# Heterogeneous Treatment Effect with Trained Kernels of the Nadaraya-Watson Regression
## 4 The TNW-CATE Description
### Study of the TNW-CATE Properties

In all figures illustrating dependencies of CATE estimators on parameters of models, dotted curves correspond to the T-learner, the S-learner, and the X-learner implemented by using the Nadaraya-Watson regression (triangle markers correspond to T-NW and S-NW; the circle marker corresponds to X-NW). Dashed curves with the same markers correspond to the same models implemented by using random forests. The solid curve with cross markers corresponds to TNW-CATE. #### 5.2.1 Experiments with Numbers of Training Data

Let us compare different CATE estimators using different numbers of the control and treatment examples. We study the estimators by numbers \(c\) of controls: 100, 250, 500, 750, and 1000. Each number of treatments is determined as 10% of each number of controls. Value \(n\) is 80 and 100; value \(m\) is 50% of \(t\). Figure 6 illustrates how MSE of the CATE values depends on the number \(c\) of controls for different estimators when different functions are used for generating examples. In fact, these experiments study how MSE depends on the entire number of the controls and treatments because the number of the treatments increases with the number of the controls. For all functions, the increase in the amount of training examples improves most estimators including TNW-CATE. These results are expected because the larger size of training data mainly leads to the better accuracy of models. It can be seen from Figure 6 that the proposed model provides better results in comparison with other models. The best results are achieved when the spiral generating function is used. The models different from TNW-CATE cannot cope with the complex structure of data in this case. However, TNW-CATE shows comparative results with the T-learner, the S-learner, and the X-learner when the indicator function is used for generating examples. The X-learner outperforms TNW-CATE in this case. The indicator function does not have a complex structure. Moreover, the corresponding outcomes linearly depend on most features (see (28) and (29)), and random forests implementing X-RF are trained better than the neural network implementing TNW-CATE. One can also see from Figure 6 that models T-NW, S-NW, and X-NW based on the Nadaraya-Watson regression with the Gaussian kernel provide close results when the logarithmic generating function is used by larger numbers of training data. This is causedby the fact that the Gaussian kernel is close to the neural network kernel implemented in TNW-CATE. Figure 7 illustrates how the number of controls separately impacts the control (the left plot) and treatment (the right plot) regressions when the power function is used for generating data. It can also be seen from Figure 7 that MSE provided by the control network is much smaller than MSE of the treatment neural network. #### 5.2.2 Experiments with Different Values of the Treatment Ratio

Another question is how the CATE estimators depend on the ratio of numbers of treatments and controls in the training set. We study the case when the number of controls \(c\) is 200 and the ratio takes values from the set \(\{0.1,0.2,0.3,0.4,0.5\}\). The coefficient \(\theta\) is taken in accordance with the certain function as described above. \[i =1,\ldots,c,\ r=1,\ldots,N. \tag{11}\]7 MSE of the CATE values as a function of the number of controls when spiral, logarithmic, power, and indicator functions are used for generating examples. \[i =1,\ldots,c,\ r=1,\ldots,N. \tag{11}\]8 MSE of the control (the **left** plot) and treatment (the **right** plot) responses as functions of the number of controls when the power function is used for generating examples. Similar results are shown in Figure 8 where plots of MSE of the CATE values as a function of the ratio of numbers of treatments in the training set by different generating functions are depicted. We again see from Figure 8 that the difference between MSE of TNW-CATE and other models is the largest when the spiral function is used.