# Heterogeneous Treatment Effect with Trained Kernels of the Nadaraya-Watson Regression
## 4 The TNW-CATE Description
TNW-CATE also provides better results in comparison with other models, except for the case of the indicator function when is TNW-CATE inferior to the X-RF. Figure 9 illustrates how the ratio of numbers of the treatments separately impacts the control (the left plot) and treatment (the right plot) regressions when the logarithmic function is used for generating data. It can be seen from Figure 9 that MSE of the treatment network is very close to MSE of other models for almost all values of the ratio, but the accuracy of the control network significantly differs from other models. #### 5.2.3 Experiments with Different Values of \(\theta\)

The next experiments allow us to investigate how the CATE estimators depend on the value of hyperparameter \(\theta\) which controls the impact of the control and treatment

\[i =1,\ldots,c,\ r=1,\ldots,N. \tag{11}\]9 MSE of the CATE values as a function of the ratio of numbers of treatments in the training set when spiral, logarithmic, power, and indicator functions are used for generating examples. \[\mathcal{T}_{j,s} =\{(\mathbf{z}_{1}^{(s)},h_{1}^{(s)}),\ldots,(\mathbf{z}_{m}^{(s) },h_{m}^{(s)})\},\]0 MSE of the control (the **left** plot) and treatment (the **right** plot) responses as functions of the ratio of numbers of treatments in the training set when the logarithmic function is used for generating examples. Numerical results are also presented in Table 1 where the MSE values and standard deviations corresponding to different models using different generating functions are given. The best results for every function are shown in bold. It can be seen from Table 1 that TNW-CATE provides the best results for the spiral, logarithmic, and power functions. Moreover, the improvement is sufficient. TNW-CATE is comparable with the X-NW and X-RF from the logarithmic and power generating functions. At the same time, the proposed model is inferior to X-RF for the indicator function. We can find many successful applications of neural networks for medical and biological tasks [97; 98]. Neural networks also are used in the CATE estimation, for example, DRNet [72], DragonNet [14], FlexTENet [66], and VCNet [69]. However, it is important to point out that many models based on neural networks have not been successful when they are trained on small datasets because the aforementioned neural networks require a large amount of data for training. Thus, the considered small datasets have led to the network overfitting. Therefore, we have studied models for comparison that are based on methods dealing with small data. ### Real Dataset

We perform numerical experiments on the popularly used datasets from the \"The Infant Health and Development Program\"(IHDP), which are designed to understand the effect of home visits by specialist doctors on the future cognitive test scores of premature infants [8]. The dataset can be regarded as a common benchmarking dataset for estimating HTE. It contains 747 subjects and 25 features (6 continuous and 19 binary features) that describe both the characteristics of the infants and the characteristics of their mothers. The datasets can be accessed at [[https://github.com/vdorie/npci](https://github.com/vdorie/npci)]([https://github.com/vdorie/npci](https://github.com/vdorie/npci)) (accessed on 19 April 2023). The number of treatments is 139. Results of numerical experiments (the MSE values and standard deviations) with IHDP by using the same models as in the previous experiments are presented in \[\mathcal{T}_{j,s} =\{(\mathbf{z}_{1}^{(s)},h_{1}^{(s)}),\ldots,(\mathbf{z}_{m}^{(s) },h_{m}^{(s)})\},\]1 We also show the MSE values and standard deviations of different numbers of treatments (70 and 35) by randomly removing a part of treatments from the dataset. We randomly split the dataset into three subsets: training, validation, and testing. We use 40% of the examples for testing. The threefold validation based on 20% of examples is used to tune the model parameters during training. It can be seen from Table 2 that TNW-CATE provides the smallest values of the MSE.