to five; \(Z_{2}\) is an exponential distribution with a mean of two; \(Z_{3}\) follows a standard normal distribution; and \(Z_{4},\ Z_{5}\) and \(Z_{6}\) are sampled from a multinomial distribution with two, three, and six categories of equal probability, respectively. For Case 2, \(Z_{1}\), \(Z_{2}\) and \(Z_{5}\) retain the same distribution as in Case 1, but \(Z_{3}\) is a linear combination of \(Z_{2}\) and a standard normal distribution scaled by 0.3. The distribution of \(Z_{4}\) is modeled as binomial distribution conditioning on \(Z_{3}\), and \(Z_{6}\) and follows a six-outcome multinomial distribution correlated with \(Z_{5}\). More details about generating correlated covariates \(Z_{4}\) and \(Z_{6}\) can be found in [54]. Lastly, another parameter considered in the simulation is the significance level used for split variable selection. This parameter determines the threshold for rejecting the null hypothesis when selecting split variables. In this simulation study, the significance level is set to 0.05, which is a standard value widely used in statistics and machine learning.